2008 SLOVAK UNIVERSITY OF TECHNOLOGY26

INTRODUCTION

Vibration-based monitoring is a branch of non-destructive testing techniques, which provides a tool that can be used in the assessment of a bridge’s condition. Research in this area has continued for more than 30 years and new computer hardware opens possibilities for more demanding calculations. A large portion of the existing bridges in Europe are 30-50 years old and the issue of maintenance is becoming more important. The aim of a vibration-based bridge monitoring system is to evaluate the integrity of a bridge structure and possibly identify the time when integrity deterioration is starting. This information can be exploited in the planning of optimal maintenance measures in terms of a benefit/cost ratio. A bridge monitoring system consists of sensors (accelerometers,

strain gauges, etc.), which provide data that is digitalized and evaluated. Evaluation algorithms extract the relevant information about the structural performance and thus dramatically reduce the amount of data. The extracted characteristics of structural vibration are the basis for damage detection. The monitoring system is ideally realised in the form of a permanent installation in order to compare current and previous structural performance.Damage detection methods are split in two basic categories: one group of methods evaluates measurement data only, while the second group uses measurement data and a computational model of the structure. The former group is historically older and computationally less demanding. It uses the fact that a change in structural stiffness or mass causes a change in the structural eigenfrequencies and mode shapes. The eigenfrequencies and mode shapes from a current

M. RALBOVSKÝ

VIBRATION-BASED DAMAGE DETECTION IN CONCRETE STRUCTURES USING MODAL FORCE RESIDUALS

KEY WORDS

• Damage detection • Model updating • Modal force residual• Permanent bridge monitoring• Measurement uncertainty

ABSTRACT

The maintenance planning of bridges can be optimized using measurements of structural performance and its development over the life-cycle of a bridge. Vibration-based permanent monitoring systems aim at detecting structural integrity and to communicating a warning message if the integrity is defective. This paper presents a damage detection method based on updating a computational model. The modal force residual method and its application for the identification of stiffness distribution along a beam are discussed. A solution of the divergence problem during optimization is proposed. A laboratory test on a concrete girder proved the applicability of the method. A method for considering the influence of measurement uncertainties in damage detection is proposed and applied on a large prestressed concrete bridge.

Marián Ralbovský

arsenal research, Business Field Transport Routes EngineeringGiefinggasse 2, 1210 Vienna, Austriae-mail: marian.ralbovsky@arsenal.ac.atresearch field: structural dynamics, vibration monitor-ing, model updating

2008/4 PAGES 26 – 34 RECEIVED 18. 6. 2008 ACCEPTED 4. 11. 2008

2008 SLOVAK UNIVERSITY OF TECHNOLOGY

Ralbovsky.indd 26 10. 12. 2008 10:08:53

2008/4 PAGES 26 — 34

27VIBRATION-BASED DAMAGE DETECTION IN CONCRETE STRUCTURES USING MODAL...

measurement are compared to a reference measurement of the structure in an undamaged condition, and the changes are evaluated. This group of methods usually performs “level 1” identification, which is detecting presence of damage. Damage detection based on eigenfrequency changes was successfully tested in a full-scale test of a progressively damaged prestressed concrete bridge (Benko, et al., 2003) and on many other structures.Methods based on updating a computational model modify the model in order to minimize the deviation between the measured and calculated performance of the bridge. In this computationally demanding inverse procedure of system identification, structural damage is identified and described by its location and extent, which is “level 3” identification. The presented method belongs to this group of methods.

DAMAGE DETECTION USING MODAL FORCE RESIDUALS

Model updating is an optimization problem. The set of model parameters that will be optimized is defined by the analyst. The optimization algorithm searches an optimal set of parameter values that minimize the objective function. Various methods differ in the definition of the objective function and the optimization algorithm used. The use of a weighted least-square definition of an objective function is widespread (Göge and Link, 2001). The objective function J is defined as:

(1)

where ∆ε is the vector of the differences between measured and calculated modal parameters,

p is the vector of the structural parameters, W, Wp are the weighting matrices.

The weighting matrices are defined by the analyst according to the importance of the particular parameters. Engineering insight and, if necessary, an iterative approach are needed for the correct determination of W. The main disadvantage is that the weighting matrix is influenced by the subjective opinion of the analyst. The derivation of the weighting matrix from the uncertainty of the modal parameters (Teughels, 2003) seems to be a more objective approach. The weighting matrix W is stated as the inverse of the covariance matrix. The modal force residual method avoids problems with the definition of the weighting matrix by a different definition of the objective function. It is defined as the sum of the modal force residual vectors (Eq.2) that are the result of the forced harmonic vibrations of the model (Eq.3). The Finite-Element model is loaded with harmonic displacements that correspond to the measured modal displacements

and they act harmonically at the measured eigenfrequencies. The reaction forces at the induced displacement locations form the modal force residual vector Rr.

(2)

(3)

The contribution of the higher modes to the modal force residual is usually larger than the contribution of the lower modes because displacements with higher curvature require larger acting forces. This property is disadvantageous because the measurement of higher modes is difficult or impossible. Another disadvantage is that the small random uncertainties of mode shapes have a large influence on the objective function value. The high sensitivity of force residuals causes convergence to poor solutions. These drawbacks can be eliminated by transforming the modal force residuals into dynamic displacement residuals (Alvin, 1996). The displacement residuals RG,r are calculated as a static solution of the structure that is loaded with modal force residuals (Eq.4) and the objective function J is defined in terms of the displacement residuals.

, (4)

The optimal set of structural parameter values that minimize the objective function is sought by a sensitivity-based optimization algorithm. The robustness of the optimization algorithm determines if the global minimum of the objective function or a realistic solution will be found. Two optimization algorithms are presented in the following text.The sensitivity vector approach of optimization takes into account the shape and magnitude of the residual vector and its sensitivity and determines the optimal linear combination of residual vector changes that will minimize the objective function (Alvin, 1996). The sensitivities of the displacement residuals are calculated by the partial derivation of the residuals with particular structural parameters (Eq.5). The parameter changes are then calculated from a linear equation system (Eq.6).

(5)

(6)

The calculation of the parameter changes has to be iteratively repeated because of the nonlinearity of the sensitivity vectors.

Ralbovsky.indd 27 10. 12. 2008 10:08:55

28 VIBRATION-BASED DAMAGE DETECTION IN CONCRETE STRUCTURES USING MODAL...

2008/4 PAGES 26 — 34

The greatest slope approach of optimization takes into account only the magnitude of the residual vector and its sensitivity. The slope of the objective function is calculated by its partial derivation with the particular parameters (Eq.7). The parameter changes are proposed in the direction of the largest negative slope of the objective function (Eq.8).

(7)

(8)

An ideal optimization algorithm does not exist, and various algorithms may perform differently depending on the type of problem being solved. The purpose of this work is to identify the stiffness distribution along a beam, which would describe the damage detection problem on most bridges. Structural parameters are chosen as the bending stiffness of particular sections of the beam. The application of a sensitivity vector approach to this type of problem leads to divergent behaviour. The sensitivity vectors of parameters corresponding to adjacent sections in the beam have a similar shape. This fact causes, together with the inherent measurement uncertainties and imperfect parameter selection, the instability of the solution. The mathematical solution leads to a physically unrealistic and largely discontinuous stiffness distribution. This behaviour can be improved by imposing boundaries for realistic parameter values. The divergence can be handled this way, and the instability of the solution can be largely reduced – but not eliminated. Similar behaviour may appear within the introduced parameter boundaries. Therefore, the use of a greatest slope optimization algorithm is proposed.The greatest slope approach provides a more stable solution of the stated optimization problem. The similarity of sensitivity vectors does not state a problem because only the vector magnitude is used in the optimization. Although the convergence is slower than with the sensitivity vector approach, the convergence stability is better. Special attention has to be paid if the damaged area is located in a section with a low sensitivity of the objective function. The greatest slope approach tends to assign changes to the more sensitive parameters. In such a case it is necessary to allow a sufficient number of iterations and to let the optimization continue even if the objective function is decreasing slowly.

LABORATORY TEST ON A CONCRETE BEAM

The proposed damage detection method has been tested in a laboratory on a reinforced concrete beam. The test was performed within the Austrian national research project “AIFIT” (see

acknowledgements). The beam had 2 spans of 2.34 m each (Fig.1) and a cross section of 20x12 cm. It was loaded in the middle of the first span in 5 load steps (Tab.1) that progressively damaged the beam.The beam was unloaded after each load step and dynamic testing was performed. Dynamic excitation was provided by two moving coil actuators acting in the middle of each span. The beam was equipped with two accelerometers as reference sensors, and the vibration displacements were measured by a laservibrometer. The laservibrometer was fixed outside of the spans and the laser ray was reflected by a surface mirror to the respective measurement point on the beam. By moving the mirror on a precise linear motion system, the measurement point was moved to particular locations. 25 measurement points in each span were measured with the laservibrometer. The amplitudes measured at different points were normalized by the amplitude of the reference accelerometer sensor, and thus the mode shapes were obtained. First, a sine sweep was performed to determine the resonant frequencies. Secondly, the excitation frequency was kept constant at the resonant frequency of interest, and the vibration displacements at 50 points were measured by the laservibrometer with a resolution better than 1µm. The vibration amplitudes ranged from 8 to 300 µm.Two modes could be reliably identified. The degradation of the modal parameters through the load steps is displayed in Figs.2 and 3.

Fig. 1 Tested concrete beam

Tab. 1 Load stepsLoad step Description Applied force

0 No damage 01 First cracks 3.5 kN2 Medium reduction in stiffness 4 kN

3Complete reduction in stiffness at maximum bending moment

position5 kN

4Cracks>0.3mm, yielding of

reinforcement6.5 kN

5 Predicted collapse 13 kN

Ralbovsky.indd 28 10. 12. 2008 10:08:58

2008/4 PAGES 26 — 34

29VIBRATION-BASED DAMAGE DETECTION IN CONCRETE STRUCTURES USING MODAL...

The damage detection calculations were implemented in a software tool that was programmed for this purpose. The software created managed the calculation process; it contained the optimization algorithm and controlled the residual and sensitivity calculations that were done by the Ansys finite-element software. The beam was divided into 26 sections, the stiffness of which was updated. The parameters described the stiffness reduction factor, which means that p=1 is full stiffness and p=0 is a complete reduction in stiffness. Both presented optimization algorithms have been used to identify the induced damages. The parameter boundary p

30 VIBRATION-BASED DAMAGE DETECTION IN CONCRETE STRUCTURES USING MODAL...

2008/4 PAGES 26 — 34

Fig. 4 Crack pattern after load step 1

Fig. 5 Identified reductions in stiffness in load step 1

Fig. 6 Crack pattern after load step 2

Fig. 7 Identified reductions in stiffness in load step 2

Ralbovsky.indd 30 10. 12. 2008 10:09:09

2008/4 PAGES 26 — 34

31VIBRATION-BASED DAMAGE DETECTION IN CONCRETE STRUCTURES USING MODAL...

identification of the damaged regions in the first load case, which was also the primary goal of the damage detection test.

MEASUREMENT UNCERTAINTIES

Unlike in laboratory conditions, the vibration measurements on real bridges suffer from a larger variance of the modal parameters. Some of the sources of the measurement uncertainties can be eliminated with correct installation of the equipment, while others are an inherent part of the structure in its changing environmental conditions and cannot be eliminated (Van der Auweraer, et al., 2005). The evaluation of modal parameter variance is important for the reliability of the measurement results and can be calculated, for example, with Monte Carlo techniques (Farrar, et al., 1998). The uncertainties of the modal parameters propagate into the damage detection and influence the results by an added uncertainty (Deix and Ralbovsky, 2005). The uncertainty of the results can be described with a Probability Density Function (PDF) of the particular resulting parameters. In order to obtain such a result it would be necessary to perform a Monte Carlo type of simulation with a damage detection algorithm. Since a damage detection algorithm is by itself already a computationally demanding procedure, performing a Monte Carlo simulation would lead to very large computation times even with the use of effective sampling techniques like Latin Hypercube Sampling.This work proposes an alternative approach to deal with uncertainties that would satisfy the primary requirement of avoiding false positive damage detection. This requirement results from the needs

of a permanent monitoring installation with an automated damage detection routine. False positive detections would repeatedly produce unwanted alarms and confuse the bridge management authority. The proposed approach can be summarized in the following steps:1. Evaluation of the PDF of the modal parameters and definition of confidence intervals2. Searching a modal parameter set within the defined confidence intervals that minimizes the objective function3. Performing standard damage detection using the modal parameter set from the previous stepThe PDF of the modal parameters can be calculated from the repeated measurements by an evaluation of the mean vector and covariance matrix of the modal parameters. The PDF can then be approximated, for example, with a multinormal probability distribution (Eq.9).

(9)

where ψ is a realization of the modal parameters, ψm is the mean of the modal parameters, σxy is the covariance matrix and det σxy its determinant, n is the length of vector ψ.

Searching a modal parameter set that minimizes the objective function can be accomplished with the same updating procedure of damage detection, but instead of structural parameters, the modal parameters are used. The defined confidence intervals then state the allowed parameter boundaries. This procedure ensures that changes

Fig. 8 “Reichsbrücke” bridge over the river Danube, Vienna, Austria

Ralbovsky.indd 31 10. 12. 2008 10:09:13

32 VIBRATION-BASED DAMAGE DETECTION IN CONCRETE STRUCTURES USING MODAL...

2008/4 PAGES 26 — 34

which could be produced by the random character of the modal parameters will not be identified as structural damage in the final step.

VIBRATION TEST ON THE “REICHSBRÜCKE” BRIDGE

The measurement uncertainties were studied on the “Reichsbrücke” bridge (Fig.8), which is located in Vienna, Austria. It is a prestressed concrete bridge that crosses the river Danube and carries 6 road lanes and 2 underground train lines. It has 5 spans with lengths from 60 to 170 m.The vibration measurements were performed under ambient excitation. Acceleration sensors were placed at 32 measurement points in the three largest spans. The sensor positions were in the maintenance corridor in the middle of the cross section. Highly sensitive sensors of Wilcoxon Research 731 type with a nominal sensitivity of 10 V/g were used. The measurement was performed in 9 set-ups with 5 biaxial sensors. The measurement time was 30-50 minutes in each set-up.The modal parameters were extracted with the MACEC software, which uses the stochastic subspace identification method (Peeters, 2000). Six modes could be identified. Table 2 lists the identified frequencies together with their variances.The variance of the modal parameters is higher for the higher modes, as can be observed from Tab. 2 and Fig. 9. Mode No. 5 was a horizontal mode; all the others are vertical bending modes.A finite-element beam model of the bridge was constructed and used for further simulations. A comparison of the calculated and measured frequencies of the vertical bending modes is listed in Tab.3. Mode shapes 1 and 3 are compared in Fig.10.The bridge is currently in good condition, so there were no

damages that could be detected. The damaged states of the bridge had to be simulated on the FE-model by reduction in stiffness at selected locations. The modal parameters of the damaged FE-model were proportionally transferred onto the measured modal parameters by the ratio of damaged and undamaged values from the FE-calculation. The damage detection was performed using these modified measured modal parameters. The measurement uncertainties were treated according to the method proposed in the previous chapter.

Tab. 2 Identified mean frequencies, their standard deviations and 90% and 99% confidence intervals (CI)

ModeFrequency

[Hz]σ [Hz] =

68.27% CI90% CI

[Hz]99% CI

[Hz]1 0.8533 ± 0.0053 ± 0.0088 ± 0.01372 1.3222 ± 0.0067 ± 0.0110 ± 0.01723 2.1716 ± 0.0125 ± 0.0205 ± 0.03224 2.7553 ± 0.0279 ± 0.0459 ± 0.07195 3.0280 ± 0.0145 ± 0.0238 ± 0.03736 3.1964 ± 0.0711 ± 0.1169 ± 0.1831

Fig. 9 Measured mode shapes 1 and 3 with their 90% confidence intervals

Tab. 3 Comparison of eigenfrequencies between measurement and reference FE-model

Mode Measured [Hz] Reference FE-model [Hz]1 0.8533 0.95452 1.3222 1.28083 2.1716 2.04274 2.7553 2.65196 3.1964 3.0940

Ralbovsky.indd 32 10. 12. 2008 10:09:16

2008/4 PAGES 26 — 34

33VIBRATION-BASED DAMAGE DETECTION IN CONCRETE STRUCTURES USING MODAL...

The stiffness distribution was modeled as a piecewise-linear function with 27 parameters. Two parameters at the right side of the second support were reduced in order to simulate the damage. Three levels of damage extent are presented: a reduction of 25%, 50% and 75% of the original stiffness. Fig. 11 summarizes the results of the damage identification. In the case of the 25% stiffness reduction, the damaged detection identified hardly any reduction of the stiffness, which means that the change of the modal parameters produced by this damage was largely within the confidence intervals of the measurement. This extent of damage cannot be reliably identified under consideration of the measurement uncertainties. Damage detection in the cases of 50% and 75% reduction in stiffness identified the damage at the location with reduced stiffness, but also showed a reduction of stiffness in other locations. This unwanted effect is due to the present differences between the measurement results and modal parameters of the reference FE-model. The second effect is the underestimation of the identified extent of damage in comparison to the introduced extent of damage. This is caused by selecting a modal parameter set that minimizes the objective function, which was done in order to exclude false positive damage detection.

CONCLUSION

The damage detection was successful in identifying the reduction in stiffness on the laboratory beam; the good results in the first load cases especially confirm that the method is suitable for the detection of cracks at an early stage. The reduction in stiffness after appearance of the first cracks was reduced by more than 50% in the affected region. The greatest slope approach of optimization provided more stable resulting parameters than the original sensitivity vector

Fig. 10 Comparison of measured and calculated mode shapes 1 and 3

Fig. 11 Damage identification results in 3 simulated damage cases

Ralbovsky.indd 33 10. 12. 2008 10:09:20

34 VIBRATION-BASED DAMAGE DETECTION IN CONCRETE STRUCTURES USING MODAL...

2008/4 PAGES 26 — 34

approach. The results from the sensitivity vector approach showed acceptable results after introducing the parameter boundaries.The uncertainty evaluation of the measurement results from the test on the “Reichsbrücke” bridge showed that the variance of the experimental modal parameters is not negligible. The higher modes had especially large deviations. The presence of the modal parameter uncertainties hinders reliable damage detection at low extents of damage. A simulated damage of 25% reduction in stiffness at a given location could not be reliably identified because the produced change of the modal parameters was largely within the confidence interval of the measurement. A reduction in stiffness of 50% – similar to the stage of the first cracks on the tested RC beam – could be identified in spite of the present measurement uncertainties, but the identified damage location was not unique.The proposed method of considering measurement uncertainties in the process of damage detection can be realized with reasonable

computational times. Meeting the primary goal of avoiding false positive damage detections has the side effect that the identified damage is of an underestimated extent.

ACKNOWLEDGEMENTS

The test on the “Reichsbrücke” bridge was done by arsenal research within the national research project “Strassenforschung – Temperaturkompensationsmodell für Bauwerksmonitoring” supported by BMVIT (Austrian Federal Ministry of Transport, Innovation and Technology).The test on the concrete beam was done under the leadership of the Department of Civil Engineering and Natural Hazards at BOKU (University of Natural Resources and Applied Life Sciences in Vienna) within the ”AIFIT“ national research project.

REFERENCES

• Alvin, K.F. (1996) Finite Element Model Update via Bayesian Estimation and Minimization of Dynamic Residuals. 14th International Modal Analysis Conference, February 12-15, 1996 Dearborn, USA, pp. 561-567

• Benko, V. – Geier, R. – Ralbovsky, M. (2003) Dynamische Untersuchung einer Segmentbrücke (Dynamic investigation of a segmental bridge). D-A-CH Tagung 2003 „Aktuelle Probleme der Brückendynamik“, Schweizerischer Ingenieur- und Architektenverein, Zürich, ISBN 3-908483-74-3, pp. 21-26

• Deix, S. – Ralbovsky, M. (2005) Uncertainties in Model Updating - Problems and Reliability. Managing Uncertainties in Noise Measurements and Prediction Conference, Le Mans, France, 2005

• Farrar, C.F. – Doebling, S.W. – Cornwell, P.J. (1998) A Comparison Study of Modal Parameter Confidence Intervals Computed Using the Monte Carlo and Bootstrap Techniques. 16th International Modal Analysis Conference, February 1998, Santa Barbara, CA, USA, pp. 936-944

• Göge, D. – Link, M. (2001) Parametric updating of finite element models by minimizing response residuals at resonances. International Conference on Structural System Identification, September 5-7, 2001 Kassel, Germany, pp. 419-430

• Peeters, B. (2000) System identification and damage detection in civil engineering. PhD thesis, Katholieke Universiteit Leuven, Belgium, pp. 105-115, ISBN 90-5682-274-8.

• Teughels, A. (2003) Inverse modelling of civil engineering structures based on operational modal data. PhD thesis, Katholieke Universiteit Leuven, Belgium, pp. 107-108, ISBN 90-5682-444-9.

• Van der Auweraer, H. – Donders, S. – Peeters, B. (2005) Importance of Uncertainty in Identifying and Using Modal Models. Managing Uncertainties in Noise Measurements and Prediction Conference, Le Mans, France, 2005

Ralbovsky.indd 34 10. 12. 2008 10:09:24