finite element model updating and structural damage ...u0044091/ij-mssp-reyn... · damage...

Finite element model updating and structural damage identification using OMAXdata

Edwin Reyndersa,∗, Anne Teughelsb, Guido De Roecka

aUniversity of Leuven (KU Leuven), Department of Civil Engineering, Kasteelpark Arenberg 40, B-3001 Leuven, BelgiumbExxonMobil, Esso Belgium, Polderdijkweg 3, B-2030 Antwerp, Belgium

Abstract

The main limitations in the finite element (FE) model updating technique lie in the ability of the FE model to representthe true behavior of the structure (modelling problem), andin the ability to identify enough modal parameters withsufficient accuracy, especially for large structures that are tested in operational conditions (identification problem).In this paper, the identification problem is solved with an OMAX approach, where an artificial force is used inoperational conditions and a structural model is identifiedthat takes both the forced and the ambient excitation intoaccount. From an extensive case study on a real three-span bridge, it is observed that, while updating the FE modelusing the experimental output-only data yields a good fit, discrepancies show up when the more extensive set ofOMAX data is used for validation, or even for updating. It canbe concluded that an OMAX approach not onlyincreases the well-posedness of the updating problem, it also allows to detect potential inaccuracies in the FE model.

Keywords: operational modal analysis; exogenous inputs; finite element model updating; damage identification.

1. Introduction

In vibration-based damage identification, changes in modaldata are used as an indicator to detect and to identifydamage. An inverse problem is solved that consists in predicting the location and severity of the damage, giventhe structural dynamic characteristics before and after the damage has occurred. Two main types of vibration-baseddamage identification techniques are discriminated: model-based and non-model-based, also called parametric andnonparametric, respectively. Parametric methods are based on a model of the structure, some parameters of whichare adjusted using vibration measurements, and their performance therefore depends on the quality of the model.Nonparametric methods do not need a detailed model of the structure, but most of them lack a solid theoretical andphysical base; exceptions are the Damage Locating Vector technique [2] and the Local Flexibility method [27].

The Finite Element (FE) model updating method belongs to theparametric class of vibration-based damage identi-fication techniques. The procedure consists in adapting theunknown properties of a FE model, so that the differencesbetween experimental modal data and the corresponding FE model predictions are minimized [10]. The structuraldamage is represented by a decrease in stiffness of the individual elements and the procedure is performed in twoupdating processes. In the first process, the initial FE model is tuned to the undamaged structure, which is used as areference model. In the second process, the reference FE model is updated to obtain a model that can reproduce theexperimental modal data of the damaged state. The correction factors of the latter process represent the damage. Inorder to reduce the number of unknown parameters, and consequently reduce the ill-posedness of the inverse problem,damage functions can be used to approximate the unknown damage pattern [32].

In civil engineering, the modal parameters to which the FE model is updated are usually determined from output-only data, since forced vibration testing of large structures in operational conditions is often impractical and expensive.The measured response is caused by unmeasured ambient forces due to for example wind or traffic loading. The un-measured, ambient forces are modelled as stochastic quantities with unknown parameters but with known behavior,

∗Corresponding author. Email: [email protected]. Tel.: +3216321677. Fax: +3216321988.Postprint submitted to Mechanical Systems and Signal ProcessingPublished version:E. Reynders, A. Teughels and G. De Roeck. Finite element model updating and structural damage identification using OMAXdata.Mechanical Systems and Signal Processing, 24(5):1306-1323, 2010. http://dx.doi.org/10.1016/j.ymssp.2010.03.014

e.g., as white noise time series with zero mean and unknown covariance [22]. This type of testing is known asOp-erational Modal Analysis(OMA). Successful applications of structural damage identification by FE model updatingusing OMA data have been reported for several types of structure such as concrete roadway bridges [28, 31], historicmasonry towers [11] and multi-storey reinforced concrete building frames [7, 16]. However, the main limitations ofthis technique lie in the ability of the FE model to representthe true behavior of the structure (modelling problem),and in the ability to identify enough modal parameters with sufficient accuracy (identification problem) [3]. Whenstructural stiffness is used as FE updating parameter, its sensitivity to changes in modal parameters is higher for modeswith a higher eigenfrequency [18]. Therefore, the accurateidentification of the higher modes is important for modelupdating, especially for damage identification applications.

In this paper, the second limitation (identification problem) is investigated in the light of recent advances in systemidentification for modal testing. In particular, it became possible to apply measured artificial forces to a structure inaddition to the unmeasured ambient excitation, and to identify a model that accounts for both excitation sources.This approach is calledOperational Modal Analysis with eXogenous forces(OMAX) [4, 25]. The crucial differencewith classical forced vibration testing is that the ambientloads are not considered as noise, but as a true part of theexcitation. Consequently, the amplitude of the artificial forces can be small compared to the amplitude of the ambientforces, and small and practical actuators can be used on relatively large structures [29]. Three other advantages ofOMAX with respect to OMA are that (i) the modes that are excited by the artificial forces, can be mass-normalized,(ii) the frequency range of excitation can be enlarged, so that more modal parameters can be identified, and (iii) theaccuracy of the identified modal parameters is generally higher, since part of the excitation is measured instead ofmodelled [24].

The paper is structured as follows. In section 2, the OMA and OMAX procedures are briefly reviewed. Theupdating procedure is outlined in section 3. Section 4 contains a detailed case study of damage identification on aprestressed concrete bridge by FE model updating. The bridge is subjected to both an OMA and an OMAX test, fromwhich 6 and10 modes are identified, respectively. Model updating resultsfor the OMA test have been presentedpreviously [31]. Here, the additional information obtained from the OMAX test is used as well, and both approachesare compared. The conclusions are presented in section 5.

2. OMA vs. OMAX testing

In classicalExperimental Modal Analysis(EMA), the structure of interest is excited by one or severalmeasureddynamic forces, the response of the structure to these forces is recorded, and the modal parameters in the frequencyrange of interest are extracted from the measurements [9, 13, 17]. During the test, the structure is often removedfrom its operating environment and tested in laboratory conditions. The excitation and the boundary conditions maytherefore differ significantly from the structure’s real-life operating conditions. Furthermore, EMA methods are ingeneral less suitable for large structures, since these areinherently tested in operational rather than in laboratoryconditions, and the contribution of the measured excitation to the total structural response is usually rather low.Output-only orOperational Modal Analysis(OMA) techniques have therefore been developed [22]. They extractthe modal parameters from the dynamic response to operational (ambient) forces, and consequently they deliver alinear (modal) model of the structure around the real working point of operation. The unmeasured, ambient forcesare usually modelled as stochastic quantities with unknownparameters but with known behavior, e.g., as white noisetime series with zero mean and unknown covariance.

The existing OMA techniques suffer from some important shortcomings, however. A first one is that output-onlymeasurements do not allow to determine a complete modal model, since the mode shapes can not be scaled in anabsolute sense, e.g., to unit modal mass. A possible solution is performing a second measurement after adding orremoving a significant amount of mass to or from the structure[20], but this is rather cumbersome for large structuressuch as civil engineering structures [19]. Secondly, the ambient excitation is sometimes confined to a narrow (low-)frequency band, and as a result only a limited number of modes can be extracted with high quality from the ambientvibration data.

For these reasons, there has been an increasing interest during the last few years towards combined experimental-operational modal testing techniques, also called hybrid vibration testing orOperational Modal Analysis with eX-ogenous inputs(OMAX) [12]. In these techniques, an artificial force is usedin operational conditions. The main

2

Experimental Modal Analysis (EMA) Operational Modal Analysis (OMA) Operational Modal Analysiswith eXogenous inputs (OMAX)

ud(t)

us(t)y(t)structure

ud(t)

us(t)y(t)structure

ud(t)

us(t)y(t)structure

Figure 1: Comparison of the EMA, OMA and OMAX approaches to modal testing. The measured forces (modeled as deterministic) are denotedby ud(t), the unmeasured forces (modeled as stochastic) byus(t) and the outputs byy(t). In EMA, the unmeasured forces are considered as anunwanted noise source that needs to be removed. In OMA, thereare no measured forces. In OMAX, both the measured and unmeasured forces aretaken into account.

difference between OMAX and the traditional EMA approach isthat the operational forces are included in the iden-tified system model: they are not considered as an unwanted noise source, but as useful excitation (fig. 1). As aconsequence, the amplitude of the artificial forces can be equal to, or even lower, than the amplitude of the unmea-sured, operational forces. This is of crucial importance for the modal testing of large structures, since it allows to useactuators that are small and practical when compared to the ones that are needed for EMA testing [29].

Two system identification methods for OMAX testing have so far been proposed and validated on real-life vibra-tion data:

• a Combined non-linear Least Squares Frequency domain methodon Input Output spectra(CLSF-IO) [5], forwhich recently the alternative nameOMAX Maximum Likelihood Estimation(MLE-OMAX) was proposed [12],and

• a Combined deterministic-stochastic Subspace Identification (CSI) approach, both in a frequency-domain vari-ant [4] and in a reference-based data-driven time-domain (CSI-data/ref) variant [25].

The CLSF-IO approach results in a non-linear minimization problem, which is solved iteratively. Local minima mayoccur, so good starting values are required for the system description. The combined subspace approach is non-iterative and numerically robust and therefore it requiresa much smaller computational effort, while the obtainedmodal parameters are also highly accurate. The subspace approach can be applied on its own, or to generate accuratestarting values for the CLSF-IO method. Of both subspace algorithms, the reference-based one is the fastest and, witha smart choice of reference outputs, also the most accurate one [25]. Therefore, the CSI-data/ref algorithm will beused to analyze the OMAX data in this paper.

3. General FE model updating procedure

In FE model updating, the numerical modal data are initiallycomputed using estimated starting values for theunknown model parameters to be updated. These parameters are then adjusted until the discrepancies between thenumerical and experimental modal data, obtained from an OMAor OMAX test, are minimized. The followingsections briefly discuss the objective function, the damagefunctions, and the optimization algorithm, that will be usedin the remainder of this paper.

3.1. Objective function

The minimization of the objective function is stated as a nonlinear least squares problem, which is defined as asum of squared differences:

f(θ) = 1

2

m∑

j=1

rj(θ)2 = 1

2‖r(θ)‖2, (1)

3

where each residuerj(θ) represents a distance between a measured and computed modalparameter that is a nonlinearfunction of theoptimization variablesθ ∈ R

n. ‖�‖ denotes the Euclidean norm. In order to obtain a unique solution,the numberm of residuals should be greater than the numbern of unknownsθ.

The residual vectorr(θ) : Rn → Rm is split into a frequency residual vectorrf , a mode shape residual vectorrs

and, for OMAX data, a modal mass residual vectorrm:

r(θ) =

rf (θ)rs(θ)rm(θ)

. (2)

The residuals are defined as:

rf,j(θ) =f2

j (θ)− f2

j

f2

j

, j ∈ {1, . . . ,mf}

rs,j(θ) = MSF

(

φj ,φj

φrj

)

φj −φj

φrj

, j ∈ {1, . . . ,ms}

rm,j(θ) =∣

∣

∣ln(

MSF(

φj , φj

))∣

∣

∣, j ∈ {1, . . . ,mm},

(3)

wherefj, andφj denote the numerical undamped eigenfrequency and the mode shape vector, respectively;fj andφj the identified undamped eigenfrequency and mode shape vector, respectively; andmf , ms andmm the number ofidentified eigenfrequencies, mode shapes and modal masses,respectively, used in the updating process.φr

j denotes

a particular,referencedegree of freedom of mode shapeφj TheModal Scale Factor(MSF) between a numericallycomputed mode shapeφj and an identified oneφj is the scale factor that minimizes the difference between bothvectors in a least-squares sense [1]:

MSF(

φj , φj

)

=φ∗

jφj

‖φj‖2, (4)

where�∗ denotes complex conjugate transpose.

Relative differences are taken inrf in order to obtain a similar weight for each frequency residual. In rs theexperimental mode shapes are scaled to1 in a reference componentφr –which is a component with large amplitude–,and the numerical mode shapes are fitted to the experimental ones in a least-squares sense. Inrm, both the exper-imental and the numerical mode shapes are mass-normalized;if the modal scaling factor between them equals1,the corresponding modal masses are equal and the residual iszero. Thanks to the absolute value of the logarithm in

rm, the residual remains unchanged whenMSF(

φj , φj

)

is replaced withMSF(

φj ,φj

)

. Since mass-normalized

experimental mode shapes are needed inrm, the residual is only useful for OMAX (or EMA) data.The least-squares problem formulation allows the residuals to be weighted separately corresponding to their im-

portance and/or the accuracy of the experimental values. The relative proportion of the weighting factors is important,not their absolute values. The following weighted least squares problem is solved:

min 1

2‖W

1

2 r(θ)‖2 , (5)

whereW is the weighting matrix, which is often diagonal, i.e.,

W = diag(. . . , w2

j , . . .), (6)

with wj the weighting factor of the residualrj .Note that, as an alternative to the residuals defined in (3), also absolute differences between experimental and

numerical mass-normalized mode shapes could be taken if theexperimental values are identified from an OMAX (orEMA) test. However, the accuracy of experimental unscaled mode shape values is usually higher than the accuracy ofthe corresponding modal masses. The formulation as in (3) allows to separate their influence on the objective function,and apply different weighting factors according to their accuracy.

4

3.2. Optimization variablesθ

One or more unknown physical propertiesX (e.g. the Young’s modulus) are updated in each elemente of thenumerical FE model. A dimensionless correction factorae expresses the relative difference of the updated value ofpropertyX with respect to its initial valueXe

0 , in elemente:

Xe = Xe0(1− ae). (7)

The correction factors can affect one element or may be assigned to an element group. If the unknown physicalproperty is linearly related to the stiffness matrix of the element (group), we have:

Ke = Ke0(1− ae) (8)

K = Ku +

ne∑

e=1

Ke0(1− ae), (9)

whereKe0 andKe are the initial and updated element stiffness matrix respectively, K is the global stiffness matrix

andKu is the stiffness matrix of the element (group) whose properties remain unchanged.ne is the number ofelements (groups) that are updated.

Adjusting the model property of all elements separately, i.e., choosing the elements of the vector with updat-ing variablesθ to be the element correction factorsae, would result in a high number of updating variables. Thiswould cause the sensitivity matrix of the residual vectorr(θ) with respect to the updating variablesθ, to becomeill-conditioned. Therefore, the distribution of the correction factorsae –which define on their turn the distribution ofthe updated physical propertiesX over the FE model– is approximated by using global damage functionsN (x, θ) asa regularisation measure [32].

An efficient damage function is a piecewise linear function,which is obtained by combining triangular shapefunctionsNi that differ from zero only over a limited area of the FE model (figure 2):

N (x, θ) = N (x, θ) =

n∑

i=1

θiNi(x). (10)

A mesh ofdamage elementsis defined on top of the mesh of finite elements, with each damage element simplyconsisting of a set of neighbouring finite elements (figure 2-bottom). The functionsNi are defined with respect toa node of this mesh and differ from zero only in the adjacent damage elements and equal zero in the other damageelements.

x

iDam

age

func

tion

θ

0

1

x

damage elements

Ni

Figure 2: A piecewise linear damage functionN (x,θ), which is obtained by combining triangular shape functionsdefined on a damage elementmesh. It is illustrated here on a beam model.

The optimization variablesθi are the correction factorsae of the elements that constitute the nodes of the damageelements. The correction factorsae of the elements that are interior to a damage element are computed by interpolationusing the damage functions (10).

5

3.3. Optimization algorithm: Trust region Gauss-Newton method

The nonlinear least squares function (Eq. (1)) is solved with the Gauss-Newton method, which is an iterativesensitivity-based optimization method that exploits the special structure of the least squares problem. Namely, thegradient and the Hessian of the objective function (Eq. (1))have the following special structure:

∇f(θ) =m∑

j=1

rj(θ)∇rj(θ) = J(θ)T r(θ) (11)

∇2f(θ) = J(θ)TJ(θ) +m∑

j=1

rj(θ)∇2rj(θ) (12)

with J the Jacobian matrix, containing the first partial derivatives of the residualsrj with respect to the optimizationvariablesθi. In the Gauss-Newton method, the Hessian is approximated with the first order term in Eq. (12). Thestandard Gauss-Newton method is further stabilized by implementing it with thetrust region strategywhich enhancesthe optimization process to converge [6]. In FE model updating, the trust region strategy is an additional measure toimprove the robustness of the updating procedure. The most effective measure to treat the ill-posedness of the inverseproblem, however, is provided by the damage functions.

4. Application to the Z24 bridge

4.1. Description of the structure

The Z24 bridge was located in the canton Bern near Solothurn,Switzerland. It was part of the road connectionbetween the villages of Koppigen and Utzenstorf, over-passing the highway A1 between Bern and Zurich. The bridgewas a post-tensioned two-cell concrete box girder bridge with a main span of30 m and two side-spans of14 m.Both abutments consist of three concrete columns connectedwith hinges to the girder. Both intermediate supportsare concrete piers clamped into the girder. All supports arerotated with respect to the longitudinal axis which yieldsa skew bridge. The bridge, which dated from 1963, was demolished in 1998 because a new railway, adjacent to thehighway, required a new bridge with a larger side-span.

In the framework of the Brite EuRam Program CT96 0277 SIMCES [8], the bridge was progressively damagedin a number of damage scenarios, before complete demolition. Before and after each applied damage scenario, thebridge was subjected to a full forced and ambient vibration test. A vibration monitoring system was installed on thebridge before and during the progressive damage test, for a total duration of one year, with the aim of quantifyingthe influence of the temperature on the bridge dynamics. A full description of the monitoring system and the damagescenarios applied to the bridge is given by Kramer et al. [15]. Both the monitoring data and the vibration data fromthe full vibration tests have been used as a benchmark for operational modal analysis algorithms and for algorithmsfor structural health monitoring and damage identification. A literature review of the benchmark results is providedby Reynders and De Roeck [26].

The damage scenario that is studied here is the settlement ofthe foundation of one of the supporting piers (at44 m). It was simulated by lowering the pier by95 mm (figure 3a), thereby inducing cracks in the bridge girder.The analysis is based on the full scale forced and ambient vibration tests that have been carried out before and afterlowering the pier. The continuous monitoring system enabled to verify that the influence of the temperature on thebridge dynamics during the experiment is negligible [23]; this influence is therefore not taken into account.

4.2. Finite element model

The bridge is modelled with a beam model (6 degrees of freedomin each node), see figure 4. Equivalent valuesfor the cross section area, the bending and torsional momentof inertia of the box section of the main girder (figure3c) are computed. The girder’s cross-section and moment of inertia are the highest above the supporting piers (figure8a,b).

The girder is modelled with82 beam elements;44 beam elements are used to model the piers, columns andabutments. The concrete is considered to be homogeneous, with an initial value for the Young’s modulus ofE0 =37.5 GPa andG0 = 16 GPa for the shear modulus. The principal axes of the piers are rotated to model the skewness

6

(a)

(b)

(c)

8.60

1.1

04

.50

(d)

Figure 3: Highway bridge Z24: (a) elevation, (b) top view with measurement grid indicated, (c) cross section and (d) crack pattern in the bridgegirder, above the lowered pier.

of the bridge, and the width of the piers is taken into accountby using constraint equations. Mass elements are used forthe cross girders and foundations. Both concentrated translational mass and rotary inertial components are considered.

In order to account for the influence of the soil, springs are included at the pier and column foundations, at theend abutments and around the columns (figure 4). The initial values of the soil stiffness parameters are:Kv,p =180 × 106 N

m3 , Kh,p = 210 × 106 Nm3 (under the piers, atx = 14 and44 m); Kv,c = Kh,c = 100 × 106 N

m3

(under the columns, atx = 0 and58 m); Kv,a = 180 × 106 Nm3 , Kh,a = 200 × 106 N

m3 (at the abutments) andKv,ac = Kh,ac = 100× 106 N

m3 (around the columns).The eigenfrequencies and MAC values calculated with the initial FE model are listed in tables 4 and 5.

4.3. OMA and OMAX analysis

Testing the structure..Before lowering one of the intermediate piers – but after theinstallation of the pier settlementsystem – and after lowering it by95 mm, the bridge was subjected to a full-scale forced and ambientvibrationtest. Since ambient forces such as wind excitation or trafficunder the bridge could not be excluded, all tests can beconsidered as operational, with or without the use of artificial forces. In the former case, the test is an OMAX test; inthe latter case, it is a pure OMA test.

7

soil springs

82 beam elements in the bridge girder

XY

Z

Figure 4: FE model of the bridge Z24.82 beam elements are used to model the girder. The soil springs at the supports are indicated.

With a measurement grid consisting of a regular3 × 45 grid on top of the bridge deck and a2 × 8 grid on eachof the two intermediate pillars,291 degrees of freedom have been measured: all three acceleration components on thepillars, and mainly vertical and lateral accelerations on the bridge deck. Because of the limited number of availableaccelerometers and data acquisition channels, the data were collected in9 setups using5 reference channels (see figure5). The forced excitation was applied by two vertical shakers of EMPA, Switzerland: a1 kN RMS shaker was placedon the middle span, and a0.5 kN RMS shaker on the Koppigen side span (see figure 5). The shaker input signalswere generated using an inverse FFT algorithm, resulting ina fairly flat force spectrum between3 and30 Hz. In eachmodal test, nearly11 minutes of data were recorded at a sampling rate of100 Hz, using an anti-aliasing filter with a30 Hz cut-off frequency.

XY

Z

KoppigenUtzenstorf

R1

R2

R3

1kN

0.5kN

Figure 5: Z24 bridge: Measurement grid, reference positions and shaker positions.

System identification..For the ambient data, theREFerence-based COVariance-driven Stochastic Subspace Identifi-cation(SSI-cov/ref) algorithm [21] was used to identify the modalparameters. As a parameter choice, half the numberof block rows of the data Hankel matrix was set toı = 50. For the shaker data, theREFerence-based DATA-drivenCombined deterministic-stochastic Subspace Identification (CSI-data/ref) algorithm [25] was used withı = 20. Ac-cording to the rule of thumb proposed in [25], modes with an eigenfrequency higher than2.5 Hz can be well identifiedwith the chosen values forı; here the first natural frequency lies around3.9 Hz. Both for the SSI-cov/ref and the CSI-data/ref algorithms, the channels that are common to all9 setups were chosen as reference outputs:R1z, R2x, R2y,R2z, andR3z (see figure 5). A stabilization diagram was constructed for model orders of2 till 120 in steps of2.Before identification, the data were filtered with a fourth order high-pass Butterworth filter with a cut-on frequency of1 Hz, both in the forward and reverse directions, in order to remove the low-frequency drift.

Results for the undamaged structure.Table 1 lists the modal parameters identified for the undamaged structure, bothfor the OMA and OMAX data. Besides the undamped eigenfrequency and the damping ratio, also theModal PhaseCollinearity (MPC) [14] is listed for each mode. MPC values lie between0 and1; a value close to1 indicates a realnormal mode, i.e., a mode for which the phase difference between all complex mode shape components is close to

8

mode OMA OMAXf [Hz] ξ [%] MPC [−] f [Hz] ξ [%] MPC [−] MP [◦]

1 3.88 0.8 1.00 3.89 0.9 1.00 0.82 5.02 1.5 0.98 4.93 1.6 1.00 36.46 9.80 1.4 0.95 9.80 1.9 0.95 8.07 10.29 1.4 0.98 10.34 2.2 0.99 4.18 12.73 2.6 0.98 12.62 3.3 0.98 5.89 13.46 4.2 0.93 13.34 5.2 0.98 7.410 17.20 3.4 0.80 13.511 19.28 2.5 0.85 21.112 19.76 5.8 0.91 21.613 26.47 3.3 0.87 19.715 37.11 4.0 0.76 46.2

Table 1: Z24 bridge, undamaged structure: OMA modes identified with SSI-cov/ref, and OMAX modes identified with CSI-data/ref. In both cases,the undamped eigenfrequencyf , the damping ratioξ, and the modal phase collinearity (MPC) are listed. For the OMAX modes, also the meanphase (MP) of the mass-normalized mode shapes is listed.

180◦ or−180◦. Since for the Z24 bridge, there are no double modes in the frequency range of interest, nor localizeddampers, an MPC value close to1 is expected for all identified modes. For the OMAX data, also the Mean Phasevalues of the mass-normalized mode shapes are listed; they should be close to zero for lowly damped modes [24,p. 73].

Figure 6 plots the real and imaginary part of the mass-normalized OMAX mode shapes; only for modes2 and15,where the imaginary part is large (cfr. the high MP values in table 1), the amplitude is plotted instead. Modes1, 8, 9and12 are pure bending modes. Mode2 is a lateral mode. Modes6 and7 consist of a mixture of bending and torsion;this is due to the skewness of the bridge supports with respect to the longitudinal axis of the bridge. Modes10, 11, 13and15 are pure torsion modes.

The high MP value for the lateral mode in table 1 comes as no surprise, since this mode is almost exclusivelyexcited by the ambient forces and not by the shakers. However, using the OMAX point of view, the mode could bevery well identified, also from the shaker data: only the mass-normalization is inaccurate. The identification of mode15 at37.11 Hz is remarkable since the cut-off frequency of the analog anti-aliasing filter was set to30 Hz. It indicatesthat the CSI-data/ref method for OMAX analysis is able to identify modes that are only very weakly present in thedata. The large MP value in table 1, however, indicates that this mode could not be properly mass-normalized.

Finally, it can be noted that modes3, 4, 5 and14 were not identified in this data set, most probably because theywere not well excited: modes3 and5 are lateral modes, mode4 is a longitudinal mode, and mode14 is a bendingmode with an eigenfrequency above the cut-off frequency of the anti-aliasing filter. These modes, however, show upin other data sets of the progressive damage test, see [24, section 7.4].

Results for the damaged structure.Table 2 lists the modal parameters identified after the95 mm pier settlement, bothfor the OMA and OMAX data. With respect to the undamaged case,an extra longitudinal mode is found, both in theOMA and OMAX data set (mode 4). On the other hand, the high-order torsion mode15 is not found for this damagescenario. Compared to the reference state, the eigenfrequencies of modes11 and12 have crossed.

Table 3 compares the eigenfrequencies, identified for the damaged and undamaged cases from the OMAX data. Asignificant decrease is observed in the eigenfrequency of modes1,6,7,8,9,11,12, and13. In the OMA data, a similarobservation can be made: the pier settlement causes a significant decrease in the eigenfrequency of modes1, 6, 7,and8. In both cases, the eigenfrequency of the torsion mode10 does not seem to change. For the OMA case, theeigenfrequency of the lateral mode2 does not change significantly.

4.4. FE model updating for the undamaged structure: OMA case

Correction factors.Two different updating processes are performed, in order tomodel the reference and the damagedstate of the bridge. The bending stiffnessEI and the torsional stiffnessGIt of the beam elements of the girder are

9

mode13.89Hz - 0.9%

mode24.93Hz - 1.6%

mode69.80Hz - 1.9%

mode710.34Hz - 2.2%

mode812.62Hz - 3.3%

mode913.34Hz - 5.2%

mode1017.20Hz - 3.4%

mode1119.28Hz - 2.5%

mode1219.76Hz - 5.8%

mode1326.47Hz - 3.3%

mode1537.11Hz - 4.0%

Figure 6: Z24 bridge, undamaged structure: modes identifiedwith CSI-data/ref. For all modes, except modes2 and15, the real part (greyscale;color online) and the imaginary part (black and white) of themass-normalized mode shapes are plotted separately. For modes2 and15, only theamplitudes are plotted.

10

mode OMA OMAXf [Hz] ξ [%] MPC [−] f [Hz] ξ [%] MPC [−] MP [◦]

1 3.68 0.7 1.00 3.69 0.9 1.00 0.62 4.92 1.4 1.00 4.87 1.6 1.00 2.34 7.24 4.9 0.91 6.93 8.3 0.82 15.16 9.23 1.4 0.95 9.18 1.6 0.97 7.87 9.68 1.4 0.96 9.68 1.8 0.94 8.88 12.00 3.2 0.98 11.95 2.8 0.99 7.59 13.41 4.3 0.92 13.00 4.9 0.98 5.910 17.20 5.0 0.77 16.411 18.93 2.8 0.95 17.312 18.10 4.2 0.88 19.413 25.93 3.0 0.89 22.2

Table 2: Z24 bridge after the95 mm pier settlement: OMA modes identified with SSI-cov/ref, andOMAX modes identified with CSI-data/ref. Inboth cases, the undamped eigenfrequencyf , the damping ratioξ, and the modal phase collinearity (MPC) are listed. For the OMAX modes, alsothe mean phase (MP) of the mass-normalized mode shapes is listed.

mode 1 2 6 7 8 9 10 11 12 13

fu [Hz] 3.89 4.93 9.80 10.34 12.62 13.34 17.20 19.27 19.76 26.47σ [Hz] 0.01 0.01 0.01 0.03 0.02 0.05 0.20 0.01 0.16 0.05

fd [Hz] 3.69 4.87 9.18 9.68 11.95 13.00 17.20 18.93 18.10 25.93σ [Hz] 0.00 0.00 0.01 0.02 0.02 0.07 0.08 0.03 0.05 0.03

Table 3: Z24 bridge: eigenfrequencies, identified with CSI-data/ref, before (fu) and after (fd) lowering the pier with95 mm. The sample standarddeviationsσ, obtained over the9 different measurement setups, are shown as well.

11

both updated separately. They are adjusted by correcting the Young’s and the shear modulus,E andG.In the first updating process additionally the vertical soilstiffness under the supporting piers,Kv,p, and the hor-

izontal soil stiffness under the end abutments,Kh,a, are updated. The former influences mainly modes2, 5, and6(lateral and bending), the latter only mode2. Since the soil stiffness is not altered by the damage, the soil springs arenot updated in the second updating process.

Damage function.The bridge girder is subdivided into 8 damage elements: 4 damage elements in the middle spanand 2 damage elements in each side span (figure 7). Two (identical) piecewise linear damage functions are used foridentifying the bending and the torsional stiffness distribution, respectively.

0 14 44 58Distance along bridge girder [m]

X,1 X,2 X,3 X,4 X,5 X,6 X,7

Ni

Dam

age

func

tion

mid−spanside−span side−span

θθθθθθθ

Figure 7: Piecewise linear damage functionN (x, θ) used to identify the distribution of both sets of correctionfactors,aE andaG, for thereference and damaged state of the bridge Z24 (X denotes eitherE or G). The bridge girder is subdivided into 8 damage elements. The mesh offinite elements is also plotted on the horizontal axis.

In the first updating process the optimization problem contains 16 (= 2 × 7 + 2) optimization variablesθi,corresponding to the parameters of both damage functions (2× 7) and the two correction factors for the soil springs.In the second process only 14 (= 2× 7) variables need to be updated.

For the updating to the reference state, a trust region strategy is applied where the optimization variablesθi areallowed to change within[−0.3, 0.3], except for the variables that determine the soil spring stiffness; these are allowedto change within[−0.9, 0.9]. For the updating to the damaged state, the values forθi obtained in the first updatingprocess are taken as starting values, and the optimization variables are only allowed to decrease, with a lower boundof −0.9, except forθG,1, θG,4 andθG,7, for which the lower bound is set to−0.3 in order to increase the numericalconditioning. It can be noted that in previous updating studies on the same bridge [30, 31], the optimization variablesθG,1, θG,4 andθG,7 were only allowed to change within[−0.15, 0.15] for the same reason.

Objective function.The first five vertical modes (bending and bending-torsion) and the first lateral mode of the un-damaged bridge are used to update the initial FE model to the reference undamaged state of the bridge.

The residual vector (Eq. (3)) contains6 eigenfrequency residuals and624 mode shape residuals. The verticaldisplacementsuz along the three measurement lines (3 × 39 points) and the horizontal displacementsuy along thecenterline (39 points) are used for the vertical and lateral modes, respectively. The total residual vectorr(θ) containsm = 630 residuals. The following weighting factors (Eq. (5)) are applied.

• For the eigenfrequencies,wf,j = 1 − 3σf ,j , whereσf ,j is the standard deviation of the identified undamped

eigenfrequencyfj , estimated as the sample standard deviation of the values identified for each of the9 differentmeasurement setups.

• For the mode shapes,ws,j = MPC(

φj

)

/10. The MPC values are listed in table 1 for the undamaged case,

and in table 2 for the damaged case. The additional division by a factor of10 is performed to create a balancebetween the eigenfrequency and mode shape residuals in the objective function.

For the optimization process, convergence is assumed when the change in objective function, relative to the initialvalue, is smaller than10−6. In the first updating process,20 trust-region Gauss-Newton iterations are needed, and the

12

objective function decreases from0.048 to 0.012. In the second updating process,10 iterations are needed, and theobjective function decreases from0.128 to 0.022.

Updating results.The updated values of the vertical soil stiffness under the piers and the horizontal stiffness underthe abutments are:Kv,p = 235× 106 N

m3 andKh,a = 261× 106 Nm3 . These values are used in the FE model when

identifying the damage.The stiffness distribution of the bridge girder is plotted in figure 8a,b, for bending as well as for torsion. The initial

and the updated values for the reference and damaged state are shown. After updating, the initial bending stiffness isincreased by almost30 % above the left pier, and decreased by almost20 % above the right pier. The initial torsionstiffness is decreased by29% at the center of all three spans.

(a) Bending stiffnessEI

0 14 44 580

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

10

Distance along bridge girder [m]

EI y [N

m2 ]

Initial FE modelUndamaged (reference)Damaged

(b) Torsional stiffnessGIt

0 14 44 580

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

10


GI t [N

m2 ]


(c) Correction factorsaE,dam

0 14 44 58

−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

00.10.20.30.40.50.60.70.80.9


a E,d

am [−

]

(d) Correction factorsaG,dam

0 14 44 58

−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

00.10.20.30.40.50.60.70.80.9


a E,d

am [−

]

Figure 8: Identified parameters for the OMA case: (a) bendingstiffness distributionEIdam = EIref(1 − aE,dam); (b) torsional stiffnessdistributionGIt,dam = GIt,ref (1− aG,dam) and their correction factors, (c)aE and (d)aG, for the damaged bridge.

In the damaged state, a decrease in the stiffness around the pier at44 m, is clearly visible. This decrease is dueto lowering the pier, which induced cracks in the beam girderat that location (figure 3d). The corresponding damagepattern, defined by the reduction factorsaE andaG, is plotted in figure 8c,d. The bending and torsional stiffness arereduced with a maximum of42 % and21 % respectively, located in the expected cracked zone.

Modal parameters.Table 4 lists the initial and updated modal data for the undamaged as well as for the damagedbridge. Figures 9 and 10 compare the experimental and computed mode shapes for the undamaged and damagedcases, respectively.

13

Undamaged DamagedExperiment Initial Updated Exp. Reference Updated

FE model FE modelMode Eigenfrequencies[Hz] Eigenfrequencies[Hz]

1 3.88 3.73 3.86 3.68 3.86 3.652 5.02 5.14 5.04 4.92 5.04 4.876 9.80 9.64 9.72 9.23 9.72 9.097 10.29 10.25 10.33 9.68 10.33 9.748 12.73 12.52 12.74 12.01 12.74 11.999 13.46 13.35 13.52 13.41 13.52 13.15

MAC values[%] MAC values[%]

1 99.93 99.91 99.82 99.78

2 98.24 98.47 97.59 97.87

6 96.01 98.45 88.26 98.62

7 95.27 98.97 85.81 97.60

8 93.08 98.10 92.43 98.18

9 93.61 97.11 87.18 90.97

Table 4: Z24 bridge, updating using OMA data: experimental,initial and updated eigenfrequencies and MAC values for theundamaged anddamaged case.

mode1 mode2 mode6

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

mode7 mode8 mode9

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

Figure 9: Z24 bridge, undamaged structure, OMA data: experimental mode shapes (full lines), mode shapes computed with the initial FE model(dotted lines), and mode shapes computed with the updated FEmodel (dashed lines). The mode shape components with negative and positivetransversal coordinates are marked with the symbols x and +,respectively; the ones at the center are not marked. The experimental mode shapesare scaled to unit modal displacement in one of the DOFs. The computed mode shapes are least-squares fitted to the experimental ones.

By updating the initial FE model to the reference state, the numerical and experimental eigenfrequencies corre-spond much better (the maximum relative difference drops from 3.9 % to 0.8 %) and a clear improvement for themode shapes and MAC values can be observed (the minimum MAC increases from93.08 to 97.11).

14

mode1 mode2 mode6

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1

Distance along bridge girder [m]D

ispl

acem

ent [

m]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

mode7 mode8 mode9

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]Figure 10: Z24 bridge, damaged structure, OMA data: experimental mode shapes (full lines), mode shapes computed with the updated FE modelfor the reference state (dotted lines), and mode shapes computed with the updated FE model for the damaged state (dashed lines). The mode shapecomponents with negative and positive transversal coordinates are marked with the symbols x and +, respectively; the ones at the center are notmarked. The experimental mode shapes are scaled to unit modal displacement in one of the DOFs. The computed mode shapes are least-squaresfitted to the experimental ones.

For the damaged bridge, the correlation between the numerical and experimental eigenfrequencies is also improvedvery well with the updated FE model. The maximum relative difference in eigenfrequency drops from6.7 % to 1.9 %,and the minimum MAC increases from87.18 to 90.97.

4.5. FE model updating for the undamaged structure: OMAX case

Correction factors and damage functions..The parameters to be updated are the same as in the OMA case: theYoung’s and shear moduli,E andG, are updated using the damage functions of fig. 7, and in the first updatingprocess (reference state), also the soil stiffness under the supporting piers,Kv,p, and the horizontal soil stiffness underthe end abutments,Kh,a, are updated. WhileKh,a only influences mode2, Kv,p influences also modes3, 5, 6, and8.

Objective function.All identified modes are used to update the FE model to the reference and the damaged state ofthe bridge, except for mode10, which could not be related to any of the modes computed with the initial FE model inthe frequency range0 − 40 Hz, and modes4 and15, which were not found in the undamaged or the damaged case.The residual vector (Eq. (3)) in the updating process contains9 frequency residuals,975 mode shape residuals, and5 modal mass residuals. The vertical displacementsuz along the three measurement lines (3 × 39 points) and thehorizontal displacementsuy along the centerline (39 points) are used for the mode shape residuals of the verticalandlateral modes, respectively. The modal mass residuals are only computed for the5 modes for which the mean phaseof the experimental mode shape is smaller than10◦ (see table 1). The total residual vectorr(θ) containsm = 989residuals. The following weighting factors (Eq. (5)) are applied.

• For the eigenfrequencies and mode shapes,wf,j = 1 − 3σf ,j andws,j = MPC(

φj

)

/10, respectively, as in

the OMA case.

15

• For the modal masses,wm,j =(

1−MP(

φj

)

/10)

/5. TheMP values are listed in table 1 for the undamaged

case, and in table 2 for the damaged case. The additional division by a factor5 is performed to create a balancebetween the eigenfrequency, mode shape, and modal mass residuals in the objective function.

For the optimization process, convergence is assumed when the change in objective function, relative to the initialvalue, is smaller than10−6. In the first updating process,28 trust-region Gauss-Newton iterations are needed, and theobjective function decreases from0.245 to 0.083. In the second updating process,48 iterations are needed, and theobjective function decreases from0.251 to 0.134. During the optimization process, the eigenfrequencies ofseveralmodes crossed. Therefore, the mode pairing between the experimental and numerical modes was based on the modalassurance criterion (MAC).

Updating results.The updated values of the vertical soil stiffness under the piers and the horizontal stiffness underthe abutments are the same as for the OMA case:Kv,p = 235 × 106 N

m3 andKh,a = 261 × 106 Nm3 . The stiffness

distribution of the bridge girder is plotted in figure 11a,b,for bending as well as for torsion. The initial and the updatedvalues for the reference and damaged states are shown. Afterupdating, the maximum increase and decrease in initialbending stiffness are both28 %. For the torsion stiffness, the maximum increase and decrease are both30 %, whichis the maximum value allowed by the defined trust region.

(a) Bending stiffnessEI

0 14 44 580

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

10


EI y [N

m2 ]


(b) Torsional stiffnessGIt

0 14 44 580

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

10


GI t [N

m2 ]


(c) Correction factorsaE,dam

0 14 44 58

−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

00.10.20.30.40.50.60.70.80.9


a E,d

am [−

]

(d) Correction factorsaG,dam

0 14 44 58

−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

00.10.20.30.40.50.60.70.80.9


a E,d

am [−

]

Figure 11: Identified parameters for the OMAX case: (a) bending stiffness distributionEIdam = EIref (1 − aE,dam); (b) torsional stiffnessdistributionGIt,dam = GIt,ref (1− aG,dam) and their correction factors, (c)aE and (d)aG, for the damaged bridge.

In the damaged state a decrease in girder stiffness, around the pier at44m, is clearly visible. This decrease is dueto lowering the pier, which induced cracks in the beam girderat that location (figure 3d). The corresponding damage

16

pattern, defined by the reduction factorsaE andaG, is plotted in figure 11c,d. The bending and the torsional stiffnessare reduced with a maximum of46 % and15 % respectively, located in the expected cracked zone.

At the left part of the bridge, a non-physical increase in torsion stiffness can be observed. A similar observationwas made for the bending stiffness in the OMA case (see figure 8a). This might be caused by several factors, ora combination of them, such as modelling errors (torsion warping and shear lag effects are not accounted for, theinteraction with the soil is modelled very approximately, the chosen damage functions might be too coarse, etc.) andidentification errors in the modal data that have been extracted from the acceleration measurements.

Modal parameters.Table 5 lists the initial and updated modal data for the undamaged as well as for the damagedbridge. For comparison purposes, the results obtained withthe FE models, that were updated using the OMA data,are shown as well. Figures 12, and 13 compare the experimental and computed mode shapes for the undamaged anddamaged cases, respectively.

Undamaged DamagedExperiment Initial Updated OMA Exp. Reference Updated OMA

FE model FE modelMode Eigenfrequencies[Hz] Eigenfrequencies[Hz]

1 3.89 3.73 3.83 3.86 3.69 3.83 3.58 3.652 4.93 5.14 5.01 5.04 4.87 5.01 4.79 4.876 9.80 9.64 9.48 9.72 9.18 9.48 8.81 9.097 10.34 10.25 10.08 10.33 9.68 10.08 9.55 9.748 12.62 12.52 12.69 12.74 11.95 12.69 12.11 11.999 13.34 13.35 13.61 13.52 13.00 13.61 13.36 13.2511 19.27 22.53 20.44 21.55 18.92 20.44 19.94 21.0212 19.76 19.32 19.60 19.33 18.10 19.60 18.62 18.6813 26.47 28.91 27.65 27.62 25.93 27.65 27.08 26.89

MAC values[%] MAC values[%]

1 99.84 99.84 99.84 99.20 99.39 99.44

2 97.89 98.07 98.07 97.38 97.50 97.76

6 96.76 98.09 97.87 87.75 98.42 97.44

7 97.79 97.53 96.01 78.89 95.12 93.42

8 96.59 96.33 96.98 90.06 97.41 98.35

9 85.42 94.99 95.50 92.28 97.79 97.19

11 89.93 89.41 87.90 90.47 88.86 87.18

12 87.19 91.13 89.09 74.60 59.36 71.02

13 82.22 86.10 83.79 87.72 86.80 84.43

MSF values[%] MSF values[%]

1 92.7 92.8 92.8 171.4 171.6 171.7

6 115.8 113.5 110.9 123.6 142.0 138.0

7 89.2 91.5 92.7 223.8 223.9 228.6

8 127.8 126.9 127.3 149.4 156.9 157.6

9 84.3 87.7 84.9 164.3 160.2 91.7

Table 5: Z24 bridge, updating using OMAX data: experimental, initial and updated eigenfrequencies, MAC values, and MSFvalues for theundamaged and damaged case. The results obtained with the “OMA” FE models, that were updated using the OMA data, are shownas well.

By updating the initial FE model to the reference state, the numerical and experimental eigenfrequencies corre-spond much better (the maximum relative difference drops from 16.9 % to 6.1 %) and a clear improvement for themode shapes and MAC values can be observed (the minimum MAC increases from82.22 to 86.10). With the FEmodal that was updated with the OMA data, the maximum relative frequency difference remains11.8 %, and theminimum MAC value83.79 %.

For the damaged bridge, the correlation between the numerical and experimental eigenfrequencies is also improved

17

mode1 mode2 mode6

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

mode7 mode8 mode9

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

mode11 mode12 mode13

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

Figure 12: Z24 bridge, undamaged structure, OMAX data: experimental mode shapes (full lines), mode shapes computed with the initial FE model(dotted lines), and mode shapes computed with the updated FEmodel (dashed lines). The mode shape components with negative and positivetransversal coordinates are marked with the symbols x and +,respectively; the ones at the center are not marked. The experimental mode shapesare scaled to unit modal displacement in one of the DOFs. The computed mode shapes are least-squares fitted to the experimental ones.

18

mode1 mode2 mode6

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

mode7 mode8 mode9

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

mode11 mode12 mode13

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

0 14 44 58

−1

0

1


Dis

plac

emen

t [m

]

Figure 13: Z24 bridge, damaged structure, OMAX data: experimental mode shapes (full lines), mode shapes computed with the updated FE modelfor the reference state (dotted lines), and mode shapes computed with the updated FE model for the damaged state (dashed lines). The mode shapecomponents with negative and positive transversal coordinates are marked with the symbols x and +, respectively; the ones at the center are notmarked. The experimental mode shapes are scaled to unit modal displacement in one of the DOFs. The computed mode shapes are least-squaresfitted to the experimental ones.

19

with the updated FE model: the maximum relative difference drops from8.3 % to 5.4 %. For the MAC values,however, an increase for modes11, 12 and13 is observed. Particularly for mode12, the discrepancy becomes larger,since in the computed mode shape, a torsion component appears which is not observed in the experimental data. Forthe FE model that was updated with the OMA data, the MAC value for mode12 is better but still not satisfactory, andthe maximum relative eigenfrequency difference is11.1 %.

For the modal scale factors (MSF values), which in theory should equal100 % when both the experimental andnumerical mass-normalized mode shapes are identical, no particular improvement is noted after updating. In practice,the uncertainty on experimental mass-normalized mode shapes is usually high, so that MSF values between80 % and120 % are deemed acceptable [13]. The large MSF values that remainafter updating for the damaged case, denotea rather large discrepancy between the modal masses in the numerical modes and the corresponding experimentalmodes.

5. Conclusions

The main limitations in the finite element (FE) model updating technique lie in the ability of the FE model torepresent the true behavior of the structure (modelling problem), and in the ability to identify enough modal param-eters with sufficient accuracy (identification problem). Inthis paper, the identification problem, which is particularlyimportant for large structures that are tested in operational conditions, was treated. It was shown that with an OMAXapproach, not only more modes can be identified than with a pure output-only (OMA) approach, but the modes thatare (partly) excited by the measured forces can also be mass-normalized.

When only few experimental modal data are available, an updated FE model may predict the modal parametersthat were used for the updating very well, but it may be physically not (entirely) correct and therefore it should beused with care e.g. for damage identification. When more experimental data are available, which can be achievedby performing an OMAX instead of an OMA test, the ill-posedness of the problem becomes smaller and potentialmodelling limitations are more easily detected.

This was illustrated here for the Z24 bridge: when updating abeam model with soil springs using the 6 modesidentified from the OMA test, a very good agreement between the updated FE model and the experimental modal datais observed after updating. However, when 3 additional higher modes, obtained from the OMAX test, are used forvalidation of the FE model, large discrepancies are detected. Updating the FE model with the OMAX instead of theOMA data did not result in a satisfactory agreement between the measured and computed modal parameters either.This indicates potential modelling problems in the FE model, which may be due to several factors, or a combinationof them, such as: torsion warping and shear lag effects not being accounted for, very approximate modelling of thesoil-structure interaction, rather coarse damage functions, etc. The solution of the modelling problem, however, fallsoutside the scope of this paper.

References

[1] R.J. Allemang and D.L. Brown. A correlation coefficient for modal vector analysis. InProceedings of the 1st International Modal AnalysisConference, pages 110–116, Orlando, FL, 1982.

[2] D. Bernal. Load vectors for damage localization.ASCE Journal of Engineering Mechanics, 128(1):7–14, 2002.[3] J. M. W. Brownjohn, P.-Q. Xia, H. Hao, and Y. Xia. Civil structure condition assessment by FE model updating: methodology and case

studies.Finite Elements in Analysis and Design, 37(10):761–775, 2001.[4] B. Cauberghe.Applied frequency-domain system identification in the fieldof experimental and operational modal analysis. PhD thesis, Vrije

Universiteit Brussel, 2004.[5] B. Cauberghe, P. Guillaume, P. Verboven, and E. Parloo. Identification of modal parameters including unmeasured forces and transient effects.

Journal of Sound and Vibration, 265(3):609–625, 2003.[6] T. F. Coleman and Y. Li. An interior, trust region approach for nonlinear minimization subject to bounds.SIAM Journal on Optimization,

6(2):418–445, 1996.[7] J.P. Conte, B. Moaveni, X. He, and A.R. Barbosa. System and damage identification of a seven-story reinforced concrete building structure

tested on the UCSD-NEES shake table. InProceedings of the 2nd International Conference on Experimental Vibration Analysis for CivilEngineering Structures: Evaces 2007, Oporto, Portugal, October 2007. Keynote paper.

[8] G. De Roeck. The state-of-the-art of damage detection byvibration monitoring: the SIMCES experience.Journal of Structural Control,10:127–143, 2003.

[9] D. J. Ewins.Modal testing. Research Studies Press, Baldock, U.K., second edition, 2000.

20

[10] M.I. Friswell and J.E. Mottershead.Finite element model updating in structural dynamics. Kluwer Academic Publishers, Dordrecht, TheNetherlands, 1995.

[11] C. Gentile and A. Saisi. Ambient vibration testing of historic masonry towers for structural identification and damage assessment.Construc-tion and Building Materials, 21(6):1311–1321, 2007.

[12] P. Guillaume, T. De Troyer, C. Devriendt, and G. De Sitter. OMAX - a combined experimental-operational modal analysis approach. InP. Sas and M. De Munck, editors,Proceedings of ISMA2006 International Conference on Noiseand Vibration Engineering, pages 2985–2996, Leuven, Belgium, September 2006.

[13] W. Heylen, S. Lammens, and P. Sas.Modal analysis theory and testing. Department of Mechanical Engineering, Katholieke UniversiteitLeuven, Leuven, Belgium, 1997.

[14] J.-N. Juang and R.S. Pappa. An eigensystem realizationalgorithm for modal parameter identification and model reduction. Journal ofGuidance, Control and Dynamics, 8(5):620–627, 1985.

[15] C. Kramer, C.A.M. de Smet, and G. De Roeck. Z24 bridge damage detection tests. InProceedings of IMAC 17, the International ModalAnalysis Conference, pages 1023–1029, Kissimmee, FL, USA, February 1999.

[16] G. Lombaert, B. Moaveni, X. He, and J.P. Conte. Damage identification of a seven-story reinforced concrete shear wall building usingbayesian model updating. InProceedings of IMAC 27, the International Modal Analysis Conference, Orlando, FL, February 2009. CD-ROM.

[17] N.M.M. Maia and J.M.M. Silva.Theoretical and experimental modal analysis. Research Studies Press, Taunton, U.K., 1997.[18] B. Moaveni, J.P. Conte, and F.M. Hemez. Uncertainty andsensitivity analysis of damage identification results obtained using finite element

model updating.Computer-Aided Civil and Infrastructure Engineering, 24(5):320–334, 2009.[19] E. Parloo, B. Cauberghe, F. Benedettini, R. Alaggio, and P. Guillaume. Sensitivity-based operational mode shape normalization: application

to a bridge.Mechanical Systems and Signal Processing, 19(1):43–55, 2005.[20] E. Parloo, P. Verboven, P. an Guillaume, and M. Van Overmeire. Sensitivity-based operational mode shape normalization. Mechanical

Systems and Signal Processing, 16(5):757–767, 2002.[21] B. Peeters and G. De Roeck. Reference-based stochasticsubspace identification for output-only modal analysis.Mechanical Systems and

Signal Processing, 13(6):855–878, 1999.[22] B. Peeters and G. De Roeck. Stochastic system identification for operational modal analysis: A review.ASME Journal of Dynamic Systems,

Measurement, and Control, 123(4):659–667, 2001.[23] B. Peeters, J. Maeck, and G. De Roeck. Vibration-based damage detection in civil engineering: excitation sources and temperature effects.

Smart Materials and Structures, 10(3):518–527, 2001.[24] E. Reynders.System identification and modal analysis in structural mechanics. PhD thesis, Department of Civil Engineering, K.U.Leuven,

2009.[25] E. Reynders and G. De Roeck. Reference-based combined deterministic-stochastic subspace identification for experimental and operational

modal analysis.Mechanical Systems and Signal Processing, 22(3):617–637, 2008.[26] E. Reynders and G. De Roeck. Continuous vibration monitoring and progressive damage testing on the Z24 bridge. In C.Boller, F.K. Chang,

and Y. Fujino, editors,Encyclopedia of Structural Health Monitoring, pages 2149–2158. John Wiley & Sons, New York, NY, 2009.[27] E. Reynders and G. De Roeck. A local flexibility method for vibration-based damage localization and quantification.Journal of Sound and

Vibration, 329(12):2367–2383, 2010.[28] E. Reynders, G. De Roeck, P.G. Bakir, and C. Sauvage. Damage identification on the Tilff bridge by vibration monitoring using optical fibre

strain sensors.ASCE Journal of Engineering Mechanics, 133(2):185–193, 2007.[29] E. Reynders, D. Degrauwe, G. De Roeck, F. Magalhaes, and E. Caetano. Combined modal testing of footbridges.ASCE Journal of Engi-

neering Mechanics, 136(6):687-696, 2010.[30] A. Teughels.Inverse modelling of civil engineering structures based onoperational modal data. PhD thesis, Department of Civil Engineering,

K.U.Leuven, 2003.[31] A. Teughels and G. De Roeck. Structural damage identification of the highway bridge Z24 by FE model updating.Journal of Sound and

Vibration, 278(3):589–610, 2004.[32] A. Teughels, J. Maeck, and G. De Roeck. Damage assessment by FE model updating using damage functions.Computers and Structures,

80(25):1869–1879, 2002.

21

finite element model updating and structural damage ...u0044091/ij-mssp-reyn... · damage...

Documents