1 INTRODUCTION

Structural health monitoring (SHM) can be broadly defined as a process involving firstly, tracking any aspect of structural performance or health by meas-uring data and secondly, interpreting changes so that structural condition and reliability can be quantified objectively (Aktan et al. 2002). During the past dec-ade research into SHM has received much attention. Despite this interest a robust and reliable SHM sys-tem has yet to be presented. In seismically active re-gions, such as New Zealand, the ability to monitor structural condition in real-time is an exciting pro-spect. In the event of an earthquake the extent and location of damage to a structure could be quickly assessed allowing safe structures to be immediately reoccupied and prioritizing resource allocation for repair or demolition of damaged structures.

SHM methods can be broadly divided into several paradigms. Vibration based methods appear to be the most promising. These methods use features ex-tracted from the structure’s dynamic response to identify damage. Various techniques have been pro-posed including frequency domain (Wu et al. 1992, Zang et al. 2001), time domain (Nichols et al. 2003, Masri et al. 2000) and wavelet analysis (Hou et al. 2000, Moyo et al. 2002, Sun et al. 2004). The analyt-ical methods of time series analysis have not been used extensively in SHM. In the authors’ opinion such methods offer a promising alternative. Studies by Sohn et al. (2000) and Sohn et al. (2001) on a concrete bridge pier and naval patrol boat, respect-ively, used Autoregressive (AR) models to fit the

dynamic response of two mechanical systems. By performing statistical analysis on the AR coeffi-cients the authors were able to distinguish the re-sponses from damaged and undamaged systems. However, no attempt was made to locate and quantify damage.

In this investigation, a SHM system utilizing AR models and Artificial Neural Networks (ANNs) is developed. AR models were applied to time histories obtained from computer simulations of a 3-storey shear building model excited by earthquakes. Dam-age in the building was simulated by a reduction in lateral stiffness at each storey. Coefficients of the AR models were considered to be damage sensitive features and were used as ANN inputs. The ANN was able to interpret the changes in these coeffi-cients and both identify and quantify the level of damage at each storey.

2 THEORY

2.1 AR ModelsAR models are used in the analysis of stationary time series processes. A stationary process is a stochastic process, i.e. one that obeys probabilistic laws, in which the mean, variance and higher order moments are time invariant. The structure’s response was assumed to be stationary. This is a reasonable assumption given the input into the structure can be considered as a random series of impulses.

An univariate time series xt can be modelled as an AR process of order p or an AR(p) model:

Analysis of seismic damage using time series methods and Artificial Neural Networks

O.R. de LautourUniversity of Auckland, Auckland, New Zealand

P. OmenzetterUniversity of Auckland, Auckland, New Zealand

ABSTRACT: In the past decade research into SHM systems has received much attention. One of the emer-ging and promising approaches is the use of time series analysis. This study develops a method for the assess-ment of earthquake-induced damage in buildings utilising Autoregressive (AR) time series models and Artifi-cial Neural Networks (ANNs). AR models were applied to the computer simulated seismic response of a lin-ear, lumped mass model of a 3-storey building under different damage conditions. Damage was simulated by a reduction in lateral stiffness at each storey. The AR coefficients were considered to be damage sensitive fea-tures of the building’s response. ANNs were trained to recognize changes in the patterns of the AR coeffi -cients caused by damage and hence identify and quantify the level of damage at each storey.

xt =xt−1a1 + xt−2a2 + ...+ xt−pap + et (1)

where xt-1… xt-p are the previous values of the series and et is Gaussian white noise with zero mean and variance 2. The AR coefficients a1… ap can be evaluated using a variety of methods (Wei 2006). In this study, the coefficients were determined using a least squares solution. Given a series of n points Equation 1 can be rewritten into matrix form:

et

Men

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

xtMxn

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥−

xt−1 L xt−pM M Mxn−1 L xn−p

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

a1Map

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

(2)

ore =y−Xa (3)

where e is an error vector, y is a vector containing the current data points, X is a matrix of the previous data points and a is a vector containing the AR coef-ficients. A least squares solution seeks to minimize:

eT e = y−Xa( )T y−Xa( ) (4)

Differentiating Equation 4 with respect to a gives:

∂eT e∂a

= −2XT y + 2XT Xa (5)

Forcing Equation 5 to be equal to zero gives the val-ues of the AR coefficients:

a = X TX( )−1X Ty (6)

If the time series xt contains multiple measure-ments at each time step, a multivartiate AR model of order p can be used:

xt =B1xt−1 + B2xt−2 + ...+ Bpxt−p + et (7)

where xt-1… xt-p are the vectors of previous values of the series and et is a multivariate Gaussian white noise with zero mean and covariance matrix ∑. The AR coefficients B1… Bp are matrices and can be evaluated using a least squares solution, as in this study. With n measurements Equation 7 can be re-written into matrix form:

etT

Men

T

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

xtT

MxnT

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥−

xt−1T L xt−p

T

M M Mxn−1T L xn−p

T

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

B1

MBp

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

(8)

or

E =Y−XB (9)where E is an error matrix, Y is a matrix containing the current values of the series and X is a matrix of the pervious values. By performing a similar proce-dure as described above for the univariate case, the AR coefficients can be estimated using:

B = X TX( )−1X TY (10)

2.2 Artificial Neural NetworksANNs are structures deliberately designed to utilize the organizational principles found in the brain (An-derson et al. 1988). ANNs are capable of pattern re-cognition, classification and function approximation, and have been used extensively in the civil engineer-ing field (Adeli 2001). ANNs utilising the super-vised error Back-propagation (BP) training al-gorithm (Rumelhart et al. 1986) are commonly re-ferred to as BP neural networks. BP networks are the most popular type of neural network employed and have been used in this study. The structure of a single hidden layer BP network is shown in Figure 1 where x is the input vector and o is the output vec-tor. The bias inputs into the hidden and output layers have been represented by solid squares and both have the value of +1. The weights for the whole net-work, denoted by vector w, store information as in the brain and are learnt during the training process.

The basic function of a single neuron, Figure 2, in either the hidden or output layers is to calculate the weighted sum of all inputs u:

u =vTx (11)

and compute the output y:

y =f u( ) (12)

where the weights vector for the single neuron has been denoted by v to avoid confusion with w that contains the weights for all neurons, and f is the neuron’s activation function.

The error E in the network is a function of the weights and can be written as:

E w( )=12e w( )T e w( ) (13)

Figure 1. A single hidden layer network.

Figure 2. The function of a single neuron.

where e(w) is an error vector defined by:e(w) =d −o(w) (14)

and d is the vector of target values or desired net-work outputs. In the training phase, the network cal-culates the output for a given input and the error is propagated backwards from the output layer to the preceding layers using the back-propagation al-gorithm. In the original algorithm, the error is min-imized by changing the weights using a gradient descent method. In this study a modified algorithm was used in which the weights are changed using the Levenberg-Marquardt algorithm (Marquardt 1963), a quasi-Newton method developed specifically for the sum of errors squared error function. The al-gorithm’s application to the original error back-propagation supervised learning algorithm is de-scribed in Hagan et al. (1996). Introducing the Jac-obian matrix J defined by:

J w( )=∂e∂w

(15)

the new weights can be found through the applica-tion of the following iterative process:

wk+1 =w k − JkTJ+ λkI⎡⎣ ⎤⎦

−1JkTek (16)

where the parameter λk is a scalar that controls con-vergence properties. If λk is equal to zero the Leven-berg-Marquardt algorithm becomes the Gauss-New-ton method. The subscript k denotes the iteration step.

3 APPLICATION TO A 3-STOREY BUILDING

In the proposed method, AR models are used to fit the structure’s response and the coefficients of these models are used as inputs into an ANN. The output of the ANN is damage at each storey quantified as the remaining stiffness. The method was applied to a simple, linear, lumped mass model of a 3-storey shear building shown in Figure 3. Damping was set at 5% critical damping for all modes and the lumped masses, shown as grey circles, were 7645 kg each. Additionally, all columns and beams had a distrib-uted mass of 612 kg/m. The stiffness k1, k2

Figure 3. Proposed SHM system applied to a 3-storey building.

and k3 for the healthy structure were set at 78.4×106

N/m. The storey height was 3.5m and bay width was 5.0m. The fundamental, second and third natural periods of the building, in the undamaged state, were 0.304s, 0.105s and 0.067s, respectively.

Damage at each storey in the building was simu-lated by a reduction in the lateral stiffness k1, k2 and k3. Three damage severities were considered; these were 1.00, 0.75 and 0.50 times the original stiffness. Any simultaneous combination of these damage states was allowed, e.g. 1st storey 1.00, 2nd storey 0.75 and 3rd storey 0.50 of the initial stiffness, re-spectively, etc.

All data used is this study was computer simu-lated. The AR coefficients were obtained from fit-ting AR models to the total acceleration time history, xt in Figure 3, of each storey when excited by an earthquake signal. To allow for some generalization, three different earthquake records were used. These records were from the 1940 Imperial Valley, 1995 Kobe and 1986 Taiwanese earthquakes (PEER Strong Motion Database 2006). Three different mon-itoring and data processing situations for the build-ing were investigated; 1) the building is monitored with a single sensor on the third storey and an uni-variate AR model is used to fit the measurements, 2) the building is monitored with three sensors, one at each storey, and a multivariate AR model is used, and 3) the building is monitored with three sensors and three independent univariate AR models are used. The three scenarios were studied to obtain an insight into the efficiency of the proposed method depending on the number of sensors and to investig-ate if capturing correlations between multiple signals in a multivariate AR model yield additional benefits. The structural acceleration time histories were win-dowed into 1000-point segments with a 250-point overlap. This allowed more AR coefficients to be obtained. Figure 4 shows the distribution of AR

coefficients from the undamaged and damaged (1st

storey 0.5, 2nd storey 0.5 and 3rd storey 0.75 of the initial stiffness, respectively) building using monit-oring method 1. Clearly, there are significant differ-ences between statistical distributions of the AR coefficients for the undamaged and damaged struc-ture, such as different means and standard devi-ations. The identification of the location and extent of the damage from these data is, however, not an obvious and straightforward task and requires more investigation.

Figure 4. Distribution of AR coefficients for damaged and un-damaged buildings.

3.1 Single sensorIn this situation the acceleration time history of the third storey was fitted using an AR(6) model. This model gave both a small number of parameters and a good fit. A data set of 189 input-output pairs was randomly divided into 160 for training and the re-

maining 29 for testing the ANNs. The best results were obtained using a single hidden layer ANN with 6 neurons in the hidden layer. In Figure 5, the results of damage prediction for each storey have been graphed individually, with the actual remaining stiff-ness plotted against the detected stiffness. The figure shows combined results for all test data. For per-fectly accurate predictions all the data in the plot would collapse into three points (0.50, 0.50), (0.75, 0.75) and (1.00, 1.00). In this case the results are, however, scattered indicating relatively large errors produced by the ANN.

Figure 5. Simulated vs. detected damage using a single sensor.

3.2 Three sensors with a multivariate AR modelTo investigate the effect of additional measurements, all three acceleration time histories were used in conjunction with a multivariate, or vector, AR(3) model. A total of 195 input-output pairs were gener-ated and randomly divided into 160 for training and 35 for testing the ANNs. Again a single hidden layer network with 6 neurons in the hidden layer was used. Figure 6 shows the results are considerably better than the previous result with only a small amount of scatter.

3.3 Three sensors and three univariate AR modelsInstead of using a multivariate AR model three uni-variate separate AR models could be used, one for each storey in this case. This would reduce the num-ber of AR coefficients, and hence computational burden, but hopefully retain a similar level of per-formance. Three AR(6) models were used in com-bination with a single hidden layer network with 6

neurons in the hidden layer. Again, a total of 195 in-put-output pairs was generated and randomly di-vided into 160 for training and 35 for testing the ANNs. The results are slightly worse than the mul-tivariate approach and are shown graphically in Fig-ure 7.

Figure 6. Simulated vs. detected damage using three sensors and a multivariate AR model.

Figure 7. Simulated vs. detected damage using three univariate AR models.

4 CONCLUSIONS

In this study a new method of SHM using combina-tion of AR time series models and ANNs has been presented. AR models were used to fit the structure’s seismic response. The coefficients of these models were chosen as the damage sensitive features and were fed into an ANN trained to predict the damage at each storey in the building.

It is clear that a combination of AR models and ANNs are capable of detecting, quantifying and loc-ating damage in structures. The three investigated monitoring approaches produced excellent results, except when only a single sensor was used. Compu-tational requirements for both multiple sensor ar-rangements were similar. However, the multivariate AR model gave slightly better results and hence is the best option for monitoring the building in this study. In more complex structures a significant dif-ference in either performance or computation re-quirements may develop. This would obviously de-termine which multiple sensor approach is appropri-ate.

In this preliminary study the number of sensors was limited by the simplicity of the building. How-ever in real buildings the number of sensors and the optimal placement of them will need to be ad-dressed. Due to the formulation of the proposed damage detection method, the optimal placement of sensors is likely to be in areas where damage is ex-pected to occur.

Future analytical research will investigate the sensitivity of the proposed method to measurement noise and changing operating conditions. Noise is expected to have only a small effect on predictions because the AR models contain an error term. Future reasearch will also focus on extension of the tools presented into an online method where seismic dam-age may be traced as it gradually develops in real time. The authors also intend to prove both the cap-ability and practicality of the proposed SHM system experimentally using a test structure and a shake table in the near future.

REFERENCES

Adeli, H. 2001. Neural Networks in Civil Engineering: 1989-2000. Computer-Aided Civil and Infrastructure Engineer-ing 16: 126-142.

Aktan, A. E., Catbas, F. N., Grimmelsman, K., Pervizpour, M., Curtis, J., Shen, K., Qin, X. 2002. Health monitoring for ef-fective management of infrastructure, San Diego, CA, United States, The International Society for Optical Engin-eering.

Anderson, J. A. and Rosenfeld, E., Eds. 1988. Neurocomput-ing: Foundations of Research. Cambridge, Massachusetts., MIT Press.

Hagan, M. T., Demuth, H. B., Beale, M.H. 1996. Neural net-work design. Boston, PWS Pub.

Hou, Z., Noori, M., et al. 2000. Wavelet-based approach for structural damage detection. Journal of Engineering Mech-anics 126 (7): 677-683.

Marquardt, D. 1963. An Algorithm for Least Squares Estima-tion of Nonlinear Parameters. Journal of the Society for In-dustrial and Applied Mathematics 11 (2): 431-441.

Masri, S. F., Smyth, A. W., Chassiakos, A.G., Caughey, T.K., Hunter, N.F. 2000. Application of Neural Networks for De-tection of Changes in Nonlinear Systems. Journal of Engin-eering Mechanics 126 (7): 666-676.

Moyo, P. and Brownjohn, J. M. W. 2002. Detection of anomal-ous structural behaviour using wavelet analysis. Mechan-ical Systems and Signal Processing 16 (2-3): 429-445.

Nichols, J. M., Virgin, L. N., Todd, M.D., Nichols, J.D. 2003. On the use of attractor dimension as a feature in structural health monitoring. Mechanical Systems and Signal Pro-cessing 17 (6): 1305-1320.

PEER Strong Motion Database 2006. http://peer.berkeley.edu/smcat/index.html. Accessed on 28/2/2006.

Rumelhart, D. E., Hinton, G. E., Williams, R.J. 1986. Learning internal representations by error propagation. Parallel Dis-tributed Processing: Explorations in the Mircostructures of Cognition. Rumelhart, D. E. and McClelland, J. L. Cam-bridge, MA., MIT Press. 1.

Sohn, H., Czarnecki, J. A., Farrar, C.R. 2000. Structural Health Monitoring Using Statistical Process Control. Journal of Structural Engineering 126 (11): 1356-1363.

Sohn, H., Farrar, C. R., Hunter, N.F., Worden, K. (2001). Ap-plying the LANL statistical pattern recognition paradigm for structural health monitoring to data from a surface-ef-fect fast patrol. Los Alamos, Los Alamos National Laborat-ory.

Sun, Z. and Chang, C. C. 2004. Statistical wavelet-based method for structural health monitoring. Journal of Struc-tural Engineering 130 (7): 1055-1062.

Wei, W. W. S. 2006. Time Series Analysis: Univariate and Multivariate Methods. Boston, Pearson.

Wu, X., Ghaboussi, J., Garrett, J.H. 1992. Use of neural net-works in detection of structural damage. Computers and Structures 42 (4): 649-659.

Zang, C. and Imregun, M. 2001. Structural damage detection using artificial neural networks and measured FRF data re-duced via principal component projection. Journal of Sound and Vibration 242 (5): 813-827.