The Finite Element Model Updating: A Powerful Tool

The Finite Element Model Updating: A Powerful Tool for Structural Health Monitoring Andrea Mordini; Konstantin Savov; Helmut Wenzel; VCE–Vienna Consulting Engineers, Vienna, Austria

eral different approaches can be found in the technical literature.

Summary The request of non-destructive methods for evaluation of the structural reliability is nowadays constantly increasing. The VCUPDATE solution for finite element model updating is presented. VCUPDATE is a Scilab code performing an iterative updating algorithm interfaced with a finite element code (OpenSees or ANSYS) executing the numerical analysis. The code has been verified by many different applications on beams and stay cables. In this contribution, the code is applied to two real cases using OpenSees as well as ANSYS. Keywords: damage detection; finite element model updating; optimization algorithms; structural health monitoring.

Introduction Structural health monitoring (SHM) is nowadays a field of great concern. The request for damage detection by means of non-destructive methods is greatly increasing in order to reduce the maintenance costs and to increase the safety level of the structures. To this purpose, the use of numerous experimental and numerical techniques is required. Dynamic testing of structures is nowadays a well-established methodology for SHM, improving the data reliability and overcoming the limitations of traditional visual inspection methods. The obtained structural response can be used in conjunction with several different numerical methods, which can be grouped into two main categories: model-based and parameter-based. The first relies on a reference structural model, e.g. finite element (FE) model, whereas the main characteristic of the second is a general mathematical description of specified parameters or system features, e.g. analysis of time series details by means of wavelet theory. This contribution is focused on the finite element model updating (FEMU) – a model-based approach – applied on the SHM of civil engineering strucPeer-reviewed by international experts and accepted for publication by SEI Editorial Board Paper received: January 10, 2007 Paper accepted: August 9, 2007

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tures. The basic idea is to use the recorded structural response to update some selected structural parameters of the numerical model (such as stiffness and internal forces) as well as some boundary conditions (such as translational or rotational springs), until an adequate agreement between numerical and experimental results is achieved. Since the damage is an intentional or unintentional change to the material and geometric properties of a given system which affects the system performance,1 the parameter distributions obtained as outcomes can provide useful information about the possible structural damage. In this effort, the FEMU is a very powerful tool for SHM. The updating procedure is based on a mathematical optimization problem: the difference between numerical and experimental eigenfrequencies and mode shapes should be minimized through iterative modal analysis. The practical experience of the authors has shown that the vibration response represented by accelerations provides suitable and sensitive structural information to this purpose. The recorded acceleration time series should be processed, that is, a transformation from the time-domain to the frequencydomain is required. The general problem of extracting eigenfrequencies and corresponding mode shapes regarding the effects of noise, excitation sources and number of instrumented points are nowadays widely discussed in the civil engineering community and sev-

From the theoretical point of view, a modal analysis depends on two categories of structural parameters: mass and stiffness. In the presented work, the structural mass is assumed to be constant while some selected parameters related to the stiffness are taken as updating parameters. In this way, many model parameters of interest for the civil engineers can be evaluated: degradation of the elastic modulus, changes in the boundary conditions and in the structural forces (e.g. geometrical stiffness). Consequently, the presence of weakness (such as cracks, plastic hinges, corrosion) in the civil structures can be detected. The FE method is nowadays a very powerful computer aided simulation technique, allowing the user to study any kind of problem virtually without limitation in model size and complexity. It has to be considered, however, that the solution of an optimization problem by iterative procedures often requires a large number of calculation steps. Moreover, the practical engineers and the owner of the structure are interested in answering the basic questions: “Is there any damage and if yes, where is it located?”. The damage extent and the remaining load-carrying capacity of the structure (levels 3 and 4 in the damage detection procedure as reported by2) are of course important, but currently not essential from the practical point of view. Considering these aspects, the use of simplified models and procedures is well suitable within the FEMU in order to correctly describe the structural behaviour and to effectively detect and locate the damage without requiring excessive computational resources. The authors discuss in this work the application of FEMU to damage detection in beam and stay cable structures from the practical point of view. Both eigenfrequencies and mode shapes are used as input data for the updating procedure. The eigenfrequencies allow

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damage detection, whereas the mode shapes are necessary to evaluate the damage distribution. For stay cables, the estimated eigenfrequencies are sufficient to detect the loss of prestressing force and therefore, only one sensor is needed to record the structural response. Several stay cable structures have been already investigated by the authors.3 A calculation technique has been implemented in VCUPDATE consisting in an iterative updating algorithm developed in Scilab4 and interfaced with the FE codes OpenSees5 and ANSYS.6 The use of two different FE codes provides VCUPDATE with a wider flexibility. In fact, the commercial code ANSYS has very powerful modeling capabilities and allows the graphical visualization of the model and of the results, including the outcomes from the updating procedure. On the other hand, the updating procedure using OpenSees is much faster and therefore, when possible (i.e. when the model is simple), the use of OpenSees is preferred. In addiction, it has to be considered that while many different commercial FEMU solutions are available on the market at high licensing prices, both Scilab and OpenSees are open source and free of charge software. This difference can improve the accessibility of the FEMU to a wide range of engineering companies which are interested in the method but cannot afford the costs of a commercial software solution (optimization + FE code).

Theoretical Basics The FEMU can be performed using direct or iterative methods.7 The former has the advantages of not requiring iteration and of reproducing measured data exactly. However, if the measured data are inaccurate, an FE model with no physical meaning could be obtained. On the contrary, within iterative methods, a non-linear penalty function is minimized through subsequent linear steps and therefore, more computational time is required. In this work, the authors present a sensitivitybased iterative method using frequencies and mode shapes as experimental data. In Fig. 1 the main steps of the code are shown. Sensitivity Analysis The sensitivity analysis is performed to select the most sensitive parameters for the FE model. Moreover, the sensitivity matrix is used also in

Scilab (updating code) START

1 FE run (with starting parameters)

Computing the sensitivity matrix Experimental data

yes

n FE runs (with modified parameters)

Updating parameters

1 FE run (with updated parameters)

Convergence ? no

Computing the sensitivity matrix

n FE runs (with modified parameters)

End

Fig. 1 : VCUPDATE flow chart

the updating algorithm. At the jth iteration, the sensitivity matrix can be written as Sj =

∂d j ∂pj

(1)

,

where ∂pj is the perturbation in the parameters and ∂d j is the difference between experimental and numerical data. The matrix S can be computed by analytical methods (direct derivation) as well as by numerical methods (perturbation techniques). If direct derivation is chosen, the differentiation of the structural eigenvalue problem with respect to the parameters must be performed, where the system stiffness and mass matrices are required. On the contrary, perturbation techniques can be carried out using results from multiple FE analyses only, without knowledge of the system matrices. In this work, the perturbation approach is chosen: the sensitivity matrix is computed by using the forward difference of the function with respect to each parameter. In particular, if the problem has n parameters, n + 1 FE runs are required: the first one with the starting values for the parameters and n different runs perturbing one parameter only in each of them. The forward difference method has the form8 ∂d ∆d d ( p + ∆p) − d ( p) ≈ = ; ∆p ∂p ∆p ∆D ∆pi = p − pi 100 i

(

)

(

)

ence step size and pi − pi is the difference between the upper and lower bound for the ith parameter. Each vector ∆d ∆p gives one column of the sensitivity matrix. Sensitivity Matrix and Mode Shape Scaling Using different structural properties with very different values as parameters may lead to numerical problems in the iterative solution.9 This problem arises from the ill-conditioning of the sensitivity matrix and is more probable when both frequencies and mode shapes are used as data. An appropriate scaling of the sensitivity matrix can improve the stability and speed up the convergence. In this work, according to,9 the sensitivity matrix is scaled by normalizing parameters and data by their initial values. This method can lead to numerical problems when the mode shape values are very small or zero. In this case a different way to scale the sensitivity matrix has to be used. Experimental and numerical mode shapes should be scaled consistently to properly perform the updating algorithm. Moreover, experimental and numerical mode shapes can be 180° out of phase. To solve these problems, within VCUPDATE, the modal scale factor (MSF) is used according to7 and.10 Updating Algorithm

S=

(2)

where ∆pi is the perturbation in the ith parameter, ∆D is the forward differ-


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OpenSees / ANSYS (FE code)

The updating problem can be represented as the minimization of a penalty function J ( ∆p) subjected to the constraint ∆d = S ∆p.7 Minimizing J ( ∆p) with respect to ∆p and inserting the weighting matrices Wd and Wp Reports

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for data and parameters respectively, the updated parameter values can be written as −1

pj +1 = pj + ⎡⎣ STj Wd S j + Wp ⎤⎦ STj Wd ∆d j , (3) (at jth iteration). The sensitivity matrix should be computed at each iteration, but this can lead to an excessive computational effort. VCUPDATE allows the user to choose the frequency with which the sensitivity matrix has to be updated. For some extremely time consuming cases, the initial matrix can be used for the entire analysis. According to experience, not all the experimental data are measured with the same accuracy. Usually lower frequency data are more reliable than higher ones. In order to express the user confidence in measured data, weighting matrices are used. Parameters can also be weighted separately. Using weighting matrices is very powerful, but engineering insight is required. Five different ways to create the Wp matrix according to9 are implemented in VCUPDATE. Several different tests were performed in order to verify the methods: in particular some of them should be used carefully since they provide, in some cases, an increased convergence speed but, in other cases, the obtained results are worse. Convergence Criteria The iterative scheme presented above is repeated until a convergence criterion is satisfied. The criterion can be based on parameters or on data. Two different convergence criteria are implemented within VCUPDATE according to.11 The first is based on the frequency deviation CCabs, j =

1 n fexp , i − f j, i ∑ f n i =1 exp , i

(at jth iteration),

(4)

where fexp , i is the ith experimental frequency and f j, i is the ith numerical frequency at jth iteration. A different way to check the convergence includes also the correlation between experimental and numerical mode shapes. The most important indices for this purpose are the modal assurance criterion (MAC)10 and the normalized modal difference (NMD)12 which, for the two mode shapes ϕ j and ϕ k , are 354

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(

) (ϕ

(

)

MAC ϕ j ,ϕ k =

NMD ϕ j ,ϕ k =

2

ϕ Tj ϕ k T k

(

ϕ k ) ϕ Tj ϕ j

(

)

and

1 − MAC ϕ j ,ϕ k

(

MAC ϕ j ,ϕ k

)

) (5)

respectively. The closer the MAC is to one, the better the correlation is. Since the NMD is much more sensitive than the MAC to the differences in similar vectors, it can be used when the mode shapes are highly correlated, while it is less useful for uncorrelated vectors. The closer the NMD is to zero, the better the correlation is. The second convergence criterion implemented in VCUPDATE includes the MAC index: 1 = 2 nWmax

CCtot, j

⎛ fexp , i − f j, i ⎞ ⎜ Wf ⎟ fexp , i ∑⎜ ⎟ i =1 ⎜⎝ +W (1 − MAC )⎟⎠ Φ i n

(at jth iteration),

(6)

where Wf and WΦ are the weights for the frequency and MAC deviation respectively and Wmax = max Wf , WΦ . The closer the selected CC approaches to zero, the better the agreement is between experimental and numerical data. The convergence is achieved if CC ≤ ε (point A in Fig. 2). During the analysis, all the information (updated data, updating parameters, CC) is stored and if the convergence is not attained, the updating procedure stops when a maximum number of iterations is reached. Then, the information corresponding to the iteration with the minimum value of CC (point B) is used as output.

)

(

Application on Beams – A Prestressed Reinforced Concrete Beam

by VCUPDATE. In general, the damage in prestressed elements is difficult to be detected since, after the load is removed, the cracks tend to close and this leads to relatively small changes in modal data. The experimental data as well as the test description are taken from a previous study.13 The results obtained by VCUPDATE are compared, in terms of damage distribution, with those obtained in that study. The investigated beam is 17,60 m long and 0,8 m high. Along the member, the cross section has an I-shape while, at the ends, it has a solid square section (Fig. 3). The beam is prestressed by an inclined post-tensioned cable. The beam was artificially damaged by increasing static load levels in the configuration shown in Fig. 4(a). The external load Q is applied in six different steps with an increasing amplitude of 45, 67,5, 125, 140, 150 and 154 kN respectively. The first visible crack on the beam corresponds to Q = 67,5 kN while the reinforcement yielding is achieved for Q = 150 kN. After each step, the beam was dynamically tested in a free-free configuration (Fig. 4(b)). The structure is excited by the impact of a falling weight at its first end. During the dynamic test procedures, 74 sensors were placed on the structure, at both longitudinal upper sides of the beam with a distance of 0,5 m between them. In most load steps the (a)

0,8

0,8

(b)

0,8 0,12 0,2

A prestressed reinforced concrete beam tested in the laboratory is investigated 0,8

CCtot /CCabs

0,2 0,18

Maximum iteration

B

0,12

0,5

Convergence limit e A Iterations

Fig. 2: Convergence criterion

Fig. 3: Beam cross sections: square (a) and I-shape (b) (Units: m)


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2,12

4,30

3,96

4,30

Q

2,12

0,40

Q

(a) 17,60

The effect of the updating algorithm on the mode shapes can be clearly seen also in Fig. 5 where the outcomes of the second numerical process are shown. It can be noted that, starting from the undamaged state, the mode shapes are transformed to the damaged state which is closer to the experimental results.

impact (b)

Fig. 4: Static (a) and dynamic (b) test configuration (Units: m)

From the experimental data, it can be observed that, before the steel yielding, the changes in frequencies and mode shapes are very small. This is probably because, after the load removal, the cracks tend to close and they produce a slight effect on the dynamic behaviour. It can be observed, however, that the damage detection procedures are more important when the structure is subjected to an irrecoverable excitation. The beam is modeled in OpenSees by means of a simple Beam model. 36 uniaxial beam elements with five integration points and three degrees of freedom (DOF) for each node are used. In order to simulate the freefree boundary conditions, two springs with very low stiffness are placed at the supports. The reinforced concrete is modeled as homogeneous and isotropic. The section properties are: area A = 0,301 m2 and 0,64 m2, moment of inertia J = 0,02246 m4 and 0,03413 m4 for the I- and the square shape respectively. Adopting a mass per unit volume M = 2500 kg/m3, a mass per unit length m = 752,5 kg/m and 1600 kg/m is obtained for the two sections respectively. The value of the elastic modulus for each element is used as updating parameter with a starting value of E0 = 37500 MPa. Therefore, a total number of 36 parameters is used. Two different updating procedures are performed: with the first one, the initial FE model is updated to the undamaged state. Then, this undamaged model is updated to the damaged state corresponding to the post reinforcement yielding step (Q = 154 kN). The first four frequencies and flexural mode

shapes are used in the second updating process while, due to the poor quality, the fourth mode shape is not used in the first updating process. The outcomes from the numerical procedures in terms of frequencies and mode shapes correlation are reported in Table 1. Since the mode shapes are highly correlated, the NMD is used instead of the MAC. The effect of the updating procedures is smaller for the first step. This means that the initial model fits the undamaged one quite well, both in terms of frequencies and mode shapes. In the second step, where the beam is damaged, the effect of the updating procedures is much higher as can be seen by comparing the CCabs Mode shape 1

Mode shape 2 1 0,8 0,6 0,4 0,2 0 −0,2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 −0,4 −0,6 −0,8 −1 Axial coordinate (m)

Mode shape 3

Mode shape 4 1 0,8 0,6 0,4 0,2 0 −0,2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 −0,4 −0,6 −0,8 −1 Axial coordinate (m)

1 0,8 0,6 0,4 0,2 0 −0,2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 −0,4 −0,6 −0,8 −1 Axial coordinate (m)

Fig. 5: Experimental and numerical mode shapes for undamaged-to-damaged step (a) 1,4

(b) 1,4

1,2

1,2

1,0

1,0

0,8 0,6 0,4

Previous study VCUPDATE

Previous study VCUPDATE

0,8 0,6 0,4 0,2

0,2

0

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Axial coordinate (m)

Axial coordinate (m)

Fig. 6: Damage distribution for initial-to-undamaged step (a) and undamaged-to-damaged step (b)


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The damage distribution is reported in Fig. 6 in terms of the ratio between the elastic modulus values before and after the updating process. In the same figure, the VCUPDATE outcomes are compared with the distribution taken from the previous study.13 It should be noted that even if the static load is symmetrical, the corresponding damage distribution is not symmetrical. In this case, the mode shapes are fundamental for the damage location. From the undamaged-to-damaged step graph in Fig. 6(b) four damaged zones can be recognized: two near the beam center with a higher damage and two closer to the beam ends with a lower damage level. The position of these points

1 0,8 0,6 0,4 0,2 0 −0,2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 −0,4 −0,6 Experimental damaged −0,8 VCUPDATE initial −1 Axial coordinate (m) VCUPDATE updated

Eund /E0

dynamic behaviour of the beam was found to be pure flexural with small torsional components. Therefore, the values from two sensors placed at the same longitudinal distance are averaged in order to obtain 37 values only. In this way, the torsional components of the mode shapes are ignored.

values. The biggest frequency deviation is recorded in the first mode shape, probably due to the presence of a slight torsional component in the experimental data.

Edam/ Eund

0,40

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N. of mode

Experimental frequency f0 (Hz)

Before update

After update

fi − f0 i

Frequency f (Hz)

NMD

fi − f0 i

Frequency f (Hz)

f0 i

NMD

f0 i (%)

(%) Initial-to-undamaged step 1

10,32

10,54

2,11

0,049

10,29

–0,25

0,049

2

29,49

29,91

1,41

0,032

29,49

0,00

0,038

3

57,77

59,84

3,59

0,039

57,77

0,01

0,025

CCabs

0,0009

0,0237

CCabs

Undamaged-to-damaged step 1

6,83

10,29

50,71

0,082

6,55

–4,08

0,027

2

22,66

29,49

30,14

0,185

22,63

–0,12

0,056

3

48,45

57,77

19,25

0,209

48,44

–0,02

0,179

4

70,99

100,48

41,55

0,395

71,03

0,05

0,120

CCabs

0,3541

CCabs

0,0107

Table 1: Updating procedure results for the OpenSees analysis

corresponds to the loading points in the static test. The two main plastic hinges are situated close to the beam center where the reinforcements yielded. Between the loading points, no damage is revealed, probably because between the plastic hinges the steel does not yield and therefore, the cracks tend again to close after the load removal. The same structure is investigated with VCUPDATE by means of an ANSYS model. A single step analysis is considered: the starting model is directly updated to the damaged beam. This is, in fact, the standard way to assess the existing structures when no test data of the initial state are available. This version of the code provides the user with an increased modeling power. In fact, the FE mesh can be freely created and there is no necessity to place a node of the FE model in the same location of a sensor. In this way, the best FE model can be chosen without taking care of the sensor position. The code requires the position and the modal displacements for each sensor. Subsequently, the numerical mode shapes are computed in the sensor positions as a function of the mode shapes of the entire FE model. Both a Beam and a Plane model are used. The first model contains the BEAM3 element: a uniaxial element with three DOF for each of its two nodes. Furthermore, the Plane model is based on the PLANE42 2D element, which has four nodes with two DOF respectively. In Fig. 7 the analysis outcomes in terms of elastic modulus reduction 356

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(a)

(b)

0

4170

8330

12500

16700

20800

25000

29200

33300 37500 MPa

Fig. 7 : Elastic modulus distribution for Beam model with element size 0,4 (a) and 0,1 m (b)

(a)

(b)

0

4170

8330

12500

16700

20800

25000

29200

33300

37500 MPa

Fig. 8: Mesh and elastic modulus distribution for Plane model with element size 0,4 (a) and 0,2 m (b)

for the Beam model with two different meshes are shown. Analogously, for the Plane model, the results are shown in Fig. 8 for an element size of 0,4 and 0,2 m. The damage distribution of these figures can be compared with the one from the OpenSees analysis (Fig. 6). It can be noted that the agreement is very high. The results in terms of frequencies, NMD values and mode shapes are very satisfactory too, but they cannot be extensively reported here due to lack of space. Several different meshes are used in the calculations for both the Beam and the Plane analysis. In general, it can be noted that the outcomes are not dependent on the used mesh as long as the mesh can effectively describe the structural response.

Application on Cables – Lanaye Bridge Cables The Lanaye bridge is a cable stayed bridge situated in Belgium, between the cities of Liege and Maastricht (Fig. 9). The bridge has a single pylon and a 177 m long main span, which is balanced by a counterweight abutment anchored to the ground. There are 20 fan-shaped cables connected to the main span with internal forces ranging between 2750 and 5400 kN. Ten cables are located on the counterweight side with an internal force of about 8800 kN. In the following, the two sides of the bridge are referred to by the name of the city they face. The cable pairs are numbered from 1 to 15 starting from the side furthest away from the pylon.


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Fig. 9: Lanaye bridge

The bridge has been investigated several times by monitoring tests, since it has some damaged cables, which qualify the bridge for studies and verifications. For these reasons, this structure was inserted in the Integrated Monitoring and Assessment of Cables (IMAC) project under the Fifth European Framework Program.14 Within this research program, an extensive experimental program was performed on all the bridge cables measuring the dynamical behaviour in terms of eigenfrequencies. Three-dimensional sensors recording the acceleration were used. Subsequently, by employing the Peak Picking method, the eigenfrequencies of the cables were evaluated. Three cables are investigated by VCUPDATE: cable M1 is known to be damaged by corrosion and it was surrounded by a security structure to catch it if it collapses (Fig. 10). After a visual inspection of cable L1, a damaged support system was detected. Therefore, changes in the dynamical behaviour were expected. Actually, in the test results, no significant changes could be observed. This means that the cable is not damaged and can carry its Cable

full load. Cable L2 is undamaged and presents a standard behaviour. The cables are modeled with 60 beam elements which can be properly used to model stay cables3 (Fig. 11). To eliminate the effects of sag and self weight, the out-of-plane frequencies are used. Different analyses are carried out using the first, the first five and the first ten experimental frequencies. The axial force is used as updating parameter with a value of 5 × 10–4 for the convergence limit. The computation of the moment of inertia of a cable is not a straightforward issue. In this work, the bending stiffness is obtained from a different software based on the measured frequencies. The comparison with this code is described in.3 The code provides, as a result, the axial force and the bending stiffness. Since cable M1 is damaged, two different values of the moment of inertia are obtained for cables M1 and L1 as reported in Table 2. This is an unphysical assumption since the two cables are equal. In order to evaluate the effect of this imprecision, a preliminary sensitivity analysis was performed and,

Fig. 10: Damage on cable M1 N L

Fig. 11: Cable model with hinged ends

in the investigated cases, a very small relevance was found for the bending stiffness parameters (Fig. 12). This means that the unphysical assumption has little influence on the results. The damage in cable M1 is clearly detected by a strong axial force reduction (Table 2). The cable L1 presents a slight increment in axial force that could be due to the loss in cable M1 and to the consequent stress redistribution, since M1 is the corresponding cable to L1 on the other side of the bridge. The increment in axial force is a signal of the integrity of the cable. Cable L2 is also undamaged as it has the axial force almost equal to the design force. These obtained results are compared, in terms of axial force, with those from the already mentioned software based on the standard method of estimating

Cable properties Length (m)

Mass per unit length (kg/m)

Moment of inertia (mm4)

Analysis results Design force ND (kN)

NVCUPDATE (kN)

NVCUPDATE – N D ND (%)

Maastricht 1 (M1)

165,07

43,80

3 927 262

2668

1944

–27,15

Liege 1 (L1)

165,07

43,80

2 000 850

2668

2826

5,93

Liege 2 (L2)

150,44

78,20

7 125 824

5042

5079

0,74

Table 2 : Cable properties and analysis results using ten frequencies as experimental data


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Cable M1

0,7 0,6 0,580

Cable L1

0,7

0,574

0,6

0,6

0,5

0,5 0,483

0,4

0,4

0,4

0,3

0,3

0,3

0,2

0,2

0,2

0,1 0

0,479

0,5

0,1 0,000

Mode 1

0,006

Mode 10

0

detect the damage through a reduction of the axial force.

Cable L2

0,7

0,496

The results of the presented applications were very satisfactory and were discussed in detail for each case. Since the investigated cases are taken from real structures, it can be concluded that this approach can be a valuable tool for SHM.

0,488

0,1 0,000

Mode 1

0,003

0

Mode 8

0,000

Mode 1

0,006

Mode 10

References

Axial force sensitivity

[1] Inman DJ, Farrar CR (eds). Damage Prognosis. John Wiley and Sons: Chichester, England, 2005.

Elastic modulus and moment of inertia sensitivity

Fig. 12: Lanaye bridge cable structural sensitivity

Conclusions In this work, the application of FEMU to the SHM of civil engineering structures is presented. The updating procedures were implemented in the software VCUPDATE, a Scilab code interfaced with the FE codes OpenSees and ANSYS. The first application was the study of a prestressed reinforced concrete beam tested in the laboratory by combining

Cable L1

8

1000

0

10 5

1 5 10 1 5 10 Used frequencies

0

20

20

19

15 2000 10 1000

0

6000

25

4


5

25 20

Axial force N

15

3000 Axial force N

-26,42% -27,25% -27,15%

Axial force N

2000

20 17

Convergence iterations

21

3000

Cable L2

4000

25

5,73% 5,68% 5,93%

4000


Cable M1

Afterwards, by using eigenfrequencies only, three stay cables from the Lanaye bridge in Belgium were investigated. One of the cables presented a strong damage which could be detected by a considerable reduction in the axial force. Moreover, in order to evaluate the influence of the unknown structural properties, a sensitivity analysis was performed, finding out that, in the investigated cases, the effect of the bending stiffness is negligible. This outcome is very important in the practical engineering applications since the cable properties are in general not well known. By investigating two equal cables (damaged and undamaged), it was demonstrated that, even with a low confidence in the knowledge of the structural parameters, it is possible to

0

4000 15 11

10 2000

6 3

0


Updated axial force Number of iterations to convergence

Fig. 13: Lanaye bridge cable analysis results as a function of the number of used frequencies 358

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5 0


In addition, from Fig. 13, it must be noted that the results are not influenced by the number of used frequencies i.e. using the first, the first five or the first ten frequencies. The percentage indicates the difference to the design force. This is a sign of a good measurement quality in the field of higher frequencies also. In the same figure, the number of iterations necessary to obtain the convergence is reported: as a general trend, the updating procedure is faster using more experimental data.

eigenfrequencies and mode shapes. In this case, even if the structure configuration was symmetrical, it was found that the damage distribution was not symmetrical and therefore, the use of mode shapes is fundamental to localize the damage. The experimental structural behaviour could be captured with very high accuracy by using different FE codes, different model dimensionality and different meshes.

0,90% 0,42% 0,74%

cable force from frequencies. The agreement between the two codes is very high, leading to a maximum discrepancy of 2,6%.

[2] Rytter A. Vibration based inspection of civil engineering structures, PhD thesis, Department of Building Technology and Structural Engineering, Aalborg University: Denmark, 1993. [3] Mordini A, Savov K, Wenzel H. Damage Detection on Stay Cables using an Open SourceBased Framework for Finite Element Model Updating, accepted for publication in Structural Health Monitoring, 2007. [4] Scilab Consortium. Scilab, A Free Scientific Software Package, version 4.1, 2006, http://www. scilab.org [5] Pacific Earthquake Engineering Research Center. OpenSees, Open System for Earthquake Engineering Simulation, version 1.7.3, 2006, http://opensees.berkeley.edu [6] SAS IP Inc. ANSYS, version 10.0, 2005, http:// www.ansys.com [7] Friswell MI, Mottershead JE. Finite Element Model Updating in Structural Dynamics. Kluwer Academic: Dordrecht, The Netherlands, 1995. [8] Jaishi B, Ren W. Structural Finite Element Model Updating Using Ambient Vibration Test Results. ASCE J. Struct. Eng. 2005; 4: 617–628. [9] Dascotte E, Strobbe J, Hua H. Sensitivitybased model updating using multiple types of simultaneous state variables. Proceedings of the 13th International Modal Analysis Conference (IMAC), Nashville, Tennessee, February 1995. [10] Allemang RJ, Brown DL. A Correlation coefficient for modal vector analysis. Proceedings of the 1st International Modal Analysis Conference (IMAC), Orlando, Florida, November 1982. [11] Dascotte E, Vanhonacker P. Development of an automatic mathematical model updating program. Proceedings of the 7th International Modal Analysis Conference (IMAC), Las Vegas, Nevada, February 1989. [12] Maya NMM, Silva JMM (eds). Theoretical and Experimental Modal Analysis. Research Studies Press Ltd: Taunton, England, 1997. [13] Teughels A. Inverse modeling of civil engineering structures based on operational modal data, Ph.D. thesis, Katholieke Universiteit Leuven, December 2003. http://www.kuleuven. be/bwm/pub/phdt.htm. [14] IMAC Consortium. IMAC–Integrated Monitoring and Assessment of Cables, 2004, http: //www.vce.at/imac/imac.htm.


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