Research ArticleA Comparative Study on Sensitivity-Based Damage DetectionMethods in Bridges
Akbar Mirzaee Reza Abbasnia and Mohsenali Shayanfar
Department of Civil Engineering Iran University of Science and Technology Narmak Tehran 16846 13114 Iran
Correspondence should be addressed to Reza Abbasnia abbasniaiustacir
Received 25 September 2014 Accepted 22 January 2015
Academic Editor Yaguo Lei
Copyright copy 2015 Akbar Mirzaee et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper provides a comparative study on four different sensitivity-based damage detection methods for bridges The methodsinvestigated in this study are approximation approach semianalytical discrete approach and analytical discrete approach whichincludes direct differential and adjoint variable methods These sensitivity-based methods utilize finite element model updatingprocedure and allow a wide choice of physically meaningful parameters leading to vast range of applications in damage detectionThe most important difficulty in these methods is calculation of sensitivity matrix Calculation of this massive matrix is repeatedin each iteration and has a significant effect on the efficiency of method In this study the acceleration measurements are simulatedfrom the solution to the forward problem using finite element method under moving load with various speeds along with theaddition of artificially produced measurement noise Various damaged structures with different damage patterns including singlemultiple and random damage are considered and efficiency of four sensitivity methods is compared Moreover various possiblesources of error such as the effects of measurement noise as well as initial assumption error in stability of the methods are alsodiscussed
1 Introduction
Structural health monitoring is the implementation of adamage identification strategy to different types of structuresSHM is vital to evaluate the fitness of a structure in var-ious disciplines including aerospace mechanical and civilengineering to perform its prescribed tasks properly Thenecessity of SHM is highlighted when it is recognized thatthe performance of structures may change due to a gradualor sudden change in states load conditions or responsemechanisms
Bridges are truly the flagships of civil engineering whichattract the greatest attention within the engineering commu-nity This is due to their small safety margins and their greatexposure to the public [1]
The main objective of developing the SHM system forbridges is to enhance structural safety However in bridgesSHM serves other economic benefits such as increasedmission reliability extended life of life-limited componentsreduced tests reduction in ldquodown timerdquo increased equipmentreliability customization of maintenance actions and greater
awareness of operating personnel resulting in fewer acci-dents SHM also promises to help in reducing maintenancecosts [2]
SHM is an inverse problem wherein the flaws in thestructure are characterized using the measured data for someknown inputs [3] Hence SHM can be thought of as a systemidentification problem and classified into two categoriesnamely the diagnosis and prognosisThrough diagnosis onecan determine the presence of flaws their locations and theirextents along with the possibility of looking at the delayingthe propagation of flaws in the structure The prognosis partuses the information of the diagnosis part and determinesthe remaining life of the structure Therefore the SHM canbe broadly divided into five levels as follows [4]
level 1 confirming the presence of damagelevel 2 determination of location and orientation ofthe damagelevel 3 evaluation of the severity of the damagelevel 4 possibility of controlling or delaying thegrowth of damage
Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 120630 19 pageshttpdxdoiorg1011552015120630
2 Shock and Vibration
level 5 determining the remaining life in the structure(prognosis)
Dynamics-based SHM techniques assess the state of healthof a structural component on the basis of the detection andanalysis of its dynamic response Such techniques can beclassified on the basis of the type of response being consideredfor the investigations on the frequency or time domainof investigation and on the modality used to excite thecomponent [5]
The developments in the field of structural damage detec-tion (DD) using vibration data of civil engineering structureshave been recently studied by several authors some of themare briefly described herein
Doebling et al [6 7] have presented comprehensivereview of literature mainly focusing on frequency-domainmethods for damage detection in linear structures anddeclared that sufficient evidence exists to promote the useof measured vibration data for the detection of damage instructures using both forced-response testing and long-termmonitoring of ambient signals and there is a significant needin this field for research on the integration of theoreticalalgorithms with application-specific knowledge bases andpractical experimental constraints Another discussion onmethods of damage detection and location using naturalfrequency changes has been presented by Salawu [8] andhis study showed that damage detection using vibrationfrequencies is not very reliable Zou et al [9] summarizedthemethods on vibration-based damage detection and healthmonitoring for composite structures especially in delamina-tion modeling techniques and delamination detection
Alampalli and Fu [10] and Alampalli et al [11] conductedlaboratory and field studies on bridge structures to investigatethe feasibility of measuring bridge vibration for inspec-tion and evaluation These studies focused on sensitivityof measured modal parameters to damage Cross diagno-sis using multiple signatures involving natural frequenciesmode shapesmodal assurance criteria and coordinatemodalassurance criteria was shown to be necessary to detect thedamage Casas and Aparicio [12] studied concrete bridgestructures and investigated dynamic response as an inspec-tion tool to assess bearing conditions and girder crackingTheir study showed the need to investigate more than onenatural frequency and also to determine mode shapes inorder that the damage could be successfully detected andlocated
Damage detection usually requires amathematical modelon the structure in conjunction with experimental modalparameters of the structureThe identification approaches aremainly based on the change in the natural frequencies [13ndash15]mode shapes [16ndash18] ormeasuredmodal flexibility [7 19ndash21]
The frequency-domain DD algorithms have been morewidely developed and applied as the amount ofmeasured datais reduced dramatically after the transform thus they can behandled easily Unfortunately the effects of local damage onthe natural frequencies and mode shapes of higher modesare greater than lower ones but they are usually difficult tomeasure from experiments In addition structural dampingproperties cannot be identified in frequency domain DD
The time-domain DD may be an attractive one to over-come the drawbacks of the frequency-domain DD For time-domain DD the forced vibration responses of the structureare needed in the identification However in some cases itis either impractical or impossible to use artificial inputs toexcite the civil engineering structures so natural excitationmust be measured along with the structural responses toassess the dynamic characteristics [22 23] In recent yearssome researchers have investigated both the problem of loadidentification (moving load and impact load) and modalparameters identification under operational conditions [2425] In addition identification of the structural parametersapplying a moving load has been considered in many papersLaw et al [26] presented a novel moving force and prestressidentification method based on the finite element and thewavelet-based methods for a bridge-vehicle system Jianget al [27] identified the parameter of a vehicle moving onmultispan continuous bridges Zhu and Law [28] presented amethod for damage detection of a simply supported concretebridge structure in time domain using the interaction forcesfrom the moving vehicles as excitation
Sensitivity-based methods allow a wide choice of phys-ically meaningful parameters and this advantage has led totheir widespread use in damage detection Calculation ofsensitivity matrix has a significant effect on the efficiency ofthese methods Despite the high importance of calculationmethod of sensitivity matrix and optimizing its performancein DD procedure there is not literature on this regardIn this paper computational methods for sensitivity matrixare discussed and a novel sensitivity base damage detectionmethod in time-domain referred to as ldquoadjoint variablemethodrdquo is introduced Fundamental principles of proposedmethod are presented and its performance is compared withthe conventional methods and it is shown that the numericalcost is considerably reduced by using the concept of adjointvariable
The outline of the work is as follows inverse prob-lems along with model updating are briefly introduced inSection 2 Different methods for sensitivity analysis alongwith introduced method (adjoint variable method) areaddressed in Section 3 Numerical simulation is presented inSection 4 and comparison studies are presented in Section 5with studies on the effect of different factors whichmay affectthe accuracy and efficiency of different methods and finallyconclusion will be drawn in the last section
2 Finite Element Model Updating andInverse Problem
A key step in model-based damage identification is theupdating of the finite element model of the structure in sucha way that the measured responses can be reproduced by theFE model A general flowchart of this operation is given inFigure 1The identification procedure presented in this paperis a sensitivity-based model updating routine Sensitivitycoefficients are the derivatives of the system responses withrespect to the physical parameters or input excitation forceand are needed in the cost function of the flowchart ofFigure 1
Shock and Vibration 3
Starting values
Numerical modelupdating model parameters
Experimenttest specimen
Improved parameters
Minimization cost function
Identified model parameters
Figure 1 General flowchart of a FEM-updating [31]
21 Finite ElementModeling of Bridge Vibration underMovingLoads For a general finite element model of a linear elastictime-invariant structure the equation of motion is given by
Mz119905119905
+ Cz119905
+ Kz = Bf (1)
where M and K are mass and stiffness matrices and Cis damping matrix z
119905119905
and z119905
and z are the respectiveacceleration velocity and displacement vectors for the wholestructure and f is a vector of applied forces with matrix Bmapping these forces to the associated DOFs of the structureProportional damping is assumed to show the effect ofdamping ratio on the dynamic magnification factor Rayleighdamping in which the dampingmatrix is proportional to thecombination of the mass and stiffness matrices is used
C = 1198860
M + 1198861
K (2)
where 1198860
and 1198861
are constants to be determined fromtwo modal damping ratios The dynamic responses of thestructures can be obtained by direct numerical integrationusing Newmark method
22 Objective Functions The approach minimizes the differ-ence between response quantities (acceleration response) ofthe measured data and model predictions This problemmaybe expressed as the minimization of j where
and z(120572) are the measured and computed responsevectors 120572 is a vector of all unknown parameters and 120598 is theresponse residual vector
23 Nonlinear Model Updating for Damage Detection Whenthe parameters of a model are unknown they must be esti-mated using measured data Since the relationship betweenthe acceleration responses z
119894
and the fractional stiffnessparameter 120572 is nonlinear a nonlinear model updating tech-nique like the Gauss-Newton method is required This kindof method has the advantage that the second derivativeswhich can be challenging to compute are not required TheGauss-Newton method in the damage detection procedurecan be described in terms of the acceleration response at the119894th DOF of the structure as
z119889119897
(120572119889
) = z119906119897
(1205720
) + S (1205720) times Δ1205721
+ S (1205720 + Δ1205721) times Δ1205722 + sdot sdot sdot (4)
The superscripts 0 1 and 2 denote the iteration numbersIndex 119906 denotes the initial state or state 0 while index 119889denotes the final damage state z
119889119897
and z119906119897
are vectors of theacceleration response at the 119894th DOF of the damaged andintact states respectivelyThe damage identification equationfor (119896 + 1)th iteration is
Δz119896 = S119896 times Δ120572119896+1 (5)
where S119896 and Δz119896 are obtained from the 119896th iteration Theiteration in (5) starts with an initial value 1205720 leading to Δz0 =z119889119897
minus z119906119897
(1205720
) and S0 = S(1205720) The parameter vector 120572119896 =1205720
+sum119896
119894=1
Δ120572119894 Sensitivity matrix S119896 = S(120572119896) and the residual
vectorΔz119896 = z119889119897
minus z119906119897
(1205720
)minussum119896minus1
119894=0
S119894Δ120572119894+1 (119896 = 1 2 ) of thenext iteration is then computed from results in the previousiterations
The acceleration response vector z119906119897
from the physicalintact structure is computed in general from the associatedanalytical model via dynamic analysis z
119889119897
is the accelerationresponse of the model of the damaged structure In generalthe measured acceleration responses (including measure-ment errors) from the damaged structure are obtained for z
119889119897
The iteration is terminated when a preselected criterion is
met The final identified damaged vector becomes [17]
Δ120572 = Δ1205721
+ Δ1205722
+ sdot sdot sdot + Δ120572119899
(6)
where 119899 is the number of iterations
24 Regularization Like many other inverse problems thesolution of (5) is often ill-conditioned and regularizationtechniques are needed to provide bounds to the solutionTheaim of regularization in the inverse analysis is to promote cer-tain regions of parameter space where the model realizationshould exist The two most widely used regularization meth-ods are Tikhonov regularization [14] and truncated singularvalue decomposition [29] In the Tikhonov regularizationthe new cost function is defined as
119895 (Δ120572119896+1
120582) =
10038171003817100381710038171003817S119896 sdot Δ120572119896+1 minus Δz1198961003817100381710038171003817
The regularization parameter 120582 ge 0 controls the extent ofcontribution of the two errors to the cost function in (7) and
4 Shock and Vibration
the fractional stiffness change incrementΔ120572119896+1 is obtained byminimizing the cost function in (7)
The regularized solution fromminimizing the function in(7) can be written in the following form as
Δ120572119896+1
= ((S119896)119879
S119896 + 1205822I)minus1
(S119896)119879
Δz119896 (8)
To express the contribution of the singular values and thecorresponding vectors in the solution clearly and to showhowthe regularization parameter plays the role as the filter factorthe sensitivity matrix is singular value decomposed andsingular value decomposition (SVD) applies to the sensitivitymatrix S119896 to obtain
S119896 = UΣV119879 (9)
where U isin 119877119899119905times119899119905 and V isin 119877
119898times119898 are orthogonal matricessatisfying U119879U = I
119899119905
and V119879V = I119898
and matrix Σ has thesize of 119899119905 times 119898 with the singular values 120590
119894
(119894 = 1 2 119898) onthe diagonal arranged in a decreasing order such that 120590
1
ge
1205902
ge sdot sdot sdot ge 120590119898
ge 0 and zeros elsewhereThe regularized solution in (8) can be written as
Δ120572119896+1
=
119898
sum
119894=1
119891119894
U119879119894
Δz119896
120590119894
V119894
(10)
where 119891119894
= 1205902
119894
(1205902
119894
+ 1205822
) (119894 = 1 2 119898) are referred as filterfactors The solution norm Δ120572
119896+1
2
2
and the residual normS119896 sdot Δ120572119896+1 minus Δz1198962
10038171003817100381710038171003817S119896 sdot Δ120572119896+1 minus Δz1198961003817100381710038171003817
1003817
2
2
=
119898
sum
119894=1
(
1205822
1205902
119894
+ 1205822
U119879119894
Δz119896)2
(11)
These two quantities represent the smoothness and goodness-of-fit of the solution and they should be balanced by choosingan appropriate regularization parameter
25 Element Damage Index In the inverse problem of dam-age identification it is assumed that the stiffness matrix ofthe whole element decreases uniformly with damage andthe flexural rigidity EI
119894
of the 119894th finite element of the beambecomes 120573
119894
EI119894
when there is damage The fractional changein stiffness of an element can be expressed as [30]
ΔK119887119894
= (K119887119894
minus K119887119894
) = (1 minus 120573119894
)K119887119894
(12)
where K119887119894
and K119887119894
are the 119894th element stiffness matrices ofthe undamaged and damaged beam respectively ΔK
119887119894
is thestiffness reduction of the element A positive value of 120573
119894
isin
[0 1] will indicate a loss in the element stiffness The 119894thelement is undamaged when 120573
119894
= 1 and the stiffness of the119894th element is completely lost when 120573
119894
= 0
The stiffness matrix of the damaged structure is theassemblage of the entire element stiffness matrix K
119887119894
K119887
=
119873
sum
119894=1
A119879119894
K119887119894
A119894
=
119873
sum
119894=1
120573119894
A119879119894
K119887119894
A119894
(13)
where A119894
is the extended matrix of element nodal displace-ment that facilitates assembling of global stiffness matrixfrom the constituent element stiffness matrix
The objective of sensitivity analysis is to quantify the effectsof parameter variations on calculated results Terms such asinfluence importance ranking by importance and domi-nance are all related to sensitivity analysis
The simplest and most common procedure for assessingthe effects of parameter variations on a model is to varyselected input parameters rerun the code and record thecorresponding changes in the results or responses calculatedby the code The model parameters responsible for thelargest relative changes in a response are then consideredto be the most important for the respective response Forcomplexmodels though the large amount of computing timeneeded by such recalculations severely restricts the scope ofthis sensitivity analysis method in practice therefore themodeler who uses this method can investigate only a fewparameters that he judges a priori to be important [32]
When the parameter variations are small the traditionalway to assess their effects on calculated responses is byusing perturbation theory either directly or indirectly viavariational principles The basic aim of perturbation theoryis to predict the effects of small parameter variations withoutactually calculating the perturbed configuration but rather byusing solely unperturbed quantities
31 Methods of Structural Sensitivity Analysis Various meth-ods employed in design sensitivity analysis are listed inFigure 2 Three approaches are used to obtain the designsensitivity the approximation discrete and continuumapproaches In the approximation approach design sensi-tivity is obtained by either the forward finite difference orthe central finite difference method In the discrete methoddesign sensitivity is obtained by taking design derivativesof the discrete governing equation For this process it isnecessary to take the design derivative of the stiffness matrixIf this derivative is obtained analytically using the explicitexpression of the stiffness matrix with respect to the designvariable it is an analytical method since the analyticalexpressions of stiffness matrix are used However if thederivative is obtained using a finite difference method themethod is called a semianalytical method
In the continuum approach the design derivative ofthe variational equation is taken before it is discretized Ifthe structural problem and sensitivity equations are solvedas a continuum problem then it is called the continuum-continuum method However only very simple classical
Shock and Vibration 5
Sensitivity methods
Approximation approach
Forward finite difference
Centeral finite difference
Discrete approach
method
Analytical
Semianalytical
method
differential
Adjoint variable
Direct
method
Continuum approach
Continuum-discrete method
Continuum-continuum
method
method
Figure 2 Approaches to design sensitivity analysis
problems can be solved analytically Thus the continuumsensitivity equation is solved by discretization in the sameway that structural problems are solved Since differentiationis taken at the continuum domain and is then followed bydiscretization this method is called the continuum-discretemethod
32 Approximation Approach The easiest way to computesensitivity information of the performance measure is byusing the finite difference method Different designs yielddifferent analysis results and thus different performancevaluesThe finite differencemethod actually computes designsensitivity of performance by evaluating performance mea-sures at different stages in the design process If u is thecurrent design then the analysis results provide the valueof performance measure 120595(u) In addition if the design isperturbed to u + Δu where Δu represents a small change inthe design then the sensitivity of 120595(u) can be approximatedas
119889120595
119889uasymp
120595 (u + Δu) minus 120595 (u)Δu
(14)
Equation (14) is called the forward difference method sincethe design is perturbed in the direction of +Δu If minusΔuis substituted in (14) for Δu then the equation is definedas the backward difference method Additionally in centraldifference method the design is perturbed in both directionsand the design sensitivity is approximated by
119889120595
119889uasymp
120595 (u + Δu) minus 120595 (u minus Δu)2Δu
(15)
Then the equation is defined as the central differencemethod
33 Discrete Approach A structural problem is often dis-cretized in finite dimensional space in order to solve complex
problems The discrete method computes the performancedesign sensitivity of the discretized problem
The design represents a structural parameter that canaffect the results of the analysis
The design sensitivity information of a general perfor-mance measure can be computed either with the directdifferentiation method or with the adjoint variable method
331 Direct Differentiation Method The direct differentia-tion method (DDM) is a general accurate and efficientmethod to compute FE response sensitivities to the modelparameters This method directly solves for the designdependency of a state variable and then computes perfor-mance sensitivity using the chain rule of differentiationThis method clearly shows the implicit dependence on thedesign and a very simple sensitivity expression can beobtained
Consider a structure in which the generalized stiffnessand mass matrices have been reduced by accounting forboundary conditions Let the damping force be representedin the form of C(119887)z
119905
where z119905
= 119889z119889119905 denotes thevelocity vector Under these conditions Lagrangersquos equationof motion becomes the second-order differential equation as[33]
M (119887) z119905119905
+ C (119887) z119905
+ K (119887) z = f (119905 119887) (16)
With the initial conditions
z (0) = z0 z119905
(0) = z0119905
(17)
If design parameters are just related to stiffness matrix weobtain
M120597z119905119905
120597119887119894
+ C120597z119905
120597119887119894
+ K 120597z120597119887119894
= minus
120597K120597119887119894
z minus 1205722
120597K120597119887119894
z119905
(18)
6 Shock and Vibration
in which 120597z120597b119894 120597z119905
120597b119894 and 120597z119905119905
120597b119894 are sensitivity vectorsof displacement velocity and acceleration with respect todesign parameter 119887
119894
respectively Assume that
y119905119905
=
120597z119905119905
120597119887119894
(19a)
y119905
=
120597z119905
120597119887119894
(19b)
y = 120597z120597119887119894
(19c)
So by replacing (19a) (19b) and (19c) into (18) we have
My119905119905
+ Cy119905
+ Ky = minus120597K120597119887119894
z minus 1205722
120597K120597119887119894
z119905
(20)
Right side of (20) can be considered as an equivalent force so(20) is similar to (16) and sensitivity vectors can be obtainedby Newmark method
332 Adjoint Variable Method Mirzaee et al [34] provedthat sensitivity analysis can be performed very efficientlyby using deterministic methods based on adjoint functionsThis method constructs an adjoint problem that solves forthe adjoint variable which contains all implicitly dependentterms
The adjoint variablemethod yields a terminal-value prob-lem compared with the initial-value problem of responseanalysis
For the dynamic structure the following formof a generalperformance measure can be considered
120595 = 119892 (z (119879) 119887) + int119879
0
119866 (z 119887) 119889119905 (21)
General form of sensitivity measure relative to the designparameter (b) is as follows [33]
1205951015840
= [
120597119892
120597zminus
120597119892
120597zz119905
(119879) + 119866 (z (119879) 119887) 1Ω119905
120597Ω
120597119911
] z1015840 (119879)
minus [
120597119892
120597119911
z119905
(119879) + 119866 (z (119879) 119887)] 1
Ω119905
120597Ω
120597119911119905
z1015840119905
(119879)
+ int
119879
0
[
120597119866
120597zz1015840 + 120597119866
120597119887
120575119887] 119889119905 +
120597119892
120597119887
120575119887
minus [
120597119892
120597119911
z119905
(119879) + 119866 (119911 (119879) 119887)]
1
Ω119905
120597Ω
120597119887
120575119887
(22)
Note that1205951015840 depends on z1015840 and z1015840119905
at119879 as well as on z1015840 withinthe integration
Mirzaee et el [34] showed that while structural vibrationresponses are used for damage detection solving (22) inknown time 119879 is as follows
119889120595
119889119887
(119879) = int
119879
0
(minus120582119879
1198860
120597K120597119887
z119905
minus120582119879120597K120597119887
z)119889119905 (23)
120582119879
(119905) = P119879
119891 (119905) +Q119879
119892 (119905) (24)
in which
P119879
= 119890minus120585120596119879
120582119905
(119879)
120596119863
cos (120596119863
119879)
119891 (119905) = 119890120585120596119905 sin (120596
119863
119905)
Q119879
= minus119890minus120585120596119879
120582119905
(119879)
120596119863
sin (120596119863
119879)
119892 (119905) = 119890120585120596119905 cos (120596
119863
119905)
(25)
It is worth noting that terminal values for different values of119879 are not similar and adjoint equation should be calculatedfor corresponding 119879 separately
Mirzaee et al [34] presented a quite efficient incrementalsolution for solving (24) which significantly improves theproposed method
34 Sensitivity Method Selection In this paper four differentmethods are presented to compute sensitivity information ofthe performance measure for damage detection procedureFirst method is the finite Difference method (FD) whichis classified as an approximation approach Second it issemianalytical discrete method which uses a finite differencescheme to calculate stiffness matrix in direct differentialmethod (DDMFD) and third and fourthmethods are two dif-ferent analytical discrete methods direct differential method(DDM) and adjoint variable method (ADM)
The first three methods including FD DDMFD andDDM are widely used by other authors [35 36] and the lastmethod (ADM) is proposed by the authors (Mirzaee et al[34])
35 Procedure of Iteration for Damage Detection The initialanalytical model of a structure deviates from the true modeland measurement from the initial intact structure is used toupdate the analytical model The improved model is thentreated as a reference model and measurement from thedamaged structure will be used to update the referencemodel
When response measurement from the intact state of thestructure is obtained the sensitivities are computed frompresented methods based on the analytical model of thestructure and the well knowing input force and velocityThe vector of parameter increments is then obtained usingthe computed and experimentally obtained responses Theanalytical model is then updated and the correspondingresponse and its sensitivity are again computed for the nextiteration When measurement from the damaged state isobtained the updated analyticalmodel is used in the iterationin the same way as that using measurement from the intactstate Convergence is considered to be achieved when thefollowing criteria are met
1003817100381710038171003817E119894+1
minus E119894
1003817100381710038171003817
1003817100381710038171003817E119894
1003817100381710038171003817
times 100 le Tol1
1003817100381710038171003817Response
119894+1
minus Response119894
1003817100381710038171003817
1003817100381710038171003817Response
119894
1003817100381710038171003817
times 100 le Tol2
(26)
Shock and Vibration 7
The final vector of identified parameter increments corre-sponds to the changes occurring in between the two statesof the structure The tolerance is set equal to 1 times 10minus5 in thisstudy except otherwise specified
4 Numerical Results
In order to evaluate the efficiency and competency of intro-duced methods analysis of two FE models with extensiveavailable numerical studies has been carried out Represen-tative examples are presented to demonstrate the effects ofspeed of loads measurement noise level and initial error onthe accuracy and effectiveness of the methods
The relative percentage error (RPE) in the identifiedresults is calculated from (27) where sdot is the norm ofmatrix and z
119905119905identifiedand z119905119905true
are the identified and the trueacceleration time history respectively [26]
Since the true value of elastic modulus is unknown RPE canjust be used for evaluating the efficiency of the method
41 Single-Span Bridge A single-span bridge is studied tocompare different methods This bridge consists of 10 Euler-Bernoulli beam elements with 11 nodes each with two DOFsTotal length of bridge is 10m and height and width ofthe frame section are 200mm Rayleigh damping model isadopted with the damping ratios of the first two modes takenequal to 005
Structural properties are summarized in Table 1The transverse point load 119875 has a constant velocity V =
119871119879 where 119879 is the traveling time across the bridge and 119871 isthe total length of the bridge
For the forced vibration analysis an implicit time integra-tion method called as ldquothe Newmark integration methodrdquo isused with the integration parameters 120573 = 14 and 120574 = 12which leads to constant-average acceleration approximation
Speed parameter is defined as
120572 =
119881
119881cr (28)
in which119881cr is critical speed (119881cr = (120587119897)radicEI120588)119881 is movingload speed and 120588 is mass per unit length of the beams
411 Damage Scenarios Six damage scenarios of singlemultiple and random damage along with initial error in thebridge without measurement noise are studied and shown inTable 2
Local damage is simulated with a reduction in the elasticmodulus of material of an element The sampling rate is15000Hz and 250 data of the acceleration response (degree ofindeterminacy is 25) collected along the z-direction at nodes2 and 7 are used in the identification
412 Analysis Results Tables 3ndash6 show the results of differentdetection methods including solution time (ST) number of
loops (NL) and RPE These are the most important param-eters to compare the efficiency and accuracy of differentmethods
413 Effect of Noise Noise is the random fluctuation in thevalue of measured or input that causes random fluctuation inthe output value Noise at the sensor output is due to eitherinternal noise sources such as resistors at finite temperaturesor externally generatedmechanical and electromagnetic fluc-tuations [10]
To evaluate the sensitivity of results to suchmeasurementnoise noise-polluted measurements are simulated by addingto the noise-free acceleration vector a corresponding noisevector whose root-mean-square (RMS) value is equal toa certain percentage of the RMS value of the noise-freedata vector The components of all the noise vectors are ofGaussian distribution uncorrelated andwith a zeromean andunit standard deviation Then on the basis of the noise-freeacceleration z
119905119905nfthe noise-polluted acceleration z
119905119905npof the
bridge at location 119909 can be simulated by
z119905119905np
= z119905119905nf
+ RMS (z119905119905nf) times 119873level times 119873unit (29)
where RMS (z119905119905nf
) is the RMS value of the noise-free accel-eration vector z
119905119905nftimes 119873level is the noise level and 119873unit is a
randomly generated noise vector with zero mean and unitstandard deviation [36]
In order to study effects of noise in stability of differentsensitivity methods scenario 3 (speed ratio of moving loadis considered to be fixed and equal to 05) is considered anddifferent levels of noise pollution are investigated and alsoRPE change with increasing the number of loops in iterativeprocedure has been studied The results are illustrated inFigure 3 for all presented methods
414 Effect of Initial Error In order to study effects of initialerror in stability of sensitivity methods scenario 6 (speedratio ofmoving load is considered to be fixed and equal to 05)is considered This scenario consists of no simulated damagein the structure but the initial elastic modulus of materialof all the elements is underestimated and different levels ofinitial error are investigated RPE changes with increasing thenumber of loops in iterative procedure have been studiedTheresults are illustrated in Figure 4 for all presented methods
42 Plane Grid Model A plane grid model of bridge isstudied as another numerical example to investigate theefficiency and accuracy of discrete methods It is notable thataccording to complexity of this model approximation andsemidiscrete methods are not useable and diverged in allscenarios in all speed ranges
The structure is modeled by 46 frame elements and 32nodes with three DOFs at each node for the translation androtational deformations Rayleigh damping model is adoptedwith the damping ratios of the first two modes taken equalto 005 The finite element model of the structure is shownin Figure 5 and structural properties are summarized inTable 1
8 Shock and Vibration
Table1Prop
ertie
sofd
ifferentm
odels
Mod
elname
Num
bero
felements
Num
bero
fnod
esMassd
ensity
Mod
ulus
ofelasticity
Rayleigh
coeffi
cients
Und
amaged
naturalfrequ
ency
ofintactmod
elkgm
m3
Nm
m2
1198860
1198861
Firstm
ode
Second
mod
eTh
irdmod
eFo
urth
mod
eFifth
mod
eMultispan
1011
78times10minus9
21times
10minus5
01
2860times10minus5
296
1183
2662
4738
7421
Grid
4632
78times10minus9
21times
10minus5
01
2364times10minus5
456
928
1817
2597
3991
Shock and Vibration 9
Table 2 Damage scenarios for single-span bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 4 17 NilM1-2 Multiple 2 7 4 21 NilM1-3 Multiple 3 5 6 and 8 12 6 5 and 2 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
Table 3 Solution time number of loops and RPE of ADMmethod for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
421 Damage Scenarios Six damage scenarios of singlemultiple and randomdamage in the bridgewithoutmeasure-ment noise are studied and shown in Table 7
The sampling rate is 14000Hz and 460 data of the acceler-ation response (degree of indeterminacy is 20) collected alongthe z-direction at nodes 4 11 21 and 27 are used
422 Analysis Results Tables 8 and 9 show the results ofdifferent detection methods including solution time (ST)
and RPE Using both described methods including ADMand DDM method the damage locations and amount areidentified correctly in all the scenarios (Figure 6)
423 Effect of Noise In order to study effects of noisescenario 3 is considered and different levels of noise pollutionare investigated and also RPE changes with increasing thenumber of loops for iterative procedure have been studiedThe results are presented in Figure 7
10 Shock and Vibration
Table 6 Solution time loops and RPE of DDMFD method for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
Figure 3 RPE contours with respect to noise level and loops in model 1
Shock and Vibration 11
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
10
20
30
40
50
60
70
(b) DDMmethod
Loops
Initi
al er
ror
FD method
5 10 15 20 25 300
005
01
015
02
025
03
10
20
30
40
50
60
70
80
(c) FD method
Loops
Initi
al er
ror
DDMFD method
5 10 15 20 25 300
005
01
015
02
025
03
0
10
20
30
40
50
60
70
80
(d) DDMFD method
Figure 4 RPE contours with respect to initial error and loops in model 1
424 Effect of Initial Error Scenario 6 (speed ratio ofmovingload is considered to be fixed and equal to 05) is considered tostudy effects of initial error in stability of sensitivity methodsRPE changes with increasing the number of loops for iterativeprocedure have been studied The results are illustrated inFigure 8
5 Comparison of Different Methods
According to the results obtained in the previous sectiondifferent sensitivity methods considered in this study arecompared in this section Iteration number relative percent-age of error (RPE) solution time noise level and initialerror effect in stability of damage detection procedure areeffective parameters to compare efficiency and accuracy of themethods
51 Iteration Number Average number of iterations fordifferent scenarios with respect to the speed parameter isillustrated in Figure 9
As seen in Figure 9(a) the number of iterations in FDand DDMFD methods is considerably larger than thoseof analytical discrete methods (DDM and ADM) It hasbeen observed that in all cases DDMFD method has feweriterations than FD method Average number of iterations is20757 and 251467 for DDMFD and FD methods respec-tively The larger the speed ratio of moving vehicle is thelarger the number of iterations in approximation methodsis In FD method for speed ratios more than 05 even byusing maximum number of iterations (300) convergence wasnot achieved while the ratio used in DDMFD method is09
Figure 9(b) shows comparison along with analytical dis-crete methods It can be seen that DDM has smaller iterationnumber in all cases Average number of iterations is 146and 11167 for ADM and DDM methods respectively Bothanalytical discrete methods converged in all cases withmaximum iterations equal to 17 and increment in speed ratiodoes not affect the number of iterations
12 Shock and Vibration
Sensors
Element number
25
17
9
2622
15
8
1
23
16
9
2
24
17
10
3
25
18
11
4
2619
12
5
2720
13
6
2821
147
4118
3510
29
2
2742
1936
1130
3
2843
2037
1231
4
29 4421
3813
325
30 4522 39
14
16
336
3132
46 2324
40 1534 7
8
Direction of measured response for identification
Node number
P V
Moving vehicle
Y Z X
7000 (mm)
3000 (mm)
Figure 5 Plane grid bridge model used in detection procedure
Table 7 Damage scenarios for multispan bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 23 5 NilM1-2 Multiple 8 13 and 29 11 4 and 7 NilM1-3 Multiple 3 7 19 25 and 28 12 6 5 2 and 18 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
52 Relative Percentage of Error (RPE) Using the sameconvergence tolerance equal to 1 times 10minus5 relative percentageof error in different methods is illustrated in Figure 10
As seen in this figure with a maximum number ofiterations equal to 300 RPE in FD and DDMFD methodsare considerably larger than DDM and ADM methodsDDMFD method has lower RPE than FD method in mostcases Average of RPE is 026329 for DDMFD compared to027766 in FD method Again the larger the speed ratio ofmoving vehicle is the larger the relative error percentage inapproximation methods is
Figure 10(b) shows that relative error percentage in ADMmethod is lower than DDM and speed ratio does not affectthe relative error
Average of RPE is 0000731 and 0002037 for ADM andDDM methods respectively Therefore ADM is nearly 28times more accurate than DDM
53 Solution Time In order to investigate computational costof sensitivity methods using the same amounts of RAMand CPU resources for the presented numerical simulationssolution time of different damaged models is evaluated andshown in Figure 11
As seen in Figure 11(a) solution times in FD and DDMFDmethods are considerably larger than analytical discretemethods and the FDmethod with an average of 200616 (s) isthemost time consumingmethod Average time for DDMFDmethod is 150849 (s)
Figure 11(b) compares the time spent by DDM and ADMAs illustrated in this figure ADM method has lower solutiontime than DDMmethod in all cases and speed ratio does notaffect the solution time
Average of solution time is 2956 (s) and 7794 (s) forADM and DDM methods respectively consequently ADMis nearly 26 times faster than DDM
54 Efficiency Parameter In order to compare and quantifythe performance of different methods relative efficiencyparameters of methods ldquo119894rdquo and ldquo119895rdquo (REP
119894119895
) are defined as
REP119894119895
= radic
ST119895
ST119894
times
RPE119895
RPE119894
(30)
in which ST is the solution time of system identificationmethod that represents the computational cost of themethodand RPE is the relative error which represents the accuracy ofthe method
Shock and Vibration 13
Table 8 Solution time number of loops and RPE of ADMmethod for model 2
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE ()
541 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to DDM Figure 12 shows REPadjddm changeswith respect to speed parameter in different scenarios
Table 10 shows that in different scenarios and for differentspeed parameters the efficiency parameter is between 19713and 55478 for model 1 and 15659 and 29893 for model2 and its average is 28948 and 22248 for models 1 and2 respectively Compared to DDM the adjoint variablemethod is more efficient and needs 609 less computationaleffort
Changes in the average of REPadjddm in different scenarioswith speed ratio are shown in Figure 13 As shown in thisfigure REPadjddm increases with speed ratios lightly Thisindicates that efficiency of ADM with respect to DDMincreases with speed ratio
542 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to Approximation and Semianalytical MethodsFigure 14 shows REPadjFD and REPadjDDMFD changes withrespect to speed parameter in different scenarios
Table 11 shows the relative efficiency parameter for differ-ent speed parameters in different scenarios It can be seen thatthe efficiency parameter is between 3193 and 78307 for finitedifference method and 4916 and 50542 for DDMFDmethodwith the average of 16714 and 14569 for FD and DDMFDmethods respectively Therefore the adjoint variable methodis extremely successful and the computational cost of thismethod is just 06 and 069 of FD and DDMFDmethodsrespectively
Changes in the average of REPadjfdampddmfd in differentscenarios with speed ratio are shown in Figure 15 As shownin this figure REPadjfdampddmfd rapidly increases with speedratio which indicates the superiority of ADM over FD andDDMFDmethods increases with speed ratio
0 5 10 15 20 25 30 35 40 45 500
051
152
25
Element number
Mod
ule o
f ela
stici
ty
0 5 10 15 20 25 30 35 40 45 5005
101520
Dam
age i
ndex
Element number
Original modelDetected model
times105
Figure 6 Detection of damage location and amount in elements 57 12 15 24 and 37 and distribution of error in different elementswith ADM scheme
543 Relative Efficiency Parameter of DDMFD with respectto FD Method Figure 16 shows REPddmfdfd changes withrespect to speed parameter in different scenarios
14 Shock and Vibration
Table 10 REPadjddm ranges in different scenarios
Damage scenario Max REP Min REP AverageModel 1 Model 2 Model 1 Model 2 Model 1 Model 2
Figure 7 RPE contours with respect to noise level and loops in model 2
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(b) DDMmethod
Figure 8 RPE contours with respect to initial error and loops in model 2
Shock and Vibration 15
0
50
100
150
200
250
300
350
0103050709
DDMFDFD
DDMADM
Itera
tion
num
ber
Speed ratio
(a) All methods
0
2
4
6
8
10
12
14
16
18
0103050709
Itera
tion
num
ber
Speed ratio
DDMADM
(b) Analytical discrete methods
Figure 9 Comparison of iteration number between different sensitivity methods
DDMFD
RPE
FDDDMADM
0
01
02
03
04
05
06
07
0103050709Speed ratio
(a) All methods
DDMADM
0103050709
RPE
Speed ratio
00E + 00
50E minus 04
10E minus 03
15E minus 03
20E minus 03
25E minus 03
(b) Analytical discrete methods
Figure 10 Comparison of RPE between different sensitivity methods
Table 12 shows the relative efficiency parameter for dif-ferent speed parameters in different scenarios Again theefficiency parameter is between 0343 and 1726 with theaverage of 1044Thus compared to FDmethod theDDMFDmethod is slightly more effective
Changes in the average of REPddmfdfd in different sce-narios with speed ratio are shown in Figure 17 As shown inthis figure REPddmfdfd increases with speed ratio For speedratio lower than 05 efficiency parameter is lower than 1 whichmeans FD method is more effective for this range
55 Stability Figures 3 and 7 show that all describedmethodsare sensitive to noise and if noise level exceeds the valuesstated in Table 11 these methods lose their efficiency and arenot able to detect damage properly Hence in cases with noise
level larger than the values stated in Table 13 a denoisingtool like wavelet transforms should be used along with thesensitivity methodsWavelet transforms are mainly attractivebecause of their ability to compress and encode informationto reduce noise or to detect any local singular behavior of asignal [37]
Stability comparison of different methods shows that FDis the most stable method and stability of analytical methodsis slightly lower than that
Figures 4 and 8 show stability of sensitivity methodswith respect to the initial error Summary of results forstability investigation is presented in Table 13 which suggeststhat stability of numerical methods is better than analyticalmethods
16 Shock and Vibration
DDMFDFD
DDMADM
0
50
100
150
200
250
300
0103050709
Solu
tion
time
Speed ratio
(a) All methods
DDMADM
0
2
4
6
8
10
0103050709
Solu
tion
time
Speed ratio
(b) Analytical discrete methods
Figure 11 Comparison of solution time between different sensitivity methods
5-64-5
3-4
2-31-20-1
120572 = 01
120572 = 05
120572 = 09
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
6
123456
0
(a) Model 1
120572 = 01
120572 = 05
120572 = 09
25ndash32ndash2515ndash2
1ndash1505ndash10ndash05
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
63252151050
(b) Model 2
Figure 12 REPadjddm changes in different scenarios with respect to speed parameter
Table 11 REPadjadm ranges in different scenarios
Damage scenario Max REP Min REP AverageFD DDMFD FD DDMFD FD DDMFD
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
level 5 determining the remaining life in the structure(prognosis)
Dynamics-based SHM techniques assess the state of healthof a structural component on the basis of the detection andanalysis of its dynamic response Such techniques can beclassified on the basis of the type of response being consideredfor the investigations on the frequency or time domainof investigation and on the modality used to excite thecomponent [5]
The developments in the field of structural damage detec-tion (DD) using vibration data of civil engineering structureshave been recently studied by several authors some of themare briefly described herein
Doebling et al [6 7] have presented comprehensivereview of literature mainly focusing on frequency-domainmethods for damage detection in linear structures anddeclared that sufficient evidence exists to promote the useof measured vibration data for the detection of damage instructures using both forced-response testing and long-termmonitoring of ambient signals and there is a significant needin this field for research on the integration of theoreticalalgorithms with application-specific knowledge bases andpractical experimental constraints Another discussion onmethods of damage detection and location using naturalfrequency changes has been presented by Salawu [8] andhis study showed that damage detection using vibrationfrequencies is not very reliable Zou et al [9] summarizedthemethods on vibration-based damage detection and healthmonitoring for composite structures especially in delamina-tion modeling techniques and delamination detection
Alampalli and Fu [10] and Alampalli et al [11] conductedlaboratory and field studies on bridge structures to investigatethe feasibility of measuring bridge vibration for inspec-tion and evaluation These studies focused on sensitivityof measured modal parameters to damage Cross diagno-sis using multiple signatures involving natural frequenciesmode shapesmodal assurance criteria and coordinatemodalassurance criteria was shown to be necessary to detect thedamage Casas and Aparicio [12] studied concrete bridgestructures and investigated dynamic response as an inspec-tion tool to assess bearing conditions and girder crackingTheir study showed the need to investigate more than onenatural frequency and also to determine mode shapes inorder that the damage could be successfully detected andlocated
Damage detection usually requires amathematical modelon the structure in conjunction with experimental modalparameters of the structureThe identification approaches aremainly based on the change in the natural frequencies [13ndash15]mode shapes [16ndash18] ormeasuredmodal flexibility [7 19ndash21]
The frequency-domain DD algorithms have been morewidely developed and applied as the amount ofmeasured datais reduced dramatically after the transform thus they can behandled easily Unfortunately the effects of local damage onthe natural frequencies and mode shapes of higher modesare greater than lower ones but they are usually difficult tomeasure from experiments In addition structural dampingproperties cannot be identified in frequency domain DD
The time-domain DD may be an attractive one to over-come the drawbacks of the frequency-domain DD For time-domain DD the forced vibration responses of the structureare needed in the identification However in some cases itis either impractical or impossible to use artificial inputs toexcite the civil engineering structures so natural excitationmust be measured along with the structural responses toassess the dynamic characteristics [22 23] In recent yearssome researchers have investigated both the problem of loadidentification (moving load and impact load) and modalparameters identification under operational conditions [2425] In addition identification of the structural parametersapplying a moving load has been considered in many papersLaw et al [26] presented a novel moving force and prestressidentification method based on the finite element and thewavelet-based methods for a bridge-vehicle system Jianget al [27] identified the parameter of a vehicle moving onmultispan continuous bridges Zhu and Law [28] presented amethod for damage detection of a simply supported concretebridge structure in time domain using the interaction forcesfrom the moving vehicles as excitation
Sensitivity-based methods allow a wide choice of phys-ically meaningful parameters and this advantage has led totheir widespread use in damage detection Calculation ofsensitivity matrix has a significant effect on the efficiency ofthese methods Despite the high importance of calculationmethod of sensitivity matrix and optimizing its performancein DD procedure there is not literature on this regardIn this paper computational methods for sensitivity matrixare discussed and a novel sensitivity base damage detectionmethod in time-domain referred to as ldquoadjoint variablemethodrdquo is introduced Fundamental principles of proposedmethod are presented and its performance is compared withthe conventional methods and it is shown that the numericalcost is considerably reduced by using the concept of adjointvariable
The outline of the work is as follows inverse prob-lems along with model updating are briefly introduced inSection 2 Different methods for sensitivity analysis alongwith introduced method (adjoint variable method) areaddressed in Section 3 Numerical simulation is presented inSection 4 and comparison studies are presented in Section 5with studies on the effect of different factors whichmay affectthe accuracy and efficiency of different methods and finallyconclusion will be drawn in the last section
2 Finite Element Model Updating andInverse Problem
A key step in model-based damage identification is theupdating of the finite element model of the structure in sucha way that the measured responses can be reproduced by theFE model A general flowchart of this operation is given inFigure 1The identification procedure presented in this paperis a sensitivity-based model updating routine Sensitivitycoefficients are the derivatives of the system responses withrespect to the physical parameters or input excitation forceand are needed in the cost function of the flowchart ofFigure 1
Shock and Vibration 3
Starting values
Numerical modelupdating model parameters
Experimenttest specimen
Improved parameters
Minimization cost function
Identified model parameters
Figure 1 General flowchart of a FEM-updating [31]
21 Finite ElementModeling of Bridge Vibration underMovingLoads For a general finite element model of a linear elastictime-invariant structure the equation of motion is given by
Mz119905119905
+ Cz119905
+ Kz = Bf (1)
where M and K are mass and stiffness matrices and Cis damping matrix z
119905119905
and z119905
and z are the respectiveacceleration velocity and displacement vectors for the wholestructure and f is a vector of applied forces with matrix Bmapping these forces to the associated DOFs of the structureProportional damping is assumed to show the effect ofdamping ratio on the dynamic magnification factor Rayleighdamping in which the dampingmatrix is proportional to thecombination of the mass and stiffness matrices is used
C = 1198860
M + 1198861
K (2)
where 1198860
and 1198861
are constants to be determined fromtwo modal damping ratios The dynamic responses of thestructures can be obtained by direct numerical integrationusing Newmark method
22 Objective Functions The approach minimizes the differ-ence between response quantities (acceleration response) ofthe measured data and model predictions This problemmaybe expressed as the minimization of j where
and z(120572) are the measured and computed responsevectors 120572 is a vector of all unknown parameters and 120598 is theresponse residual vector
23 Nonlinear Model Updating for Damage Detection Whenthe parameters of a model are unknown they must be esti-mated using measured data Since the relationship betweenthe acceleration responses z
119894
and the fractional stiffnessparameter 120572 is nonlinear a nonlinear model updating tech-nique like the Gauss-Newton method is required This kindof method has the advantage that the second derivativeswhich can be challenging to compute are not required TheGauss-Newton method in the damage detection procedurecan be described in terms of the acceleration response at the119894th DOF of the structure as
z119889119897
(120572119889
) = z119906119897
(1205720
) + S (1205720) times Δ1205721
+ S (1205720 + Δ1205721) times Δ1205722 + sdot sdot sdot (4)
The superscripts 0 1 and 2 denote the iteration numbersIndex 119906 denotes the initial state or state 0 while index 119889denotes the final damage state z
119889119897
and z119906119897
are vectors of theacceleration response at the 119894th DOF of the damaged andintact states respectivelyThe damage identification equationfor (119896 + 1)th iteration is
Δz119896 = S119896 times Δ120572119896+1 (5)
where S119896 and Δz119896 are obtained from the 119896th iteration Theiteration in (5) starts with an initial value 1205720 leading to Δz0 =z119889119897
minus z119906119897
(1205720
) and S0 = S(1205720) The parameter vector 120572119896 =1205720
+sum119896
119894=1
Δ120572119894 Sensitivity matrix S119896 = S(120572119896) and the residual
vectorΔz119896 = z119889119897
minus z119906119897
(1205720
)minussum119896minus1
119894=0
S119894Δ120572119894+1 (119896 = 1 2 ) of thenext iteration is then computed from results in the previousiterations
The acceleration response vector z119906119897
from the physicalintact structure is computed in general from the associatedanalytical model via dynamic analysis z
119889119897
is the accelerationresponse of the model of the damaged structure In generalthe measured acceleration responses (including measure-ment errors) from the damaged structure are obtained for z
119889119897
The iteration is terminated when a preselected criterion is
met The final identified damaged vector becomes [17]
Δ120572 = Δ1205721
+ Δ1205722
+ sdot sdot sdot + Δ120572119899
(6)
where 119899 is the number of iterations
24 Regularization Like many other inverse problems thesolution of (5) is often ill-conditioned and regularizationtechniques are needed to provide bounds to the solutionTheaim of regularization in the inverse analysis is to promote cer-tain regions of parameter space where the model realizationshould exist The two most widely used regularization meth-ods are Tikhonov regularization [14] and truncated singularvalue decomposition [29] In the Tikhonov regularizationthe new cost function is defined as
119895 (Δ120572119896+1
120582) =
10038171003817100381710038171003817S119896 sdot Δ120572119896+1 minus Δz1198961003817100381710038171003817
The regularization parameter 120582 ge 0 controls the extent ofcontribution of the two errors to the cost function in (7) and
4 Shock and Vibration
the fractional stiffness change incrementΔ120572119896+1 is obtained byminimizing the cost function in (7)
The regularized solution fromminimizing the function in(7) can be written in the following form as
Δ120572119896+1
= ((S119896)119879
S119896 + 1205822I)minus1
(S119896)119879
Δz119896 (8)
To express the contribution of the singular values and thecorresponding vectors in the solution clearly and to showhowthe regularization parameter plays the role as the filter factorthe sensitivity matrix is singular value decomposed andsingular value decomposition (SVD) applies to the sensitivitymatrix S119896 to obtain
S119896 = UΣV119879 (9)
where U isin 119877119899119905times119899119905 and V isin 119877
119898times119898 are orthogonal matricessatisfying U119879U = I
119899119905
and V119879V = I119898
and matrix Σ has thesize of 119899119905 times 119898 with the singular values 120590
119894
(119894 = 1 2 119898) onthe diagonal arranged in a decreasing order such that 120590
1
ge
1205902
ge sdot sdot sdot ge 120590119898
ge 0 and zeros elsewhereThe regularized solution in (8) can be written as
Δ120572119896+1
=
119898
sum
119894=1
119891119894
U119879119894
Δz119896
120590119894
V119894
(10)
where 119891119894
= 1205902
119894
(1205902
119894
+ 1205822
) (119894 = 1 2 119898) are referred as filterfactors The solution norm Δ120572
119896+1
2
2
and the residual normS119896 sdot Δ120572119896+1 minus Δz1198962
10038171003817100381710038171003817S119896 sdot Δ120572119896+1 minus Δz1198961003817100381710038171003817
1003817
2
2
=
119898
sum
119894=1
(
1205822
1205902
119894
+ 1205822
U119879119894
Δz119896)2
(11)
These two quantities represent the smoothness and goodness-of-fit of the solution and they should be balanced by choosingan appropriate regularization parameter
25 Element Damage Index In the inverse problem of dam-age identification it is assumed that the stiffness matrix ofthe whole element decreases uniformly with damage andthe flexural rigidity EI
119894
of the 119894th finite element of the beambecomes 120573
119894
EI119894
when there is damage The fractional changein stiffness of an element can be expressed as [30]
ΔK119887119894
= (K119887119894
minus K119887119894
) = (1 minus 120573119894
)K119887119894
(12)
where K119887119894
and K119887119894
are the 119894th element stiffness matrices ofthe undamaged and damaged beam respectively ΔK
119887119894
is thestiffness reduction of the element A positive value of 120573
119894
isin
[0 1] will indicate a loss in the element stiffness The 119894thelement is undamaged when 120573
119894
= 1 and the stiffness of the119894th element is completely lost when 120573
119894
= 0
The stiffness matrix of the damaged structure is theassemblage of the entire element stiffness matrix K
119887119894
K119887
=
119873
sum
119894=1
A119879119894
K119887119894
A119894
=
119873
sum
119894=1
120573119894
A119879119894
K119887119894
A119894
(13)
where A119894
is the extended matrix of element nodal displace-ment that facilitates assembling of global stiffness matrixfrom the constituent element stiffness matrix
The objective of sensitivity analysis is to quantify the effectsof parameter variations on calculated results Terms such asinfluence importance ranking by importance and domi-nance are all related to sensitivity analysis
The simplest and most common procedure for assessingthe effects of parameter variations on a model is to varyselected input parameters rerun the code and record thecorresponding changes in the results or responses calculatedby the code The model parameters responsible for thelargest relative changes in a response are then consideredto be the most important for the respective response Forcomplexmodels though the large amount of computing timeneeded by such recalculations severely restricts the scope ofthis sensitivity analysis method in practice therefore themodeler who uses this method can investigate only a fewparameters that he judges a priori to be important [32]
When the parameter variations are small the traditionalway to assess their effects on calculated responses is byusing perturbation theory either directly or indirectly viavariational principles The basic aim of perturbation theoryis to predict the effects of small parameter variations withoutactually calculating the perturbed configuration but rather byusing solely unperturbed quantities
31 Methods of Structural Sensitivity Analysis Various meth-ods employed in design sensitivity analysis are listed inFigure 2 Three approaches are used to obtain the designsensitivity the approximation discrete and continuumapproaches In the approximation approach design sensi-tivity is obtained by either the forward finite difference orthe central finite difference method In the discrete methoddesign sensitivity is obtained by taking design derivativesof the discrete governing equation For this process it isnecessary to take the design derivative of the stiffness matrixIf this derivative is obtained analytically using the explicitexpression of the stiffness matrix with respect to the designvariable it is an analytical method since the analyticalexpressions of stiffness matrix are used However if thederivative is obtained using a finite difference method themethod is called a semianalytical method
In the continuum approach the design derivative ofthe variational equation is taken before it is discretized Ifthe structural problem and sensitivity equations are solvedas a continuum problem then it is called the continuum-continuum method However only very simple classical
Shock and Vibration 5
Sensitivity methods
Approximation approach
Forward finite difference
Centeral finite difference
Discrete approach
method
Analytical
Semianalytical
method
differential
Adjoint variable
Direct
method
Continuum approach
Continuum-discrete method
Continuum-continuum
method
method
Figure 2 Approaches to design sensitivity analysis
problems can be solved analytically Thus the continuumsensitivity equation is solved by discretization in the sameway that structural problems are solved Since differentiationis taken at the continuum domain and is then followed bydiscretization this method is called the continuum-discretemethod
32 Approximation Approach The easiest way to computesensitivity information of the performance measure is byusing the finite difference method Different designs yielddifferent analysis results and thus different performancevaluesThe finite differencemethod actually computes designsensitivity of performance by evaluating performance mea-sures at different stages in the design process If u is thecurrent design then the analysis results provide the valueof performance measure 120595(u) In addition if the design isperturbed to u + Δu where Δu represents a small change inthe design then the sensitivity of 120595(u) can be approximatedas
119889120595
119889uasymp
120595 (u + Δu) minus 120595 (u)Δu
(14)
Equation (14) is called the forward difference method sincethe design is perturbed in the direction of +Δu If minusΔuis substituted in (14) for Δu then the equation is definedas the backward difference method Additionally in centraldifference method the design is perturbed in both directionsand the design sensitivity is approximated by
119889120595
119889uasymp
120595 (u + Δu) minus 120595 (u minus Δu)2Δu
(15)
Then the equation is defined as the central differencemethod
33 Discrete Approach A structural problem is often dis-cretized in finite dimensional space in order to solve complex
problems The discrete method computes the performancedesign sensitivity of the discretized problem
The design represents a structural parameter that canaffect the results of the analysis
The design sensitivity information of a general perfor-mance measure can be computed either with the directdifferentiation method or with the adjoint variable method
331 Direct Differentiation Method The direct differentia-tion method (DDM) is a general accurate and efficientmethod to compute FE response sensitivities to the modelparameters This method directly solves for the designdependency of a state variable and then computes perfor-mance sensitivity using the chain rule of differentiationThis method clearly shows the implicit dependence on thedesign and a very simple sensitivity expression can beobtained
Consider a structure in which the generalized stiffnessand mass matrices have been reduced by accounting forboundary conditions Let the damping force be representedin the form of C(119887)z
119905
where z119905
= 119889z119889119905 denotes thevelocity vector Under these conditions Lagrangersquos equationof motion becomes the second-order differential equation as[33]
M (119887) z119905119905
+ C (119887) z119905
+ K (119887) z = f (119905 119887) (16)
With the initial conditions
z (0) = z0 z119905
(0) = z0119905
(17)
If design parameters are just related to stiffness matrix weobtain
M120597z119905119905
120597119887119894
+ C120597z119905
120597119887119894
+ K 120597z120597119887119894
= minus
120597K120597119887119894
z minus 1205722
120597K120597119887119894
z119905
(18)
6 Shock and Vibration
in which 120597z120597b119894 120597z119905
120597b119894 and 120597z119905119905
120597b119894 are sensitivity vectorsof displacement velocity and acceleration with respect todesign parameter 119887
119894
respectively Assume that
y119905119905
=
120597z119905119905
120597119887119894
(19a)
y119905
=
120597z119905
120597119887119894
(19b)
y = 120597z120597119887119894
(19c)
So by replacing (19a) (19b) and (19c) into (18) we have
My119905119905
+ Cy119905
+ Ky = minus120597K120597119887119894
z minus 1205722
120597K120597119887119894
z119905
(20)
Right side of (20) can be considered as an equivalent force so(20) is similar to (16) and sensitivity vectors can be obtainedby Newmark method
332 Adjoint Variable Method Mirzaee et al [34] provedthat sensitivity analysis can be performed very efficientlyby using deterministic methods based on adjoint functionsThis method constructs an adjoint problem that solves forthe adjoint variable which contains all implicitly dependentterms
The adjoint variablemethod yields a terminal-value prob-lem compared with the initial-value problem of responseanalysis
For the dynamic structure the following formof a generalperformance measure can be considered
120595 = 119892 (z (119879) 119887) + int119879
0
119866 (z 119887) 119889119905 (21)
General form of sensitivity measure relative to the designparameter (b) is as follows [33]
1205951015840
= [
120597119892
120597zminus
120597119892
120597zz119905
(119879) + 119866 (z (119879) 119887) 1Ω119905
120597Ω
120597119911
] z1015840 (119879)
minus [
120597119892
120597119911
z119905
(119879) + 119866 (z (119879) 119887)] 1
Ω119905
120597Ω
120597119911119905
z1015840119905
(119879)
+ int
119879
0
[
120597119866
120597zz1015840 + 120597119866
120597119887
120575119887] 119889119905 +
120597119892
120597119887
120575119887
minus [
120597119892
120597119911
z119905
(119879) + 119866 (119911 (119879) 119887)]
1
Ω119905
120597Ω
120597119887
120575119887
(22)
Note that1205951015840 depends on z1015840 and z1015840119905
at119879 as well as on z1015840 withinthe integration
Mirzaee et el [34] showed that while structural vibrationresponses are used for damage detection solving (22) inknown time 119879 is as follows
119889120595
119889119887
(119879) = int
119879
0
(minus120582119879
1198860
120597K120597119887
z119905
minus120582119879120597K120597119887
z)119889119905 (23)
120582119879
(119905) = P119879
119891 (119905) +Q119879
119892 (119905) (24)
in which
P119879
= 119890minus120585120596119879
120582119905
(119879)
120596119863
cos (120596119863
119879)
119891 (119905) = 119890120585120596119905 sin (120596
119863
119905)
Q119879
= minus119890minus120585120596119879
120582119905
(119879)
120596119863
sin (120596119863
119879)
119892 (119905) = 119890120585120596119905 cos (120596
119863
119905)
(25)
It is worth noting that terminal values for different values of119879 are not similar and adjoint equation should be calculatedfor corresponding 119879 separately
Mirzaee et al [34] presented a quite efficient incrementalsolution for solving (24) which significantly improves theproposed method
34 Sensitivity Method Selection In this paper four differentmethods are presented to compute sensitivity information ofthe performance measure for damage detection procedureFirst method is the finite Difference method (FD) whichis classified as an approximation approach Second it issemianalytical discrete method which uses a finite differencescheme to calculate stiffness matrix in direct differentialmethod (DDMFD) and third and fourthmethods are two dif-ferent analytical discrete methods direct differential method(DDM) and adjoint variable method (ADM)
The first three methods including FD DDMFD andDDM are widely used by other authors [35 36] and the lastmethod (ADM) is proposed by the authors (Mirzaee et al[34])
35 Procedure of Iteration for Damage Detection The initialanalytical model of a structure deviates from the true modeland measurement from the initial intact structure is used toupdate the analytical model The improved model is thentreated as a reference model and measurement from thedamaged structure will be used to update the referencemodel
When response measurement from the intact state of thestructure is obtained the sensitivities are computed frompresented methods based on the analytical model of thestructure and the well knowing input force and velocityThe vector of parameter increments is then obtained usingthe computed and experimentally obtained responses Theanalytical model is then updated and the correspondingresponse and its sensitivity are again computed for the nextiteration When measurement from the damaged state isobtained the updated analyticalmodel is used in the iterationin the same way as that using measurement from the intactstate Convergence is considered to be achieved when thefollowing criteria are met
1003817100381710038171003817E119894+1
minus E119894
1003817100381710038171003817
1003817100381710038171003817E119894
1003817100381710038171003817
times 100 le Tol1
1003817100381710038171003817Response
119894+1
minus Response119894
1003817100381710038171003817
1003817100381710038171003817Response
119894
1003817100381710038171003817
times 100 le Tol2
(26)
Shock and Vibration 7
The final vector of identified parameter increments corre-sponds to the changes occurring in between the two statesof the structure The tolerance is set equal to 1 times 10minus5 in thisstudy except otherwise specified
4 Numerical Results
In order to evaluate the efficiency and competency of intro-duced methods analysis of two FE models with extensiveavailable numerical studies has been carried out Represen-tative examples are presented to demonstrate the effects ofspeed of loads measurement noise level and initial error onthe accuracy and effectiveness of the methods
The relative percentage error (RPE) in the identifiedresults is calculated from (27) where sdot is the norm ofmatrix and z
119905119905identifiedand z119905119905true
are the identified and the trueacceleration time history respectively [26]
Since the true value of elastic modulus is unknown RPE canjust be used for evaluating the efficiency of the method
41 Single-Span Bridge A single-span bridge is studied tocompare different methods This bridge consists of 10 Euler-Bernoulli beam elements with 11 nodes each with two DOFsTotal length of bridge is 10m and height and width ofthe frame section are 200mm Rayleigh damping model isadopted with the damping ratios of the first two modes takenequal to 005
Structural properties are summarized in Table 1The transverse point load 119875 has a constant velocity V =
119871119879 where 119879 is the traveling time across the bridge and 119871 isthe total length of the bridge
For the forced vibration analysis an implicit time integra-tion method called as ldquothe Newmark integration methodrdquo isused with the integration parameters 120573 = 14 and 120574 = 12which leads to constant-average acceleration approximation
Speed parameter is defined as
120572 =
119881
119881cr (28)
in which119881cr is critical speed (119881cr = (120587119897)radicEI120588)119881 is movingload speed and 120588 is mass per unit length of the beams
411 Damage Scenarios Six damage scenarios of singlemultiple and random damage along with initial error in thebridge without measurement noise are studied and shown inTable 2
Local damage is simulated with a reduction in the elasticmodulus of material of an element The sampling rate is15000Hz and 250 data of the acceleration response (degree ofindeterminacy is 25) collected along the z-direction at nodes2 and 7 are used in the identification
412 Analysis Results Tables 3ndash6 show the results of differentdetection methods including solution time (ST) number of
loops (NL) and RPE These are the most important param-eters to compare the efficiency and accuracy of differentmethods
413 Effect of Noise Noise is the random fluctuation in thevalue of measured or input that causes random fluctuation inthe output value Noise at the sensor output is due to eitherinternal noise sources such as resistors at finite temperaturesor externally generatedmechanical and electromagnetic fluc-tuations [10]
To evaluate the sensitivity of results to suchmeasurementnoise noise-polluted measurements are simulated by addingto the noise-free acceleration vector a corresponding noisevector whose root-mean-square (RMS) value is equal toa certain percentage of the RMS value of the noise-freedata vector The components of all the noise vectors are ofGaussian distribution uncorrelated andwith a zeromean andunit standard deviation Then on the basis of the noise-freeacceleration z
119905119905nfthe noise-polluted acceleration z
119905119905npof the
bridge at location 119909 can be simulated by
z119905119905np
= z119905119905nf
+ RMS (z119905119905nf) times 119873level times 119873unit (29)
where RMS (z119905119905nf
) is the RMS value of the noise-free accel-eration vector z
119905119905nftimes 119873level is the noise level and 119873unit is a
randomly generated noise vector with zero mean and unitstandard deviation [36]
In order to study effects of noise in stability of differentsensitivity methods scenario 3 (speed ratio of moving loadis considered to be fixed and equal to 05) is considered anddifferent levels of noise pollution are investigated and alsoRPE change with increasing the number of loops in iterativeprocedure has been studied The results are illustrated inFigure 3 for all presented methods
414 Effect of Initial Error In order to study effects of initialerror in stability of sensitivity methods scenario 6 (speedratio ofmoving load is considered to be fixed and equal to 05)is considered This scenario consists of no simulated damagein the structure but the initial elastic modulus of materialof all the elements is underestimated and different levels ofinitial error are investigated RPE changes with increasing thenumber of loops in iterative procedure have been studiedTheresults are illustrated in Figure 4 for all presented methods
42 Plane Grid Model A plane grid model of bridge isstudied as another numerical example to investigate theefficiency and accuracy of discrete methods It is notable thataccording to complexity of this model approximation andsemidiscrete methods are not useable and diverged in allscenarios in all speed ranges
The structure is modeled by 46 frame elements and 32nodes with three DOFs at each node for the translation androtational deformations Rayleigh damping model is adoptedwith the damping ratios of the first two modes taken equalto 005 The finite element model of the structure is shownin Figure 5 and structural properties are summarized inTable 1
8 Shock and Vibration
Table1Prop
ertie
sofd
ifferentm
odels
Mod
elname
Num
bero
felements
Num
bero
fnod
esMassd
ensity
Mod
ulus
ofelasticity
Rayleigh
coeffi
cients
Und
amaged
naturalfrequ
ency
ofintactmod
elkgm
m3
Nm
m2
1198860
1198861
Firstm
ode
Second
mod
eTh
irdmod
eFo
urth
mod
eFifth
mod
eMultispan
1011
78times10minus9
21times
10minus5
01
2860times10minus5
296
1183
2662
4738
7421
Grid
4632
78times10minus9
21times
10minus5
01
2364times10minus5
456
928
1817
2597
3991
Shock and Vibration 9
Table 2 Damage scenarios for single-span bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 4 17 NilM1-2 Multiple 2 7 4 21 NilM1-3 Multiple 3 5 6 and 8 12 6 5 and 2 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
Table 3 Solution time number of loops and RPE of ADMmethod for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
421 Damage Scenarios Six damage scenarios of singlemultiple and randomdamage in the bridgewithoutmeasure-ment noise are studied and shown in Table 7
The sampling rate is 14000Hz and 460 data of the acceler-ation response (degree of indeterminacy is 20) collected alongthe z-direction at nodes 4 11 21 and 27 are used
422 Analysis Results Tables 8 and 9 show the results ofdifferent detection methods including solution time (ST)
and RPE Using both described methods including ADMand DDM method the damage locations and amount areidentified correctly in all the scenarios (Figure 6)
423 Effect of Noise In order to study effects of noisescenario 3 is considered and different levels of noise pollutionare investigated and also RPE changes with increasing thenumber of loops for iterative procedure have been studiedThe results are presented in Figure 7
10 Shock and Vibration
Table 6 Solution time loops and RPE of DDMFD method for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
Figure 3 RPE contours with respect to noise level and loops in model 1
Shock and Vibration 11
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
10
20
30
40
50
60
70
(b) DDMmethod
Loops
Initi
al er
ror
FD method
5 10 15 20 25 300
005
01
015
02
025
03
10
20
30
40
50
60
70
80
(c) FD method
Loops
Initi
al er
ror
DDMFD method
5 10 15 20 25 300
005
01
015
02
025
03
0
10
20
30
40
50
60
70
80
(d) DDMFD method
Figure 4 RPE contours with respect to initial error and loops in model 1
424 Effect of Initial Error Scenario 6 (speed ratio ofmovingload is considered to be fixed and equal to 05) is considered tostudy effects of initial error in stability of sensitivity methodsRPE changes with increasing the number of loops for iterativeprocedure have been studied The results are illustrated inFigure 8
5 Comparison of Different Methods
According to the results obtained in the previous sectiondifferent sensitivity methods considered in this study arecompared in this section Iteration number relative percent-age of error (RPE) solution time noise level and initialerror effect in stability of damage detection procedure areeffective parameters to compare efficiency and accuracy of themethods
51 Iteration Number Average number of iterations fordifferent scenarios with respect to the speed parameter isillustrated in Figure 9
As seen in Figure 9(a) the number of iterations in FDand DDMFD methods is considerably larger than thoseof analytical discrete methods (DDM and ADM) It hasbeen observed that in all cases DDMFD method has feweriterations than FD method Average number of iterations is20757 and 251467 for DDMFD and FD methods respec-tively The larger the speed ratio of moving vehicle is thelarger the number of iterations in approximation methodsis In FD method for speed ratios more than 05 even byusing maximum number of iterations (300) convergence wasnot achieved while the ratio used in DDMFD method is09
Figure 9(b) shows comparison along with analytical dis-crete methods It can be seen that DDM has smaller iterationnumber in all cases Average number of iterations is 146and 11167 for ADM and DDM methods respectively Bothanalytical discrete methods converged in all cases withmaximum iterations equal to 17 and increment in speed ratiodoes not affect the number of iterations
12 Shock and Vibration
Sensors
Element number
25
17
9
2622
15
8
1
23
16
9
2
24
17
10
3
25
18
11
4
2619
12
5
2720
13
6
2821
147
4118
3510
29
2
2742
1936
1130
3
2843
2037
1231
4
29 4421
3813
325
30 4522 39
14
16
336
3132
46 2324
40 1534 7
8
Direction of measured response for identification
Node number
P V
Moving vehicle
Y Z X
7000 (mm)
3000 (mm)
Figure 5 Plane grid bridge model used in detection procedure
Table 7 Damage scenarios for multispan bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 23 5 NilM1-2 Multiple 8 13 and 29 11 4 and 7 NilM1-3 Multiple 3 7 19 25 and 28 12 6 5 2 and 18 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
52 Relative Percentage of Error (RPE) Using the sameconvergence tolerance equal to 1 times 10minus5 relative percentageof error in different methods is illustrated in Figure 10
As seen in this figure with a maximum number ofiterations equal to 300 RPE in FD and DDMFD methodsare considerably larger than DDM and ADM methodsDDMFD method has lower RPE than FD method in mostcases Average of RPE is 026329 for DDMFD compared to027766 in FD method Again the larger the speed ratio ofmoving vehicle is the larger the relative error percentage inapproximation methods is
Figure 10(b) shows that relative error percentage in ADMmethod is lower than DDM and speed ratio does not affectthe relative error
Average of RPE is 0000731 and 0002037 for ADM andDDM methods respectively Therefore ADM is nearly 28times more accurate than DDM
53 Solution Time In order to investigate computational costof sensitivity methods using the same amounts of RAMand CPU resources for the presented numerical simulationssolution time of different damaged models is evaluated andshown in Figure 11
As seen in Figure 11(a) solution times in FD and DDMFDmethods are considerably larger than analytical discretemethods and the FDmethod with an average of 200616 (s) isthemost time consumingmethod Average time for DDMFDmethod is 150849 (s)
Figure 11(b) compares the time spent by DDM and ADMAs illustrated in this figure ADM method has lower solutiontime than DDMmethod in all cases and speed ratio does notaffect the solution time
Average of solution time is 2956 (s) and 7794 (s) forADM and DDM methods respectively consequently ADMis nearly 26 times faster than DDM
54 Efficiency Parameter In order to compare and quantifythe performance of different methods relative efficiencyparameters of methods ldquo119894rdquo and ldquo119895rdquo (REP
119894119895
) are defined as
REP119894119895
= radic
ST119895
ST119894
times
RPE119895
RPE119894
(30)
in which ST is the solution time of system identificationmethod that represents the computational cost of themethodand RPE is the relative error which represents the accuracy ofthe method
Shock and Vibration 13
Table 8 Solution time number of loops and RPE of ADMmethod for model 2
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE ()
541 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to DDM Figure 12 shows REPadjddm changeswith respect to speed parameter in different scenarios
Table 10 shows that in different scenarios and for differentspeed parameters the efficiency parameter is between 19713and 55478 for model 1 and 15659 and 29893 for model2 and its average is 28948 and 22248 for models 1 and2 respectively Compared to DDM the adjoint variablemethod is more efficient and needs 609 less computationaleffort
Changes in the average of REPadjddm in different scenarioswith speed ratio are shown in Figure 13 As shown in thisfigure REPadjddm increases with speed ratios lightly Thisindicates that efficiency of ADM with respect to DDMincreases with speed ratio
542 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to Approximation and Semianalytical MethodsFigure 14 shows REPadjFD and REPadjDDMFD changes withrespect to speed parameter in different scenarios
Table 11 shows the relative efficiency parameter for differ-ent speed parameters in different scenarios It can be seen thatthe efficiency parameter is between 3193 and 78307 for finitedifference method and 4916 and 50542 for DDMFDmethodwith the average of 16714 and 14569 for FD and DDMFDmethods respectively Therefore the adjoint variable methodis extremely successful and the computational cost of thismethod is just 06 and 069 of FD and DDMFDmethodsrespectively
Changes in the average of REPadjfdampddmfd in differentscenarios with speed ratio are shown in Figure 15 As shownin this figure REPadjfdampddmfd rapidly increases with speedratio which indicates the superiority of ADM over FD andDDMFDmethods increases with speed ratio
0 5 10 15 20 25 30 35 40 45 500
051
152
25
Element number
Mod
ule o
f ela
stici
ty
0 5 10 15 20 25 30 35 40 45 5005
101520
Dam
age i
ndex
Element number
Original modelDetected model
times105
Figure 6 Detection of damage location and amount in elements 57 12 15 24 and 37 and distribution of error in different elementswith ADM scheme
543 Relative Efficiency Parameter of DDMFD with respectto FD Method Figure 16 shows REPddmfdfd changes withrespect to speed parameter in different scenarios
14 Shock and Vibration
Table 10 REPadjddm ranges in different scenarios
Damage scenario Max REP Min REP AverageModel 1 Model 2 Model 1 Model 2 Model 1 Model 2
Figure 7 RPE contours with respect to noise level and loops in model 2
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(b) DDMmethod
Figure 8 RPE contours with respect to initial error and loops in model 2
Shock and Vibration 15
0
50
100
150
200
250
300
350
0103050709
DDMFDFD
DDMADM
Itera
tion
num
ber
Speed ratio
(a) All methods
0
2
4
6
8
10
12
14
16
18
0103050709
Itera
tion
num
ber
Speed ratio
DDMADM
(b) Analytical discrete methods
Figure 9 Comparison of iteration number between different sensitivity methods
DDMFD
RPE
FDDDMADM
0
01
02
03
04
05
06
07
0103050709Speed ratio
(a) All methods
DDMADM
0103050709
RPE
Speed ratio
00E + 00
50E minus 04
10E minus 03
15E minus 03
20E minus 03
25E minus 03
(b) Analytical discrete methods
Figure 10 Comparison of RPE between different sensitivity methods
Table 12 shows the relative efficiency parameter for dif-ferent speed parameters in different scenarios Again theefficiency parameter is between 0343 and 1726 with theaverage of 1044Thus compared to FDmethod theDDMFDmethod is slightly more effective
Changes in the average of REPddmfdfd in different sce-narios with speed ratio are shown in Figure 17 As shown inthis figure REPddmfdfd increases with speed ratio For speedratio lower than 05 efficiency parameter is lower than 1 whichmeans FD method is more effective for this range
55 Stability Figures 3 and 7 show that all describedmethodsare sensitive to noise and if noise level exceeds the valuesstated in Table 11 these methods lose their efficiency and arenot able to detect damage properly Hence in cases with noise
level larger than the values stated in Table 13 a denoisingtool like wavelet transforms should be used along with thesensitivity methodsWavelet transforms are mainly attractivebecause of their ability to compress and encode informationto reduce noise or to detect any local singular behavior of asignal [37]
Stability comparison of different methods shows that FDis the most stable method and stability of analytical methodsis slightly lower than that
Figures 4 and 8 show stability of sensitivity methodswith respect to the initial error Summary of results forstability investigation is presented in Table 13 which suggeststhat stability of numerical methods is better than analyticalmethods
16 Shock and Vibration
DDMFDFD
DDMADM
0
50
100
150
200
250
300
0103050709
Solu
tion
time
Speed ratio
(a) All methods
DDMADM
0
2
4
6
8
10
0103050709
Solu
tion
time
Speed ratio
(b) Analytical discrete methods
Figure 11 Comparison of solution time between different sensitivity methods
5-64-5
3-4
2-31-20-1
120572 = 01
120572 = 05
120572 = 09
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
6
123456
0
(a) Model 1
120572 = 01
120572 = 05
120572 = 09
25ndash32ndash2515ndash2
1ndash1505ndash10ndash05
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
63252151050
(b) Model 2
Figure 12 REPadjddm changes in different scenarios with respect to speed parameter
Table 11 REPadjadm ranges in different scenarios
Damage scenario Max REP Min REP AverageFD DDMFD FD DDMFD FD DDMFD
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
21 Finite ElementModeling of Bridge Vibration underMovingLoads For a general finite element model of a linear elastictime-invariant structure the equation of motion is given by
Mz119905119905
+ Cz119905
+ Kz = Bf (1)
where M and K are mass and stiffness matrices and Cis damping matrix z
119905119905
and z119905
and z are the respectiveacceleration velocity and displacement vectors for the wholestructure and f is a vector of applied forces with matrix Bmapping these forces to the associated DOFs of the structureProportional damping is assumed to show the effect ofdamping ratio on the dynamic magnification factor Rayleighdamping in which the dampingmatrix is proportional to thecombination of the mass and stiffness matrices is used
C = 1198860
M + 1198861
K (2)
where 1198860
and 1198861
are constants to be determined fromtwo modal damping ratios The dynamic responses of thestructures can be obtained by direct numerical integrationusing Newmark method
22 Objective Functions The approach minimizes the differ-ence between response quantities (acceleration response) ofthe measured data and model predictions This problemmaybe expressed as the minimization of j where
and z(120572) are the measured and computed responsevectors 120572 is a vector of all unknown parameters and 120598 is theresponse residual vector
23 Nonlinear Model Updating for Damage Detection Whenthe parameters of a model are unknown they must be esti-mated using measured data Since the relationship betweenthe acceleration responses z
119894
and the fractional stiffnessparameter 120572 is nonlinear a nonlinear model updating tech-nique like the Gauss-Newton method is required This kindof method has the advantage that the second derivativeswhich can be challenging to compute are not required TheGauss-Newton method in the damage detection procedurecan be described in terms of the acceleration response at the119894th DOF of the structure as
z119889119897
(120572119889
) = z119906119897
(1205720
) + S (1205720) times Δ1205721
+ S (1205720 + Δ1205721) times Δ1205722 + sdot sdot sdot (4)
The superscripts 0 1 and 2 denote the iteration numbersIndex 119906 denotes the initial state or state 0 while index 119889denotes the final damage state z
119889119897
and z119906119897
are vectors of theacceleration response at the 119894th DOF of the damaged andintact states respectivelyThe damage identification equationfor (119896 + 1)th iteration is
Δz119896 = S119896 times Δ120572119896+1 (5)
where S119896 and Δz119896 are obtained from the 119896th iteration Theiteration in (5) starts with an initial value 1205720 leading to Δz0 =z119889119897
minus z119906119897
(1205720
) and S0 = S(1205720) The parameter vector 120572119896 =1205720
+sum119896
119894=1
Δ120572119894 Sensitivity matrix S119896 = S(120572119896) and the residual
vectorΔz119896 = z119889119897
minus z119906119897
(1205720
)minussum119896minus1
119894=0
S119894Δ120572119894+1 (119896 = 1 2 ) of thenext iteration is then computed from results in the previousiterations
The acceleration response vector z119906119897
from the physicalintact structure is computed in general from the associatedanalytical model via dynamic analysis z
119889119897
is the accelerationresponse of the model of the damaged structure In generalthe measured acceleration responses (including measure-ment errors) from the damaged structure are obtained for z
119889119897
The iteration is terminated when a preselected criterion is
met The final identified damaged vector becomes [17]
Δ120572 = Δ1205721
+ Δ1205722
+ sdot sdot sdot + Δ120572119899
(6)
where 119899 is the number of iterations
24 Regularization Like many other inverse problems thesolution of (5) is often ill-conditioned and regularizationtechniques are needed to provide bounds to the solutionTheaim of regularization in the inverse analysis is to promote cer-tain regions of parameter space where the model realizationshould exist The two most widely used regularization meth-ods are Tikhonov regularization [14] and truncated singularvalue decomposition [29] In the Tikhonov regularizationthe new cost function is defined as
119895 (Δ120572119896+1
120582) =
10038171003817100381710038171003817S119896 sdot Δ120572119896+1 minus Δz1198961003817100381710038171003817
The regularization parameter 120582 ge 0 controls the extent ofcontribution of the two errors to the cost function in (7) and
4 Shock and Vibration
the fractional stiffness change incrementΔ120572119896+1 is obtained byminimizing the cost function in (7)
The regularized solution fromminimizing the function in(7) can be written in the following form as
Δ120572119896+1
= ((S119896)119879
S119896 + 1205822I)minus1
(S119896)119879
Δz119896 (8)
To express the contribution of the singular values and thecorresponding vectors in the solution clearly and to showhowthe regularization parameter plays the role as the filter factorthe sensitivity matrix is singular value decomposed andsingular value decomposition (SVD) applies to the sensitivitymatrix S119896 to obtain
S119896 = UΣV119879 (9)
where U isin 119877119899119905times119899119905 and V isin 119877
119898times119898 are orthogonal matricessatisfying U119879U = I
119899119905
and V119879V = I119898
and matrix Σ has thesize of 119899119905 times 119898 with the singular values 120590
119894
(119894 = 1 2 119898) onthe diagonal arranged in a decreasing order such that 120590
1
ge
1205902
ge sdot sdot sdot ge 120590119898
ge 0 and zeros elsewhereThe regularized solution in (8) can be written as
Δ120572119896+1
=
119898
sum
119894=1
119891119894
U119879119894
Δz119896
120590119894
V119894
(10)
where 119891119894
= 1205902
119894
(1205902
119894
+ 1205822
) (119894 = 1 2 119898) are referred as filterfactors The solution norm Δ120572
119896+1
2
2
and the residual normS119896 sdot Δ120572119896+1 minus Δz1198962
10038171003817100381710038171003817S119896 sdot Δ120572119896+1 minus Δz1198961003817100381710038171003817
1003817
2
2
=
119898
sum
119894=1
(
1205822
1205902
119894
+ 1205822
U119879119894
Δz119896)2
(11)
These two quantities represent the smoothness and goodness-of-fit of the solution and they should be balanced by choosingan appropriate regularization parameter
25 Element Damage Index In the inverse problem of dam-age identification it is assumed that the stiffness matrix ofthe whole element decreases uniformly with damage andthe flexural rigidity EI
119894
of the 119894th finite element of the beambecomes 120573
119894
EI119894
when there is damage The fractional changein stiffness of an element can be expressed as [30]
ΔK119887119894
= (K119887119894
minus K119887119894
) = (1 minus 120573119894
)K119887119894
(12)
where K119887119894
and K119887119894
are the 119894th element stiffness matrices ofthe undamaged and damaged beam respectively ΔK
119887119894
is thestiffness reduction of the element A positive value of 120573
119894
isin
[0 1] will indicate a loss in the element stiffness The 119894thelement is undamaged when 120573
119894
= 1 and the stiffness of the119894th element is completely lost when 120573
119894
= 0
The stiffness matrix of the damaged structure is theassemblage of the entire element stiffness matrix K
119887119894
K119887
=
119873
sum
119894=1
A119879119894
K119887119894
A119894
=
119873
sum
119894=1
120573119894
A119879119894
K119887119894
A119894
(13)
where A119894
is the extended matrix of element nodal displace-ment that facilitates assembling of global stiffness matrixfrom the constituent element stiffness matrix
The objective of sensitivity analysis is to quantify the effectsof parameter variations on calculated results Terms such asinfluence importance ranking by importance and domi-nance are all related to sensitivity analysis
The simplest and most common procedure for assessingthe effects of parameter variations on a model is to varyselected input parameters rerun the code and record thecorresponding changes in the results or responses calculatedby the code The model parameters responsible for thelargest relative changes in a response are then consideredto be the most important for the respective response Forcomplexmodels though the large amount of computing timeneeded by such recalculations severely restricts the scope ofthis sensitivity analysis method in practice therefore themodeler who uses this method can investigate only a fewparameters that he judges a priori to be important [32]
When the parameter variations are small the traditionalway to assess their effects on calculated responses is byusing perturbation theory either directly or indirectly viavariational principles The basic aim of perturbation theoryis to predict the effects of small parameter variations withoutactually calculating the perturbed configuration but rather byusing solely unperturbed quantities
31 Methods of Structural Sensitivity Analysis Various meth-ods employed in design sensitivity analysis are listed inFigure 2 Three approaches are used to obtain the designsensitivity the approximation discrete and continuumapproaches In the approximation approach design sensi-tivity is obtained by either the forward finite difference orthe central finite difference method In the discrete methoddesign sensitivity is obtained by taking design derivativesof the discrete governing equation For this process it isnecessary to take the design derivative of the stiffness matrixIf this derivative is obtained analytically using the explicitexpression of the stiffness matrix with respect to the designvariable it is an analytical method since the analyticalexpressions of stiffness matrix are used However if thederivative is obtained using a finite difference method themethod is called a semianalytical method
In the continuum approach the design derivative ofthe variational equation is taken before it is discretized Ifthe structural problem and sensitivity equations are solvedas a continuum problem then it is called the continuum-continuum method However only very simple classical
Shock and Vibration 5
Sensitivity methods
Approximation approach
Forward finite difference
Centeral finite difference
Discrete approach
method
Analytical
Semianalytical
method
differential
Adjoint variable
Direct
method
Continuum approach
Continuum-discrete method
Continuum-continuum
method
method
Figure 2 Approaches to design sensitivity analysis
problems can be solved analytically Thus the continuumsensitivity equation is solved by discretization in the sameway that structural problems are solved Since differentiationis taken at the continuum domain and is then followed bydiscretization this method is called the continuum-discretemethod
32 Approximation Approach The easiest way to computesensitivity information of the performance measure is byusing the finite difference method Different designs yielddifferent analysis results and thus different performancevaluesThe finite differencemethod actually computes designsensitivity of performance by evaluating performance mea-sures at different stages in the design process If u is thecurrent design then the analysis results provide the valueof performance measure 120595(u) In addition if the design isperturbed to u + Δu where Δu represents a small change inthe design then the sensitivity of 120595(u) can be approximatedas
119889120595
119889uasymp
120595 (u + Δu) minus 120595 (u)Δu
(14)
Equation (14) is called the forward difference method sincethe design is perturbed in the direction of +Δu If minusΔuis substituted in (14) for Δu then the equation is definedas the backward difference method Additionally in centraldifference method the design is perturbed in both directionsand the design sensitivity is approximated by
119889120595
119889uasymp
120595 (u + Δu) minus 120595 (u minus Δu)2Δu
(15)
Then the equation is defined as the central differencemethod
33 Discrete Approach A structural problem is often dis-cretized in finite dimensional space in order to solve complex
problems The discrete method computes the performancedesign sensitivity of the discretized problem
The design represents a structural parameter that canaffect the results of the analysis
The design sensitivity information of a general perfor-mance measure can be computed either with the directdifferentiation method or with the adjoint variable method
331 Direct Differentiation Method The direct differentia-tion method (DDM) is a general accurate and efficientmethod to compute FE response sensitivities to the modelparameters This method directly solves for the designdependency of a state variable and then computes perfor-mance sensitivity using the chain rule of differentiationThis method clearly shows the implicit dependence on thedesign and a very simple sensitivity expression can beobtained
Consider a structure in which the generalized stiffnessand mass matrices have been reduced by accounting forboundary conditions Let the damping force be representedin the form of C(119887)z
119905
where z119905
= 119889z119889119905 denotes thevelocity vector Under these conditions Lagrangersquos equationof motion becomes the second-order differential equation as[33]
M (119887) z119905119905
+ C (119887) z119905
+ K (119887) z = f (119905 119887) (16)
With the initial conditions
z (0) = z0 z119905
(0) = z0119905
(17)
If design parameters are just related to stiffness matrix weobtain
M120597z119905119905
120597119887119894
+ C120597z119905
120597119887119894
+ K 120597z120597119887119894
= minus
120597K120597119887119894
z minus 1205722
120597K120597119887119894
z119905
(18)
6 Shock and Vibration
in which 120597z120597b119894 120597z119905
120597b119894 and 120597z119905119905
120597b119894 are sensitivity vectorsof displacement velocity and acceleration with respect todesign parameter 119887
119894
respectively Assume that
y119905119905
=
120597z119905119905
120597119887119894
(19a)
y119905
=
120597z119905
120597119887119894
(19b)
y = 120597z120597119887119894
(19c)
So by replacing (19a) (19b) and (19c) into (18) we have
My119905119905
+ Cy119905
+ Ky = minus120597K120597119887119894
z minus 1205722
120597K120597119887119894
z119905
(20)
Right side of (20) can be considered as an equivalent force so(20) is similar to (16) and sensitivity vectors can be obtainedby Newmark method
332 Adjoint Variable Method Mirzaee et al [34] provedthat sensitivity analysis can be performed very efficientlyby using deterministic methods based on adjoint functionsThis method constructs an adjoint problem that solves forthe adjoint variable which contains all implicitly dependentterms
The adjoint variablemethod yields a terminal-value prob-lem compared with the initial-value problem of responseanalysis
For the dynamic structure the following formof a generalperformance measure can be considered
120595 = 119892 (z (119879) 119887) + int119879
0
119866 (z 119887) 119889119905 (21)
General form of sensitivity measure relative to the designparameter (b) is as follows [33]
1205951015840
= [
120597119892
120597zminus
120597119892
120597zz119905
(119879) + 119866 (z (119879) 119887) 1Ω119905
120597Ω
120597119911
] z1015840 (119879)
minus [
120597119892
120597119911
z119905
(119879) + 119866 (z (119879) 119887)] 1
Ω119905
120597Ω
120597119911119905
z1015840119905
(119879)
+ int
119879
0
[
120597119866
120597zz1015840 + 120597119866
120597119887
120575119887] 119889119905 +
120597119892
120597119887
120575119887
minus [
120597119892
120597119911
z119905
(119879) + 119866 (119911 (119879) 119887)]
1
Ω119905
120597Ω
120597119887
120575119887
(22)
Note that1205951015840 depends on z1015840 and z1015840119905
at119879 as well as on z1015840 withinthe integration
Mirzaee et el [34] showed that while structural vibrationresponses are used for damage detection solving (22) inknown time 119879 is as follows
119889120595
119889119887
(119879) = int
119879
0
(minus120582119879
1198860
120597K120597119887
z119905
minus120582119879120597K120597119887
z)119889119905 (23)
120582119879
(119905) = P119879
119891 (119905) +Q119879
119892 (119905) (24)
in which
P119879
= 119890minus120585120596119879
120582119905
(119879)
120596119863
cos (120596119863
119879)
119891 (119905) = 119890120585120596119905 sin (120596
119863
119905)
Q119879
= minus119890minus120585120596119879
120582119905
(119879)
120596119863
sin (120596119863
119879)
119892 (119905) = 119890120585120596119905 cos (120596
119863
119905)
(25)
It is worth noting that terminal values for different values of119879 are not similar and adjoint equation should be calculatedfor corresponding 119879 separately
Mirzaee et al [34] presented a quite efficient incrementalsolution for solving (24) which significantly improves theproposed method
34 Sensitivity Method Selection In this paper four differentmethods are presented to compute sensitivity information ofthe performance measure for damage detection procedureFirst method is the finite Difference method (FD) whichis classified as an approximation approach Second it issemianalytical discrete method which uses a finite differencescheme to calculate stiffness matrix in direct differentialmethod (DDMFD) and third and fourthmethods are two dif-ferent analytical discrete methods direct differential method(DDM) and adjoint variable method (ADM)
The first three methods including FD DDMFD andDDM are widely used by other authors [35 36] and the lastmethod (ADM) is proposed by the authors (Mirzaee et al[34])
35 Procedure of Iteration for Damage Detection The initialanalytical model of a structure deviates from the true modeland measurement from the initial intact structure is used toupdate the analytical model The improved model is thentreated as a reference model and measurement from thedamaged structure will be used to update the referencemodel
When response measurement from the intact state of thestructure is obtained the sensitivities are computed frompresented methods based on the analytical model of thestructure and the well knowing input force and velocityThe vector of parameter increments is then obtained usingthe computed and experimentally obtained responses Theanalytical model is then updated and the correspondingresponse and its sensitivity are again computed for the nextiteration When measurement from the damaged state isobtained the updated analyticalmodel is used in the iterationin the same way as that using measurement from the intactstate Convergence is considered to be achieved when thefollowing criteria are met
1003817100381710038171003817E119894+1
minus E119894
1003817100381710038171003817
1003817100381710038171003817E119894
1003817100381710038171003817
times 100 le Tol1
1003817100381710038171003817Response
119894+1
minus Response119894
1003817100381710038171003817
1003817100381710038171003817Response
119894
1003817100381710038171003817
times 100 le Tol2
(26)
Shock and Vibration 7
The final vector of identified parameter increments corre-sponds to the changes occurring in between the two statesof the structure The tolerance is set equal to 1 times 10minus5 in thisstudy except otherwise specified
4 Numerical Results
In order to evaluate the efficiency and competency of intro-duced methods analysis of two FE models with extensiveavailable numerical studies has been carried out Represen-tative examples are presented to demonstrate the effects ofspeed of loads measurement noise level and initial error onthe accuracy and effectiveness of the methods
The relative percentage error (RPE) in the identifiedresults is calculated from (27) where sdot is the norm ofmatrix and z
119905119905identifiedand z119905119905true
are the identified and the trueacceleration time history respectively [26]
Since the true value of elastic modulus is unknown RPE canjust be used for evaluating the efficiency of the method
41 Single-Span Bridge A single-span bridge is studied tocompare different methods This bridge consists of 10 Euler-Bernoulli beam elements with 11 nodes each with two DOFsTotal length of bridge is 10m and height and width ofthe frame section are 200mm Rayleigh damping model isadopted with the damping ratios of the first two modes takenequal to 005
Structural properties are summarized in Table 1The transverse point load 119875 has a constant velocity V =
119871119879 where 119879 is the traveling time across the bridge and 119871 isthe total length of the bridge
For the forced vibration analysis an implicit time integra-tion method called as ldquothe Newmark integration methodrdquo isused with the integration parameters 120573 = 14 and 120574 = 12which leads to constant-average acceleration approximation
Speed parameter is defined as
120572 =
119881
119881cr (28)
in which119881cr is critical speed (119881cr = (120587119897)radicEI120588)119881 is movingload speed and 120588 is mass per unit length of the beams
411 Damage Scenarios Six damage scenarios of singlemultiple and random damage along with initial error in thebridge without measurement noise are studied and shown inTable 2
Local damage is simulated with a reduction in the elasticmodulus of material of an element The sampling rate is15000Hz and 250 data of the acceleration response (degree ofindeterminacy is 25) collected along the z-direction at nodes2 and 7 are used in the identification
412 Analysis Results Tables 3ndash6 show the results of differentdetection methods including solution time (ST) number of
loops (NL) and RPE These are the most important param-eters to compare the efficiency and accuracy of differentmethods
413 Effect of Noise Noise is the random fluctuation in thevalue of measured or input that causes random fluctuation inthe output value Noise at the sensor output is due to eitherinternal noise sources such as resistors at finite temperaturesor externally generatedmechanical and electromagnetic fluc-tuations [10]
To evaluate the sensitivity of results to suchmeasurementnoise noise-polluted measurements are simulated by addingto the noise-free acceleration vector a corresponding noisevector whose root-mean-square (RMS) value is equal toa certain percentage of the RMS value of the noise-freedata vector The components of all the noise vectors are ofGaussian distribution uncorrelated andwith a zeromean andunit standard deviation Then on the basis of the noise-freeacceleration z
119905119905nfthe noise-polluted acceleration z
119905119905npof the
bridge at location 119909 can be simulated by
z119905119905np
= z119905119905nf
+ RMS (z119905119905nf) times 119873level times 119873unit (29)
where RMS (z119905119905nf
) is the RMS value of the noise-free accel-eration vector z
119905119905nftimes 119873level is the noise level and 119873unit is a
randomly generated noise vector with zero mean and unitstandard deviation [36]
In order to study effects of noise in stability of differentsensitivity methods scenario 3 (speed ratio of moving loadis considered to be fixed and equal to 05) is considered anddifferent levels of noise pollution are investigated and alsoRPE change with increasing the number of loops in iterativeprocedure has been studied The results are illustrated inFigure 3 for all presented methods
414 Effect of Initial Error In order to study effects of initialerror in stability of sensitivity methods scenario 6 (speedratio ofmoving load is considered to be fixed and equal to 05)is considered This scenario consists of no simulated damagein the structure but the initial elastic modulus of materialof all the elements is underestimated and different levels ofinitial error are investigated RPE changes with increasing thenumber of loops in iterative procedure have been studiedTheresults are illustrated in Figure 4 for all presented methods
42 Plane Grid Model A plane grid model of bridge isstudied as another numerical example to investigate theefficiency and accuracy of discrete methods It is notable thataccording to complexity of this model approximation andsemidiscrete methods are not useable and diverged in allscenarios in all speed ranges
The structure is modeled by 46 frame elements and 32nodes with three DOFs at each node for the translation androtational deformations Rayleigh damping model is adoptedwith the damping ratios of the first two modes taken equalto 005 The finite element model of the structure is shownin Figure 5 and structural properties are summarized inTable 1
8 Shock and Vibration
Table1Prop
ertie
sofd
ifferentm
odels
Mod
elname
Num
bero
felements
Num
bero
fnod
esMassd
ensity
Mod
ulus
ofelasticity
Rayleigh
coeffi
cients
Und
amaged
naturalfrequ
ency
ofintactmod
elkgm
m3
Nm
m2
1198860
1198861
Firstm
ode
Second
mod
eTh
irdmod
eFo
urth
mod
eFifth
mod
eMultispan
1011
78times10minus9
21times
10minus5
01
2860times10minus5
296
1183
2662
4738
7421
Grid
4632
78times10minus9
21times
10minus5
01
2364times10minus5
456
928
1817
2597
3991
Shock and Vibration 9
Table 2 Damage scenarios for single-span bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 4 17 NilM1-2 Multiple 2 7 4 21 NilM1-3 Multiple 3 5 6 and 8 12 6 5 and 2 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
Table 3 Solution time number of loops and RPE of ADMmethod for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
421 Damage Scenarios Six damage scenarios of singlemultiple and randomdamage in the bridgewithoutmeasure-ment noise are studied and shown in Table 7
The sampling rate is 14000Hz and 460 data of the acceler-ation response (degree of indeterminacy is 20) collected alongthe z-direction at nodes 4 11 21 and 27 are used
422 Analysis Results Tables 8 and 9 show the results ofdifferent detection methods including solution time (ST)
and RPE Using both described methods including ADMand DDM method the damage locations and amount areidentified correctly in all the scenarios (Figure 6)
423 Effect of Noise In order to study effects of noisescenario 3 is considered and different levels of noise pollutionare investigated and also RPE changes with increasing thenumber of loops for iterative procedure have been studiedThe results are presented in Figure 7
10 Shock and Vibration
Table 6 Solution time loops and RPE of DDMFD method for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
Figure 3 RPE contours with respect to noise level and loops in model 1
Shock and Vibration 11
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
10
20
30
40
50
60
70
(b) DDMmethod
Loops
Initi
al er
ror
FD method
5 10 15 20 25 300
005
01
015
02
025
03
10
20
30
40
50
60
70
80
(c) FD method
Loops
Initi
al er
ror
DDMFD method
5 10 15 20 25 300
005
01
015
02
025
03
0
10
20
30
40
50
60
70
80
(d) DDMFD method
Figure 4 RPE contours with respect to initial error and loops in model 1
424 Effect of Initial Error Scenario 6 (speed ratio ofmovingload is considered to be fixed and equal to 05) is considered tostudy effects of initial error in stability of sensitivity methodsRPE changes with increasing the number of loops for iterativeprocedure have been studied The results are illustrated inFigure 8
5 Comparison of Different Methods
According to the results obtained in the previous sectiondifferent sensitivity methods considered in this study arecompared in this section Iteration number relative percent-age of error (RPE) solution time noise level and initialerror effect in stability of damage detection procedure areeffective parameters to compare efficiency and accuracy of themethods
51 Iteration Number Average number of iterations fordifferent scenarios with respect to the speed parameter isillustrated in Figure 9
As seen in Figure 9(a) the number of iterations in FDand DDMFD methods is considerably larger than thoseof analytical discrete methods (DDM and ADM) It hasbeen observed that in all cases DDMFD method has feweriterations than FD method Average number of iterations is20757 and 251467 for DDMFD and FD methods respec-tively The larger the speed ratio of moving vehicle is thelarger the number of iterations in approximation methodsis In FD method for speed ratios more than 05 even byusing maximum number of iterations (300) convergence wasnot achieved while the ratio used in DDMFD method is09
Figure 9(b) shows comparison along with analytical dis-crete methods It can be seen that DDM has smaller iterationnumber in all cases Average number of iterations is 146and 11167 for ADM and DDM methods respectively Bothanalytical discrete methods converged in all cases withmaximum iterations equal to 17 and increment in speed ratiodoes not affect the number of iterations
12 Shock and Vibration
Sensors
Element number
25
17
9
2622
15
8
1
23
16
9
2
24
17
10
3
25
18
11
4
2619
12
5
2720
13
6
2821
147
4118
3510
29
2
2742
1936
1130
3
2843
2037
1231
4
29 4421
3813
325
30 4522 39
14
16
336
3132
46 2324
40 1534 7
8
Direction of measured response for identification
Node number
P V
Moving vehicle
Y Z X
7000 (mm)
3000 (mm)
Figure 5 Plane grid bridge model used in detection procedure
Table 7 Damage scenarios for multispan bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 23 5 NilM1-2 Multiple 8 13 and 29 11 4 and 7 NilM1-3 Multiple 3 7 19 25 and 28 12 6 5 2 and 18 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
52 Relative Percentage of Error (RPE) Using the sameconvergence tolerance equal to 1 times 10minus5 relative percentageof error in different methods is illustrated in Figure 10
As seen in this figure with a maximum number ofiterations equal to 300 RPE in FD and DDMFD methodsare considerably larger than DDM and ADM methodsDDMFD method has lower RPE than FD method in mostcases Average of RPE is 026329 for DDMFD compared to027766 in FD method Again the larger the speed ratio ofmoving vehicle is the larger the relative error percentage inapproximation methods is
Figure 10(b) shows that relative error percentage in ADMmethod is lower than DDM and speed ratio does not affectthe relative error
Average of RPE is 0000731 and 0002037 for ADM andDDM methods respectively Therefore ADM is nearly 28times more accurate than DDM
53 Solution Time In order to investigate computational costof sensitivity methods using the same amounts of RAMand CPU resources for the presented numerical simulationssolution time of different damaged models is evaluated andshown in Figure 11
As seen in Figure 11(a) solution times in FD and DDMFDmethods are considerably larger than analytical discretemethods and the FDmethod with an average of 200616 (s) isthemost time consumingmethod Average time for DDMFDmethod is 150849 (s)
Figure 11(b) compares the time spent by DDM and ADMAs illustrated in this figure ADM method has lower solutiontime than DDMmethod in all cases and speed ratio does notaffect the solution time
Average of solution time is 2956 (s) and 7794 (s) forADM and DDM methods respectively consequently ADMis nearly 26 times faster than DDM
54 Efficiency Parameter In order to compare and quantifythe performance of different methods relative efficiencyparameters of methods ldquo119894rdquo and ldquo119895rdquo (REP
119894119895
) are defined as
REP119894119895
= radic
ST119895
ST119894
times
RPE119895
RPE119894
(30)
in which ST is the solution time of system identificationmethod that represents the computational cost of themethodand RPE is the relative error which represents the accuracy ofthe method
Shock and Vibration 13
Table 8 Solution time number of loops and RPE of ADMmethod for model 2
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE ()
541 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to DDM Figure 12 shows REPadjddm changeswith respect to speed parameter in different scenarios
Table 10 shows that in different scenarios and for differentspeed parameters the efficiency parameter is between 19713and 55478 for model 1 and 15659 and 29893 for model2 and its average is 28948 and 22248 for models 1 and2 respectively Compared to DDM the adjoint variablemethod is more efficient and needs 609 less computationaleffort
Changes in the average of REPadjddm in different scenarioswith speed ratio are shown in Figure 13 As shown in thisfigure REPadjddm increases with speed ratios lightly Thisindicates that efficiency of ADM with respect to DDMincreases with speed ratio
542 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to Approximation and Semianalytical MethodsFigure 14 shows REPadjFD and REPadjDDMFD changes withrespect to speed parameter in different scenarios
Table 11 shows the relative efficiency parameter for differ-ent speed parameters in different scenarios It can be seen thatthe efficiency parameter is between 3193 and 78307 for finitedifference method and 4916 and 50542 for DDMFDmethodwith the average of 16714 and 14569 for FD and DDMFDmethods respectively Therefore the adjoint variable methodis extremely successful and the computational cost of thismethod is just 06 and 069 of FD and DDMFDmethodsrespectively
Changes in the average of REPadjfdampddmfd in differentscenarios with speed ratio are shown in Figure 15 As shownin this figure REPadjfdampddmfd rapidly increases with speedratio which indicates the superiority of ADM over FD andDDMFDmethods increases with speed ratio
0 5 10 15 20 25 30 35 40 45 500
051
152
25
Element number
Mod
ule o
f ela
stici
ty
0 5 10 15 20 25 30 35 40 45 5005
101520
Dam
age i
ndex
Element number
Original modelDetected model
times105
Figure 6 Detection of damage location and amount in elements 57 12 15 24 and 37 and distribution of error in different elementswith ADM scheme
543 Relative Efficiency Parameter of DDMFD with respectto FD Method Figure 16 shows REPddmfdfd changes withrespect to speed parameter in different scenarios
14 Shock and Vibration
Table 10 REPadjddm ranges in different scenarios
Damage scenario Max REP Min REP AverageModel 1 Model 2 Model 1 Model 2 Model 1 Model 2
Figure 7 RPE contours with respect to noise level and loops in model 2
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(b) DDMmethod
Figure 8 RPE contours with respect to initial error and loops in model 2
Shock and Vibration 15
0
50
100
150
200
250
300
350
0103050709
DDMFDFD
DDMADM
Itera
tion
num
ber
Speed ratio
(a) All methods
0
2
4
6
8
10
12
14
16
18
0103050709
Itera
tion
num
ber
Speed ratio
DDMADM
(b) Analytical discrete methods
Figure 9 Comparison of iteration number between different sensitivity methods
DDMFD
RPE
FDDDMADM
0
01
02
03
04
05
06
07
0103050709Speed ratio
(a) All methods
DDMADM
0103050709
RPE
Speed ratio
00E + 00
50E minus 04
10E minus 03
15E minus 03
20E minus 03
25E minus 03
(b) Analytical discrete methods
Figure 10 Comparison of RPE between different sensitivity methods
Table 12 shows the relative efficiency parameter for dif-ferent speed parameters in different scenarios Again theefficiency parameter is between 0343 and 1726 with theaverage of 1044Thus compared to FDmethod theDDMFDmethod is slightly more effective
Changes in the average of REPddmfdfd in different sce-narios with speed ratio are shown in Figure 17 As shown inthis figure REPddmfdfd increases with speed ratio For speedratio lower than 05 efficiency parameter is lower than 1 whichmeans FD method is more effective for this range
55 Stability Figures 3 and 7 show that all describedmethodsare sensitive to noise and if noise level exceeds the valuesstated in Table 11 these methods lose their efficiency and arenot able to detect damage properly Hence in cases with noise
level larger than the values stated in Table 13 a denoisingtool like wavelet transforms should be used along with thesensitivity methodsWavelet transforms are mainly attractivebecause of their ability to compress and encode informationto reduce noise or to detect any local singular behavior of asignal [37]
Stability comparison of different methods shows that FDis the most stable method and stability of analytical methodsis slightly lower than that
Figures 4 and 8 show stability of sensitivity methodswith respect to the initial error Summary of results forstability investigation is presented in Table 13 which suggeststhat stability of numerical methods is better than analyticalmethods
16 Shock and Vibration
DDMFDFD
DDMADM
0
50
100
150
200
250
300
0103050709
Solu
tion
time
Speed ratio
(a) All methods
DDMADM
0
2
4
6
8
10
0103050709
Solu
tion
time
Speed ratio
(b) Analytical discrete methods
Figure 11 Comparison of solution time between different sensitivity methods
5-64-5
3-4
2-31-20-1
120572 = 01
120572 = 05
120572 = 09
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
6
123456
0
(a) Model 1
120572 = 01
120572 = 05
120572 = 09
25ndash32ndash2515ndash2
1ndash1505ndash10ndash05
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
63252151050
(b) Model 2
Figure 12 REPadjddm changes in different scenarios with respect to speed parameter
Table 11 REPadjadm ranges in different scenarios
Damage scenario Max REP Min REP AverageFD DDMFD FD DDMFD FD DDMFD
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
the fractional stiffness change incrementΔ120572119896+1 is obtained byminimizing the cost function in (7)
The regularized solution fromminimizing the function in(7) can be written in the following form as
Δ120572119896+1
= ((S119896)119879
S119896 + 1205822I)minus1
(S119896)119879
Δz119896 (8)
To express the contribution of the singular values and thecorresponding vectors in the solution clearly and to showhowthe regularization parameter plays the role as the filter factorthe sensitivity matrix is singular value decomposed andsingular value decomposition (SVD) applies to the sensitivitymatrix S119896 to obtain
S119896 = UΣV119879 (9)
where U isin 119877119899119905times119899119905 and V isin 119877
119898times119898 are orthogonal matricessatisfying U119879U = I
119899119905
and V119879V = I119898
and matrix Σ has thesize of 119899119905 times 119898 with the singular values 120590
119894
(119894 = 1 2 119898) onthe diagonal arranged in a decreasing order such that 120590
1
ge
1205902
ge sdot sdot sdot ge 120590119898
ge 0 and zeros elsewhereThe regularized solution in (8) can be written as
Δ120572119896+1
=
119898
sum
119894=1
119891119894
U119879119894
Δz119896
120590119894
V119894
(10)
where 119891119894
= 1205902
119894
(1205902
119894
+ 1205822
) (119894 = 1 2 119898) are referred as filterfactors The solution norm Δ120572
119896+1
2
2
and the residual normS119896 sdot Δ120572119896+1 minus Δz1198962
10038171003817100381710038171003817S119896 sdot Δ120572119896+1 minus Δz1198961003817100381710038171003817
1003817
2
2
=
119898
sum
119894=1
(
1205822
1205902
119894
+ 1205822
U119879119894
Δz119896)2
(11)
These two quantities represent the smoothness and goodness-of-fit of the solution and they should be balanced by choosingan appropriate regularization parameter
25 Element Damage Index In the inverse problem of dam-age identification it is assumed that the stiffness matrix ofthe whole element decreases uniformly with damage andthe flexural rigidity EI
119894
of the 119894th finite element of the beambecomes 120573
119894
EI119894
when there is damage The fractional changein stiffness of an element can be expressed as [30]
ΔK119887119894
= (K119887119894
minus K119887119894
) = (1 minus 120573119894
)K119887119894
(12)
where K119887119894
and K119887119894
are the 119894th element stiffness matrices ofthe undamaged and damaged beam respectively ΔK
119887119894
is thestiffness reduction of the element A positive value of 120573
119894
isin
[0 1] will indicate a loss in the element stiffness The 119894thelement is undamaged when 120573
119894
= 1 and the stiffness of the119894th element is completely lost when 120573
119894
= 0
The stiffness matrix of the damaged structure is theassemblage of the entire element stiffness matrix K
119887119894
K119887
=
119873
sum
119894=1
A119879119894
K119887119894
A119894
=
119873
sum
119894=1
120573119894
A119879119894
K119887119894
A119894
(13)
where A119894
is the extended matrix of element nodal displace-ment that facilitates assembling of global stiffness matrixfrom the constituent element stiffness matrix
The objective of sensitivity analysis is to quantify the effectsof parameter variations on calculated results Terms such asinfluence importance ranking by importance and domi-nance are all related to sensitivity analysis
The simplest and most common procedure for assessingthe effects of parameter variations on a model is to varyselected input parameters rerun the code and record thecorresponding changes in the results or responses calculatedby the code The model parameters responsible for thelargest relative changes in a response are then consideredto be the most important for the respective response Forcomplexmodels though the large amount of computing timeneeded by such recalculations severely restricts the scope ofthis sensitivity analysis method in practice therefore themodeler who uses this method can investigate only a fewparameters that he judges a priori to be important [32]
When the parameter variations are small the traditionalway to assess their effects on calculated responses is byusing perturbation theory either directly or indirectly viavariational principles The basic aim of perturbation theoryis to predict the effects of small parameter variations withoutactually calculating the perturbed configuration but rather byusing solely unperturbed quantities
31 Methods of Structural Sensitivity Analysis Various meth-ods employed in design sensitivity analysis are listed inFigure 2 Three approaches are used to obtain the designsensitivity the approximation discrete and continuumapproaches In the approximation approach design sensi-tivity is obtained by either the forward finite difference orthe central finite difference method In the discrete methoddesign sensitivity is obtained by taking design derivativesof the discrete governing equation For this process it isnecessary to take the design derivative of the stiffness matrixIf this derivative is obtained analytically using the explicitexpression of the stiffness matrix with respect to the designvariable it is an analytical method since the analyticalexpressions of stiffness matrix are used However if thederivative is obtained using a finite difference method themethod is called a semianalytical method
In the continuum approach the design derivative ofthe variational equation is taken before it is discretized Ifthe structural problem and sensitivity equations are solvedas a continuum problem then it is called the continuum-continuum method However only very simple classical
Shock and Vibration 5
Sensitivity methods
Approximation approach
Forward finite difference
Centeral finite difference
Discrete approach
method
Analytical
Semianalytical
method
differential
Adjoint variable
Direct
method
Continuum approach
Continuum-discrete method
Continuum-continuum
method
method
Figure 2 Approaches to design sensitivity analysis
problems can be solved analytically Thus the continuumsensitivity equation is solved by discretization in the sameway that structural problems are solved Since differentiationis taken at the continuum domain and is then followed bydiscretization this method is called the continuum-discretemethod
32 Approximation Approach The easiest way to computesensitivity information of the performance measure is byusing the finite difference method Different designs yielddifferent analysis results and thus different performancevaluesThe finite differencemethod actually computes designsensitivity of performance by evaluating performance mea-sures at different stages in the design process If u is thecurrent design then the analysis results provide the valueof performance measure 120595(u) In addition if the design isperturbed to u + Δu where Δu represents a small change inthe design then the sensitivity of 120595(u) can be approximatedas
119889120595
119889uasymp
120595 (u + Δu) minus 120595 (u)Δu
(14)
Equation (14) is called the forward difference method sincethe design is perturbed in the direction of +Δu If minusΔuis substituted in (14) for Δu then the equation is definedas the backward difference method Additionally in centraldifference method the design is perturbed in both directionsand the design sensitivity is approximated by
119889120595
119889uasymp
120595 (u + Δu) minus 120595 (u minus Δu)2Δu
(15)
Then the equation is defined as the central differencemethod
33 Discrete Approach A structural problem is often dis-cretized in finite dimensional space in order to solve complex
problems The discrete method computes the performancedesign sensitivity of the discretized problem
The design represents a structural parameter that canaffect the results of the analysis
The design sensitivity information of a general perfor-mance measure can be computed either with the directdifferentiation method or with the adjoint variable method
331 Direct Differentiation Method The direct differentia-tion method (DDM) is a general accurate and efficientmethod to compute FE response sensitivities to the modelparameters This method directly solves for the designdependency of a state variable and then computes perfor-mance sensitivity using the chain rule of differentiationThis method clearly shows the implicit dependence on thedesign and a very simple sensitivity expression can beobtained
Consider a structure in which the generalized stiffnessand mass matrices have been reduced by accounting forboundary conditions Let the damping force be representedin the form of C(119887)z
119905
where z119905
= 119889z119889119905 denotes thevelocity vector Under these conditions Lagrangersquos equationof motion becomes the second-order differential equation as[33]
M (119887) z119905119905
+ C (119887) z119905
+ K (119887) z = f (119905 119887) (16)
With the initial conditions
z (0) = z0 z119905
(0) = z0119905
(17)
If design parameters are just related to stiffness matrix weobtain
M120597z119905119905
120597119887119894
+ C120597z119905
120597119887119894
+ K 120597z120597119887119894
= minus
120597K120597119887119894
z minus 1205722
120597K120597119887119894
z119905
(18)
6 Shock and Vibration
in which 120597z120597b119894 120597z119905
120597b119894 and 120597z119905119905
120597b119894 are sensitivity vectorsof displacement velocity and acceleration with respect todesign parameter 119887
119894
respectively Assume that
y119905119905
=
120597z119905119905
120597119887119894
(19a)
y119905
=
120597z119905
120597119887119894
(19b)
y = 120597z120597119887119894
(19c)
So by replacing (19a) (19b) and (19c) into (18) we have
My119905119905
+ Cy119905
+ Ky = minus120597K120597119887119894
z minus 1205722
120597K120597119887119894
z119905
(20)
Right side of (20) can be considered as an equivalent force so(20) is similar to (16) and sensitivity vectors can be obtainedby Newmark method
332 Adjoint Variable Method Mirzaee et al [34] provedthat sensitivity analysis can be performed very efficientlyby using deterministic methods based on adjoint functionsThis method constructs an adjoint problem that solves forthe adjoint variable which contains all implicitly dependentterms
The adjoint variablemethod yields a terminal-value prob-lem compared with the initial-value problem of responseanalysis
For the dynamic structure the following formof a generalperformance measure can be considered
120595 = 119892 (z (119879) 119887) + int119879
0
119866 (z 119887) 119889119905 (21)
General form of sensitivity measure relative to the designparameter (b) is as follows [33]
1205951015840
= [
120597119892
120597zminus
120597119892
120597zz119905
(119879) + 119866 (z (119879) 119887) 1Ω119905
120597Ω
120597119911
] z1015840 (119879)
minus [
120597119892
120597119911
z119905
(119879) + 119866 (z (119879) 119887)] 1
Ω119905
120597Ω
120597119911119905
z1015840119905
(119879)
+ int
119879
0
[
120597119866
120597zz1015840 + 120597119866
120597119887
120575119887] 119889119905 +
120597119892
120597119887
120575119887
minus [
120597119892
120597119911
z119905
(119879) + 119866 (119911 (119879) 119887)]
1
Ω119905
120597Ω
120597119887
120575119887
(22)
Note that1205951015840 depends on z1015840 and z1015840119905
at119879 as well as on z1015840 withinthe integration
Mirzaee et el [34] showed that while structural vibrationresponses are used for damage detection solving (22) inknown time 119879 is as follows
119889120595
119889119887
(119879) = int
119879
0
(minus120582119879
1198860
120597K120597119887
z119905
minus120582119879120597K120597119887
z)119889119905 (23)
120582119879
(119905) = P119879
119891 (119905) +Q119879
119892 (119905) (24)
in which
P119879
= 119890minus120585120596119879
120582119905
(119879)
120596119863
cos (120596119863
119879)
119891 (119905) = 119890120585120596119905 sin (120596
119863
119905)
Q119879
= minus119890minus120585120596119879
120582119905
(119879)
120596119863
sin (120596119863
119879)
119892 (119905) = 119890120585120596119905 cos (120596
119863
119905)
(25)
It is worth noting that terminal values for different values of119879 are not similar and adjoint equation should be calculatedfor corresponding 119879 separately
Mirzaee et al [34] presented a quite efficient incrementalsolution for solving (24) which significantly improves theproposed method
34 Sensitivity Method Selection In this paper four differentmethods are presented to compute sensitivity information ofthe performance measure for damage detection procedureFirst method is the finite Difference method (FD) whichis classified as an approximation approach Second it issemianalytical discrete method which uses a finite differencescheme to calculate stiffness matrix in direct differentialmethod (DDMFD) and third and fourthmethods are two dif-ferent analytical discrete methods direct differential method(DDM) and adjoint variable method (ADM)
The first three methods including FD DDMFD andDDM are widely used by other authors [35 36] and the lastmethod (ADM) is proposed by the authors (Mirzaee et al[34])
35 Procedure of Iteration for Damage Detection The initialanalytical model of a structure deviates from the true modeland measurement from the initial intact structure is used toupdate the analytical model The improved model is thentreated as a reference model and measurement from thedamaged structure will be used to update the referencemodel
When response measurement from the intact state of thestructure is obtained the sensitivities are computed frompresented methods based on the analytical model of thestructure and the well knowing input force and velocityThe vector of parameter increments is then obtained usingthe computed and experimentally obtained responses Theanalytical model is then updated and the correspondingresponse and its sensitivity are again computed for the nextiteration When measurement from the damaged state isobtained the updated analyticalmodel is used in the iterationin the same way as that using measurement from the intactstate Convergence is considered to be achieved when thefollowing criteria are met
1003817100381710038171003817E119894+1
minus E119894
1003817100381710038171003817
1003817100381710038171003817E119894
1003817100381710038171003817
times 100 le Tol1
1003817100381710038171003817Response
119894+1
minus Response119894
1003817100381710038171003817
1003817100381710038171003817Response
119894
1003817100381710038171003817
times 100 le Tol2
(26)
Shock and Vibration 7
The final vector of identified parameter increments corre-sponds to the changes occurring in between the two statesof the structure The tolerance is set equal to 1 times 10minus5 in thisstudy except otherwise specified
4 Numerical Results
In order to evaluate the efficiency and competency of intro-duced methods analysis of two FE models with extensiveavailable numerical studies has been carried out Represen-tative examples are presented to demonstrate the effects ofspeed of loads measurement noise level and initial error onthe accuracy and effectiveness of the methods
The relative percentage error (RPE) in the identifiedresults is calculated from (27) where sdot is the norm ofmatrix and z
119905119905identifiedand z119905119905true
are the identified and the trueacceleration time history respectively [26]
Since the true value of elastic modulus is unknown RPE canjust be used for evaluating the efficiency of the method
41 Single-Span Bridge A single-span bridge is studied tocompare different methods This bridge consists of 10 Euler-Bernoulli beam elements with 11 nodes each with two DOFsTotal length of bridge is 10m and height and width ofthe frame section are 200mm Rayleigh damping model isadopted with the damping ratios of the first two modes takenequal to 005
Structural properties are summarized in Table 1The transverse point load 119875 has a constant velocity V =
119871119879 where 119879 is the traveling time across the bridge and 119871 isthe total length of the bridge
For the forced vibration analysis an implicit time integra-tion method called as ldquothe Newmark integration methodrdquo isused with the integration parameters 120573 = 14 and 120574 = 12which leads to constant-average acceleration approximation
Speed parameter is defined as
120572 =
119881
119881cr (28)
in which119881cr is critical speed (119881cr = (120587119897)radicEI120588)119881 is movingload speed and 120588 is mass per unit length of the beams
411 Damage Scenarios Six damage scenarios of singlemultiple and random damage along with initial error in thebridge without measurement noise are studied and shown inTable 2
Local damage is simulated with a reduction in the elasticmodulus of material of an element The sampling rate is15000Hz and 250 data of the acceleration response (degree ofindeterminacy is 25) collected along the z-direction at nodes2 and 7 are used in the identification
412 Analysis Results Tables 3ndash6 show the results of differentdetection methods including solution time (ST) number of
loops (NL) and RPE These are the most important param-eters to compare the efficiency and accuracy of differentmethods
413 Effect of Noise Noise is the random fluctuation in thevalue of measured or input that causes random fluctuation inthe output value Noise at the sensor output is due to eitherinternal noise sources such as resistors at finite temperaturesor externally generatedmechanical and electromagnetic fluc-tuations [10]
To evaluate the sensitivity of results to suchmeasurementnoise noise-polluted measurements are simulated by addingto the noise-free acceleration vector a corresponding noisevector whose root-mean-square (RMS) value is equal toa certain percentage of the RMS value of the noise-freedata vector The components of all the noise vectors are ofGaussian distribution uncorrelated andwith a zeromean andunit standard deviation Then on the basis of the noise-freeacceleration z
119905119905nfthe noise-polluted acceleration z
119905119905npof the
bridge at location 119909 can be simulated by
z119905119905np
= z119905119905nf
+ RMS (z119905119905nf) times 119873level times 119873unit (29)
where RMS (z119905119905nf
) is the RMS value of the noise-free accel-eration vector z
119905119905nftimes 119873level is the noise level and 119873unit is a
randomly generated noise vector with zero mean and unitstandard deviation [36]
In order to study effects of noise in stability of differentsensitivity methods scenario 3 (speed ratio of moving loadis considered to be fixed and equal to 05) is considered anddifferent levels of noise pollution are investigated and alsoRPE change with increasing the number of loops in iterativeprocedure has been studied The results are illustrated inFigure 3 for all presented methods
414 Effect of Initial Error In order to study effects of initialerror in stability of sensitivity methods scenario 6 (speedratio ofmoving load is considered to be fixed and equal to 05)is considered This scenario consists of no simulated damagein the structure but the initial elastic modulus of materialof all the elements is underestimated and different levels ofinitial error are investigated RPE changes with increasing thenumber of loops in iterative procedure have been studiedTheresults are illustrated in Figure 4 for all presented methods
42 Plane Grid Model A plane grid model of bridge isstudied as another numerical example to investigate theefficiency and accuracy of discrete methods It is notable thataccording to complexity of this model approximation andsemidiscrete methods are not useable and diverged in allscenarios in all speed ranges
The structure is modeled by 46 frame elements and 32nodes with three DOFs at each node for the translation androtational deformations Rayleigh damping model is adoptedwith the damping ratios of the first two modes taken equalto 005 The finite element model of the structure is shownin Figure 5 and structural properties are summarized inTable 1
8 Shock and Vibration
Table1Prop
ertie
sofd
ifferentm
odels
Mod
elname
Num
bero
felements
Num
bero
fnod
esMassd
ensity
Mod
ulus
ofelasticity
Rayleigh
coeffi
cients
Und
amaged
naturalfrequ
ency
ofintactmod
elkgm
m3
Nm
m2
1198860
1198861
Firstm
ode
Second
mod
eTh
irdmod
eFo
urth
mod
eFifth
mod
eMultispan
1011
78times10minus9
21times
10minus5
01
2860times10minus5
296
1183
2662
4738
7421
Grid
4632
78times10minus9
21times
10minus5
01
2364times10minus5
456
928
1817
2597
3991
Shock and Vibration 9
Table 2 Damage scenarios for single-span bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 4 17 NilM1-2 Multiple 2 7 4 21 NilM1-3 Multiple 3 5 6 and 8 12 6 5 and 2 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
Table 3 Solution time number of loops and RPE of ADMmethod for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
421 Damage Scenarios Six damage scenarios of singlemultiple and randomdamage in the bridgewithoutmeasure-ment noise are studied and shown in Table 7
The sampling rate is 14000Hz and 460 data of the acceler-ation response (degree of indeterminacy is 20) collected alongthe z-direction at nodes 4 11 21 and 27 are used
422 Analysis Results Tables 8 and 9 show the results ofdifferent detection methods including solution time (ST)
and RPE Using both described methods including ADMand DDM method the damage locations and amount areidentified correctly in all the scenarios (Figure 6)
423 Effect of Noise In order to study effects of noisescenario 3 is considered and different levels of noise pollutionare investigated and also RPE changes with increasing thenumber of loops for iterative procedure have been studiedThe results are presented in Figure 7
10 Shock and Vibration
Table 6 Solution time loops and RPE of DDMFD method for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
Figure 3 RPE contours with respect to noise level and loops in model 1
Shock and Vibration 11
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
10
20
30
40
50
60
70
(b) DDMmethod
Loops
Initi
al er
ror
FD method
5 10 15 20 25 300
005
01
015
02
025
03
10
20
30
40
50
60
70
80
(c) FD method
Loops
Initi
al er
ror
DDMFD method
5 10 15 20 25 300
005
01
015
02
025
03
0
10
20
30
40
50
60
70
80
(d) DDMFD method
Figure 4 RPE contours with respect to initial error and loops in model 1
424 Effect of Initial Error Scenario 6 (speed ratio ofmovingload is considered to be fixed and equal to 05) is considered tostudy effects of initial error in stability of sensitivity methodsRPE changes with increasing the number of loops for iterativeprocedure have been studied The results are illustrated inFigure 8
5 Comparison of Different Methods
According to the results obtained in the previous sectiondifferent sensitivity methods considered in this study arecompared in this section Iteration number relative percent-age of error (RPE) solution time noise level and initialerror effect in stability of damage detection procedure areeffective parameters to compare efficiency and accuracy of themethods
51 Iteration Number Average number of iterations fordifferent scenarios with respect to the speed parameter isillustrated in Figure 9
As seen in Figure 9(a) the number of iterations in FDand DDMFD methods is considerably larger than thoseof analytical discrete methods (DDM and ADM) It hasbeen observed that in all cases DDMFD method has feweriterations than FD method Average number of iterations is20757 and 251467 for DDMFD and FD methods respec-tively The larger the speed ratio of moving vehicle is thelarger the number of iterations in approximation methodsis In FD method for speed ratios more than 05 even byusing maximum number of iterations (300) convergence wasnot achieved while the ratio used in DDMFD method is09
Figure 9(b) shows comparison along with analytical dis-crete methods It can be seen that DDM has smaller iterationnumber in all cases Average number of iterations is 146and 11167 for ADM and DDM methods respectively Bothanalytical discrete methods converged in all cases withmaximum iterations equal to 17 and increment in speed ratiodoes not affect the number of iterations
12 Shock and Vibration
Sensors
Element number
25
17
9
2622
15
8
1
23
16
9
2
24
17
10
3
25
18
11
4
2619
12
5
2720
13
6
2821
147
4118
3510
29
2
2742
1936
1130
3
2843
2037
1231
4
29 4421
3813
325
30 4522 39
14
16
336
3132
46 2324
40 1534 7
8
Direction of measured response for identification
Node number
P V
Moving vehicle
Y Z X
7000 (mm)
3000 (mm)
Figure 5 Plane grid bridge model used in detection procedure
Table 7 Damage scenarios for multispan bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 23 5 NilM1-2 Multiple 8 13 and 29 11 4 and 7 NilM1-3 Multiple 3 7 19 25 and 28 12 6 5 2 and 18 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
52 Relative Percentage of Error (RPE) Using the sameconvergence tolerance equal to 1 times 10minus5 relative percentageof error in different methods is illustrated in Figure 10
As seen in this figure with a maximum number ofiterations equal to 300 RPE in FD and DDMFD methodsare considerably larger than DDM and ADM methodsDDMFD method has lower RPE than FD method in mostcases Average of RPE is 026329 for DDMFD compared to027766 in FD method Again the larger the speed ratio ofmoving vehicle is the larger the relative error percentage inapproximation methods is
Figure 10(b) shows that relative error percentage in ADMmethod is lower than DDM and speed ratio does not affectthe relative error
Average of RPE is 0000731 and 0002037 for ADM andDDM methods respectively Therefore ADM is nearly 28times more accurate than DDM
53 Solution Time In order to investigate computational costof sensitivity methods using the same amounts of RAMand CPU resources for the presented numerical simulationssolution time of different damaged models is evaluated andshown in Figure 11
As seen in Figure 11(a) solution times in FD and DDMFDmethods are considerably larger than analytical discretemethods and the FDmethod with an average of 200616 (s) isthemost time consumingmethod Average time for DDMFDmethod is 150849 (s)
Figure 11(b) compares the time spent by DDM and ADMAs illustrated in this figure ADM method has lower solutiontime than DDMmethod in all cases and speed ratio does notaffect the solution time
Average of solution time is 2956 (s) and 7794 (s) forADM and DDM methods respectively consequently ADMis nearly 26 times faster than DDM
54 Efficiency Parameter In order to compare and quantifythe performance of different methods relative efficiencyparameters of methods ldquo119894rdquo and ldquo119895rdquo (REP
119894119895
) are defined as
REP119894119895
= radic
ST119895
ST119894
times
RPE119895
RPE119894
(30)
in which ST is the solution time of system identificationmethod that represents the computational cost of themethodand RPE is the relative error which represents the accuracy ofthe method
Shock and Vibration 13
Table 8 Solution time number of loops and RPE of ADMmethod for model 2
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE ()
541 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to DDM Figure 12 shows REPadjddm changeswith respect to speed parameter in different scenarios
Table 10 shows that in different scenarios and for differentspeed parameters the efficiency parameter is between 19713and 55478 for model 1 and 15659 and 29893 for model2 and its average is 28948 and 22248 for models 1 and2 respectively Compared to DDM the adjoint variablemethod is more efficient and needs 609 less computationaleffort
Changes in the average of REPadjddm in different scenarioswith speed ratio are shown in Figure 13 As shown in thisfigure REPadjddm increases with speed ratios lightly Thisindicates that efficiency of ADM with respect to DDMincreases with speed ratio
542 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to Approximation and Semianalytical MethodsFigure 14 shows REPadjFD and REPadjDDMFD changes withrespect to speed parameter in different scenarios
Table 11 shows the relative efficiency parameter for differ-ent speed parameters in different scenarios It can be seen thatthe efficiency parameter is between 3193 and 78307 for finitedifference method and 4916 and 50542 for DDMFDmethodwith the average of 16714 and 14569 for FD and DDMFDmethods respectively Therefore the adjoint variable methodis extremely successful and the computational cost of thismethod is just 06 and 069 of FD and DDMFDmethodsrespectively
Changes in the average of REPadjfdampddmfd in differentscenarios with speed ratio are shown in Figure 15 As shownin this figure REPadjfdampddmfd rapidly increases with speedratio which indicates the superiority of ADM over FD andDDMFDmethods increases with speed ratio
0 5 10 15 20 25 30 35 40 45 500
051
152
25
Element number
Mod
ule o
f ela
stici
ty
0 5 10 15 20 25 30 35 40 45 5005
101520
Dam
age i
ndex
Element number
Original modelDetected model
times105
Figure 6 Detection of damage location and amount in elements 57 12 15 24 and 37 and distribution of error in different elementswith ADM scheme
543 Relative Efficiency Parameter of DDMFD with respectto FD Method Figure 16 shows REPddmfdfd changes withrespect to speed parameter in different scenarios
14 Shock and Vibration
Table 10 REPadjddm ranges in different scenarios
Damage scenario Max REP Min REP AverageModel 1 Model 2 Model 1 Model 2 Model 1 Model 2
Figure 7 RPE contours with respect to noise level and loops in model 2
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(b) DDMmethod
Figure 8 RPE contours with respect to initial error and loops in model 2
Shock and Vibration 15
0
50
100
150
200
250
300
350
0103050709
DDMFDFD
DDMADM
Itera
tion
num
ber
Speed ratio
(a) All methods
0
2
4
6
8
10
12
14
16
18
0103050709
Itera
tion
num
ber
Speed ratio
DDMADM
(b) Analytical discrete methods
Figure 9 Comparison of iteration number between different sensitivity methods
DDMFD
RPE
FDDDMADM
0
01
02
03
04
05
06
07
0103050709Speed ratio
(a) All methods
DDMADM
0103050709
RPE
Speed ratio
00E + 00
50E minus 04
10E minus 03
15E minus 03
20E minus 03
25E minus 03
(b) Analytical discrete methods
Figure 10 Comparison of RPE between different sensitivity methods
Table 12 shows the relative efficiency parameter for dif-ferent speed parameters in different scenarios Again theefficiency parameter is between 0343 and 1726 with theaverage of 1044Thus compared to FDmethod theDDMFDmethod is slightly more effective
Changes in the average of REPddmfdfd in different sce-narios with speed ratio are shown in Figure 17 As shown inthis figure REPddmfdfd increases with speed ratio For speedratio lower than 05 efficiency parameter is lower than 1 whichmeans FD method is more effective for this range
55 Stability Figures 3 and 7 show that all describedmethodsare sensitive to noise and if noise level exceeds the valuesstated in Table 11 these methods lose their efficiency and arenot able to detect damage properly Hence in cases with noise
level larger than the values stated in Table 13 a denoisingtool like wavelet transforms should be used along with thesensitivity methodsWavelet transforms are mainly attractivebecause of their ability to compress and encode informationto reduce noise or to detect any local singular behavior of asignal [37]
Stability comparison of different methods shows that FDis the most stable method and stability of analytical methodsis slightly lower than that
Figures 4 and 8 show stability of sensitivity methodswith respect to the initial error Summary of results forstability investigation is presented in Table 13 which suggeststhat stability of numerical methods is better than analyticalmethods
16 Shock and Vibration
DDMFDFD
DDMADM
0
50
100
150
200
250
300
0103050709
Solu
tion
time
Speed ratio
(a) All methods
DDMADM
0
2
4
6
8
10
0103050709
Solu
tion
time
Speed ratio
(b) Analytical discrete methods
Figure 11 Comparison of solution time between different sensitivity methods
5-64-5
3-4
2-31-20-1
120572 = 01
120572 = 05
120572 = 09
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
6
123456
0
(a) Model 1
120572 = 01
120572 = 05
120572 = 09
25ndash32ndash2515ndash2
1ndash1505ndash10ndash05
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
63252151050
(b) Model 2
Figure 12 REPadjddm changes in different scenarios with respect to speed parameter
Table 11 REPadjadm ranges in different scenarios
Damage scenario Max REP Min REP AverageFD DDMFD FD DDMFD FD DDMFD
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
Figure 2 Approaches to design sensitivity analysis
problems can be solved analytically Thus the continuumsensitivity equation is solved by discretization in the sameway that structural problems are solved Since differentiationis taken at the continuum domain and is then followed bydiscretization this method is called the continuum-discretemethod
32 Approximation Approach The easiest way to computesensitivity information of the performance measure is byusing the finite difference method Different designs yielddifferent analysis results and thus different performancevaluesThe finite differencemethod actually computes designsensitivity of performance by evaluating performance mea-sures at different stages in the design process If u is thecurrent design then the analysis results provide the valueof performance measure 120595(u) In addition if the design isperturbed to u + Δu where Δu represents a small change inthe design then the sensitivity of 120595(u) can be approximatedas
119889120595
119889uasymp
120595 (u + Δu) minus 120595 (u)Δu
(14)
Equation (14) is called the forward difference method sincethe design is perturbed in the direction of +Δu If minusΔuis substituted in (14) for Δu then the equation is definedas the backward difference method Additionally in centraldifference method the design is perturbed in both directionsand the design sensitivity is approximated by
119889120595
119889uasymp
120595 (u + Δu) minus 120595 (u minus Δu)2Δu
(15)
Then the equation is defined as the central differencemethod
33 Discrete Approach A structural problem is often dis-cretized in finite dimensional space in order to solve complex
problems The discrete method computes the performancedesign sensitivity of the discretized problem
The design represents a structural parameter that canaffect the results of the analysis
The design sensitivity information of a general perfor-mance measure can be computed either with the directdifferentiation method or with the adjoint variable method
331 Direct Differentiation Method The direct differentia-tion method (DDM) is a general accurate and efficientmethod to compute FE response sensitivities to the modelparameters This method directly solves for the designdependency of a state variable and then computes perfor-mance sensitivity using the chain rule of differentiationThis method clearly shows the implicit dependence on thedesign and a very simple sensitivity expression can beobtained
Consider a structure in which the generalized stiffnessand mass matrices have been reduced by accounting forboundary conditions Let the damping force be representedin the form of C(119887)z
119905
where z119905
= 119889z119889119905 denotes thevelocity vector Under these conditions Lagrangersquos equationof motion becomes the second-order differential equation as[33]
M (119887) z119905119905
+ C (119887) z119905
+ K (119887) z = f (119905 119887) (16)
With the initial conditions
z (0) = z0 z119905
(0) = z0119905
(17)
If design parameters are just related to stiffness matrix weobtain
M120597z119905119905
120597119887119894
+ C120597z119905
120597119887119894
+ K 120597z120597119887119894
= minus
120597K120597119887119894
z minus 1205722
120597K120597119887119894
z119905
(18)
6 Shock and Vibration
in which 120597z120597b119894 120597z119905
120597b119894 and 120597z119905119905
120597b119894 are sensitivity vectorsof displacement velocity and acceleration with respect todesign parameter 119887
119894
respectively Assume that
y119905119905
=
120597z119905119905
120597119887119894
(19a)
y119905
=
120597z119905
120597119887119894
(19b)
y = 120597z120597119887119894
(19c)
So by replacing (19a) (19b) and (19c) into (18) we have
My119905119905
+ Cy119905
+ Ky = minus120597K120597119887119894
z minus 1205722
120597K120597119887119894
z119905
(20)
Right side of (20) can be considered as an equivalent force so(20) is similar to (16) and sensitivity vectors can be obtainedby Newmark method
332 Adjoint Variable Method Mirzaee et al [34] provedthat sensitivity analysis can be performed very efficientlyby using deterministic methods based on adjoint functionsThis method constructs an adjoint problem that solves forthe adjoint variable which contains all implicitly dependentterms
The adjoint variablemethod yields a terminal-value prob-lem compared with the initial-value problem of responseanalysis
For the dynamic structure the following formof a generalperformance measure can be considered
120595 = 119892 (z (119879) 119887) + int119879
0
119866 (z 119887) 119889119905 (21)
General form of sensitivity measure relative to the designparameter (b) is as follows [33]
1205951015840
= [
120597119892
120597zminus
120597119892
120597zz119905
(119879) + 119866 (z (119879) 119887) 1Ω119905
120597Ω
120597119911
] z1015840 (119879)
minus [
120597119892
120597119911
z119905
(119879) + 119866 (z (119879) 119887)] 1
Ω119905
120597Ω
120597119911119905
z1015840119905
(119879)
+ int
119879
0
[
120597119866
120597zz1015840 + 120597119866
120597119887
120575119887] 119889119905 +
120597119892
120597119887
120575119887
minus [
120597119892
120597119911
z119905
(119879) + 119866 (119911 (119879) 119887)]
1
Ω119905
120597Ω
120597119887
120575119887
(22)
Note that1205951015840 depends on z1015840 and z1015840119905
at119879 as well as on z1015840 withinthe integration
Mirzaee et el [34] showed that while structural vibrationresponses are used for damage detection solving (22) inknown time 119879 is as follows
119889120595
119889119887
(119879) = int
119879
0
(minus120582119879
1198860
120597K120597119887
z119905
minus120582119879120597K120597119887
z)119889119905 (23)
120582119879
(119905) = P119879
119891 (119905) +Q119879
119892 (119905) (24)
in which
P119879
= 119890minus120585120596119879
120582119905
(119879)
120596119863
cos (120596119863
119879)
119891 (119905) = 119890120585120596119905 sin (120596
119863
119905)
Q119879
= minus119890minus120585120596119879
120582119905
(119879)
120596119863
sin (120596119863
119879)
119892 (119905) = 119890120585120596119905 cos (120596
119863
119905)
(25)
It is worth noting that terminal values for different values of119879 are not similar and adjoint equation should be calculatedfor corresponding 119879 separately
Mirzaee et al [34] presented a quite efficient incrementalsolution for solving (24) which significantly improves theproposed method
34 Sensitivity Method Selection In this paper four differentmethods are presented to compute sensitivity information ofthe performance measure for damage detection procedureFirst method is the finite Difference method (FD) whichis classified as an approximation approach Second it issemianalytical discrete method which uses a finite differencescheme to calculate stiffness matrix in direct differentialmethod (DDMFD) and third and fourthmethods are two dif-ferent analytical discrete methods direct differential method(DDM) and adjoint variable method (ADM)
The first three methods including FD DDMFD andDDM are widely used by other authors [35 36] and the lastmethod (ADM) is proposed by the authors (Mirzaee et al[34])
35 Procedure of Iteration for Damage Detection The initialanalytical model of a structure deviates from the true modeland measurement from the initial intact structure is used toupdate the analytical model The improved model is thentreated as a reference model and measurement from thedamaged structure will be used to update the referencemodel
When response measurement from the intact state of thestructure is obtained the sensitivities are computed frompresented methods based on the analytical model of thestructure and the well knowing input force and velocityThe vector of parameter increments is then obtained usingthe computed and experimentally obtained responses Theanalytical model is then updated and the correspondingresponse and its sensitivity are again computed for the nextiteration When measurement from the damaged state isobtained the updated analyticalmodel is used in the iterationin the same way as that using measurement from the intactstate Convergence is considered to be achieved when thefollowing criteria are met
1003817100381710038171003817E119894+1
minus E119894
1003817100381710038171003817
1003817100381710038171003817E119894
1003817100381710038171003817
times 100 le Tol1
1003817100381710038171003817Response
119894+1
minus Response119894
1003817100381710038171003817
1003817100381710038171003817Response
119894
1003817100381710038171003817
times 100 le Tol2
(26)
Shock and Vibration 7
The final vector of identified parameter increments corre-sponds to the changes occurring in between the two statesof the structure The tolerance is set equal to 1 times 10minus5 in thisstudy except otherwise specified
4 Numerical Results
In order to evaluate the efficiency and competency of intro-duced methods analysis of two FE models with extensiveavailable numerical studies has been carried out Represen-tative examples are presented to demonstrate the effects ofspeed of loads measurement noise level and initial error onthe accuracy and effectiveness of the methods
The relative percentage error (RPE) in the identifiedresults is calculated from (27) where sdot is the norm ofmatrix and z
119905119905identifiedand z119905119905true
are the identified and the trueacceleration time history respectively [26]
Since the true value of elastic modulus is unknown RPE canjust be used for evaluating the efficiency of the method
41 Single-Span Bridge A single-span bridge is studied tocompare different methods This bridge consists of 10 Euler-Bernoulli beam elements with 11 nodes each with two DOFsTotal length of bridge is 10m and height and width ofthe frame section are 200mm Rayleigh damping model isadopted with the damping ratios of the first two modes takenequal to 005
Structural properties are summarized in Table 1The transverse point load 119875 has a constant velocity V =
119871119879 where 119879 is the traveling time across the bridge and 119871 isthe total length of the bridge
For the forced vibration analysis an implicit time integra-tion method called as ldquothe Newmark integration methodrdquo isused with the integration parameters 120573 = 14 and 120574 = 12which leads to constant-average acceleration approximation
Speed parameter is defined as
120572 =
119881
119881cr (28)
in which119881cr is critical speed (119881cr = (120587119897)radicEI120588)119881 is movingload speed and 120588 is mass per unit length of the beams
411 Damage Scenarios Six damage scenarios of singlemultiple and random damage along with initial error in thebridge without measurement noise are studied and shown inTable 2
Local damage is simulated with a reduction in the elasticmodulus of material of an element The sampling rate is15000Hz and 250 data of the acceleration response (degree ofindeterminacy is 25) collected along the z-direction at nodes2 and 7 are used in the identification
412 Analysis Results Tables 3ndash6 show the results of differentdetection methods including solution time (ST) number of
loops (NL) and RPE These are the most important param-eters to compare the efficiency and accuracy of differentmethods
413 Effect of Noise Noise is the random fluctuation in thevalue of measured or input that causes random fluctuation inthe output value Noise at the sensor output is due to eitherinternal noise sources such as resistors at finite temperaturesor externally generatedmechanical and electromagnetic fluc-tuations [10]
To evaluate the sensitivity of results to suchmeasurementnoise noise-polluted measurements are simulated by addingto the noise-free acceleration vector a corresponding noisevector whose root-mean-square (RMS) value is equal toa certain percentage of the RMS value of the noise-freedata vector The components of all the noise vectors are ofGaussian distribution uncorrelated andwith a zeromean andunit standard deviation Then on the basis of the noise-freeacceleration z
119905119905nfthe noise-polluted acceleration z
119905119905npof the
bridge at location 119909 can be simulated by
z119905119905np
= z119905119905nf
+ RMS (z119905119905nf) times 119873level times 119873unit (29)
where RMS (z119905119905nf
) is the RMS value of the noise-free accel-eration vector z
119905119905nftimes 119873level is the noise level and 119873unit is a
randomly generated noise vector with zero mean and unitstandard deviation [36]
In order to study effects of noise in stability of differentsensitivity methods scenario 3 (speed ratio of moving loadis considered to be fixed and equal to 05) is considered anddifferent levels of noise pollution are investigated and alsoRPE change with increasing the number of loops in iterativeprocedure has been studied The results are illustrated inFigure 3 for all presented methods
414 Effect of Initial Error In order to study effects of initialerror in stability of sensitivity methods scenario 6 (speedratio ofmoving load is considered to be fixed and equal to 05)is considered This scenario consists of no simulated damagein the structure but the initial elastic modulus of materialof all the elements is underestimated and different levels ofinitial error are investigated RPE changes with increasing thenumber of loops in iterative procedure have been studiedTheresults are illustrated in Figure 4 for all presented methods
42 Plane Grid Model A plane grid model of bridge isstudied as another numerical example to investigate theefficiency and accuracy of discrete methods It is notable thataccording to complexity of this model approximation andsemidiscrete methods are not useable and diverged in allscenarios in all speed ranges
The structure is modeled by 46 frame elements and 32nodes with three DOFs at each node for the translation androtational deformations Rayleigh damping model is adoptedwith the damping ratios of the first two modes taken equalto 005 The finite element model of the structure is shownin Figure 5 and structural properties are summarized inTable 1
8 Shock and Vibration
Table1Prop
ertie
sofd
ifferentm
odels
Mod
elname
Num
bero
felements
Num
bero
fnod
esMassd
ensity
Mod
ulus
ofelasticity
Rayleigh
coeffi
cients
Und
amaged
naturalfrequ
ency
ofintactmod
elkgm
m3
Nm
m2
1198860
1198861
Firstm
ode
Second
mod
eTh
irdmod
eFo
urth
mod
eFifth
mod
eMultispan
1011
78times10minus9
21times
10minus5
01
2860times10minus5
296
1183
2662
4738
7421
Grid
4632
78times10minus9
21times
10minus5
01
2364times10minus5
456
928
1817
2597
3991
Shock and Vibration 9
Table 2 Damage scenarios for single-span bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 4 17 NilM1-2 Multiple 2 7 4 21 NilM1-3 Multiple 3 5 6 and 8 12 6 5 and 2 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
Table 3 Solution time number of loops and RPE of ADMmethod for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
421 Damage Scenarios Six damage scenarios of singlemultiple and randomdamage in the bridgewithoutmeasure-ment noise are studied and shown in Table 7
The sampling rate is 14000Hz and 460 data of the acceler-ation response (degree of indeterminacy is 20) collected alongthe z-direction at nodes 4 11 21 and 27 are used
422 Analysis Results Tables 8 and 9 show the results ofdifferent detection methods including solution time (ST)
and RPE Using both described methods including ADMand DDM method the damage locations and amount areidentified correctly in all the scenarios (Figure 6)
423 Effect of Noise In order to study effects of noisescenario 3 is considered and different levels of noise pollutionare investigated and also RPE changes with increasing thenumber of loops for iterative procedure have been studiedThe results are presented in Figure 7
10 Shock and Vibration
Table 6 Solution time loops and RPE of DDMFD method for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
Figure 3 RPE contours with respect to noise level and loops in model 1
Shock and Vibration 11
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
10
20
30
40
50
60
70
(b) DDMmethod
Loops
Initi
al er
ror
FD method
5 10 15 20 25 300
005
01
015
02
025
03
10
20
30
40
50
60
70
80
(c) FD method
Loops
Initi
al er
ror
DDMFD method
5 10 15 20 25 300
005
01
015
02
025
03
0
10
20
30
40
50
60
70
80
(d) DDMFD method
Figure 4 RPE contours with respect to initial error and loops in model 1
424 Effect of Initial Error Scenario 6 (speed ratio ofmovingload is considered to be fixed and equal to 05) is considered tostudy effects of initial error in stability of sensitivity methodsRPE changes with increasing the number of loops for iterativeprocedure have been studied The results are illustrated inFigure 8
5 Comparison of Different Methods
According to the results obtained in the previous sectiondifferent sensitivity methods considered in this study arecompared in this section Iteration number relative percent-age of error (RPE) solution time noise level and initialerror effect in stability of damage detection procedure areeffective parameters to compare efficiency and accuracy of themethods
51 Iteration Number Average number of iterations fordifferent scenarios with respect to the speed parameter isillustrated in Figure 9
As seen in Figure 9(a) the number of iterations in FDand DDMFD methods is considerably larger than thoseof analytical discrete methods (DDM and ADM) It hasbeen observed that in all cases DDMFD method has feweriterations than FD method Average number of iterations is20757 and 251467 for DDMFD and FD methods respec-tively The larger the speed ratio of moving vehicle is thelarger the number of iterations in approximation methodsis In FD method for speed ratios more than 05 even byusing maximum number of iterations (300) convergence wasnot achieved while the ratio used in DDMFD method is09
Figure 9(b) shows comparison along with analytical dis-crete methods It can be seen that DDM has smaller iterationnumber in all cases Average number of iterations is 146and 11167 for ADM and DDM methods respectively Bothanalytical discrete methods converged in all cases withmaximum iterations equal to 17 and increment in speed ratiodoes not affect the number of iterations
12 Shock and Vibration
Sensors
Element number
25
17
9
2622
15
8
1
23
16
9
2
24
17
10
3
25
18
11
4
2619
12
5
2720
13
6
2821
147
4118
3510
29
2
2742
1936
1130
3
2843
2037
1231
4
29 4421
3813
325
30 4522 39
14
16
336
3132
46 2324
40 1534 7
8
Direction of measured response for identification
Node number
P V
Moving vehicle
Y Z X
7000 (mm)
3000 (mm)
Figure 5 Plane grid bridge model used in detection procedure
Table 7 Damage scenarios for multispan bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 23 5 NilM1-2 Multiple 8 13 and 29 11 4 and 7 NilM1-3 Multiple 3 7 19 25 and 28 12 6 5 2 and 18 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
52 Relative Percentage of Error (RPE) Using the sameconvergence tolerance equal to 1 times 10minus5 relative percentageof error in different methods is illustrated in Figure 10
As seen in this figure with a maximum number ofiterations equal to 300 RPE in FD and DDMFD methodsare considerably larger than DDM and ADM methodsDDMFD method has lower RPE than FD method in mostcases Average of RPE is 026329 for DDMFD compared to027766 in FD method Again the larger the speed ratio ofmoving vehicle is the larger the relative error percentage inapproximation methods is
Figure 10(b) shows that relative error percentage in ADMmethod is lower than DDM and speed ratio does not affectthe relative error
Average of RPE is 0000731 and 0002037 for ADM andDDM methods respectively Therefore ADM is nearly 28times more accurate than DDM
53 Solution Time In order to investigate computational costof sensitivity methods using the same amounts of RAMand CPU resources for the presented numerical simulationssolution time of different damaged models is evaluated andshown in Figure 11
As seen in Figure 11(a) solution times in FD and DDMFDmethods are considerably larger than analytical discretemethods and the FDmethod with an average of 200616 (s) isthemost time consumingmethod Average time for DDMFDmethod is 150849 (s)
Figure 11(b) compares the time spent by DDM and ADMAs illustrated in this figure ADM method has lower solutiontime than DDMmethod in all cases and speed ratio does notaffect the solution time
Average of solution time is 2956 (s) and 7794 (s) forADM and DDM methods respectively consequently ADMis nearly 26 times faster than DDM
54 Efficiency Parameter In order to compare and quantifythe performance of different methods relative efficiencyparameters of methods ldquo119894rdquo and ldquo119895rdquo (REP
119894119895
) are defined as
REP119894119895
= radic
ST119895
ST119894
times
RPE119895
RPE119894
(30)
in which ST is the solution time of system identificationmethod that represents the computational cost of themethodand RPE is the relative error which represents the accuracy ofthe method
Shock and Vibration 13
Table 8 Solution time number of loops and RPE of ADMmethod for model 2
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE ()
541 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to DDM Figure 12 shows REPadjddm changeswith respect to speed parameter in different scenarios
Table 10 shows that in different scenarios and for differentspeed parameters the efficiency parameter is between 19713and 55478 for model 1 and 15659 and 29893 for model2 and its average is 28948 and 22248 for models 1 and2 respectively Compared to DDM the adjoint variablemethod is more efficient and needs 609 less computationaleffort
Changes in the average of REPadjddm in different scenarioswith speed ratio are shown in Figure 13 As shown in thisfigure REPadjddm increases with speed ratios lightly Thisindicates that efficiency of ADM with respect to DDMincreases with speed ratio
542 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to Approximation and Semianalytical MethodsFigure 14 shows REPadjFD and REPadjDDMFD changes withrespect to speed parameter in different scenarios
Table 11 shows the relative efficiency parameter for differ-ent speed parameters in different scenarios It can be seen thatthe efficiency parameter is between 3193 and 78307 for finitedifference method and 4916 and 50542 for DDMFDmethodwith the average of 16714 and 14569 for FD and DDMFDmethods respectively Therefore the adjoint variable methodis extremely successful and the computational cost of thismethod is just 06 and 069 of FD and DDMFDmethodsrespectively
Changes in the average of REPadjfdampddmfd in differentscenarios with speed ratio are shown in Figure 15 As shownin this figure REPadjfdampddmfd rapidly increases with speedratio which indicates the superiority of ADM over FD andDDMFDmethods increases with speed ratio
0 5 10 15 20 25 30 35 40 45 500
051
152
25
Element number
Mod
ule o
f ela
stici
ty
0 5 10 15 20 25 30 35 40 45 5005
101520
Dam
age i
ndex
Element number
Original modelDetected model
times105
Figure 6 Detection of damage location and amount in elements 57 12 15 24 and 37 and distribution of error in different elementswith ADM scheme
543 Relative Efficiency Parameter of DDMFD with respectto FD Method Figure 16 shows REPddmfdfd changes withrespect to speed parameter in different scenarios
14 Shock and Vibration
Table 10 REPadjddm ranges in different scenarios
Damage scenario Max REP Min REP AverageModel 1 Model 2 Model 1 Model 2 Model 1 Model 2
Figure 7 RPE contours with respect to noise level and loops in model 2
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(b) DDMmethod
Figure 8 RPE contours with respect to initial error and loops in model 2
Shock and Vibration 15
0
50
100
150
200
250
300
350
0103050709
DDMFDFD
DDMADM
Itera
tion
num
ber
Speed ratio
(a) All methods
0
2
4
6
8
10
12
14
16
18
0103050709
Itera
tion
num
ber
Speed ratio
DDMADM
(b) Analytical discrete methods
Figure 9 Comparison of iteration number between different sensitivity methods
DDMFD
RPE
FDDDMADM
0
01
02
03
04
05
06
07
0103050709Speed ratio
(a) All methods
DDMADM
0103050709
RPE
Speed ratio
00E + 00
50E minus 04
10E minus 03
15E minus 03
20E minus 03
25E minus 03
(b) Analytical discrete methods
Figure 10 Comparison of RPE between different sensitivity methods
Table 12 shows the relative efficiency parameter for dif-ferent speed parameters in different scenarios Again theefficiency parameter is between 0343 and 1726 with theaverage of 1044Thus compared to FDmethod theDDMFDmethod is slightly more effective
Changes in the average of REPddmfdfd in different sce-narios with speed ratio are shown in Figure 17 As shown inthis figure REPddmfdfd increases with speed ratio For speedratio lower than 05 efficiency parameter is lower than 1 whichmeans FD method is more effective for this range
55 Stability Figures 3 and 7 show that all describedmethodsare sensitive to noise and if noise level exceeds the valuesstated in Table 11 these methods lose their efficiency and arenot able to detect damage properly Hence in cases with noise
level larger than the values stated in Table 13 a denoisingtool like wavelet transforms should be used along with thesensitivity methodsWavelet transforms are mainly attractivebecause of their ability to compress and encode informationto reduce noise or to detect any local singular behavior of asignal [37]
Stability comparison of different methods shows that FDis the most stable method and stability of analytical methodsis slightly lower than that
Figures 4 and 8 show stability of sensitivity methodswith respect to the initial error Summary of results forstability investigation is presented in Table 13 which suggeststhat stability of numerical methods is better than analyticalmethods
16 Shock and Vibration
DDMFDFD
DDMADM
0
50
100
150
200
250
300
0103050709
Solu
tion
time
Speed ratio
(a) All methods
DDMADM
0
2
4
6
8
10
0103050709
Solu
tion
time
Speed ratio
(b) Analytical discrete methods
Figure 11 Comparison of solution time between different sensitivity methods
5-64-5
3-4
2-31-20-1
120572 = 01
120572 = 05
120572 = 09
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
6
123456
0
(a) Model 1
120572 = 01
120572 = 05
120572 = 09
25ndash32ndash2515ndash2
1ndash1505ndash10ndash05
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
63252151050
(b) Model 2
Figure 12 REPadjddm changes in different scenarios with respect to speed parameter
Table 11 REPadjadm ranges in different scenarios
Damage scenario Max REP Min REP AverageFD DDMFD FD DDMFD FD DDMFD
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
120597b119894 are sensitivity vectorsof displacement velocity and acceleration with respect todesign parameter 119887
119894
respectively Assume that
y119905119905
=
120597z119905119905
120597119887119894
(19a)
y119905
=
120597z119905
120597119887119894
(19b)
y = 120597z120597119887119894
(19c)
So by replacing (19a) (19b) and (19c) into (18) we have
My119905119905
+ Cy119905
+ Ky = minus120597K120597119887119894
z minus 1205722
120597K120597119887119894
z119905
(20)
Right side of (20) can be considered as an equivalent force so(20) is similar to (16) and sensitivity vectors can be obtainedby Newmark method
332 Adjoint Variable Method Mirzaee et al [34] provedthat sensitivity analysis can be performed very efficientlyby using deterministic methods based on adjoint functionsThis method constructs an adjoint problem that solves forthe adjoint variable which contains all implicitly dependentterms
The adjoint variablemethod yields a terminal-value prob-lem compared with the initial-value problem of responseanalysis
For the dynamic structure the following formof a generalperformance measure can be considered
120595 = 119892 (z (119879) 119887) + int119879
0
119866 (z 119887) 119889119905 (21)
General form of sensitivity measure relative to the designparameter (b) is as follows [33]
1205951015840
= [
120597119892
120597zminus
120597119892
120597zz119905
(119879) + 119866 (z (119879) 119887) 1Ω119905
120597Ω
120597119911
] z1015840 (119879)
minus [
120597119892
120597119911
z119905
(119879) + 119866 (z (119879) 119887)] 1
Ω119905
120597Ω
120597119911119905
z1015840119905
(119879)
+ int
119879
0
[
120597119866
120597zz1015840 + 120597119866
120597119887
120575119887] 119889119905 +
120597119892
120597119887
120575119887
minus [
120597119892
120597119911
z119905
(119879) + 119866 (119911 (119879) 119887)]
1
Ω119905
120597Ω
120597119887
120575119887
(22)
Note that1205951015840 depends on z1015840 and z1015840119905
at119879 as well as on z1015840 withinthe integration
Mirzaee et el [34] showed that while structural vibrationresponses are used for damage detection solving (22) inknown time 119879 is as follows
119889120595
119889119887
(119879) = int
119879
0
(minus120582119879
1198860
120597K120597119887
z119905
minus120582119879120597K120597119887
z)119889119905 (23)
120582119879
(119905) = P119879
119891 (119905) +Q119879
119892 (119905) (24)
in which
P119879
= 119890minus120585120596119879
120582119905
(119879)
120596119863
cos (120596119863
119879)
119891 (119905) = 119890120585120596119905 sin (120596
119863
119905)
Q119879
= minus119890minus120585120596119879
120582119905
(119879)
120596119863
sin (120596119863
119879)
119892 (119905) = 119890120585120596119905 cos (120596
119863
119905)
(25)
It is worth noting that terminal values for different values of119879 are not similar and adjoint equation should be calculatedfor corresponding 119879 separately
Mirzaee et al [34] presented a quite efficient incrementalsolution for solving (24) which significantly improves theproposed method
34 Sensitivity Method Selection In this paper four differentmethods are presented to compute sensitivity information ofthe performance measure for damage detection procedureFirst method is the finite Difference method (FD) whichis classified as an approximation approach Second it issemianalytical discrete method which uses a finite differencescheme to calculate stiffness matrix in direct differentialmethod (DDMFD) and third and fourthmethods are two dif-ferent analytical discrete methods direct differential method(DDM) and adjoint variable method (ADM)
The first three methods including FD DDMFD andDDM are widely used by other authors [35 36] and the lastmethod (ADM) is proposed by the authors (Mirzaee et al[34])
35 Procedure of Iteration for Damage Detection The initialanalytical model of a structure deviates from the true modeland measurement from the initial intact structure is used toupdate the analytical model The improved model is thentreated as a reference model and measurement from thedamaged structure will be used to update the referencemodel
When response measurement from the intact state of thestructure is obtained the sensitivities are computed frompresented methods based on the analytical model of thestructure and the well knowing input force and velocityThe vector of parameter increments is then obtained usingthe computed and experimentally obtained responses Theanalytical model is then updated and the correspondingresponse and its sensitivity are again computed for the nextiteration When measurement from the damaged state isobtained the updated analyticalmodel is used in the iterationin the same way as that using measurement from the intactstate Convergence is considered to be achieved when thefollowing criteria are met
1003817100381710038171003817E119894+1
minus E119894
1003817100381710038171003817
1003817100381710038171003817E119894
1003817100381710038171003817
times 100 le Tol1
1003817100381710038171003817Response
119894+1
minus Response119894
1003817100381710038171003817
1003817100381710038171003817Response
119894
1003817100381710038171003817
times 100 le Tol2
(26)
Shock and Vibration 7
The final vector of identified parameter increments corre-sponds to the changes occurring in between the two statesof the structure The tolerance is set equal to 1 times 10minus5 in thisstudy except otherwise specified
4 Numerical Results
In order to evaluate the efficiency and competency of intro-duced methods analysis of two FE models with extensiveavailable numerical studies has been carried out Represen-tative examples are presented to demonstrate the effects ofspeed of loads measurement noise level and initial error onthe accuracy and effectiveness of the methods
The relative percentage error (RPE) in the identifiedresults is calculated from (27) where sdot is the norm ofmatrix and z
119905119905identifiedand z119905119905true
are the identified and the trueacceleration time history respectively [26]
Since the true value of elastic modulus is unknown RPE canjust be used for evaluating the efficiency of the method
41 Single-Span Bridge A single-span bridge is studied tocompare different methods This bridge consists of 10 Euler-Bernoulli beam elements with 11 nodes each with two DOFsTotal length of bridge is 10m and height and width ofthe frame section are 200mm Rayleigh damping model isadopted with the damping ratios of the first two modes takenequal to 005
Structural properties are summarized in Table 1The transverse point load 119875 has a constant velocity V =
119871119879 where 119879 is the traveling time across the bridge and 119871 isthe total length of the bridge
For the forced vibration analysis an implicit time integra-tion method called as ldquothe Newmark integration methodrdquo isused with the integration parameters 120573 = 14 and 120574 = 12which leads to constant-average acceleration approximation
Speed parameter is defined as
120572 =
119881
119881cr (28)
in which119881cr is critical speed (119881cr = (120587119897)radicEI120588)119881 is movingload speed and 120588 is mass per unit length of the beams
411 Damage Scenarios Six damage scenarios of singlemultiple and random damage along with initial error in thebridge without measurement noise are studied and shown inTable 2
Local damage is simulated with a reduction in the elasticmodulus of material of an element The sampling rate is15000Hz and 250 data of the acceleration response (degree ofindeterminacy is 25) collected along the z-direction at nodes2 and 7 are used in the identification
412 Analysis Results Tables 3ndash6 show the results of differentdetection methods including solution time (ST) number of
loops (NL) and RPE These are the most important param-eters to compare the efficiency and accuracy of differentmethods
413 Effect of Noise Noise is the random fluctuation in thevalue of measured or input that causes random fluctuation inthe output value Noise at the sensor output is due to eitherinternal noise sources such as resistors at finite temperaturesor externally generatedmechanical and electromagnetic fluc-tuations [10]
To evaluate the sensitivity of results to suchmeasurementnoise noise-polluted measurements are simulated by addingto the noise-free acceleration vector a corresponding noisevector whose root-mean-square (RMS) value is equal toa certain percentage of the RMS value of the noise-freedata vector The components of all the noise vectors are ofGaussian distribution uncorrelated andwith a zeromean andunit standard deviation Then on the basis of the noise-freeacceleration z
119905119905nfthe noise-polluted acceleration z
119905119905npof the
bridge at location 119909 can be simulated by
z119905119905np
= z119905119905nf
+ RMS (z119905119905nf) times 119873level times 119873unit (29)
where RMS (z119905119905nf
) is the RMS value of the noise-free accel-eration vector z
119905119905nftimes 119873level is the noise level and 119873unit is a
randomly generated noise vector with zero mean and unitstandard deviation [36]
In order to study effects of noise in stability of differentsensitivity methods scenario 3 (speed ratio of moving loadis considered to be fixed and equal to 05) is considered anddifferent levels of noise pollution are investigated and alsoRPE change with increasing the number of loops in iterativeprocedure has been studied The results are illustrated inFigure 3 for all presented methods
414 Effect of Initial Error In order to study effects of initialerror in stability of sensitivity methods scenario 6 (speedratio ofmoving load is considered to be fixed and equal to 05)is considered This scenario consists of no simulated damagein the structure but the initial elastic modulus of materialof all the elements is underestimated and different levels ofinitial error are investigated RPE changes with increasing thenumber of loops in iterative procedure have been studiedTheresults are illustrated in Figure 4 for all presented methods
42 Plane Grid Model A plane grid model of bridge isstudied as another numerical example to investigate theefficiency and accuracy of discrete methods It is notable thataccording to complexity of this model approximation andsemidiscrete methods are not useable and diverged in allscenarios in all speed ranges
The structure is modeled by 46 frame elements and 32nodes with three DOFs at each node for the translation androtational deformations Rayleigh damping model is adoptedwith the damping ratios of the first two modes taken equalto 005 The finite element model of the structure is shownin Figure 5 and structural properties are summarized inTable 1
8 Shock and Vibration
Table1Prop
ertie
sofd
ifferentm
odels
Mod
elname
Num
bero
felements
Num
bero
fnod
esMassd
ensity
Mod
ulus
ofelasticity
Rayleigh
coeffi
cients
Und
amaged
naturalfrequ
ency
ofintactmod
elkgm
m3
Nm
m2
1198860
1198861
Firstm
ode
Second
mod
eTh
irdmod
eFo
urth
mod
eFifth
mod
eMultispan
1011
78times10minus9
21times
10minus5
01
2860times10minus5
296
1183
2662
4738
7421
Grid
4632
78times10minus9
21times
10minus5
01
2364times10minus5
456
928
1817
2597
3991
Shock and Vibration 9
Table 2 Damage scenarios for single-span bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 4 17 NilM1-2 Multiple 2 7 4 21 NilM1-3 Multiple 3 5 6 and 8 12 6 5 and 2 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
Table 3 Solution time number of loops and RPE of ADMmethod for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
421 Damage Scenarios Six damage scenarios of singlemultiple and randomdamage in the bridgewithoutmeasure-ment noise are studied and shown in Table 7
The sampling rate is 14000Hz and 460 data of the acceler-ation response (degree of indeterminacy is 20) collected alongthe z-direction at nodes 4 11 21 and 27 are used
422 Analysis Results Tables 8 and 9 show the results ofdifferent detection methods including solution time (ST)
and RPE Using both described methods including ADMand DDM method the damage locations and amount areidentified correctly in all the scenarios (Figure 6)
423 Effect of Noise In order to study effects of noisescenario 3 is considered and different levels of noise pollutionare investigated and also RPE changes with increasing thenumber of loops for iterative procedure have been studiedThe results are presented in Figure 7
10 Shock and Vibration
Table 6 Solution time loops and RPE of DDMFD method for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
Figure 3 RPE contours with respect to noise level and loops in model 1
Shock and Vibration 11
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
10
20
30
40
50
60
70
(b) DDMmethod
Loops
Initi
al er
ror
FD method
5 10 15 20 25 300
005
01
015
02
025
03
10
20
30
40
50
60
70
80
(c) FD method
Loops
Initi
al er
ror
DDMFD method
5 10 15 20 25 300
005
01
015
02
025
03
0
10
20
30
40
50
60
70
80
(d) DDMFD method
Figure 4 RPE contours with respect to initial error and loops in model 1
424 Effect of Initial Error Scenario 6 (speed ratio ofmovingload is considered to be fixed and equal to 05) is considered tostudy effects of initial error in stability of sensitivity methodsRPE changes with increasing the number of loops for iterativeprocedure have been studied The results are illustrated inFigure 8
5 Comparison of Different Methods
According to the results obtained in the previous sectiondifferent sensitivity methods considered in this study arecompared in this section Iteration number relative percent-age of error (RPE) solution time noise level and initialerror effect in stability of damage detection procedure areeffective parameters to compare efficiency and accuracy of themethods
51 Iteration Number Average number of iterations fordifferent scenarios with respect to the speed parameter isillustrated in Figure 9
As seen in Figure 9(a) the number of iterations in FDand DDMFD methods is considerably larger than thoseof analytical discrete methods (DDM and ADM) It hasbeen observed that in all cases DDMFD method has feweriterations than FD method Average number of iterations is20757 and 251467 for DDMFD and FD methods respec-tively The larger the speed ratio of moving vehicle is thelarger the number of iterations in approximation methodsis In FD method for speed ratios more than 05 even byusing maximum number of iterations (300) convergence wasnot achieved while the ratio used in DDMFD method is09
Figure 9(b) shows comparison along with analytical dis-crete methods It can be seen that DDM has smaller iterationnumber in all cases Average number of iterations is 146and 11167 for ADM and DDM methods respectively Bothanalytical discrete methods converged in all cases withmaximum iterations equal to 17 and increment in speed ratiodoes not affect the number of iterations
12 Shock and Vibration
Sensors
Element number
25
17
9
2622
15
8
1
23
16
9
2
24
17
10
3
25
18
11
4
2619
12
5
2720
13
6
2821
147
4118
3510
29
2
2742
1936
1130
3
2843
2037
1231
4
29 4421
3813
325
30 4522 39
14
16
336
3132
46 2324
40 1534 7
8
Direction of measured response for identification
Node number
P V
Moving vehicle
Y Z X
7000 (mm)
3000 (mm)
Figure 5 Plane grid bridge model used in detection procedure
Table 7 Damage scenarios for multispan bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 23 5 NilM1-2 Multiple 8 13 and 29 11 4 and 7 NilM1-3 Multiple 3 7 19 25 and 28 12 6 5 2 and 18 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
52 Relative Percentage of Error (RPE) Using the sameconvergence tolerance equal to 1 times 10minus5 relative percentageof error in different methods is illustrated in Figure 10
As seen in this figure with a maximum number ofiterations equal to 300 RPE in FD and DDMFD methodsare considerably larger than DDM and ADM methodsDDMFD method has lower RPE than FD method in mostcases Average of RPE is 026329 for DDMFD compared to027766 in FD method Again the larger the speed ratio ofmoving vehicle is the larger the relative error percentage inapproximation methods is
Figure 10(b) shows that relative error percentage in ADMmethod is lower than DDM and speed ratio does not affectthe relative error
Average of RPE is 0000731 and 0002037 for ADM andDDM methods respectively Therefore ADM is nearly 28times more accurate than DDM
53 Solution Time In order to investigate computational costof sensitivity methods using the same amounts of RAMand CPU resources for the presented numerical simulationssolution time of different damaged models is evaluated andshown in Figure 11
As seen in Figure 11(a) solution times in FD and DDMFDmethods are considerably larger than analytical discretemethods and the FDmethod with an average of 200616 (s) isthemost time consumingmethod Average time for DDMFDmethod is 150849 (s)
Figure 11(b) compares the time spent by DDM and ADMAs illustrated in this figure ADM method has lower solutiontime than DDMmethod in all cases and speed ratio does notaffect the solution time
Average of solution time is 2956 (s) and 7794 (s) forADM and DDM methods respectively consequently ADMis nearly 26 times faster than DDM
54 Efficiency Parameter In order to compare and quantifythe performance of different methods relative efficiencyparameters of methods ldquo119894rdquo and ldquo119895rdquo (REP
119894119895
) are defined as
REP119894119895
= radic
ST119895
ST119894
times
RPE119895
RPE119894
(30)
in which ST is the solution time of system identificationmethod that represents the computational cost of themethodand RPE is the relative error which represents the accuracy ofthe method
Shock and Vibration 13
Table 8 Solution time number of loops and RPE of ADMmethod for model 2
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE ()
541 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to DDM Figure 12 shows REPadjddm changeswith respect to speed parameter in different scenarios
Table 10 shows that in different scenarios and for differentspeed parameters the efficiency parameter is between 19713and 55478 for model 1 and 15659 and 29893 for model2 and its average is 28948 and 22248 for models 1 and2 respectively Compared to DDM the adjoint variablemethod is more efficient and needs 609 less computationaleffort
Changes in the average of REPadjddm in different scenarioswith speed ratio are shown in Figure 13 As shown in thisfigure REPadjddm increases with speed ratios lightly Thisindicates that efficiency of ADM with respect to DDMincreases with speed ratio
542 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to Approximation and Semianalytical MethodsFigure 14 shows REPadjFD and REPadjDDMFD changes withrespect to speed parameter in different scenarios
Table 11 shows the relative efficiency parameter for differ-ent speed parameters in different scenarios It can be seen thatthe efficiency parameter is between 3193 and 78307 for finitedifference method and 4916 and 50542 for DDMFDmethodwith the average of 16714 and 14569 for FD and DDMFDmethods respectively Therefore the adjoint variable methodis extremely successful and the computational cost of thismethod is just 06 and 069 of FD and DDMFDmethodsrespectively
Changes in the average of REPadjfdampddmfd in differentscenarios with speed ratio are shown in Figure 15 As shownin this figure REPadjfdampddmfd rapidly increases with speedratio which indicates the superiority of ADM over FD andDDMFDmethods increases with speed ratio
0 5 10 15 20 25 30 35 40 45 500
051
152
25
Element number
Mod
ule o
f ela
stici
ty
0 5 10 15 20 25 30 35 40 45 5005
101520
Dam
age i
ndex
Element number
Original modelDetected model
times105
Figure 6 Detection of damage location and amount in elements 57 12 15 24 and 37 and distribution of error in different elementswith ADM scheme
543 Relative Efficiency Parameter of DDMFD with respectto FD Method Figure 16 shows REPddmfdfd changes withrespect to speed parameter in different scenarios
14 Shock and Vibration
Table 10 REPadjddm ranges in different scenarios
Damage scenario Max REP Min REP AverageModel 1 Model 2 Model 1 Model 2 Model 1 Model 2
Figure 7 RPE contours with respect to noise level and loops in model 2
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(b) DDMmethod
Figure 8 RPE contours with respect to initial error and loops in model 2
Shock and Vibration 15
0
50
100
150
200
250
300
350
0103050709
DDMFDFD
DDMADM
Itera
tion
num
ber
Speed ratio
(a) All methods
0
2
4
6
8
10
12
14
16
18
0103050709
Itera
tion
num
ber
Speed ratio
DDMADM
(b) Analytical discrete methods
Figure 9 Comparison of iteration number between different sensitivity methods
DDMFD
RPE
FDDDMADM
0
01
02
03
04
05
06
07
0103050709Speed ratio
(a) All methods
DDMADM
0103050709
RPE
Speed ratio
00E + 00
50E minus 04
10E minus 03
15E minus 03
20E minus 03
25E minus 03
(b) Analytical discrete methods
Figure 10 Comparison of RPE between different sensitivity methods
Table 12 shows the relative efficiency parameter for dif-ferent speed parameters in different scenarios Again theefficiency parameter is between 0343 and 1726 with theaverage of 1044Thus compared to FDmethod theDDMFDmethod is slightly more effective
Changes in the average of REPddmfdfd in different sce-narios with speed ratio are shown in Figure 17 As shown inthis figure REPddmfdfd increases with speed ratio For speedratio lower than 05 efficiency parameter is lower than 1 whichmeans FD method is more effective for this range
55 Stability Figures 3 and 7 show that all describedmethodsare sensitive to noise and if noise level exceeds the valuesstated in Table 11 these methods lose their efficiency and arenot able to detect damage properly Hence in cases with noise
level larger than the values stated in Table 13 a denoisingtool like wavelet transforms should be used along with thesensitivity methodsWavelet transforms are mainly attractivebecause of their ability to compress and encode informationto reduce noise or to detect any local singular behavior of asignal [37]
Stability comparison of different methods shows that FDis the most stable method and stability of analytical methodsis slightly lower than that
Figures 4 and 8 show stability of sensitivity methodswith respect to the initial error Summary of results forstability investigation is presented in Table 13 which suggeststhat stability of numerical methods is better than analyticalmethods
16 Shock and Vibration
DDMFDFD
DDMADM
0
50
100
150
200
250
300
0103050709
Solu
tion
time
Speed ratio
(a) All methods
DDMADM
0
2
4
6
8
10
0103050709
Solu
tion
time
Speed ratio
(b) Analytical discrete methods
Figure 11 Comparison of solution time between different sensitivity methods
5-64-5
3-4
2-31-20-1
120572 = 01
120572 = 05
120572 = 09
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
6
123456
0
(a) Model 1
120572 = 01
120572 = 05
120572 = 09
25ndash32ndash2515ndash2
1ndash1505ndash10ndash05
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
63252151050
(b) Model 2
Figure 12 REPadjddm changes in different scenarios with respect to speed parameter
Table 11 REPadjadm ranges in different scenarios
Damage scenario Max REP Min REP AverageFD DDMFD FD DDMFD FD DDMFD
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
The final vector of identified parameter increments corre-sponds to the changes occurring in between the two statesof the structure The tolerance is set equal to 1 times 10minus5 in thisstudy except otherwise specified
4 Numerical Results
In order to evaluate the efficiency and competency of intro-duced methods analysis of two FE models with extensiveavailable numerical studies has been carried out Represen-tative examples are presented to demonstrate the effects ofspeed of loads measurement noise level and initial error onthe accuracy and effectiveness of the methods
The relative percentage error (RPE) in the identifiedresults is calculated from (27) where sdot is the norm ofmatrix and z
119905119905identifiedand z119905119905true
are the identified and the trueacceleration time history respectively [26]
Since the true value of elastic modulus is unknown RPE canjust be used for evaluating the efficiency of the method
41 Single-Span Bridge A single-span bridge is studied tocompare different methods This bridge consists of 10 Euler-Bernoulli beam elements with 11 nodes each with two DOFsTotal length of bridge is 10m and height and width ofthe frame section are 200mm Rayleigh damping model isadopted with the damping ratios of the first two modes takenequal to 005
Structural properties are summarized in Table 1The transverse point load 119875 has a constant velocity V =
119871119879 where 119879 is the traveling time across the bridge and 119871 isthe total length of the bridge
For the forced vibration analysis an implicit time integra-tion method called as ldquothe Newmark integration methodrdquo isused with the integration parameters 120573 = 14 and 120574 = 12which leads to constant-average acceleration approximation
Speed parameter is defined as
120572 =
119881
119881cr (28)
in which119881cr is critical speed (119881cr = (120587119897)radicEI120588)119881 is movingload speed and 120588 is mass per unit length of the beams
411 Damage Scenarios Six damage scenarios of singlemultiple and random damage along with initial error in thebridge without measurement noise are studied and shown inTable 2
Local damage is simulated with a reduction in the elasticmodulus of material of an element The sampling rate is15000Hz and 250 data of the acceleration response (degree ofindeterminacy is 25) collected along the z-direction at nodes2 and 7 are used in the identification
412 Analysis Results Tables 3ndash6 show the results of differentdetection methods including solution time (ST) number of
loops (NL) and RPE These are the most important param-eters to compare the efficiency and accuracy of differentmethods
413 Effect of Noise Noise is the random fluctuation in thevalue of measured or input that causes random fluctuation inthe output value Noise at the sensor output is due to eitherinternal noise sources such as resistors at finite temperaturesor externally generatedmechanical and electromagnetic fluc-tuations [10]
To evaluate the sensitivity of results to suchmeasurementnoise noise-polluted measurements are simulated by addingto the noise-free acceleration vector a corresponding noisevector whose root-mean-square (RMS) value is equal toa certain percentage of the RMS value of the noise-freedata vector The components of all the noise vectors are ofGaussian distribution uncorrelated andwith a zeromean andunit standard deviation Then on the basis of the noise-freeacceleration z
119905119905nfthe noise-polluted acceleration z
119905119905npof the
bridge at location 119909 can be simulated by
z119905119905np
= z119905119905nf
+ RMS (z119905119905nf) times 119873level times 119873unit (29)
where RMS (z119905119905nf
) is the RMS value of the noise-free accel-eration vector z
119905119905nftimes 119873level is the noise level and 119873unit is a
randomly generated noise vector with zero mean and unitstandard deviation [36]
In order to study effects of noise in stability of differentsensitivity methods scenario 3 (speed ratio of moving loadis considered to be fixed and equal to 05) is considered anddifferent levels of noise pollution are investigated and alsoRPE change with increasing the number of loops in iterativeprocedure has been studied The results are illustrated inFigure 3 for all presented methods
414 Effect of Initial Error In order to study effects of initialerror in stability of sensitivity methods scenario 6 (speedratio ofmoving load is considered to be fixed and equal to 05)is considered This scenario consists of no simulated damagein the structure but the initial elastic modulus of materialof all the elements is underestimated and different levels ofinitial error are investigated RPE changes with increasing thenumber of loops in iterative procedure have been studiedTheresults are illustrated in Figure 4 for all presented methods
42 Plane Grid Model A plane grid model of bridge isstudied as another numerical example to investigate theefficiency and accuracy of discrete methods It is notable thataccording to complexity of this model approximation andsemidiscrete methods are not useable and diverged in allscenarios in all speed ranges
The structure is modeled by 46 frame elements and 32nodes with three DOFs at each node for the translation androtational deformations Rayleigh damping model is adoptedwith the damping ratios of the first two modes taken equalto 005 The finite element model of the structure is shownin Figure 5 and structural properties are summarized inTable 1
8 Shock and Vibration
Table1Prop
ertie
sofd
ifferentm
odels
Mod
elname
Num
bero
felements
Num
bero
fnod
esMassd
ensity
Mod
ulus
ofelasticity
Rayleigh
coeffi
cients
Und
amaged
naturalfrequ
ency
ofintactmod
elkgm
m3
Nm
m2
1198860
1198861
Firstm
ode
Second
mod
eTh
irdmod
eFo
urth
mod
eFifth
mod
eMultispan
1011
78times10minus9
21times
10minus5
01
2860times10minus5
296
1183
2662
4738
7421
Grid
4632
78times10minus9
21times
10minus5
01
2364times10minus5
456
928
1817
2597
3991
Shock and Vibration 9
Table 2 Damage scenarios for single-span bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 4 17 NilM1-2 Multiple 2 7 4 21 NilM1-3 Multiple 3 5 6 and 8 12 6 5 and 2 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
Table 3 Solution time number of loops and RPE of ADMmethod for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
421 Damage Scenarios Six damage scenarios of singlemultiple and randomdamage in the bridgewithoutmeasure-ment noise are studied and shown in Table 7
The sampling rate is 14000Hz and 460 data of the acceler-ation response (degree of indeterminacy is 20) collected alongthe z-direction at nodes 4 11 21 and 27 are used
422 Analysis Results Tables 8 and 9 show the results ofdifferent detection methods including solution time (ST)
and RPE Using both described methods including ADMand DDM method the damage locations and amount areidentified correctly in all the scenarios (Figure 6)
423 Effect of Noise In order to study effects of noisescenario 3 is considered and different levels of noise pollutionare investigated and also RPE changes with increasing thenumber of loops for iterative procedure have been studiedThe results are presented in Figure 7
10 Shock and Vibration
Table 6 Solution time loops and RPE of DDMFD method for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
Figure 3 RPE contours with respect to noise level and loops in model 1
Shock and Vibration 11
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
10
20
30
40
50
60
70
(b) DDMmethod
Loops
Initi
al er
ror
FD method
5 10 15 20 25 300
005
01
015
02
025
03
10
20
30
40
50
60
70
80
(c) FD method
Loops
Initi
al er
ror
DDMFD method
5 10 15 20 25 300
005
01
015
02
025
03
0
10
20
30
40
50
60
70
80
(d) DDMFD method
Figure 4 RPE contours with respect to initial error and loops in model 1
424 Effect of Initial Error Scenario 6 (speed ratio ofmovingload is considered to be fixed and equal to 05) is considered tostudy effects of initial error in stability of sensitivity methodsRPE changes with increasing the number of loops for iterativeprocedure have been studied The results are illustrated inFigure 8
5 Comparison of Different Methods
According to the results obtained in the previous sectiondifferent sensitivity methods considered in this study arecompared in this section Iteration number relative percent-age of error (RPE) solution time noise level and initialerror effect in stability of damage detection procedure areeffective parameters to compare efficiency and accuracy of themethods
51 Iteration Number Average number of iterations fordifferent scenarios with respect to the speed parameter isillustrated in Figure 9
As seen in Figure 9(a) the number of iterations in FDand DDMFD methods is considerably larger than thoseof analytical discrete methods (DDM and ADM) It hasbeen observed that in all cases DDMFD method has feweriterations than FD method Average number of iterations is20757 and 251467 for DDMFD and FD methods respec-tively The larger the speed ratio of moving vehicle is thelarger the number of iterations in approximation methodsis In FD method for speed ratios more than 05 even byusing maximum number of iterations (300) convergence wasnot achieved while the ratio used in DDMFD method is09
Figure 9(b) shows comparison along with analytical dis-crete methods It can be seen that DDM has smaller iterationnumber in all cases Average number of iterations is 146and 11167 for ADM and DDM methods respectively Bothanalytical discrete methods converged in all cases withmaximum iterations equal to 17 and increment in speed ratiodoes not affect the number of iterations
12 Shock and Vibration
Sensors
Element number
25
17
9
2622
15
8
1
23
16
9
2
24
17
10
3
25
18
11
4
2619
12
5
2720
13
6
2821
147
4118
3510
29
2
2742
1936
1130
3
2843
2037
1231
4
29 4421
3813
325
30 4522 39
14
16
336
3132
46 2324
40 1534 7
8
Direction of measured response for identification
Node number
P V
Moving vehicle
Y Z X
7000 (mm)
3000 (mm)
Figure 5 Plane grid bridge model used in detection procedure
Table 7 Damage scenarios for multispan bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 23 5 NilM1-2 Multiple 8 13 and 29 11 4 and 7 NilM1-3 Multiple 3 7 19 25 and 28 12 6 5 2 and 18 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
52 Relative Percentage of Error (RPE) Using the sameconvergence tolerance equal to 1 times 10minus5 relative percentageof error in different methods is illustrated in Figure 10
As seen in this figure with a maximum number ofiterations equal to 300 RPE in FD and DDMFD methodsare considerably larger than DDM and ADM methodsDDMFD method has lower RPE than FD method in mostcases Average of RPE is 026329 for DDMFD compared to027766 in FD method Again the larger the speed ratio ofmoving vehicle is the larger the relative error percentage inapproximation methods is
Figure 10(b) shows that relative error percentage in ADMmethod is lower than DDM and speed ratio does not affectthe relative error
Average of RPE is 0000731 and 0002037 for ADM andDDM methods respectively Therefore ADM is nearly 28times more accurate than DDM
53 Solution Time In order to investigate computational costof sensitivity methods using the same amounts of RAMand CPU resources for the presented numerical simulationssolution time of different damaged models is evaluated andshown in Figure 11
As seen in Figure 11(a) solution times in FD and DDMFDmethods are considerably larger than analytical discretemethods and the FDmethod with an average of 200616 (s) isthemost time consumingmethod Average time for DDMFDmethod is 150849 (s)
Figure 11(b) compares the time spent by DDM and ADMAs illustrated in this figure ADM method has lower solutiontime than DDMmethod in all cases and speed ratio does notaffect the solution time
Average of solution time is 2956 (s) and 7794 (s) forADM and DDM methods respectively consequently ADMis nearly 26 times faster than DDM
54 Efficiency Parameter In order to compare and quantifythe performance of different methods relative efficiencyparameters of methods ldquo119894rdquo and ldquo119895rdquo (REP
119894119895
) are defined as
REP119894119895
= radic
ST119895
ST119894
times
RPE119895
RPE119894
(30)
in which ST is the solution time of system identificationmethod that represents the computational cost of themethodand RPE is the relative error which represents the accuracy ofthe method
Shock and Vibration 13
Table 8 Solution time number of loops and RPE of ADMmethod for model 2
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE ()
541 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to DDM Figure 12 shows REPadjddm changeswith respect to speed parameter in different scenarios
Table 10 shows that in different scenarios and for differentspeed parameters the efficiency parameter is between 19713and 55478 for model 1 and 15659 and 29893 for model2 and its average is 28948 and 22248 for models 1 and2 respectively Compared to DDM the adjoint variablemethod is more efficient and needs 609 less computationaleffort
Changes in the average of REPadjddm in different scenarioswith speed ratio are shown in Figure 13 As shown in thisfigure REPadjddm increases with speed ratios lightly Thisindicates that efficiency of ADM with respect to DDMincreases with speed ratio
542 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to Approximation and Semianalytical MethodsFigure 14 shows REPadjFD and REPadjDDMFD changes withrespect to speed parameter in different scenarios
Table 11 shows the relative efficiency parameter for differ-ent speed parameters in different scenarios It can be seen thatthe efficiency parameter is between 3193 and 78307 for finitedifference method and 4916 and 50542 for DDMFDmethodwith the average of 16714 and 14569 for FD and DDMFDmethods respectively Therefore the adjoint variable methodis extremely successful and the computational cost of thismethod is just 06 and 069 of FD and DDMFDmethodsrespectively
Changes in the average of REPadjfdampddmfd in differentscenarios with speed ratio are shown in Figure 15 As shownin this figure REPadjfdampddmfd rapidly increases with speedratio which indicates the superiority of ADM over FD andDDMFDmethods increases with speed ratio
0 5 10 15 20 25 30 35 40 45 500
051
152
25
Element number
Mod
ule o
f ela
stici
ty
0 5 10 15 20 25 30 35 40 45 5005
101520
Dam
age i
ndex
Element number
Original modelDetected model
times105
Figure 6 Detection of damage location and amount in elements 57 12 15 24 and 37 and distribution of error in different elementswith ADM scheme
543 Relative Efficiency Parameter of DDMFD with respectto FD Method Figure 16 shows REPddmfdfd changes withrespect to speed parameter in different scenarios
14 Shock and Vibration
Table 10 REPadjddm ranges in different scenarios
Damage scenario Max REP Min REP AverageModel 1 Model 2 Model 1 Model 2 Model 1 Model 2
Figure 7 RPE contours with respect to noise level and loops in model 2
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(b) DDMmethod
Figure 8 RPE contours with respect to initial error and loops in model 2
Shock and Vibration 15
0
50
100
150
200
250
300
350
0103050709
DDMFDFD
DDMADM
Itera
tion
num
ber
Speed ratio
(a) All methods
0
2
4
6
8
10
12
14
16
18
0103050709
Itera
tion
num
ber
Speed ratio
DDMADM
(b) Analytical discrete methods
Figure 9 Comparison of iteration number between different sensitivity methods
DDMFD
RPE
FDDDMADM
0
01
02
03
04
05
06
07
0103050709Speed ratio
(a) All methods
DDMADM
0103050709
RPE
Speed ratio
00E + 00
50E minus 04
10E minus 03
15E minus 03
20E minus 03
25E minus 03
(b) Analytical discrete methods
Figure 10 Comparison of RPE between different sensitivity methods
Table 12 shows the relative efficiency parameter for dif-ferent speed parameters in different scenarios Again theefficiency parameter is between 0343 and 1726 with theaverage of 1044Thus compared to FDmethod theDDMFDmethod is slightly more effective
Changes in the average of REPddmfdfd in different sce-narios with speed ratio are shown in Figure 17 As shown inthis figure REPddmfdfd increases with speed ratio For speedratio lower than 05 efficiency parameter is lower than 1 whichmeans FD method is more effective for this range
55 Stability Figures 3 and 7 show that all describedmethodsare sensitive to noise and if noise level exceeds the valuesstated in Table 11 these methods lose their efficiency and arenot able to detect damage properly Hence in cases with noise
level larger than the values stated in Table 13 a denoisingtool like wavelet transforms should be used along with thesensitivity methodsWavelet transforms are mainly attractivebecause of their ability to compress and encode informationto reduce noise or to detect any local singular behavior of asignal [37]
Stability comparison of different methods shows that FDis the most stable method and stability of analytical methodsis slightly lower than that
Figures 4 and 8 show stability of sensitivity methodswith respect to the initial error Summary of results forstability investigation is presented in Table 13 which suggeststhat stability of numerical methods is better than analyticalmethods
16 Shock and Vibration
DDMFDFD
DDMADM
0
50
100
150
200
250
300
0103050709
Solu
tion
time
Speed ratio
(a) All methods
DDMADM
0
2
4
6
8
10
0103050709
Solu
tion
time
Speed ratio
(b) Analytical discrete methods
Figure 11 Comparison of solution time between different sensitivity methods
5-64-5
3-4
2-31-20-1
120572 = 01
120572 = 05
120572 = 09
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
6
123456
0
(a) Model 1
120572 = 01
120572 = 05
120572 = 09
25ndash32ndash2515ndash2
1ndash1505ndash10ndash05
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
63252151050
(b) Model 2
Figure 12 REPadjddm changes in different scenarios with respect to speed parameter
Table 11 REPadjadm ranges in different scenarios
Damage scenario Max REP Min REP AverageFD DDMFD FD DDMFD FD DDMFD
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 4 17 NilM1-2 Multiple 2 7 4 21 NilM1-3 Multiple 3 5 6 and 8 12 6 5 and 2 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
Table 3 Solution time number of loops and RPE of ADMmethod for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
421 Damage Scenarios Six damage scenarios of singlemultiple and randomdamage in the bridgewithoutmeasure-ment noise are studied and shown in Table 7
The sampling rate is 14000Hz and 460 data of the acceler-ation response (degree of indeterminacy is 20) collected alongthe z-direction at nodes 4 11 21 and 27 are used
422 Analysis Results Tables 8 and 9 show the results ofdifferent detection methods including solution time (ST)
and RPE Using both described methods including ADMand DDM method the damage locations and amount areidentified correctly in all the scenarios (Figure 6)
423 Effect of Noise In order to study effects of noisescenario 3 is considered and different levels of noise pollutionare investigated and also RPE changes with increasing thenumber of loops for iterative procedure have been studiedThe results are presented in Figure 7
10 Shock and Vibration
Table 6 Solution time loops and RPE of DDMFD method for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
Figure 3 RPE contours with respect to noise level and loops in model 1
Shock and Vibration 11
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
10
20
30
40
50
60
70
(b) DDMmethod
Loops
Initi
al er
ror
FD method
5 10 15 20 25 300
005
01
015
02
025
03
10
20
30
40
50
60
70
80
(c) FD method
Loops
Initi
al er
ror
DDMFD method
5 10 15 20 25 300
005
01
015
02
025
03
0
10
20
30
40
50
60
70
80
(d) DDMFD method
Figure 4 RPE contours with respect to initial error and loops in model 1
424 Effect of Initial Error Scenario 6 (speed ratio ofmovingload is considered to be fixed and equal to 05) is considered tostudy effects of initial error in stability of sensitivity methodsRPE changes with increasing the number of loops for iterativeprocedure have been studied The results are illustrated inFigure 8
5 Comparison of Different Methods
According to the results obtained in the previous sectiondifferent sensitivity methods considered in this study arecompared in this section Iteration number relative percent-age of error (RPE) solution time noise level and initialerror effect in stability of damage detection procedure areeffective parameters to compare efficiency and accuracy of themethods
51 Iteration Number Average number of iterations fordifferent scenarios with respect to the speed parameter isillustrated in Figure 9
As seen in Figure 9(a) the number of iterations in FDand DDMFD methods is considerably larger than thoseof analytical discrete methods (DDM and ADM) It hasbeen observed that in all cases DDMFD method has feweriterations than FD method Average number of iterations is20757 and 251467 for DDMFD and FD methods respec-tively The larger the speed ratio of moving vehicle is thelarger the number of iterations in approximation methodsis In FD method for speed ratios more than 05 even byusing maximum number of iterations (300) convergence wasnot achieved while the ratio used in DDMFD method is09
Figure 9(b) shows comparison along with analytical dis-crete methods It can be seen that DDM has smaller iterationnumber in all cases Average number of iterations is 146and 11167 for ADM and DDM methods respectively Bothanalytical discrete methods converged in all cases withmaximum iterations equal to 17 and increment in speed ratiodoes not affect the number of iterations
12 Shock and Vibration
Sensors
Element number
25
17
9
2622
15
8
1
23
16
9
2
24
17
10
3
25
18
11
4
2619
12
5
2720
13
6
2821
147
4118
3510
29
2
2742
1936
1130
3
2843
2037
1231
4
29 4421
3813
325
30 4522 39
14
16
336
3132
46 2324
40 1534 7
8
Direction of measured response for identification
Node number
P V
Moving vehicle
Y Z X
7000 (mm)
3000 (mm)
Figure 5 Plane grid bridge model used in detection procedure
Table 7 Damage scenarios for multispan bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 23 5 NilM1-2 Multiple 8 13 and 29 11 4 and 7 NilM1-3 Multiple 3 7 19 25 and 28 12 6 5 2 and 18 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
52 Relative Percentage of Error (RPE) Using the sameconvergence tolerance equal to 1 times 10minus5 relative percentageof error in different methods is illustrated in Figure 10
As seen in this figure with a maximum number ofiterations equal to 300 RPE in FD and DDMFD methodsare considerably larger than DDM and ADM methodsDDMFD method has lower RPE than FD method in mostcases Average of RPE is 026329 for DDMFD compared to027766 in FD method Again the larger the speed ratio ofmoving vehicle is the larger the relative error percentage inapproximation methods is
Figure 10(b) shows that relative error percentage in ADMmethod is lower than DDM and speed ratio does not affectthe relative error
Average of RPE is 0000731 and 0002037 for ADM andDDM methods respectively Therefore ADM is nearly 28times more accurate than DDM
53 Solution Time In order to investigate computational costof sensitivity methods using the same amounts of RAMand CPU resources for the presented numerical simulationssolution time of different damaged models is evaluated andshown in Figure 11
As seen in Figure 11(a) solution times in FD and DDMFDmethods are considerably larger than analytical discretemethods and the FDmethod with an average of 200616 (s) isthemost time consumingmethod Average time for DDMFDmethod is 150849 (s)
Figure 11(b) compares the time spent by DDM and ADMAs illustrated in this figure ADM method has lower solutiontime than DDMmethod in all cases and speed ratio does notaffect the solution time
Average of solution time is 2956 (s) and 7794 (s) forADM and DDM methods respectively consequently ADMis nearly 26 times faster than DDM
54 Efficiency Parameter In order to compare and quantifythe performance of different methods relative efficiencyparameters of methods ldquo119894rdquo and ldquo119895rdquo (REP
119894119895
) are defined as
REP119894119895
= radic
ST119895
ST119894
times
RPE119895
RPE119894
(30)
in which ST is the solution time of system identificationmethod that represents the computational cost of themethodand RPE is the relative error which represents the accuracy ofthe method
Shock and Vibration 13
Table 8 Solution time number of loops and RPE of ADMmethod for model 2
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE ()
541 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to DDM Figure 12 shows REPadjddm changeswith respect to speed parameter in different scenarios
Table 10 shows that in different scenarios and for differentspeed parameters the efficiency parameter is between 19713and 55478 for model 1 and 15659 and 29893 for model2 and its average is 28948 and 22248 for models 1 and2 respectively Compared to DDM the adjoint variablemethod is more efficient and needs 609 less computationaleffort
Changes in the average of REPadjddm in different scenarioswith speed ratio are shown in Figure 13 As shown in thisfigure REPadjddm increases with speed ratios lightly Thisindicates that efficiency of ADM with respect to DDMincreases with speed ratio
542 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to Approximation and Semianalytical MethodsFigure 14 shows REPadjFD and REPadjDDMFD changes withrespect to speed parameter in different scenarios
Table 11 shows the relative efficiency parameter for differ-ent speed parameters in different scenarios It can be seen thatthe efficiency parameter is between 3193 and 78307 for finitedifference method and 4916 and 50542 for DDMFDmethodwith the average of 16714 and 14569 for FD and DDMFDmethods respectively Therefore the adjoint variable methodis extremely successful and the computational cost of thismethod is just 06 and 069 of FD and DDMFDmethodsrespectively
Changes in the average of REPadjfdampddmfd in differentscenarios with speed ratio are shown in Figure 15 As shownin this figure REPadjfdampddmfd rapidly increases with speedratio which indicates the superiority of ADM over FD andDDMFDmethods increases with speed ratio
0 5 10 15 20 25 30 35 40 45 500
051
152
25
Element number
Mod
ule o
f ela
stici
ty
0 5 10 15 20 25 30 35 40 45 5005
101520
Dam
age i
ndex
Element number
Original modelDetected model
times105
Figure 6 Detection of damage location and amount in elements 57 12 15 24 and 37 and distribution of error in different elementswith ADM scheme
543 Relative Efficiency Parameter of DDMFD with respectto FD Method Figure 16 shows REPddmfdfd changes withrespect to speed parameter in different scenarios
14 Shock and Vibration
Table 10 REPadjddm ranges in different scenarios
Damage scenario Max REP Min REP AverageModel 1 Model 2 Model 1 Model 2 Model 1 Model 2
Figure 7 RPE contours with respect to noise level and loops in model 2
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(b) DDMmethod
Figure 8 RPE contours with respect to initial error and loops in model 2
Shock and Vibration 15
0
50
100
150
200
250
300
350
0103050709
DDMFDFD
DDMADM
Itera
tion
num
ber
Speed ratio
(a) All methods
0
2
4
6
8
10
12
14
16
18
0103050709
Itera
tion
num
ber
Speed ratio
DDMADM
(b) Analytical discrete methods
Figure 9 Comparison of iteration number between different sensitivity methods
DDMFD
RPE
FDDDMADM
0
01
02
03
04
05
06
07
0103050709Speed ratio
(a) All methods
DDMADM
0103050709
RPE
Speed ratio
00E + 00
50E minus 04
10E minus 03
15E minus 03
20E minus 03
25E minus 03
(b) Analytical discrete methods
Figure 10 Comparison of RPE between different sensitivity methods
Table 12 shows the relative efficiency parameter for dif-ferent speed parameters in different scenarios Again theefficiency parameter is between 0343 and 1726 with theaverage of 1044Thus compared to FDmethod theDDMFDmethod is slightly more effective
Changes in the average of REPddmfdfd in different sce-narios with speed ratio are shown in Figure 17 As shown inthis figure REPddmfdfd increases with speed ratio For speedratio lower than 05 efficiency parameter is lower than 1 whichmeans FD method is more effective for this range
55 Stability Figures 3 and 7 show that all describedmethodsare sensitive to noise and if noise level exceeds the valuesstated in Table 11 these methods lose their efficiency and arenot able to detect damage properly Hence in cases with noise
level larger than the values stated in Table 13 a denoisingtool like wavelet transforms should be used along with thesensitivity methodsWavelet transforms are mainly attractivebecause of their ability to compress and encode informationto reduce noise or to detect any local singular behavior of asignal [37]
Stability comparison of different methods shows that FDis the most stable method and stability of analytical methodsis slightly lower than that
Figures 4 and 8 show stability of sensitivity methodswith respect to the initial error Summary of results forstability investigation is presented in Table 13 which suggeststhat stability of numerical methods is better than analyticalmethods
16 Shock and Vibration
DDMFDFD
DDMADM
0
50
100
150
200
250
300
0103050709
Solu
tion
time
Speed ratio
(a) All methods
DDMADM
0
2
4
6
8
10
0103050709
Solu
tion
time
Speed ratio
(b) Analytical discrete methods
Figure 11 Comparison of solution time between different sensitivity methods
5-64-5
3-4
2-31-20-1
120572 = 01
120572 = 05
120572 = 09
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
6
123456
0
(a) Model 1
120572 = 01
120572 = 05
120572 = 09
25ndash32ndash2515ndash2
1ndash1505ndash10ndash05
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
63252151050
(b) Model 2
Figure 12 REPadjddm changes in different scenarios with respect to speed parameter
Table 11 REPadjadm ranges in different scenarios
Damage scenario Max REP Min REP AverageFD DDMFD FD DDMFD FD DDMFD
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 4 17 NilM1-2 Multiple 2 7 4 21 NilM1-3 Multiple 3 5 6 and 8 12 6 5 and 2 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
Table 3 Solution time number of loops and RPE of ADMmethod for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
421 Damage Scenarios Six damage scenarios of singlemultiple and randomdamage in the bridgewithoutmeasure-ment noise are studied and shown in Table 7
The sampling rate is 14000Hz and 460 data of the acceler-ation response (degree of indeterminacy is 20) collected alongthe z-direction at nodes 4 11 21 and 27 are used
422 Analysis Results Tables 8 and 9 show the results ofdifferent detection methods including solution time (ST)
and RPE Using both described methods including ADMand DDM method the damage locations and amount areidentified correctly in all the scenarios (Figure 6)
423 Effect of Noise In order to study effects of noisescenario 3 is considered and different levels of noise pollutionare investigated and also RPE changes with increasing thenumber of loops for iterative procedure have been studiedThe results are presented in Figure 7
10 Shock and Vibration
Table 6 Solution time loops and RPE of DDMFD method for model 1
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE ST (s) NL RPE
Figure 3 RPE contours with respect to noise level and loops in model 1
Shock and Vibration 11
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
10
20
30
40
50
60
70
(b) DDMmethod
Loops
Initi
al er
ror
FD method
5 10 15 20 25 300
005
01
015
02
025
03
10
20
30
40
50
60
70
80
(c) FD method
Loops
Initi
al er
ror
DDMFD method
5 10 15 20 25 300
005
01
015
02
025
03
0
10
20
30
40
50
60
70
80
(d) DDMFD method
Figure 4 RPE contours with respect to initial error and loops in model 1
424 Effect of Initial Error Scenario 6 (speed ratio ofmovingload is considered to be fixed and equal to 05) is considered tostudy effects of initial error in stability of sensitivity methodsRPE changes with increasing the number of loops for iterativeprocedure have been studied The results are illustrated inFigure 8
5 Comparison of Different Methods
According to the results obtained in the previous sectiondifferent sensitivity methods considered in this study arecompared in this section Iteration number relative percent-age of error (RPE) solution time noise level and initialerror effect in stability of damage detection procedure areeffective parameters to compare efficiency and accuracy of themethods
51 Iteration Number Average number of iterations fordifferent scenarios with respect to the speed parameter isillustrated in Figure 9
As seen in Figure 9(a) the number of iterations in FDand DDMFD methods is considerably larger than thoseof analytical discrete methods (DDM and ADM) It hasbeen observed that in all cases DDMFD method has feweriterations than FD method Average number of iterations is20757 and 251467 for DDMFD and FD methods respec-tively The larger the speed ratio of moving vehicle is thelarger the number of iterations in approximation methodsis In FD method for speed ratios more than 05 even byusing maximum number of iterations (300) convergence wasnot achieved while the ratio used in DDMFD method is09
Figure 9(b) shows comparison along with analytical dis-crete methods It can be seen that DDM has smaller iterationnumber in all cases Average number of iterations is 146and 11167 for ADM and DDM methods respectively Bothanalytical discrete methods converged in all cases withmaximum iterations equal to 17 and increment in speed ratiodoes not affect the number of iterations
12 Shock and Vibration
Sensors
Element number
25
17
9
2622
15
8
1
23
16
9
2
24
17
10
3
25
18
11
4
2619
12
5
2720
13
6
2821
147
4118
3510
29
2
2742
1936
1130
3
2843
2037
1231
4
29 4421
3813
325
30 4522 39
14
16
336
3132
46 2324
40 1534 7
8
Direction of measured response for identification
Node number
P V
Moving vehicle
Y Z X
7000 (mm)
3000 (mm)
Figure 5 Plane grid bridge model used in detection procedure
Table 7 Damage scenarios for multispan bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 23 5 NilM1-2 Multiple 8 13 and 29 11 4 and 7 NilM1-3 Multiple 3 7 19 25 and 28 12 6 5 2 and 18 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
52 Relative Percentage of Error (RPE) Using the sameconvergence tolerance equal to 1 times 10minus5 relative percentageof error in different methods is illustrated in Figure 10
As seen in this figure with a maximum number ofiterations equal to 300 RPE in FD and DDMFD methodsare considerably larger than DDM and ADM methodsDDMFD method has lower RPE than FD method in mostcases Average of RPE is 026329 for DDMFD compared to027766 in FD method Again the larger the speed ratio ofmoving vehicle is the larger the relative error percentage inapproximation methods is
Figure 10(b) shows that relative error percentage in ADMmethod is lower than DDM and speed ratio does not affectthe relative error
Average of RPE is 0000731 and 0002037 for ADM andDDM methods respectively Therefore ADM is nearly 28times more accurate than DDM
53 Solution Time In order to investigate computational costof sensitivity methods using the same amounts of RAMand CPU resources for the presented numerical simulationssolution time of different damaged models is evaluated andshown in Figure 11
As seen in Figure 11(a) solution times in FD and DDMFDmethods are considerably larger than analytical discretemethods and the FDmethod with an average of 200616 (s) isthemost time consumingmethod Average time for DDMFDmethod is 150849 (s)
Figure 11(b) compares the time spent by DDM and ADMAs illustrated in this figure ADM method has lower solutiontime than DDMmethod in all cases and speed ratio does notaffect the solution time
Average of solution time is 2956 (s) and 7794 (s) forADM and DDM methods respectively consequently ADMis nearly 26 times faster than DDM
54 Efficiency Parameter In order to compare and quantifythe performance of different methods relative efficiencyparameters of methods ldquo119894rdquo and ldquo119895rdquo (REP
119894119895
) are defined as
REP119894119895
= radic
ST119895
ST119894
times
RPE119895
RPE119894
(30)
in which ST is the solution time of system identificationmethod that represents the computational cost of themethodand RPE is the relative error which represents the accuracy ofthe method
Shock and Vibration 13
Table 8 Solution time number of loops and RPE of ADMmethod for model 2
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE ()
541 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to DDM Figure 12 shows REPadjddm changeswith respect to speed parameter in different scenarios
Table 10 shows that in different scenarios and for differentspeed parameters the efficiency parameter is between 19713and 55478 for model 1 and 15659 and 29893 for model2 and its average is 28948 and 22248 for models 1 and2 respectively Compared to DDM the adjoint variablemethod is more efficient and needs 609 less computationaleffort
Changes in the average of REPadjddm in different scenarioswith speed ratio are shown in Figure 13 As shown in thisfigure REPadjddm increases with speed ratios lightly Thisindicates that efficiency of ADM with respect to DDMincreases with speed ratio
542 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to Approximation and Semianalytical MethodsFigure 14 shows REPadjFD and REPadjDDMFD changes withrespect to speed parameter in different scenarios
Table 11 shows the relative efficiency parameter for differ-ent speed parameters in different scenarios It can be seen thatthe efficiency parameter is between 3193 and 78307 for finitedifference method and 4916 and 50542 for DDMFDmethodwith the average of 16714 and 14569 for FD and DDMFDmethods respectively Therefore the adjoint variable methodis extremely successful and the computational cost of thismethod is just 06 and 069 of FD and DDMFDmethodsrespectively
Changes in the average of REPadjfdampddmfd in differentscenarios with speed ratio are shown in Figure 15 As shownin this figure REPadjfdampddmfd rapidly increases with speedratio which indicates the superiority of ADM over FD andDDMFDmethods increases with speed ratio
0 5 10 15 20 25 30 35 40 45 500
051
152
25
Element number
Mod
ule o
f ela
stici
ty
0 5 10 15 20 25 30 35 40 45 5005
101520
Dam
age i
ndex
Element number
Original modelDetected model
times105
Figure 6 Detection of damage location and amount in elements 57 12 15 24 and 37 and distribution of error in different elementswith ADM scheme
543 Relative Efficiency Parameter of DDMFD with respectto FD Method Figure 16 shows REPddmfdfd changes withrespect to speed parameter in different scenarios
14 Shock and Vibration
Table 10 REPadjddm ranges in different scenarios
Damage scenario Max REP Min REP AverageModel 1 Model 2 Model 1 Model 2 Model 1 Model 2
Figure 7 RPE contours with respect to noise level and loops in model 2
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(b) DDMmethod
Figure 8 RPE contours with respect to initial error and loops in model 2
Shock and Vibration 15
0
50
100
150
200
250
300
350
0103050709
DDMFDFD
DDMADM
Itera
tion
num
ber
Speed ratio
(a) All methods
0
2
4
6
8
10
12
14
16
18
0103050709
Itera
tion
num
ber
Speed ratio
DDMADM
(b) Analytical discrete methods
Figure 9 Comparison of iteration number between different sensitivity methods
DDMFD
RPE
FDDDMADM
0
01
02
03
04
05
06
07
0103050709Speed ratio
(a) All methods
DDMADM
0103050709
RPE
Speed ratio
00E + 00
50E minus 04
10E minus 03
15E minus 03
20E minus 03
25E minus 03
(b) Analytical discrete methods
Figure 10 Comparison of RPE between different sensitivity methods
Table 12 shows the relative efficiency parameter for dif-ferent speed parameters in different scenarios Again theefficiency parameter is between 0343 and 1726 with theaverage of 1044Thus compared to FDmethod theDDMFDmethod is slightly more effective
Changes in the average of REPddmfdfd in different sce-narios with speed ratio are shown in Figure 17 As shown inthis figure REPddmfdfd increases with speed ratio For speedratio lower than 05 efficiency parameter is lower than 1 whichmeans FD method is more effective for this range
55 Stability Figures 3 and 7 show that all describedmethodsare sensitive to noise and if noise level exceeds the valuesstated in Table 11 these methods lose their efficiency and arenot able to detect damage properly Hence in cases with noise
level larger than the values stated in Table 13 a denoisingtool like wavelet transforms should be used along with thesensitivity methodsWavelet transforms are mainly attractivebecause of their ability to compress and encode informationto reduce noise or to detect any local singular behavior of asignal [37]
Stability comparison of different methods shows that FDis the most stable method and stability of analytical methodsis slightly lower than that
Figures 4 and 8 show stability of sensitivity methodswith respect to the initial error Summary of results forstability investigation is presented in Table 13 which suggeststhat stability of numerical methods is better than analyticalmethods
16 Shock and Vibration
DDMFDFD
DDMADM
0
50
100
150
200
250
300
0103050709
Solu
tion
time
Speed ratio
(a) All methods
DDMADM
0
2
4
6
8
10
0103050709
Solu
tion
time
Speed ratio
(b) Analytical discrete methods
Figure 11 Comparison of solution time between different sensitivity methods
5-64-5
3-4
2-31-20-1
120572 = 01
120572 = 05
120572 = 09
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
6
123456
0
(a) Model 1
120572 = 01
120572 = 05
120572 = 09
25ndash32ndash2515ndash2
1ndash1505ndash10ndash05
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
63252151050
(b) Model 2
Figure 12 REPadjddm changes in different scenarios with respect to speed parameter
Table 11 REPadjadm ranges in different scenarios
Damage scenario Max REP Min REP AverageFD DDMFD FD DDMFD FD DDMFD
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
Figure 3 RPE contours with respect to noise level and loops in model 1
Shock and Vibration 11
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
10
20
30
40
50
60
70
(b) DDMmethod
Loops
Initi
al er
ror
FD method
5 10 15 20 25 300
005
01
015
02
025
03
10
20
30
40
50
60
70
80
(c) FD method
Loops
Initi
al er
ror
DDMFD method
5 10 15 20 25 300
005
01
015
02
025
03
0
10
20
30
40
50
60
70
80
(d) DDMFD method
Figure 4 RPE contours with respect to initial error and loops in model 1
424 Effect of Initial Error Scenario 6 (speed ratio ofmovingload is considered to be fixed and equal to 05) is considered tostudy effects of initial error in stability of sensitivity methodsRPE changes with increasing the number of loops for iterativeprocedure have been studied The results are illustrated inFigure 8
5 Comparison of Different Methods
According to the results obtained in the previous sectiondifferent sensitivity methods considered in this study arecompared in this section Iteration number relative percent-age of error (RPE) solution time noise level and initialerror effect in stability of damage detection procedure areeffective parameters to compare efficiency and accuracy of themethods
51 Iteration Number Average number of iterations fordifferent scenarios with respect to the speed parameter isillustrated in Figure 9
As seen in Figure 9(a) the number of iterations in FDand DDMFD methods is considerably larger than thoseof analytical discrete methods (DDM and ADM) It hasbeen observed that in all cases DDMFD method has feweriterations than FD method Average number of iterations is20757 and 251467 for DDMFD and FD methods respec-tively The larger the speed ratio of moving vehicle is thelarger the number of iterations in approximation methodsis In FD method for speed ratios more than 05 even byusing maximum number of iterations (300) convergence wasnot achieved while the ratio used in DDMFD method is09
Figure 9(b) shows comparison along with analytical dis-crete methods It can be seen that DDM has smaller iterationnumber in all cases Average number of iterations is 146and 11167 for ADM and DDM methods respectively Bothanalytical discrete methods converged in all cases withmaximum iterations equal to 17 and increment in speed ratiodoes not affect the number of iterations
12 Shock and Vibration
Sensors
Element number
25
17
9
2622
15
8
1
23
16
9
2
24
17
10
3
25
18
11
4
2619
12
5
2720
13
6
2821
147
4118
3510
29
2
2742
1936
1130
3
2843
2037
1231
4
29 4421
3813
325
30 4522 39
14
16
336
3132
46 2324
40 1534 7
8
Direction of measured response for identification
Node number
P V
Moving vehicle
Y Z X
7000 (mm)
3000 (mm)
Figure 5 Plane grid bridge model used in detection procedure
Table 7 Damage scenarios for multispan bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 23 5 NilM1-2 Multiple 8 13 and 29 11 4 and 7 NilM1-3 Multiple 3 7 19 25 and 28 12 6 5 2 and 18 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
52 Relative Percentage of Error (RPE) Using the sameconvergence tolerance equal to 1 times 10minus5 relative percentageof error in different methods is illustrated in Figure 10
As seen in this figure with a maximum number ofiterations equal to 300 RPE in FD and DDMFD methodsare considerably larger than DDM and ADM methodsDDMFD method has lower RPE than FD method in mostcases Average of RPE is 026329 for DDMFD compared to027766 in FD method Again the larger the speed ratio ofmoving vehicle is the larger the relative error percentage inapproximation methods is
Figure 10(b) shows that relative error percentage in ADMmethod is lower than DDM and speed ratio does not affectthe relative error
Average of RPE is 0000731 and 0002037 for ADM andDDM methods respectively Therefore ADM is nearly 28times more accurate than DDM
53 Solution Time In order to investigate computational costof sensitivity methods using the same amounts of RAMand CPU resources for the presented numerical simulationssolution time of different damaged models is evaluated andshown in Figure 11
As seen in Figure 11(a) solution times in FD and DDMFDmethods are considerably larger than analytical discretemethods and the FDmethod with an average of 200616 (s) isthemost time consumingmethod Average time for DDMFDmethod is 150849 (s)
Figure 11(b) compares the time spent by DDM and ADMAs illustrated in this figure ADM method has lower solutiontime than DDMmethod in all cases and speed ratio does notaffect the solution time
Average of solution time is 2956 (s) and 7794 (s) forADM and DDM methods respectively consequently ADMis nearly 26 times faster than DDM
54 Efficiency Parameter In order to compare and quantifythe performance of different methods relative efficiencyparameters of methods ldquo119894rdquo and ldquo119895rdquo (REP
119894119895
) are defined as
REP119894119895
= radic
ST119895
ST119894
times
RPE119895
RPE119894
(30)
in which ST is the solution time of system identificationmethod that represents the computational cost of themethodand RPE is the relative error which represents the accuracy ofthe method
Shock and Vibration 13
Table 8 Solution time number of loops and RPE of ADMmethod for model 2
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE ()
541 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to DDM Figure 12 shows REPadjddm changeswith respect to speed parameter in different scenarios
Table 10 shows that in different scenarios and for differentspeed parameters the efficiency parameter is between 19713and 55478 for model 1 and 15659 and 29893 for model2 and its average is 28948 and 22248 for models 1 and2 respectively Compared to DDM the adjoint variablemethod is more efficient and needs 609 less computationaleffort
Changes in the average of REPadjddm in different scenarioswith speed ratio are shown in Figure 13 As shown in thisfigure REPadjddm increases with speed ratios lightly Thisindicates that efficiency of ADM with respect to DDMincreases with speed ratio
542 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to Approximation and Semianalytical MethodsFigure 14 shows REPadjFD and REPadjDDMFD changes withrespect to speed parameter in different scenarios
Table 11 shows the relative efficiency parameter for differ-ent speed parameters in different scenarios It can be seen thatthe efficiency parameter is between 3193 and 78307 for finitedifference method and 4916 and 50542 for DDMFDmethodwith the average of 16714 and 14569 for FD and DDMFDmethods respectively Therefore the adjoint variable methodis extremely successful and the computational cost of thismethod is just 06 and 069 of FD and DDMFDmethodsrespectively
Changes in the average of REPadjfdampddmfd in differentscenarios with speed ratio are shown in Figure 15 As shownin this figure REPadjfdampddmfd rapidly increases with speedratio which indicates the superiority of ADM over FD andDDMFDmethods increases with speed ratio
0 5 10 15 20 25 30 35 40 45 500
051
152
25
Element number
Mod
ule o
f ela
stici
ty
0 5 10 15 20 25 30 35 40 45 5005
101520
Dam
age i
ndex
Element number
Original modelDetected model
times105
Figure 6 Detection of damage location and amount in elements 57 12 15 24 and 37 and distribution of error in different elementswith ADM scheme
543 Relative Efficiency Parameter of DDMFD with respectto FD Method Figure 16 shows REPddmfdfd changes withrespect to speed parameter in different scenarios
14 Shock and Vibration
Table 10 REPadjddm ranges in different scenarios
Damage scenario Max REP Min REP AverageModel 1 Model 2 Model 1 Model 2 Model 1 Model 2
Figure 7 RPE contours with respect to noise level and loops in model 2
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(b) DDMmethod
Figure 8 RPE contours with respect to initial error and loops in model 2
Shock and Vibration 15
0
50
100
150
200
250
300
350
0103050709
DDMFDFD
DDMADM
Itera
tion
num
ber
Speed ratio
(a) All methods
0
2
4
6
8
10
12
14
16
18
0103050709
Itera
tion
num
ber
Speed ratio
DDMADM
(b) Analytical discrete methods
Figure 9 Comparison of iteration number between different sensitivity methods
DDMFD
RPE
FDDDMADM
0
01
02
03
04
05
06
07
0103050709Speed ratio
(a) All methods
DDMADM
0103050709
RPE
Speed ratio
00E + 00
50E minus 04
10E minus 03
15E minus 03
20E minus 03
25E minus 03
(b) Analytical discrete methods
Figure 10 Comparison of RPE between different sensitivity methods
Table 12 shows the relative efficiency parameter for dif-ferent speed parameters in different scenarios Again theefficiency parameter is between 0343 and 1726 with theaverage of 1044Thus compared to FDmethod theDDMFDmethod is slightly more effective
Changes in the average of REPddmfdfd in different sce-narios with speed ratio are shown in Figure 17 As shown inthis figure REPddmfdfd increases with speed ratio For speedratio lower than 05 efficiency parameter is lower than 1 whichmeans FD method is more effective for this range
55 Stability Figures 3 and 7 show that all describedmethodsare sensitive to noise and if noise level exceeds the valuesstated in Table 11 these methods lose their efficiency and arenot able to detect damage properly Hence in cases with noise
level larger than the values stated in Table 13 a denoisingtool like wavelet transforms should be used along with thesensitivity methodsWavelet transforms are mainly attractivebecause of their ability to compress and encode informationto reduce noise or to detect any local singular behavior of asignal [37]
Stability comparison of different methods shows that FDis the most stable method and stability of analytical methodsis slightly lower than that
Figures 4 and 8 show stability of sensitivity methodswith respect to the initial error Summary of results forstability investigation is presented in Table 13 which suggeststhat stability of numerical methods is better than analyticalmethods
16 Shock and Vibration
DDMFDFD
DDMADM
0
50
100
150
200
250
300
0103050709
Solu
tion
time
Speed ratio
(a) All methods
DDMADM
0
2
4
6
8
10
0103050709
Solu
tion
time
Speed ratio
(b) Analytical discrete methods
Figure 11 Comparison of solution time between different sensitivity methods
5-64-5
3-4
2-31-20-1
120572 = 01
120572 = 05
120572 = 09
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
6
123456
0
(a) Model 1
120572 = 01
120572 = 05
120572 = 09
25ndash32ndash2515ndash2
1ndash1505ndash10ndash05
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
63252151050
(b) Model 2
Figure 12 REPadjddm changes in different scenarios with respect to speed parameter
Table 11 REPadjadm ranges in different scenarios
Damage scenario Max REP Min REP AverageFD DDMFD FD DDMFD FD DDMFD
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
Figure 4 RPE contours with respect to initial error and loops in model 1
424 Effect of Initial Error Scenario 6 (speed ratio ofmovingload is considered to be fixed and equal to 05) is considered tostudy effects of initial error in stability of sensitivity methodsRPE changes with increasing the number of loops for iterativeprocedure have been studied The results are illustrated inFigure 8
5 Comparison of Different Methods
According to the results obtained in the previous sectiondifferent sensitivity methods considered in this study arecompared in this section Iteration number relative percent-age of error (RPE) solution time noise level and initialerror effect in stability of damage detection procedure areeffective parameters to compare efficiency and accuracy of themethods
51 Iteration Number Average number of iterations fordifferent scenarios with respect to the speed parameter isillustrated in Figure 9
As seen in Figure 9(a) the number of iterations in FDand DDMFD methods is considerably larger than thoseof analytical discrete methods (DDM and ADM) It hasbeen observed that in all cases DDMFD method has feweriterations than FD method Average number of iterations is20757 and 251467 for DDMFD and FD methods respec-tively The larger the speed ratio of moving vehicle is thelarger the number of iterations in approximation methodsis In FD method for speed ratios more than 05 even byusing maximum number of iterations (300) convergence wasnot achieved while the ratio used in DDMFD method is09
Figure 9(b) shows comparison along with analytical dis-crete methods It can be seen that DDM has smaller iterationnumber in all cases Average number of iterations is 146and 11167 for ADM and DDM methods respectively Bothanalytical discrete methods converged in all cases withmaximum iterations equal to 17 and increment in speed ratiodoes not affect the number of iterations
12 Shock and Vibration
Sensors
Element number
25
17
9
2622
15
8
1
23
16
9
2
24
17
10
3
25
18
11
4
2619
12
5
2720
13
6
2821
147
4118
3510
29
2
2742
1936
1130
3
2843
2037
1231
4
29 4421
3813
325
30 4522 39
14
16
336
3132
46 2324
40 1534 7
8
Direction of measured response for identification
Node number
P V
Moving vehicle
Y Z X
7000 (mm)
3000 (mm)
Figure 5 Plane grid bridge model used in detection procedure
Table 7 Damage scenarios for multispan bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 23 5 NilM1-2 Multiple 8 13 and 29 11 4 and 7 NilM1-3 Multiple 3 7 19 25 and 28 12 6 5 2 and 18 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
52 Relative Percentage of Error (RPE) Using the sameconvergence tolerance equal to 1 times 10minus5 relative percentageof error in different methods is illustrated in Figure 10
As seen in this figure with a maximum number ofiterations equal to 300 RPE in FD and DDMFD methodsare considerably larger than DDM and ADM methodsDDMFD method has lower RPE than FD method in mostcases Average of RPE is 026329 for DDMFD compared to027766 in FD method Again the larger the speed ratio ofmoving vehicle is the larger the relative error percentage inapproximation methods is
Figure 10(b) shows that relative error percentage in ADMmethod is lower than DDM and speed ratio does not affectthe relative error
Average of RPE is 0000731 and 0002037 for ADM andDDM methods respectively Therefore ADM is nearly 28times more accurate than DDM
53 Solution Time In order to investigate computational costof sensitivity methods using the same amounts of RAMand CPU resources for the presented numerical simulationssolution time of different damaged models is evaluated andshown in Figure 11
As seen in Figure 11(a) solution times in FD and DDMFDmethods are considerably larger than analytical discretemethods and the FDmethod with an average of 200616 (s) isthemost time consumingmethod Average time for DDMFDmethod is 150849 (s)
Figure 11(b) compares the time spent by DDM and ADMAs illustrated in this figure ADM method has lower solutiontime than DDMmethod in all cases and speed ratio does notaffect the solution time
Average of solution time is 2956 (s) and 7794 (s) forADM and DDM methods respectively consequently ADMis nearly 26 times faster than DDM
54 Efficiency Parameter In order to compare and quantifythe performance of different methods relative efficiencyparameters of methods ldquo119894rdquo and ldquo119895rdquo (REP
119894119895
) are defined as
REP119894119895
= radic
ST119895
ST119894
times
RPE119895
RPE119894
(30)
in which ST is the solution time of system identificationmethod that represents the computational cost of themethodand RPE is the relative error which represents the accuracy ofthe method
Shock and Vibration 13
Table 8 Solution time number of loops and RPE of ADMmethod for model 2
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE ()
541 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to DDM Figure 12 shows REPadjddm changeswith respect to speed parameter in different scenarios
Table 10 shows that in different scenarios and for differentspeed parameters the efficiency parameter is between 19713and 55478 for model 1 and 15659 and 29893 for model2 and its average is 28948 and 22248 for models 1 and2 respectively Compared to DDM the adjoint variablemethod is more efficient and needs 609 less computationaleffort
Changes in the average of REPadjddm in different scenarioswith speed ratio are shown in Figure 13 As shown in thisfigure REPadjddm increases with speed ratios lightly Thisindicates that efficiency of ADM with respect to DDMincreases with speed ratio
542 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to Approximation and Semianalytical MethodsFigure 14 shows REPadjFD and REPadjDDMFD changes withrespect to speed parameter in different scenarios
Table 11 shows the relative efficiency parameter for differ-ent speed parameters in different scenarios It can be seen thatthe efficiency parameter is between 3193 and 78307 for finitedifference method and 4916 and 50542 for DDMFDmethodwith the average of 16714 and 14569 for FD and DDMFDmethods respectively Therefore the adjoint variable methodis extremely successful and the computational cost of thismethod is just 06 and 069 of FD and DDMFDmethodsrespectively
Changes in the average of REPadjfdampddmfd in differentscenarios with speed ratio are shown in Figure 15 As shownin this figure REPadjfdampddmfd rapidly increases with speedratio which indicates the superiority of ADM over FD andDDMFDmethods increases with speed ratio
0 5 10 15 20 25 30 35 40 45 500
051
152
25
Element number
Mod
ule o
f ela
stici
ty
0 5 10 15 20 25 30 35 40 45 5005
101520
Dam
age i
ndex
Element number
Original modelDetected model
times105
Figure 6 Detection of damage location and amount in elements 57 12 15 24 and 37 and distribution of error in different elementswith ADM scheme
543 Relative Efficiency Parameter of DDMFD with respectto FD Method Figure 16 shows REPddmfdfd changes withrespect to speed parameter in different scenarios
14 Shock and Vibration
Table 10 REPadjddm ranges in different scenarios
Damage scenario Max REP Min REP AverageModel 1 Model 2 Model 1 Model 2 Model 1 Model 2
Figure 7 RPE contours with respect to noise level and loops in model 2
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(b) DDMmethod
Figure 8 RPE contours with respect to initial error and loops in model 2
Shock and Vibration 15
0
50
100
150
200
250
300
350
0103050709
DDMFDFD
DDMADM
Itera
tion
num
ber
Speed ratio
(a) All methods
0
2
4
6
8
10
12
14
16
18
0103050709
Itera
tion
num
ber
Speed ratio
DDMADM
(b) Analytical discrete methods
Figure 9 Comparison of iteration number between different sensitivity methods
DDMFD
RPE
FDDDMADM
0
01
02
03
04
05
06
07
0103050709Speed ratio
(a) All methods
DDMADM
0103050709
RPE
Speed ratio
00E + 00
50E minus 04
10E minus 03
15E minus 03
20E minus 03
25E minus 03
(b) Analytical discrete methods
Figure 10 Comparison of RPE between different sensitivity methods
Table 12 shows the relative efficiency parameter for dif-ferent speed parameters in different scenarios Again theefficiency parameter is between 0343 and 1726 with theaverage of 1044Thus compared to FDmethod theDDMFDmethod is slightly more effective
Changes in the average of REPddmfdfd in different sce-narios with speed ratio are shown in Figure 17 As shown inthis figure REPddmfdfd increases with speed ratio For speedratio lower than 05 efficiency parameter is lower than 1 whichmeans FD method is more effective for this range
55 Stability Figures 3 and 7 show that all describedmethodsare sensitive to noise and if noise level exceeds the valuesstated in Table 11 these methods lose their efficiency and arenot able to detect damage properly Hence in cases with noise
level larger than the values stated in Table 13 a denoisingtool like wavelet transforms should be used along with thesensitivity methodsWavelet transforms are mainly attractivebecause of their ability to compress and encode informationto reduce noise or to detect any local singular behavior of asignal [37]
Stability comparison of different methods shows that FDis the most stable method and stability of analytical methodsis slightly lower than that
Figures 4 and 8 show stability of sensitivity methodswith respect to the initial error Summary of results forstability investigation is presented in Table 13 which suggeststhat stability of numerical methods is better than analyticalmethods
16 Shock and Vibration
DDMFDFD
DDMADM
0
50
100
150
200
250
300
0103050709
Solu
tion
time
Speed ratio
(a) All methods
DDMADM
0
2
4
6
8
10
0103050709
Solu
tion
time
Speed ratio
(b) Analytical discrete methods
Figure 11 Comparison of solution time between different sensitivity methods
5-64-5
3-4
2-31-20-1
120572 = 01
120572 = 05
120572 = 09
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
6
123456
0
(a) Model 1
120572 = 01
120572 = 05
120572 = 09
25ndash32ndash2515ndash2
1ndash1505ndash10ndash05
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
63252151050
(b) Model 2
Figure 12 REPadjddm changes in different scenarios with respect to speed parameter
Table 11 REPadjadm ranges in different scenarios
Damage scenario Max REP Min REP AverageFD DDMFD FD DDMFD FD DDMFD
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
Figure 5 Plane grid bridge model used in detection procedure
Table 7 Damage scenarios for multispan bridge
Damage scenario Damage type Damage location Reduction in elastic modulus NoiseM1-1 Single 23 5 NilM1-2 Multiple 8 13 and 29 11 4 and 7 NilM1-3 Multiple 3 7 19 25 and 28 12 6 5 2 and 18 NilM1-4 Random All elements Random damage in all elements with an average of 5 NilM1-5 Random All elements Random damage in all elements with an average of 15 NilM1-6 Estimation of undamaged state All elements 5 reduction in all elements Nil
52 Relative Percentage of Error (RPE) Using the sameconvergence tolerance equal to 1 times 10minus5 relative percentageof error in different methods is illustrated in Figure 10
As seen in this figure with a maximum number ofiterations equal to 300 RPE in FD and DDMFD methodsare considerably larger than DDM and ADM methodsDDMFD method has lower RPE than FD method in mostcases Average of RPE is 026329 for DDMFD compared to027766 in FD method Again the larger the speed ratio ofmoving vehicle is the larger the relative error percentage inapproximation methods is
Figure 10(b) shows that relative error percentage in ADMmethod is lower than DDM and speed ratio does not affectthe relative error
Average of RPE is 0000731 and 0002037 for ADM andDDM methods respectively Therefore ADM is nearly 28times more accurate than DDM
53 Solution Time In order to investigate computational costof sensitivity methods using the same amounts of RAMand CPU resources for the presented numerical simulationssolution time of different damaged models is evaluated andshown in Figure 11
As seen in Figure 11(a) solution times in FD and DDMFDmethods are considerably larger than analytical discretemethods and the FDmethod with an average of 200616 (s) isthemost time consumingmethod Average time for DDMFDmethod is 150849 (s)
Figure 11(b) compares the time spent by DDM and ADMAs illustrated in this figure ADM method has lower solutiontime than DDMmethod in all cases and speed ratio does notaffect the solution time
Average of solution time is 2956 (s) and 7794 (s) forADM and DDM methods respectively consequently ADMis nearly 26 times faster than DDM
54 Efficiency Parameter In order to compare and quantifythe performance of different methods relative efficiencyparameters of methods ldquo119894rdquo and ldquo119895rdquo (REP
119894119895
) are defined as
REP119894119895
= radic
ST119895
ST119894
times
RPE119895
RPE119894
(30)
in which ST is the solution time of system identificationmethod that represents the computational cost of themethodand RPE is the relative error which represents the accuracy ofthe method
Shock and Vibration 13
Table 8 Solution time number of loops and RPE of ADMmethod for model 2
Damage scenarioSpeed parameter
01 03 05 07 09ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE () ST (s) RPE ()
541 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to DDM Figure 12 shows REPadjddm changeswith respect to speed parameter in different scenarios
Table 10 shows that in different scenarios and for differentspeed parameters the efficiency parameter is between 19713and 55478 for model 1 and 15659 and 29893 for model2 and its average is 28948 and 22248 for models 1 and2 respectively Compared to DDM the adjoint variablemethod is more efficient and needs 609 less computationaleffort
Changes in the average of REPadjddm in different scenarioswith speed ratio are shown in Figure 13 As shown in thisfigure REPadjddm increases with speed ratios lightly Thisindicates that efficiency of ADM with respect to DDMincreases with speed ratio
542 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to Approximation and Semianalytical MethodsFigure 14 shows REPadjFD and REPadjDDMFD changes withrespect to speed parameter in different scenarios
Table 11 shows the relative efficiency parameter for differ-ent speed parameters in different scenarios It can be seen thatthe efficiency parameter is between 3193 and 78307 for finitedifference method and 4916 and 50542 for DDMFDmethodwith the average of 16714 and 14569 for FD and DDMFDmethods respectively Therefore the adjoint variable methodis extremely successful and the computational cost of thismethod is just 06 and 069 of FD and DDMFDmethodsrespectively
Changes in the average of REPadjfdampddmfd in differentscenarios with speed ratio are shown in Figure 15 As shownin this figure REPadjfdampddmfd rapidly increases with speedratio which indicates the superiority of ADM over FD andDDMFDmethods increases with speed ratio
0 5 10 15 20 25 30 35 40 45 500
051
152
25
Element number
Mod
ule o
f ela
stici
ty
0 5 10 15 20 25 30 35 40 45 5005
101520
Dam
age i
ndex
Element number
Original modelDetected model
times105
Figure 6 Detection of damage location and amount in elements 57 12 15 24 and 37 and distribution of error in different elementswith ADM scheme
543 Relative Efficiency Parameter of DDMFD with respectto FD Method Figure 16 shows REPddmfdfd changes withrespect to speed parameter in different scenarios
14 Shock and Vibration
Table 10 REPadjddm ranges in different scenarios
Damage scenario Max REP Min REP AverageModel 1 Model 2 Model 1 Model 2 Model 1 Model 2
Figure 7 RPE contours with respect to noise level and loops in model 2
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(b) DDMmethod
Figure 8 RPE contours with respect to initial error and loops in model 2
Shock and Vibration 15
0
50
100
150
200
250
300
350
0103050709
DDMFDFD
DDMADM
Itera
tion
num
ber
Speed ratio
(a) All methods
0
2
4
6
8
10
12
14
16
18
0103050709
Itera
tion
num
ber
Speed ratio
DDMADM
(b) Analytical discrete methods
Figure 9 Comparison of iteration number between different sensitivity methods
DDMFD
RPE
FDDDMADM
0
01
02
03
04
05
06
07
0103050709Speed ratio
(a) All methods
DDMADM
0103050709
RPE
Speed ratio
00E + 00
50E minus 04
10E minus 03
15E minus 03
20E minus 03
25E minus 03
(b) Analytical discrete methods
Figure 10 Comparison of RPE between different sensitivity methods
Table 12 shows the relative efficiency parameter for dif-ferent speed parameters in different scenarios Again theefficiency parameter is between 0343 and 1726 with theaverage of 1044Thus compared to FDmethod theDDMFDmethod is slightly more effective
Changes in the average of REPddmfdfd in different sce-narios with speed ratio are shown in Figure 17 As shown inthis figure REPddmfdfd increases with speed ratio For speedratio lower than 05 efficiency parameter is lower than 1 whichmeans FD method is more effective for this range
55 Stability Figures 3 and 7 show that all describedmethodsare sensitive to noise and if noise level exceeds the valuesstated in Table 11 these methods lose their efficiency and arenot able to detect damage properly Hence in cases with noise
level larger than the values stated in Table 13 a denoisingtool like wavelet transforms should be used along with thesensitivity methodsWavelet transforms are mainly attractivebecause of their ability to compress and encode informationto reduce noise or to detect any local singular behavior of asignal [37]
Stability comparison of different methods shows that FDis the most stable method and stability of analytical methodsis slightly lower than that
Figures 4 and 8 show stability of sensitivity methodswith respect to the initial error Summary of results forstability investigation is presented in Table 13 which suggeststhat stability of numerical methods is better than analyticalmethods
16 Shock and Vibration
DDMFDFD
DDMADM
0
50
100
150
200
250
300
0103050709
Solu
tion
time
Speed ratio
(a) All methods
DDMADM
0
2
4
6
8
10
0103050709
Solu
tion
time
Speed ratio
(b) Analytical discrete methods
Figure 11 Comparison of solution time between different sensitivity methods
5-64-5
3-4
2-31-20-1
120572 = 01
120572 = 05
120572 = 09
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
6
123456
0
(a) Model 1
120572 = 01
120572 = 05
120572 = 09
25ndash32ndash2515ndash2
1ndash1505ndash10ndash05
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
63252151050
(b) Model 2
Figure 12 REPadjddm changes in different scenarios with respect to speed parameter
Table 11 REPadjadm ranges in different scenarios
Damage scenario Max REP Min REP AverageFD DDMFD FD DDMFD FD DDMFD
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
541 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to DDM Figure 12 shows REPadjddm changeswith respect to speed parameter in different scenarios
Table 10 shows that in different scenarios and for differentspeed parameters the efficiency parameter is between 19713and 55478 for model 1 and 15659 and 29893 for model2 and its average is 28948 and 22248 for models 1 and2 respectively Compared to DDM the adjoint variablemethod is more efficient and needs 609 less computationaleffort
Changes in the average of REPadjddm in different scenarioswith speed ratio are shown in Figure 13 As shown in thisfigure REPadjddm increases with speed ratios lightly Thisindicates that efficiency of ADM with respect to DDMincreases with speed ratio
542 Relative Efficiency Parameter of Adjoint VariableMethodwith respect to Approximation and Semianalytical MethodsFigure 14 shows REPadjFD and REPadjDDMFD changes withrespect to speed parameter in different scenarios
Table 11 shows the relative efficiency parameter for differ-ent speed parameters in different scenarios It can be seen thatthe efficiency parameter is between 3193 and 78307 for finitedifference method and 4916 and 50542 for DDMFDmethodwith the average of 16714 and 14569 for FD and DDMFDmethods respectively Therefore the adjoint variable methodis extremely successful and the computational cost of thismethod is just 06 and 069 of FD and DDMFDmethodsrespectively
Changes in the average of REPadjfdampddmfd in differentscenarios with speed ratio are shown in Figure 15 As shownin this figure REPadjfdampddmfd rapidly increases with speedratio which indicates the superiority of ADM over FD andDDMFDmethods increases with speed ratio
0 5 10 15 20 25 30 35 40 45 500
051
152
25
Element number
Mod
ule o
f ela
stici
ty
0 5 10 15 20 25 30 35 40 45 5005
101520
Dam
age i
ndex
Element number
Original modelDetected model
times105
Figure 6 Detection of damage location and amount in elements 57 12 15 24 and 37 and distribution of error in different elementswith ADM scheme
543 Relative Efficiency Parameter of DDMFD with respectto FD Method Figure 16 shows REPddmfdfd changes withrespect to speed parameter in different scenarios
14 Shock and Vibration
Table 10 REPadjddm ranges in different scenarios
Damage scenario Max REP Min REP AverageModel 1 Model 2 Model 1 Model 2 Model 1 Model 2
Figure 7 RPE contours with respect to noise level and loops in model 2
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(b) DDMmethod
Figure 8 RPE contours with respect to initial error and loops in model 2
Shock and Vibration 15
0
50
100
150
200
250
300
350
0103050709
DDMFDFD
DDMADM
Itera
tion
num
ber
Speed ratio
(a) All methods
0
2
4
6
8
10
12
14
16
18
0103050709
Itera
tion
num
ber
Speed ratio
DDMADM
(b) Analytical discrete methods
Figure 9 Comparison of iteration number between different sensitivity methods
DDMFD
RPE
FDDDMADM
0
01
02
03
04
05
06
07
0103050709Speed ratio
(a) All methods
DDMADM
0103050709
RPE
Speed ratio
00E + 00
50E minus 04
10E minus 03
15E minus 03
20E minus 03
25E minus 03
(b) Analytical discrete methods
Figure 10 Comparison of RPE between different sensitivity methods
Table 12 shows the relative efficiency parameter for dif-ferent speed parameters in different scenarios Again theefficiency parameter is between 0343 and 1726 with theaverage of 1044Thus compared to FDmethod theDDMFDmethod is slightly more effective
Changes in the average of REPddmfdfd in different sce-narios with speed ratio are shown in Figure 17 As shown inthis figure REPddmfdfd increases with speed ratio For speedratio lower than 05 efficiency parameter is lower than 1 whichmeans FD method is more effective for this range
55 Stability Figures 3 and 7 show that all describedmethodsare sensitive to noise and if noise level exceeds the valuesstated in Table 11 these methods lose their efficiency and arenot able to detect damage properly Hence in cases with noise
level larger than the values stated in Table 13 a denoisingtool like wavelet transforms should be used along with thesensitivity methodsWavelet transforms are mainly attractivebecause of their ability to compress and encode informationto reduce noise or to detect any local singular behavior of asignal [37]
Stability comparison of different methods shows that FDis the most stable method and stability of analytical methodsis slightly lower than that
Figures 4 and 8 show stability of sensitivity methodswith respect to the initial error Summary of results forstability investigation is presented in Table 13 which suggeststhat stability of numerical methods is better than analyticalmethods
16 Shock and Vibration
DDMFDFD
DDMADM
0
50
100
150
200
250
300
0103050709
Solu
tion
time
Speed ratio
(a) All methods
DDMADM
0
2
4
6
8
10
0103050709
Solu
tion
time
Speed ratio
(b) Analytical discrete methods
Figure 11 Comparison of solution time between different sensitivity methods
5-64-5
3-4
2-31-20-1
120572 = 01
120572 = 05
120572 = 09
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
6
123456
0
(a) Model 1
120572 = 01
120572 = 05
120572 = 09
25ndash32ndash2515ndash2
1ndash1505ndash10ndash05
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
63252151050
(b) Model 2
Figure 12 REPadjddm changes in different scenarios with respect to speed parameter
Table 11 REPadjadm ranges in different scenarios
Damage scenario Max REP Min REP AverageFD DDMFD FD DDMFD FD DDMFD
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
Figure 7 RPE contours with respect to noise level and loops in model 2
Loops
Initi
al er
ror
ADM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
(a) ADMmethod
Loops
Initi
al er
ror
DDM method
2 4 6 8 10 12 14 16 18 200
002
004
006
008
01
012
014
016
018
02
0
20
40
60
80
100
120
140
160
180
200
(b) DDMmethod
Figure 8 RPE contours with respect to initial error and loops in model 2
Shock and Vibration 15
0
50
100
150
200
250
300
350
0103050709
DDMFDFD
DDMADM
Itera
tion
num
ber
Speed ratio
(a) All methods
0
2
4
6
8
10
12
14
16
18
0103050709
Itera
tion
num
ber
Speed ratio
DDMADM
(b) Analytical discrete methods
Figure 9 Comparison of iteration number between different sensitivity methods
DDMFD
RPE
FDDDMADM
0
01
02
03
04
05
06
07
0103050709Speed ratio
(a) All methods
DDMADM
0103050709
RPE
Speed ratio
00E + 00
50E minus 04
10E minus 03
15E minus 03
20E minus 03
25E minus 03
(b) Analytical discrete methods
Figure 10 Comparison of RPE between different sensitivity methods
Table 12 shows the relative efficiency parameter for dif-ferent speed parameters in different scenarios Again theefficiency parameter is between 0343 and 1726 with theaverage of 1044Thus compared to FDmethod theDDMFDmethod is slightly more effective
Changes in the average of REPddmfdfd in different sce-narios with speed ratio are shown in Figure 17 As shown inthis figure REPddmfdfd increases with speed ratio For speedratio lower than 05 efficiency parameter is lower than 1 whichmeans FD method is more effective for this range
55 Stability Figures 3 and 7 show that all describedmethodsare sensitive to noise and if noise level exceeds the valuesstated in Table 11 these methods lose their efficiency and arenot able to detect damage properly Hence in cases with noise
level larger than the values stated in Table 13 a denoisingtool like wavelet transforms should be used along with thesensitivity methodsWavelet transforms are mainly attractivebecause of their ability to compress and encode informationto reduce noise or to detect any local singular behavior of asignal [37]
Stability comparison of different methods shows that FDis the most stable method and stability of analytical methodsis slightly lower than that
Figures 4 and 8 show stability of sensitivity methodswith respect to the initial error Summary of results forstability investigation is presented in Table 13 which suggeststhat stability of numerical methods is better than analyticalmethods
16 Shock and Vibration
DDMFDFD
DDMADM
0
50
100
150
200
250
300
0103050709
Solu
tion
time
Speed ratio
(a) All methods
DDMADM
0
2
4
6
8
10
0103050709
Solu
tion
time
Speed ratio
(b) Analytical discrete methods
Figure 11 Comparison of solution time between different sensitivity methods
5-64-5
3-4
2-31-20-1
120572 = 01
120572 = 05
120572 = 09
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
6
123456
0
(a) Model 1
120572 = 01
120572 = 05
120572 = 09
25ndash32ndash2515ndash2
1ndash1505ndash10ndash05
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
63252151050
(b) Model 2
Figure 12 REPadjddm changes in different scenarios with respect to speed parameter
Table 11 REPadjadm ranges in different scenarios
Damage scenario Max REP Min REP AverageFD DDMFD FD DDMFD FD DDMFD
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
Figure 9 Comparison of iteration number between different sensitivity methods
DDMFD
RPE
FDDDMADM
0
01
02
03
04
05
06
07
0103050709Speed ratio
(a) All methods
DDMADM
0103050709
RPE
Speed ratio
00E + 00
50E minus 04
10E minus 03
15E minus 03
20E minus 03
25E minus 03
(b) Analytical discrete methods
Figure 10 Comparison of RPE between different sensitivity methods
Table 12 shows the relative efficiency parameter for dif-ferent speed parameters in different scenarios Again theefficiency parameter is between 0343 and 1726 with theaverage of 1044Thus compared to FDmethod theDDMFDmethod is slightly more effective
Changes in the average of REPddmfdfd in different sce-narios with speed ratio are shown in Figure 17 As shown inthis figure REPddmfdfd increases with speed ratio For speedratio lower than 05 efficiency parameter is lower than 1 whichmeans FD method is more effective for this range
55 Stability Figures 3 and 7 show that all describedmethodsare sensitive to noise and if noise level exceeds the valuesstated in Table 11 these methods lose their efficiency and arenot able to detect damage properly Hence in cases with noise
level larger than the values stated in Table 13 a denoisingtool like wavelet transforms should be used along with thesensitivity methodsWavelet transforms are mainly attractivebecause of their ability to compress and encode informationto reduce noise or to detect any local singular behavior of asignal [37]
Stability comparison of different methods shows that FDis the most stable method and stability of analytical methodsis slightly lower than that
Figures 4 and 8 show stability of sensitivity methodswith respect to the initial error Summary of results forstability investigation is presented in Table 13 which suggeststhat stability of numerical methods is better than analyticalmethods
16 Shock and Vibration
DDMFDFD
DDMADM
0
50
100
150
200
250
300
0103050709
Solu
tion
time
Speed ratio
(a) All methods
DDMADM
0
2
4
6
8
10
0103050709
Solu
tion
time
Speed ratio
(b) Analytical discrete methods
Figure 11 Comparison of solution time between different sensitivity methods
5-64-5
3-4
2-31-20-1
120572 = 01
120572 = 05
120572 = 09
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
6
123456
0
(a) Model 1
120572 = 01
120572 = 05
120572 = 09
25ndash32ndash2515ndash2
1ndash1505ndash10ndash05
Scen
ario
1
Scen
ario
2
Scen
ario
3
Scen
ario
4
Scen
ario
5
Scen
ario
63252151050
(b) Model 2
Figure 12 REPadjddm changes in different scenarios with respect to speed parameter
Table 11 REPadjadm ranges in different scenarios
Damage scenario Max REP Min REP AverageFD DDMFD FD DDMFD FD DDMFD
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
Comparing Figures 3 and 4 shows that divergence typedue to initial error is sharp unlike the noise level which occursgradually
6 Conclusion
Different sensitivity-based damage detection methods arepresented and acceleration time history data affected bya moving vehicle with specified load is used for damagedetection procedure Newmark method is used to calculatethe structural dynamic response and its dynamic responsesensitivities are calculated by four different sensitivity meth-ods (finite difference method (FD) semianalytical discretemethod (DDMFD) direct differential method (DDM) andadjoint variable method (ADM))
Different damaged structures including single multipleand random damage are considered and efficiency of fouraforementioned sensitivity methods is compared and follow-ing remarks are made
(i) The advantage of the finite difference method isobvious If structural analysis can be performed andthe performance measure can be obtained as a resultof structural analysis then FDmethod is independentof the problem types considered However sensitivitycomputation costs become the main concern in thedamage detection process for the large problemsComparison of sensitivity methods shows that FDmethod is the most expensive method among otherprocedures
(ii) Semianalytical discrete method (DDMFD) alleviatessome disadvantages of FD method and its computa-tional cost is rather lower than FD method
(iii) The major disadvantage of the finite difference basedmethods (FD and DDMFD) is the accuracy of theirsensitivity results Depending on perturbation size
sensitivity results are quite different As a result it isvery difficult to determine design perturbation sizesthat work for all problems
(iv) The computational cost of damage detection pro-cedure using these methods is too expensive andthese methods are infeasible for large-scale problemscontaining many variables for example the secondcase study (model 2)
(v) The DDM is an accurate and efficient method tocompute sensitivity matrix This method directlysolves for the design dependency of a state variableand then computes performance sensitivity using thechain rule of differentiation The comparative studyshows this has a very low computational cost methodin all cases and is more accurate than finite differencemethods
(vi) ADM calculates each element of sensitivity matrixseparately by defining an adjoint variable parameterThe main advantage is in evaluating the dynamicresponse an analytical solution exists which sig-nificantly increases the speed and accuracy of thesolutionThe comparative study shows that efficiencyparameter of ADM is 289 16714 and 14569 com-pared to DDM FD and DDMFD methods respec-tively This result indicates that ADM is extremelysuccessful and can be applied as a powerful tool inSHM
(vii) Investigations of initial assumption error in stabilityof methods show that finite difference based methodshave more enhanced stability than analytical discretemethods and all sensitivity-basedmethods havemod-erate stability against initial assumption error
(viii) The drawback of all sensitivity-basedmethods is theirlow stability against input measurement noise (about25) which can be improved by using low-passdenoising tools
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] H Wenzel Health Monitoring of Bridges John Wiley amp SonsChichester UK 2009
[2] P M Pawar and R Ganguli Structural Health Monitoring UsingGenetic Fuzzy Systems Springer London UK 2011
[3] S Gopalakrishnan M Ruzzene and S Hanagud Computa-tional Techniques for Structural Health Monitoring SpringerLondon UK 2011
[4] A Morrasi and F Vestroni Dynamic Methods for DamageDetection in Structures Springer New York NY USA 2008
[5] Z R Lu and J K Liu ldquoParameters identification for a coupledbridge-vehicle system with spring-mass attachmentsrdquo AppliedMathematics and Computation vol 219 no 17 pp 9174ndash91862013
Shock and Vibration 19
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
[6] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration characteris-tics a literature reviewrdquo Tech Rep LA-13070- MSUC-900 LosAlamos National Laboratory Los Alamos NM USA 1996
[8] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997
[9] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000
[10] S Alampalli and G Fu ldquoRemote monitoring systems forbridge conditionrdquo Client Report 94 Transportation Researchand Development Bureau New York State Department ofTransportation New York NY USA 1994
[11] S Alampalli G Fu and E W Dillon ldquoMeasuring bridgevibration for detection of structural damagerdquo Research Report165 Transportation Researchand Development Bureau NewYork State Department of Transportation 1995
[12] J R Casas andA C Aparicio ldquoStructural damage identificationfrom dynamic-test datardquo Journal of Structural EngineeringAmerican Society of Chemical Engineers vol 120 no 8 pp 2437ndash2449 1994
[13] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979
[14] M I Friswell J E T Penny and D A L Wilson ldquoUsingvibration data and statistical measures to locate damage instructuresrdquoModal Analysis vol 9 no 4 pp 239ndash254 1994
[15] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994
[16] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991
[17] C P Ratcliffe ldquoDamage detection using a modified laplacianoperator on mode shape datardquo Journal of Sound and Vibrationvol 204 no 3 pp 505ndash517 1997
[18] P F Rizos N Aspragathos and A D Dimarogonas ldquoIdentifica-tion of crack location and magnitude in a cantilever beam fromthe vibration modesrdquo Journal of Sound and Vibration vol 138no 3 pp 381ndash388 1990
[19] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994
[20] T W Lim ldquoStructural damage detection using modal test datardquoAIAA Journal vol 29 no 12 pp 2271ndash2274 1991
[21] D Wu and S S Law ldquoModel error correction from truncatedmodal flexibility sensitivity and generic parameters part Imdashsimulationrdquo Mechanical Systems and Signal Processing vol 18no 6 pp 1381ndash1399 2004
[22] K F Alvin AN RobertsonGWReich andKC Park ldquoStruc-tural system identification from reality to modelsrdquo Computersand Structures vol 81 no 12 pp 1149ndash1176 2003
[23] R Sieniawska P Sniady and S Zukowski ldquoIdentification of thestructure parameters applying a moving loadrdquo Journal of Soundand Vibration vol 319 no 1-2 pp 355ndash365 2009
[24] C Gentile and A Saisi ldquoAmbient vibration testing of historicmasonry towers for structural identification and damage assess-mentrdquo Construction and Building Materials vol 21 no 6 pp1311ndash1321 2007
[26] S S Law J Q Bu X Q Zhu and S L Chan ldquoVehicle axleloads identification using finite element methodrdquo Journal ofEngineering Structures vol 26 no 8 pp 1143ndash1153 2004
[27] R J Jiang F T K Au and Y K Cheung ldquoIdentification ofvehicles moving on continuous bridges with rough surfacerdquoJournal of Sound and Vibration vol 274 no 3ndash5 pp 1045ndash10632004
[28] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics vol 129 no 1 pp 58ndash65 2007
[29] M I Friswell and J E Mottershead Finite Element ModelUpdating in Structural Dynamics vol 38 of Solid Mechanicsand its Applications Kluwer Academic Publishers Group Dor-drecht The Netherlands 1995
[30] X Q Zhu and H Hao Damage Detection of Bridge BeamStructures under Moving Loads School of Civil and ResourceEngineering University of Western Australia 2007
[31] T Lauwagie H Sol and E Dascotte Damage Identificationin Beams Using Inverse Methods Department of MechanicalEngineering Katholieke Universiteit Leuven Leuven Belgium2002
[32] D Cacuci Sensitivity and Uncertainty Analysis Volume 1Chapman amp HallCRC Boca Raton Fla USA 2003
[33] K K Choi and N H Kim Structural Sensitivity Analysis andOptimization 1 Linear Systems Springer New York NY USA2005
[34] A Mirzaee M A Shayanfar and R Abbasnia ldquoDamagedetection of bridges using vibration data by adjoint variablemethodrdquo Shock and Vibration vol 2014 Article ID 698658 17pages 2014
[35] M Friswell J Mottershead and H Ahmadian ldquoFinite-elementmodel updating using experimental test data parametrizationand regularizationrdquo Philosophical Transactions of the RoyalSociety A vol 365 pp 393ndash410 2007
[36] X Y Li and S S Law ldquoAdaptive Tikhonov regularizationfor damage detection based on nonlinear model updatingrdquoMechanical Systems and Signal Processing vol 24 no 6 pp1646ndash1664 2010
[37] M SolısMAlgaba andPGalvın ldquoContinuouswavelet analysisof mode shapes differences for damage detectionrdquo Journal ofMechanical Systems and Signal Processing vol 40 no 2 pp 645ndash666 2013
