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DAMAGE ASSESSMENT USING NEURAL NETWORKS
Mechanical Systems and Signal Processing (2003) 17(1), 119–125 doi:10.1006/mssp.2002.1547, available online at http://www.idealibrary.com on
DAMAGE ASSESSMENT USING NEURAL NETWORKS J. L. Zapico, M. P. Gonza¤ lez Departamento de Construccio!n, Universidad de Oviedo, Campus de Viesques 7.1.16, 33203 Gijo!n, Spain. E-mail: [email protected]
and K. Worden Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, U.K. E-mail: K.Worden@sheffield.ac.uk (Received and accepted 1 October 2002) In this paper, a method of damage assessment based on neural networks (NNs) is presented and applied to the Steelquake structure. The method is intended to assess the overall damage at each floor in composite frames caused by seismic loading. A neural network is used to calibrate the initial undamaged structure, and another to predict the damage. The natural frequencies of the structure are used as inputs of the NNs. The data used to train the NNs were obtained through a finite element (FE) model. Many previous approaches have exhibited a relatively poor capacity of generalisation. In order to overcome this problem, a FE model more suitable to the definition of damage is tried herein. Further work in this paper is concerned with the validation of the method. For this end, the damage levels of the structure were obtained through the trained NNs from the available experimental modal data. Then, the stiffness matrices of the structure predicted by the method were compared with those identified from pseudo-dynamic tests. Results are excellent. The new FE model definition allows the NNs to have a much better generalisation. The obtained values of the terms of the stiffness matrix of the undamaged structure are almost exact when comparing with the experimental ones, while the absolute differences are lower than 8.6% for the damaged structure. # 2003 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
The general interest in damage identification of structures and mechanical systems has grown in the last decades due to security and economical reasons. The public’s demands and the technological advancements in computing and sensor monitoring have motivated an extensive research activity in this area. As a result of this, numerous vibration-based methods have been developed in order to detect the damage at early stages. Reference [1] gives a complete review of this researching field. Recently, neural computing has been applied successfully in many fields. Most of these applications deal with problems of pattern recognition [2]. Various researches have begun to experiment with neural networks (NNs) for damage identification purposes during the last decade [3, 4] as an alternative to the updating methods [5]. NNs have some advantages that make them very attractive: ability to treat damage mechanisms implicitly, capacity to generalise their responses and robustness in the presence of noise. The strategy of these approaches is to train a NN to recognise different damage scenarios from the measured response of the system. In these approaches, the selection of damage parameters, damage 0888–3270/03/+$35.00/0
# 2003 Elsevier Science Ltd. All rights reserved.
120
J. L. ZAPICO ET AL.
scenarios and the adjustment of the numerical model to the physical system are prerequisites for success [6]. This work deals with a methodology intended to give overall information about the localisation and the amount of damage in the Steelquake structure after the seismic loading. The previous research developed by the authors [7–10] is summarised, and a new accurate approach is included.
2. INITIAL APPROACH
The application of the method consists of the following successive steps: *
*
*
*
*
*
*
*
A simplified three-dimension (3-D) model of the structure (see Fig. 1) composed by bars and lumped masses is built (references [7, 8] give a detailed description of the modelling). The stiffness of the composite beams and the density of the concrete, which are the most uncertain physical variables, are selected as updating parameters. A database containing the two x-bending natural frequencies of the FE model is generated by varying at random the updating parameters. A 2:10:2 multi-layer perceptron (MLP) is trained with the latter database through the error back-propagation algorithm using the natural frequencies as inputs and the random parameters as outputs (details about the NN computing are shown in [7, 10]). The parameters corresponding to the updated model are obtained from the experimental frequencies of the undamaged structure by forward propagation in the trained MLP. Cracking at the ends of the longitudinal beams, which are the weakest elements under seismic loading, is chosen as a likely damage scenario. The damage at each beam is defined as the quotient of the flexural stiffness of the damaged beam to that of the undamaged one. The damage parameters are established as the average of all the beams belonging to each floor. Thus, a damage parameter equal to zero corresponds to the undamaged floor, and one to the complete damaged.
Figure 1. Initial FE model. Left: lumped masses, right: degrees of freedom.
DAMAGE ASSESSMENT USING NEURAL NETWORKS *
* * *
121
Another similar database is generated by using the updated FE model by varying at random the damage at the beams, and computing the damage parameters of each floor and the x-bending natural frequencies. A 2:10:2 MLP is trained with the first half of this database. The MLP is tested using the second half of the database. The overall damage at each floor of the damaged structure is obtained from the experimental frequencies by forward propagation in the trained MLP.
Results of testing of this approach over a database containing 1000 patterns are depicted in Fig. 2. As can be seen, the results are concentrated within a band that has maximum scatter for medium values of the damage. The correlation coefficients between the predicted and expected damages are 0.9919 and 0.9862 for the first and second floor, respectively. Moreover, damage parameters of 5 and 10% are predicted by the MLP for the undamaged structure (see reference [9] for more details). The relatively poor generalisation achieved with this approach is due to the definition of damage itself. This is because different combinations of damage in the bars lead to the same average floor damage and different natural frequencies.
Figure 2. Results of testing. Above: initial model, below: improved model.
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J. L. ZAPICO ET AL.
The comparison of the stiffness matrix predicted by this initial FE model with that obtained from experimental data for the undamaged structure, which will be described in detail in Section 4, reveals that all the elements of the analytical matrix have values rather higher than the experimental ones. This is due to the fact that the shear flexibility of the bars was not taken into account in this initial FE model [9].
3. IMPROVEMENT
In order to overcome the aforementioned problems and improve the accuracy of the method, a new approach has been tried. As the two x-bending modes have near antisymmetric shape even for the damaged structure, a 2-D model corresponding to an antisymmetric half of the previous one has been selected (see Fig. 3). The stiffness of the columns and beams in this model represent the overall stiffness of each storey of the structure. The damage parameter is now defined as the ratio of stiffness of the damaged beam to the undamaged one. Thus, each combination of natural frequencies maps only one combination of damage; that is, the uniqueness of the solution is reached. The shear flexibility of the bars was also taken into account selecting Timoshenko bars, and distributed consistent masses were selected instead the lumped ones so as to improve the modelling. As can be seen in Table 1, the obtained stiffness matrix for the undamaged structure is now closer to the experimental one with absolute differences between elements comprising between 0.04 and 0.41%. This indicates that the new model reproduces much better the physical properties of the structure. Results of testing show a better correlation between the predicted and expected damages (see Fig. 2). The correlation coefficients are almost equal to one. The damage parameters predicted by the MLP are 0.0041, 0.0019 for the undamaged structure and 0.9975, 0.9978 for the totally damaged structure. Therefore, the low and high levels of damage are accurately predicted by the improved method.
Figure 3. Error! Unknown switch argument. Improved FE model.
123
DAMAGE ASSESSMENT USING NEURAL NETWORKS
Table 1 State\MODEL
Improved FE
Undamaged Damaged
Experimental
54:48 22:31 22:31 15:52 48:46 17:55 17:55 10:862
54:29 22:29 22:40 15:51 49:21 19:19 19:13 11:80
Differ. %
0:34 0:09 0:41 0:04 1:53 8:55 8:26 7:94
4. VALIDATION
The method has been validated by comparing the predicted stiffness matrix with the experimental one. The evaluation has been extended to both the undamaged and the damaged structure. The predicted stiffness matrix was directly computed from the updated FE model by static condensation to the x-translations of the floors. For this, a damage parameter equal to zero was taken into account for the undamaged structure, and the damage parameter obtained through the trained NN for the damaged structure. The stiffness matrix corresponding to the experimental model was computed from the available data of the pseudo-dynamic tests. Namely, a seismic test was used for the undamaged structure, and a burst test for the damaged structure. The data consist of a set of n=1999 samples taken at regular intervals. They include the average x-displacement and the total force of the pistons in each floor. Since both the seismic and the burst tests had low intensity, a linear elastic time invariant response of the structure is expected. Moreover, as these pseudo-dynamic tests run very slowly, between 100 and 1000 times slower than reality [11], and the structural damping was very low, the inertial and dissipative forces are supposed to be low in these tests and they are neglected in the remainder of the analysis. Assuming the aforementioned hypothesis, which will be tested later in this section, the response of the structure can be expressed in the time domain by the following spatial model: fPðtÞg ¼ ½KfDðtÞg
ð1Þ
where [K] represents the stiffness matrix, {P}and {D} are the load and displacement vectors, respectively. If equation (1) is extended to n experimental observations, ½P ffi ½K ½D ð2nÞ
ð2Þ
ð22Þ ð2nÞ
it becomes only approximate due to the neglected effects in the model, the non-linear components and the measurements errors. The best estimate of the matrix [K], in a least squares sense, is obtained from the experimental data through the following: ½K ¼ ½P½Dþ
ð3Þ
where [D]+ is the Moore–Penrose pseudo-inverse of the experimental displacements. As the matrix [[D]T [D]] is singular, [D]+ was calculated using the singular value decomposition (see references [5,6] for more details). The computation was carried out
124
J. L. ZAPICO ET AL.
Figure 4. Comparison between predicted and experimental loads.
with the Matlab function pinv. The calculated stiffness matrixes for both the undamaged and the damaged structure are shown in Table 1. The absolute differences between the elements of the predicted stiffness matrix for the damaged structure and the experimental ones are 1.53 to 8.55%. They are inherent in the method because the modes shapes of the structure are not exactly anti-symmetric when the damage has not a symmetric distribution. Finally, the capacity of the linear model (1) to approximate the response of the structure is evaluated. In this way, the loads predicted by the estimated stiffness matrix with the experimental displacements were computed and compared with the experimental ones; results are depicted in Fig. 4. As can be seen, there is a good agreement with correlation coefficients almost equal to one in all the cases. This means that the linear low-order model (1) is able to reproduce quite exactly the structural response.
5. CONCLUSIONS
The FE model used in this work has demonstrated to be more suitable to the overall definition of damage of the method, even though it is simpler than those used in previous works. It is found that the damage predictions of the method are almost exact for the undamaged and completely damaged structure. The stiffness matrix of the partially damaged structure computed on the basis of the damage predicted by the method has shown absolute differences lower than 8.6% when comparing with that obtained from a pseudo-dynamic test. All these results allow the precision of the method to be validated from a practical point of view. In this approach, the NN has demonstrated to have a high capacity of generalisation and to give very quickly the damage levels of the structure. The methodology, therefore, is
DAMAGE ASSESSMENT USING NEURAL NETWORKS
125
very promising as an overall damage predictor, and it could be enlarged to more complex structures in future works.
REFERENCES 1. S. W. Doebling, C. R. Farrar and M. B. Prime 1996 Los Alamos National Laboratory, Report no. LA-13070-MS. Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review. 2. C. M. Bishop 1998 Neural Networks for Pattern Recognition. Oxford, U.K.: Clarendon Press. 3. A. Rytter and P. H. Kirkegaard 1997 Proceedings of DAMAS’97, University of Sheffield, UK. 97–108. Vibration based inspection using neural networks. 4. K. Worden 1996 Multi-layer Perceptron (MLP), Version 3.4, A Users Guide. Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield. 5. M. I. Friswell and J. E. Mottershead 1999 Finite Element Updating in Structural Dynamics. Dordrecht, The Netherlands: Kluwer Academic Publishers. 6. N. M. M. Maia and J. M. M. Silva 1997 Theoretical and Experimental Modal Analysis. Tauton, U.K.: Research Studies Press Ltd. 7. J. L. Zapico, K. Worden and F. J. Molina 2000 Proceedings of the European COST F3 Conference on System Identification & Structural Health Monitoring, E.T.S.I. Aeron!auticos, Universidad Polit!ecnica de Madrid, Spain, 387–396. Structural damage assessment using neural networks. 8. J. L. Zapico and K. Worden 2000 COST F3 Action, Working Group 2, Steelquake case. Damage assessment. 9. J. L. Zapico, F. J. Molina, K. Worden and M. P. Gonza¤ lez 2001 Proceedings of the European COST F3 Conference on Structural System Identification, University of Kassel, Germany, 495–503. Improvement and validation of a method of damage assessment in a steel frame using neural networks. 10. J. L. Zapico, K. Worden and F. J. Molina 2001 Smart Materials and Structures, Special Issue: Structural Health Monitoring 10, 553–559. Vibration-based damage assessment in steel frames using neural networks. 11. F. J. Molina, M. P. Gonza¤ lez, P. Pegon, H. Varum and A. Pinto 2000 Proceedings of the European COST F3 Conference on System Identification & Structural Health Monitoring, E.T.S.I. Aerona!uticos, Universidad Polite!cnica de Madrid, Spain, 213–222. Frequency and damping evolution during experimental seismic response of civil engineering structures.
DAMAGE ASSESSMENT USING NEURAL NETWORKS J. L. Zapico, M. P. Gonza¤ lez Departamento de Construccio!n, Universidad de Oviedo, Campus de Viesques 7.1.16, 33203 Gijo!n, Spain. E-mail: [email protected]
and K. Worden Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, U.K. E-mail: K.Worden@sheffield.ac.uk (Received and accepted 1 October 2002) In this paper, a method of damage assessment based on neural networks (NNs) is presented and applied to the Steelquake structure. The method is intended to assess the overall damage at each floor in composite frames caused by seismic loading. A neural network is used to calibrate the initial undamaged structure, and another to predict the damage. The natural frequencies of the structure are used as inputs of the NNs. The data used to train the NNs were obtained through a finite element (FE) model. Many previous approaches have exhibited a relatively poor capacity of generalisation. In order to overcome this problem, a FE model more suitable to the definition of damage is tried herein. Further work in this paper is concerned with the validation of the method. For this end, the damage levels of the structure were obtained through the trained NNs from the available experimental modal data. Then, the stiffness matrices of the structure predicted by the method were compared with those identified from pseudo-dynamic tests. Results are excellent. The new FE model definition allows the NNs to have a much better generalisation. The obtained values of the terms of the stiffness matrix of the undamaged structure are almost exact when comparing with the experimental ones, while the absolute differences are lower than 8.6% for the damaged structure. # 2003 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
The general interest in damage identification of structures and mechanical systems has grown in the last decades due to security and economical reasons. The public’s demands and the technological advancements in computing and sensor monitoring have motivated an extensive research activity in this area. As a result of this, numerous vibration-based methods have been developed in order to detect the damage at early stages. Reference [1] gives a complete review of this researching field. Recently, neural computing has been applied successfully in many fields. Most of these applications deal with problems of pattern recognition [2]. Various researches have begun to experiment with neural networks (NNs) for damage identification purposes during the last decade [3, 4] as an alternative to the updating methods [5]. NNs have some advantages that make them very attractive: ability to treat damage mechanisms implicitly, capacity to generalise their responses and robustness in the presence of noise. The strategy of these approaches is to train a NN to recognise different damage scenarios from the measured response of the system. In these approaches, the selection of damage parameters, damage 0888–3270/03/+$35.00/0
# 2003 Elsevier Science Ltd. All rights reserved.
120
J. L. ZAPICO ET AL.
scenarios and the adjustment of the numerical model to the physical system are prerequisites for success [6]. This work deals with a methodology intended to give overall information about the localisation and the amount of damage in the Steelquake structure after the seismic loading. The previous research developed by the authors [7–10] is summarised, and a new accurate approach is included.
2. INITIAL APPROACH
The application of the method consists of the following successive steps: *
*
*
*
*
*
*
*
A simplified three-dimension (3-D) model of the structure (see Fig. 1) composed by bars and lumped masses is built (references [7, 8] give a detailed description of the modelling). The stiffness of the composite beams and the density of the concrete, which are the most uncertain physical variables, are selected as updating parameters. A database containing the two x-bending natural frequencies of the FE model is generated by varying at random the updating parameters. A 2:10:2 multi-layer perceptron (MLP) is trained with the latter database through the error back-propagation algorithm using the natural frequencies as inputs and the random parameters as outputs (details about the NN computing are shown in [7, 10]). The parameters corresponding to the updated model are obtained from the experimental frequencies of the undamaged structure by forward propagation in the trained MLP. Cracking at the ends of the longitudinal beams, which are the weakest elements under seismic loading, is chosen as a likely damage scenario. The damage at each beam is defined as the quotient of the flexural stiffness of the damaged beam to that of the undamaged one. The damage parameters are established as the average of all the beams belonging to each floor. Thus, a damage parameter equal to zero corresponds to the undamaged floor, and one to the complete damaged.
Figure 1. Initial FE model. Left: lumped masses, right: degrees of freedom.
DAMAGE ASSESSMENT USING NEURAL NETWORKS *
* * *
121
Another similar database is generated by using the updated FE model by varying at random the damage at the beams, and computing the damage parameters of each floor and the x-bending natural frequencies. A 2:10:2 MLP is trained with the first half of this database. The MLP is tested using the second half of the database. The overall damage at each floor of the damaged structure is obtained from the experimental frequencies by forward propagation in the trained MLP.
Results of testing of this approach over a database containing 1000 patterns are depicted in Fig. 2. As can be seen, the results are concentrated within a band that has maximum scatter for medium values of the damage. The correlation coefficients between the predicted and expected damages are 0.9919 and 0.9862 for the first and second floor, respectively. Moreover, damage parameters of 5 and 10% are predicted by the MLP for the undamaged structure (see reference [9] for more details). The relatively poor generalisation achieved with this approach is due to the definition of damage itself. This is because different combinations of damage in the bars lead to the same average floor damage and different natural frequencies.
Figure 2. Results of testing. Above: initial model, below: improved model.
122
J. L. ZAPICO ET AL.
The comparison of the stiffness matrix predicted by this initial FE model with that obtained from experimental data for the undamaged structure, which will be described in detail in Section 4, reveals that all the elements of the analytical matrix have values rather higher than the experimental ones. This is due to the fact that the shear flexibility of the bars was not taken into account in this initial FE model [9].
3. IMPROVEMENT
In order to overcome the aforementioned problems and improve the accuracy of the method, a new approach has been tried. As the two x-bending modes have near antisymmetric shape even for the damaged structure, a 2-D model corresponding to an antisymmetric half of the previous one has been selected (see Fig. 3). The stiffness of the columns and beams in this model represent the overall stiffness of each storey of the structure. The damage parameter is now defined as the ratio of stiffness of the damaged beam to the undamaged one. Thus, each combination of natural frequencies maps only one combination of damage; that is, the uniqueness of the solution is reached. The shear flexibility of the bars was also taken into account selecting Timoshenko bars, and distributed consistent masses were selected instead the lumped ones so as to improve the modelling. As can be seen in Table 1, the obtained stiffness matrix for the undamaged structure is now closer to the experimental one with absolute differences between elements comprising between 0.04 and 0.41%. This indicates that the new model reproduces much better the physical properties of the structure. Results of testing show a better correlation between the predicted and expected damages (see Fig. 2). The correlation coefficients are almost equal to one. The damage parameters predicted by the MLP are 0.0041, 0.0019 for the undamaged structure and 0.9975, 0.9978 for the totally damaged structure. Therefore, the low and high levels of damage are accurately predicted by the improved method.
Figure 3. Error! Unknown switch argument. Improved FE model.
123
DAMAGE ASSESSMENT USING NEURAL NETWORKS
Table 1 State\MODEL
Improved FE
Undamaged Damaged
Experimental
54:48 22:31 22:31 15:52 48:46 17:55 17:55 10:862
54:29 22:29 22:40 15:51 49:21 19:19 19:13 11:80
Differ. %
0:34 0:09 0:41 0:04 1:53 8:55 8:26 7:94
4. VALIDATION
The method has been validated by comparing the predicted stiffness matrix with the experimental one. The evaluation has been extended to both the undamaged and the damaged structure. The predicted stiffness matrix was directly computed from the updated FE model by static condensation to the x-translations of the floors. For this, a damage parameter equal to zero was taken into account for the undamaged structure, and the damage parameter obtained through the trained NN for the damaged structure. The stiffness matrix corresponding to the experimental model was computed from the available data of the pseudo-dynamic tests. Namely, a seismic test was used for the undamaged structure, and a burst test for the damaged structure. The data consist of a set of n=1999 samples taken at regular intervals. They include the average x-displacement and the total force of the pistons in each floor. Since both the seismic and the burst tests had low intensity, a linear elastic time invariant response of the structure is expected. Moreover, as these pseudo-dynamic tests run very slowly, between 100 and 1000 times slower than reality [11], and the structural damping was very low, the inertial and dissipative forces are supposed to be low in these tests and they are neglected in the remainder of the analysis. Assuming the aforementioned hypothesis, which will be tested later in this section, the response of the structure can be expressed in the time domain by the following spatial model: fPðtÞg ¼ ½KfDðtÞg
ð1Þ
where [K] represents the stiffness matrix, {P}and {D} are the load and displacement vectors, respectively. If equation (1) is extended to n experimental observations, ½P ffi ½K ½D ð2nÞ
ð2Þ
ð22Þ ð2nÞ
it becomes only approximate due to the neglected effects in the model, the non-linear components and the measurements errors. The best estimate of the matrix [K], in a least squares sense, is obtained from the experimental data through the following: ½K ¼ ½P½Dþ
ð3Þ
where [D]+ is the Moore–Penrose pseudo-inverse of the experimental displacements. As the matrix [[D]T [D]] is singular, [D]+ was calculated using the singular value decomposition (see references [5,6] for more details). The computation was carried out
124
J. L. ZAPICO ET AL.
Figure 4. Comparison between predicted and experimental loads.
with the Matlab function pinv. The calculated stiffness matrixes for both the undamaged and the damaged structure are shown in Table 1. The absolute differences between the elements of the predicted stiffness matrix for the damaged structure and the experimental ones are 1.53 to 8.55%. They are inherent in the method because the modes shapes of the structure are not exactly anti-symmetric when the damage has not a symmetric distribution. Finally, the capacity of the linear model (1) to approximate the response of the structure is evaluated. In this way, the loads predicted by the estimated stiffness matrix with the experimental displacements were computed and compared with the experimental ones; results are depicted in Fig. 4. As can be seen, there is a good agreement with correlation coefficients almost equal to one in all the cases. This means that the linear low-order model (1) is able to reproduce quite exactly the structural response.
5. CONCLUSIONS
The FE model used in this work has demonstrated to be more suitable to the overall definition of damage of the method, even though it is simpler than those used in previous works. It is found that the damage predictions of the method are almost exact for the undamaged and completely damaged structure. The stiffness matrix of the partially damaged structure computed on the basis of the damage predicted by the method has shown absolute differences lower than 8.6% when comparing with that obtained from a pseudo-dynamic test. All these results allow the precision of the method to be validated from a practical point of view. In this approach, the NN has demonstrated to have a high capacity of generalisation and to give very quickly the damage levels of the structure. The methodology, therefore, is
DAMAGE ASSESSMENT USING NEURAL NETWORKS
125
very promising as an overall damage predictor, and it could be enlarged to more complex structures in future works.
REFERENCES 1. S. W. Doebling, C. R. Farrar and M. B. Prime 1996 Los Alamos National Laboratory, Report no. LA-13070-MS. Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review. 2. C. M. Bishop 1998 Neural Networks for Pattern Recognition. Oxford, U.K.: Clarendon Press. 3. A. Rytter and P. H. Kirkegaard 1997 Proceedings of DAMAS’97, University of Sheffield, UK. 97–108. Vibration based inspection using neural networks. 4. K. Worden 1996 Multi-layer Perceptron (MLP), Version 3.4, A Users Guide. Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield. 5. M. I. Friswell and J. E. Mottershead 1999 Finite Element Updating in Structural Dynamics. Dordrecht, The Netherlands: Kluwer Academic Publishers. 6. N. M. M. Maia and J. M. M. Silva 1997 Theoretical and Experimental Modal Analysis. Tauton, U.K.: Research Studies Press Ltd. 7. J. L. Zapico, K. Worden and F. J. Molina 2000 Proceedings of the European COST F3 Conference on System Identification & Structural Health Monitoring, E.T.S.I. Aeron!auticos, Universidad Polit!ecnica de Madrid, Spain, 387–396. Structural damage assessment using neural networks. 8. J. L. Zapico and K. Worden 2000 COST F3 Action, Working Group 2, Steelquake case. Damage assessment. 9. J. L. Zapico, F. J. Molina, K. Worden and M. P. Gonza¤ lez 2001 Proceedings of the European COST F3 Conference on Structural System Identification, University of Kassel, Germany, 495–503. Improvement and validation of a method of damage assessment in a steel frame using neural networks. 10. J. L. Zapico, K. Worden and F. J. Molina 2001 Smart Materials and Structures, Special Issue: Structural Health Monitoring 10, 553–559. Vibration-based damage assessment in steel frames using neural networks. 11. F. J. Molina, M. P. Gonza¤ lez, P. Pegon, H. Varum and A. Pinto 2000 Proceedings of the European COST F3 Conference on System Identification & Structural Health Monitoring, E.T.S.I. Aerona!uticos, Universidad Polite!cnica de Madrid, Spain, 213–222. Frequency and damping evolution during experimental seismic response of civil engineering structures.