finite element model updating for damage detection

Finite element model updating for damage detection

A. S. Kompalka ∗1 and S. Reese ∗∗1

1 Institute of Mechanics, Ruhr-University Bochum, Germany

In this contribution we derive the iterative finite element model updating algorithm. The ability of the method to detect thedamage is verified by means of a simulation with a reference finite element model.

1 Introduction

The detection of damage is still a wide field of research in civil and mechanical engineering. Many of the publications dealingwith this problem are based on a combination of an identification method to extract the characteristic information and amodelling method to simulate the characteristic behaviour of a structure. Thus, a change of this properties can be utilisedto detect the damage. In our investigations the used characteristic informations are modal parameters, eigenvalues and modeshape coordinates exclusively. This is caused by the utilised stochastic subspace system identification which is a commonmethod to extract modal data from experimental measurements. We employ an extension of this method by a state spacetransformation to obtain the undamped modal data. One method to minimise the difference between the experimental andanalytical modal data is the finite element model updating. In this paper we focus on iterative updating algorithms becausethey have the advantage to update physically meaningful model parameters which allow the definition of an ultimate limit ofstate. This is necessary for our future goal the lifetime estimation of the structure. Finally, we present the validation of theupdating algorithm on the basis of a damaged cantelever beam.

2 Finite element model updating

The aim of the updating algorithm is the minimisation of the difference between the modal data of the experiment (or areference model) and the modal data of the finite element model. This difference is defined as modal error

εk = zex − zk (1)

where the iteration index k denotes the dependency from the iteratively changing model parameters. The modal vectors of theexperiment zex and of the finite element model zk are assembled with the eigenvalues λ and the mode shape vectors φ

z =(λ1,φ

T1 , ..., λm, φT

m

). (2)

The correct pairing and scaling of the modal vectors is essential for a successful model updating and can be proven e.g. withthe Modal Assurance Criterion and the Mode Scaling Factor [1]. A Taylor expansion can approximate the modal vector

zk+1 ≈ zk + Sk (θk+1 − θk) (3)

depending of the model parameters θk+1 and θk of iteration k+1 and k. Here, the sensitivity matrix Sk represents a relativechange of a modal data by a modification of the model parameters often numerically calculated with the difference quotient.Using the Bayes estimation it is possible to derive an object function

J (θk+1) = εTk W (k)

ε εk + (θk+1 − θk)TW

(k)θ (θk+1 − θk) (4)

which includes the modal error in combination with a weighting matrix W (k)ε of the modal error and the model parameters

in combined with a weighting matrix W(k)θ of the model parameters. The minimisation of the object function (4) leads to the

iterative updating algorithm

θk+1 = θk +[ST

k W (k)ε Sk + W

(k)θ

]−1

STk W (k)

ε (zex − zk) . (5)

The shown algorithm is known in the literature as Extended Weighted Least Squares Estimation. The difficulty lies in the def-inition or calculation of the weighting matrices. The weighting matrix W (k)

ε has the task to weight the different significancesof the measured informations [1]. The weighting matrix W

(k)θ is mainly applied in case of ill-conditioned or rank deficit

problems to force the convergence of the iteration [2].

∗ e-mail: [email protected], Phone: +49 0234 32 25878, Fax: +49 0234 32 14488∗∗ e-mail: [email protected], Phone: +49 0234 32 25883, Fax: +49 0234 32 14488

PAMM · Proc. Appl. Math. Mech. 5, 505–506 (2005) / DOI 10.1002/pamm.200510229

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


3 Simulation and detection of damage

The finite element models presented here belong to the preparing investigations for our next experimental setups. The modaldata of a damaged structure are provided by a finite element model with 64 Bernoulli beam elements and has 129 degrees-of-freedom (see Fig. 1). The damage is simulated by a reduction of the cross-sectional height of 40% at two positions. The

64 beam elements129 degrees−of−freedom

Fig. 1 Damaged finite element model

updated finite element model consists of 16 Bernoulli beam elements and has 33 degrees-of-freedom (see Fig. 2). The utilised

16 beam elements116 15 14 13 12 11 10 9 2345678

33 degrees−of−freedom

Fig. 2 Updated finite element model

modal data are the first and second eigenvalue and the corresponding mode shape coordinates at the four measurement points(see Fig. 1). This means, we solve an under-determined problem with only eight informations (modal parameters, eigenvaluesand mode shape coordinates) and 16 unknowns (model parameters, cross-sectional heights). Additionally, we sum randomnumbers in size of 2% on the eigenvalues and 4% on the mode shape coordinates to validate the updating algorithm in thepresent of generated measurement noise. The percentage magnitudes are chosen by experience. In other words, the simulationincludes a certain measurement error, a small modelling error due to the deviations of the finite element models to representthe modal shapes and rounding error which can not be avoided due to the limited numbers of digits.

The results of the updating procedure is shown in Fig. 3. As we can see the damaged elements 3 and 9 converge very well.There a variations at element 11 and 15 and very small variations at element 1 and 8. A quantitative illustration of the updatedcross-sectional heights in terms of bars is displayed in Fig. 2. The experimental validation in the near future will verify thequality of the simulation and the ability of the finite element model updating algorithm.

0 500 1000

0.009

0.01

0.011

0.012

0.013

0.014

0.015

Iterations [−]

Cro

ss−

sect

iona

l hei

ght [

m]

Element 1Element 2Element 3Element 4Element 5Element 6Element 7Element 8Element 9Element 10Element 11Element 12Element 13Element 14Element 15Element 16

Fig. 3 Iteration of model parameters

Acknowledgements The presented results are the present state of research in the project B4 in the Collaborative Research Center 398“Lifetime oriented design concepts” funded by the German Science Foundation.

References

[1] M. I. Friswell, J. E. Mottershead, Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, (1995)[2] M. Link, O. Santiago, Updating and Localizing Structural Errors based on Minimisation of Equation Errors, Proceedings of the

International Conference on Spacecraft Structures and Mechanical Testing, pp. 503-510 (1991)


Section 10 506

finite element model updating for damage detection

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