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Damage detection using artificial neural network with consideration of uncertainties
Engineering Structures 29 (2007) 2806–2815 www.elsevier.com/locate/engstruct
Damage detection using artificial neural network with consideration of uncertainties Norhisham Bakhary ∗ , Hong Hao, Andrew J. Deeks School of Civil and Resource Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia Received 27 October 2006; received in revised form 8 January 2007; accepted 8 January 2007 Available online 12 March 2007
Abstract Artificial Neural Networks (ANN) have received increasing attention for use in detecting damage in structures based on vibration modal parameters. However, uncertainties existing in the finite element model used and the measured vibration data may lead to false or unreliable output result from such networks. In this study, a statistical approach is proposed to take into account the effect of uncertainties in developing an ANN model. By applying Rosenblueth’s point estimate method verified by Monte Carlo simulation, the statistics of the stiffness parameters are estimated. The probability of damage existence (PDE) is then calculated based on the probability density function of the existence of undamaged and damaged states. The developed approach is applied to detect simulated damage in a numerical steel portal frame model and also in a laboratory tested concrete slab. The effects of using different severity levels and noise levels on the damage detection results are discussed. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Damage detection; Neural networks; Uncertainties; Rosenblueth’s point estimate; Random noise; Modal data
1. Introduction Various structural health monitoring techniques have been researched in order to obtain a reliable and efficient approach to increase the safety of civil engineering structures. Many different techniques have been proposed and investigated ranging from application of electrical impedance techniques to structural dynamics approaches. Among these techniques, structural dynamics approaches have been extensively explored by many researchers, as they can provide information on unforeseen potential failure mechanisms. Since the earliest work by Cawley and Adams [1], dynamics parameters such as natural frequencies, and mode shapes have been widely applied for damage detection, as the modal parameters are functions of structural properties. This implies that any degradation of the structural properties results in changes of the modal parameters. Artificial Neural Networks (ANN) have been utilized by many researchers to identify damage location and severity from various types of input and output variables, as they provide an efficient tool for pattern recognition. Wu et al. [2] explored the use of an ANN to detect member damage in a 3-storey frame. ∗ Corresponding author. Tel.: +61 8 6488 3072; fax: +61 8 6488 1044.
E-mail address: [email protected] (N. Bakhary). c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.01.013
Pandey and Barai [3] provided a more detailed treatment of ANN architecture in their study identifying damage in a 21bar bridge truss. Zhao et al. [4] applied a counter-propagation neural network to detect damage and support movement in a continuous beam. Zapico [5] developed a procedure for damage assessment of steel structures based on ANN. Most of the studies concluded that ANNs are capable of providing correct damage identification, especially when the structural damage and the associated changes in vibration properties are simulated numerically and are error free. However, in practice uncertainties in the FE model parameters and modelling errors are inevitable. The existence of modelling error in the FE model due to the inaccuracy of physical parameters, nonideal boundary conditions, finite element discretization and nonlinear structural properties may result in the vibration parameters generated from such a FE model not exactly representing the relationship between the modal parameters and the damage parameters of the real structure. On the other hand, the existence of measurement error in the measured data that is normally used as testing data in an ANN model is also unavoidable. Since the efficiency of an ANN prediction relies on the accuracy of both components, the existence of these uncertainties may result in false and inaccurate ANN predictions. Therefore, the impact of uncertainties on the
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reliability of ANN models for structural damage detection needs to be analysed. Several studies have considered uncertainty in application of ANN [6–9], and the noise injection learning method proposed by Matsuoka [10] has been a popular method of considering the uncertainties in ANN applications. Through this method, certain amounts of noise are applied to the training data to consider either modelling error or measurement error. However, studies that consider both errors in the FE model and noises in measurement data are quite limited. The objective of this paper is to study the influence of uncertainty on damage identification using a combination of frequency and mode shape as the input variables. To consider the uncertainties in the FE modelling and the measurement data, an approach introduced by Papadopoulos and Garcia [11] is applied. Using this method, the probability of damage existence (PDE) can be estimated by comparing the probability distribution of the undamaged and damaged models. To consider the effect of FE modelling error, a statistical ANN model is trained with vibration data generated from the FE model, but smeared with random variations. To include the effect of noise in the measurement data, the testing data used as input to the statistical ANN model for damage identification are also smeared with random noises. The probability moments of the undamaged and damaged states of the structural parameters are estimated using the point estimation method and verified by Monte Carlo simulation. The Monte Carlo simulation data is also used to determine the type of probability distribution function of the structural parameters of both the undamaged and damaged states. The PDEs are determined from the probability distribution for each structural member. A numerical steel frame model and a laboratory tested two-span reinforced concrete slab are used to demonstrate the developed procedure. Some parametric calculations are also performed to investigate the influences of using different noise levels and damage severities on the damage identification results. In this study, only random errors are considered. The systematic errors which may also exist, especially in FE model are not considered in the present study. 2. Theoretical background In this study, the uncertainties in the FE model and the measurement data are assumed to be normally distributed independent random variables with zero means and specified coefficients of variation. This implies that the parameters with uncertainties are equal to true values plus the random variations. Thus, the frequencies, mode shapes and stiffness parameters for training and testing are derived as below: λi = λi0 + λi0 X λi = λi0 (1 + X λi )
(1a)
λˆ i = λˆ i0 + λˆ i0 X λi = λˆ i0 (1 + X λi )
(1b)
φi = φi0 + φi0 X φi = φi0 (1 + X φi )
(1c)
φˆ i = φˆ i0 + φˆ i0 X φi α j = α 0j + α 0j X α j
(1d)
= φˆ i0 (1 + X φi ) = α 0j (1 + X α j )
(1e)
Table 1 Training functions and testing variables used in point estimation method Model
Training function
Testing variable Input
Output
1
α j++ = fn(λi0 + σλi , φi0 + σφi )
λˆ i0 + σλi , φˆ i0 + σφi
αˆ ++
2
α j−− = fn(λi0 − σλi , φi0 − σφi ) α j+− = fn(λi0 + σλi , φi0 − σφi ) α j−+ = fn(λi0 − σλi , φi0 + σφi )
λˆ i0 − σλi , φˆ i0 − σφi
αˆ −−
λˆ i0 + σλi , φˆ i0 − σφi
αˆ +−
λˆ i0 − σλi , φˆ i0 + σφi
αˆ −+
3 4
where λi , φi and λˆ i , φˆ i are the ith frequencies and mode shapes for training and testing, respectively, and α j is a parameter related to the stiffness of the jth segment. The change of stiffness parameters is described by the reduction of Young’s modulus (E values). Stiffness Reduction Factor (SRF) defined as the ratio of E value change to the initial E value is used as the damage indicator. Superscript ‘0’ represents the corresponding mean value and X λi , X φi , X αi are the zero mean normally distributed random variables with specific COV in frequencies, mode shapes and stiffness parameters, which are assumed in this study to be the same for both training and testing data. In this study, both the training and testing data sets contain randomly varying errors or noises. Because of the randomly varying training data set, the trained ANN model will be statistical with probability distribution parameters such as mean and standard deviation for each variable. Moreover, since the testing data set is random, the ANN prediction of the condition of each variable, in this case the Young’s modulus of each structural member, is also statistical. The most direct method to obtain the probability parameters is Monte Carlo simulation. However, Monte Carlo simulation requires high computational effort and is very time consuming. Another approach is to attempt a theoretical solution, but it is very difficult to obtain a closed form solution of probability distributions for such a multi degree of freedom dynamic system, and derivation of such a solution is beyond the scope of this study. In this study the probability moments of the undamaged and damaged states of the structural parameters are approximated using the point estimation method introduced by Rosenblueth [12]. By using this method, the mean values E(α) and standard deviation σ (α) of each variable can be calculated based on the upper limit (αi+ ) and the lower limit (αi− ) of the function corresponding to the ith variable. The upper and lower limit can be obtained by using mean plus one standard deviation and mean minus one standard deviation of each random variable in training and testing of the ANN model. Since two variables are used (frequency and mode shape) in this study, two upper limits (α++ , α−+ ) and two lower limit (α−− , α+− ) need to be obtained. Thus, four ANN models are developed by considering the mean values and standard deviations (σ ) of the random noises applied to each variable. The training functions and the input and output variables for testing are listed in Table 1. The relationship between input and output for ANN model is denoted by fn(·). The expectation (mean value, µα ) and standard deviation (σα ) of α can then be approximately calculated by:
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Fig. 1. Probability density functions for α j and α 0j and probability of damage j existence, Pd .
µα = E(α) =
1 (αˆ ++ + αˆ −− + αˆ +− + αˆ −+ ) 4 1
σα = [E(α 2 ) − [E(α)]2 ] 2
(2) (3)
where E(α 2 ) is calculated using Eq. (2) with α 2 terms substituted for the α terms. Using this approach substantially reduces the computational time in deriving the statistical mean and standard deviation of each stiffness parameter for structural condition monitoring. The PDE can be estimated from statistical distributions of the stiffness parameters of the undamaged and damaged models. For example, if the stiffness parameter (α j ) of the undamaged segment j is normally distributed with mean E(α j ) and standard deviation σ (α j ), the probability density function can be obtained as illustrated in Fig. 1, where L α j is the lower bound of the healthy parameter. In this study, the confidence level is set to 95%, thus the lower bound is L α j = E(α j ) − 1.645σ (α j ), which indicates that there is a probability of 95% that the healthy stiffness parameter falls in the range of [E(α j )−1.645σ (α j ), ∞]. Similarly, for the stiffness parameter of segment j in the damaged state (α 0j ), the distribution is again assumed as normal with mean E(α 0j ) and standard deviation σ (α 0j ), and the corresponding probability density function is also plotted in Fig. 1. The PDE is defined as the probability of α 0j not being within the 95% confidence healthy interval. Thus the PDE of segment j is j
Pd = 1 − prob(L α j ≤ xα 0 ≤ ∞) = prob(−∞ ≤ xα 0 ≤ L α j ).
(4)
PDE is a value between 0 and 1, and if the PDE of a segment is close to 1, then it is most likely the element is damaged. If the PDE is close to 0, damage existing in the element is very unlikely [13]. It should be noted that the stiffness parameters of the undamaged and damaged state have normal distributions because the random variations in Eq. (1) are assumed as zero mean normally distributed random variables. This will be proved later.
Fig. 2. Finite element model of the frame.
The input data consist of natural frequencies and mode shapes, and the output layers consists of Young’s modulus (E values) to represent the stiffness parameter. The change of the stiffness parameter or the damage severity for each segment are denoted by a Stiffness Reduction Ratio (SRF), defined as SRF = 1 −
E0 E
(5)
where E is the Young’s modulus in the intact state and E 0 is that at the damage level of interest. In most ANN applications for damage detection, the training data are obtained from FE analysis, which involved generating large number of damage cases based on an initial baseline FE model. Once the ANN model is well-trained, the testing data are then applied to the ANN model to obtain the locations and severities of any damages. In most of the previous studies, both training and testing data are assumed to be free from modelling and measurement error. In practice, however, modelling error and measurement noise are inevitable. According to Xia [15], the inaccuracy due to modelling and measurement error can be overcome by taking into account the uncertainties through a statistical method. In this study, modelling error and measurement noise are assumed to be normally distributed with zero means and specific variance. The noise is applied in terms of coefficient of variations (COV). The statistical properties of E value for each segment are obtained by using Rosenblueth’s point estimation method verified by Monte Carlo simulation. A Kolmorogov–Smirnov goodness-offit test (K–S test) is then applied to verify the distribution type of E values. This is followed by calculation of the PDE of E values for each segment.
3. Methodology 4. Numerical example A multilayer perceptron with Levenberg–Marquardt backpropagation algorithm is utilized to train the ANN model in this study. Sigmoid functions are employed as nonlinear activation functions for all layers. To reduce the effect of overfitting, the early-stopping method [14] is applied during the training.
To demonstrate the proposed method, the single span steel portal frame shown in Fig. 2 is used as an example. The cross section of beam is 40.50 × 6.0 mm2 , and column is 50.50 × 6.0 mm2 . The span length and height of the frame are
N. Bakhary et al. / Engineering Structures 29 (2007) 2806–2815 Table 2 E values for scenario 1 and scenario 2 Segment
1
2
3
4
5
6
Scenario 1 Scenario 2
0.4× E 0.4× E
1.0× E 1.0× E
1.0× E 0.3× E
0.2× E 1.0× E
1.0× E 0.4× E
1.0 × E 0.3 × E
Table 3 First three frequencies values for scenario 1 and scenario 2
Mode 1 Mode 2 Mode 3
Undamaged
Scenario 1
Scenario 2
4.628 16.112 20.649
3.9373 12.567 16.491
3.530 11.269 14.891
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the actual values. The results show that the ANN model is able to provide good output, which indicates that the ANN model is reliable in detecting damage from noise-free data. To investigate the reliability of the ANN model in predicting structural damage with noisy data, 2% and 15% random noise are applied respectively to the frequencies and mode shapes of the testing data. Applying these errors implies that the measured data is no longer exact. These data are then introduced to the trained ANN model. Fig. 5 shows the ANN output. The figure shows that for scenario 1, the false damage identification occurs at segments 2 and 3, while the stiffness of segment 6 is overpredicted. The same situation occurred in scenario 2, where the ANN model falsely identifies damage at segment 2, overestimates damage at segments 1 and 3, but underestimates at segment 6. This indicates that the common ANN model trained with simulated vibration parameters from finite element model cannot give reliable structural damage prediction if the errors exist. 4.1. Statistical artificial neural network
Fig. 3. First three mode shapes for undamaged, scenario 1 and scenario 2.
both 1000 mm. Rigid connections are applied between the beam and the columns, and the supports are assumed as clamped. 11 2 The material properties used are: E = 2.1 × 10 N/m , ρ = 7.67 × 103 , v = 0.2. The frame is modelled with 10 elements in each member. To reduce the complexity of ANN training, the frame is divided to six segments as shown in the figure. Each segment consists of five elements. Modal analysis is conducted using the FE model to generate input and output data to train the ANN models. Two damage scenarios are generated to assess the ANN prediction performance. Scenario 1 consists of damage in two segments of the frame (1 & 4), and scenario 2 consists of damage in four segments (1, 3, 5 & 6). The damage is imposed by reducing the E values of each corresponding segment. Table 2 shows the E values for scenario 1 and scenario 2. The frequencies and mode shapes of the first three modes are shown in Table 3 and Fig. 3. To train the ANN model, 1200 data sets are generated from the FE model based on the Latin hypercube sampling method [16]. To apply the early-stopping method, the data are divided into three parts in a ratio of 2:1:1. A trial and error method based on Kalmorogov and Lippmann’s approach [17] was utilized to attain the best ANN topology. To reduce the ‘curse of dimensionality’ as discussed by Bishop [18], only nine mode shape points and frequencies for the first three modes were used as the input parameters. The output parameters are Young’s modulus (E values) of every segment. The numbers of neurons in the input and output layers are the same as the respective number of input and output variables. The best ANN model is obtained with 97 hidden neurons. The trained ANN model is then assessed by introducing the modal parameters of the two damage scenarios mentioned above. Fig. 4 shows the predicted E values in comparison with
As demonstrated above, noise in measured data may lead to unreliable and false prediction of structural damage. To consider the uncertainties in the FE model and in the measured data, normally distributed random noises with zero means and specific variance are added to training data. The noise levels of the frequencies and mode shapes are also assumed to be 2% and 15% respectively. These data are then used to train the ANN model. A statistical ANN model is developed using the Rosenblueth’s point estimation method. To verify the reliability of the Rosenblueth’s point estimation method, the mean values and standard deviations of every segment in the undamaged state are obtained by using both the point estimation method and Monte Carlo simulation. In Monte Carlo simulation, randomly distributed training data and testing data are generated and used to predict structural damage. The training and testing process is repeated until the mean values and standard deviations of E values converge. The Monte Carlo simulations converge after about 125 iterations. Fig. 6 shows the Monte Carlo simulation result. The Monte Carlo simulation data are also used to determine the distribution type of the E value. A Kolmorogov–Smirnov goodness-of-fit test (K–S test) is applied to verify the accuracy of the selected distribution type of E values. The results show that the E values have normal type of characteristic. Fig. 7 illustrates the cumulative distribution function (CDF) for segment 3 as compared with the theoretical one. Based on the goodness-of-fit test, a normal distribution hypothesis of the E values for all segments is accepted with a confidence level of 95%. Fig. 8 shows the mean values and COV of E values obtained from both methods. It is observed that both methods provide quite similar results, indicating the point estimation method is reliable. Using the trained ANN model with 2% and 15% random errors (COV) in frequencies and mode shapes, and the testing data with the same level of noise, the mean values and standard deviations of structural stiffness parameters corresponding to the two damage scenarios are estimated. From the normally
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Fig. 4. ANN prediction for scenario 1 and scenario 2 compared to the actual value using noise-free input data.
Fig. 5. ANN prediction for scenario 1 and scenario 2 compared to the actual value using noisy input data.
Fig. 6. Monte Carlo simulation result. Table 4 Probability of damage existence (%) for every segment of scenario 1 and scenario 2
Scenario 1 Scenario 2
Fig. 7. K–S test result for segment 3.
Segment 1 2
3
4
5
6
96.0 99.7
3.0 100.0
99.8 13.5
0.00 99.7
0.00 99.9
38.5 1.0
distributed probability density function of the damaged and undamaged states, the PDEs can be calculated. The PDEs for scenario 1 and scenario 2 are listed in Table 4. From Table 4, it is observed that in scenario 1, the PDEs of segments 1 and 4 are very high and the PDEs of the other segments are low. This indicates that it is very likely damage exists in segments 1 and 4 only. For scenario 2, the highest
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Fig. 8. Mean values and coefficients of variation (COV) of E values in undamaged state. Table 5 Probability of damage existence (%) for different damage severities Severity level (%) −25 −10 −7 −5 −2
Segment 1
2
3
4
5
6
100.0 89.1 62.6 38.0 10.0
14.3 3.5 3.2 3.0 2.7
0.0 4.5 3.9 3.5 2.9
100.0 71.2 47.1 29.3 8.9
5.8 8.9 6.5 5.1 3.4
0.0 1.5 1.8 2.0 2.3
PDE occurred at segments 1, 3, 5 and 6, which are also the true damage locations. These results show that using the proposed statistical ANN model, the damages are detected with high confidence and undamaged segments are less likely to be falsely identified, as compared to the common ANN model. However, further investigation needs to be conducted in order to see the sensitivity of different uncertainty levels to the result, so does the damage severities. 4.2. Parametric study More detailed studies are carried out in this part to investigate the sensitivity of the proposed method to different damage levels and different uncertainties levels. To assess the reliability of the proposed method at different damage levels, an analysis is performed by varying the damage level, while the uncertainty level remains unchanged. Damage is introduced to segments 1 and 4 and results generated using the FE model with SRF equal to −25%, −10%, −7%, −5% and −2%. The uncertainty level is constant at 1% for frequencies, and 10% for mode shapes. By using the same method as described above, the PDE of each segment is calculated. The PDEs are listed in Table 5. It is observed that the highest PDEs occurred at the correct damage location (segments 1 and 4) for all cases. The results also show the highest PDEs are gained with a −25% damage level and the lowest are with a −2% damage. The results indicate that when the damage is severer, it can be confidently identified. However, when the damage is not significant, the damage cannot be detected confidently, although the damaged segments always have higher PDEs than the undamaged segments. The reason is that when the damage levels are small, the changes of modal parameters are not so apparent, and the influences of modelling and measurement noises are comparatively more significant.
In order to investigate the effect of different uncertainty levels on the proposed damage identification method, four ANN models are developed. Each model is trained using the training data that is generated from the same baseline FE model but smeared with different levels of uncertainties in frequencies and mode shapes respectively. These levels are (1) 0.0% and 0.0%, (2) 0.5% and 5.0% (3)1.0% and 10.0% and (4) 2.0% and 15.0%. The trained models are then tested with testing data which are also smeared with different levels of uncertainties. In this case, −25% damage levels are applied to segments 1 and 4. Table 6(a)–(d) illustrate the results in terms of PDE of different combinations of uncertainties in training and testing data. Table 6(a) lists the PDEs of each segment when the ANN model is trained with deterministic data. It is evident that the ANN model trained with deterministic data performed poorly when tested with noisy data. For example, the highest PDEs are observed at segments 1 and 6 when the network is tested with 0.5% (frequencies) and 5% (mode shapes) uncertainties. The ANN model gets more unreliable when the testing data contain higher noise. Table 6(b) shows the results obtained when the ANN model was trained with noisy data with COV 0.5% for frequencies and 5% for mode shapes. From the results, it is observed that in most cases the highest PDEs are obtained at segments 1 and 4, which indicates that the damage is correctly identified. The results show that the highest PDEs are obtained when the testing data also has noise with COV 0.5% for frequencies and 5% for mode shapes, as might be expected. When the measured data contains a significantly higher noise level than the training data, i.e. 5% in frequency and 40% in mode shape, the method, nevertheless, fails to give correct identification of damage locations. The same situation occurs when higher uncertainties are applied to the training data. Table 6(c) and (d) show the results when the training data are smeared with 1% and 10%, and 2% and 15% noise in frequencies and mode shapes respectively. In most cases the damage can be detected correctly. The highest confidence levels are obtained when the noise level in the training and testing data are the same. It is also clear that the PDEs values decreased when the difference between the uncertainty levels in training and testing data increased. This may indicate that if the uncertainties in either the training data or the testing data are larger than each other, the true information is submerged in noise, and thus the actual damage location cannot be identified accurately.
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Table 6 Probability of damage existence (%) for different combination of uncertainties in training and testing data Uncertainties level in testing data Frequencies (%) Mode shapes (%)
Segment 1
2
3
(a) Uncertainties in training data = 0.0% (frequencies) and 0.0% (mode shapes) 0.5 5.0 91.0 1.0 10.0 88.3 2.0 15.0 32.3 5 40 2.4
10.1 43.5 81.0 13.2
47.0 42.3 3.6 0.16
(b) Uncertainties in training data = 0.5% (frequencies) and 5.0% (mode shapes) 0 0 60.5 0.5 5.0 99.9 1.0 10.0 98.6 2.0 15.0 45.3 5 40 12.2
31.0 3.2 11.9 53.3 49.2
(c) Uncertainties in training data = 1.0% (frequencies) and 10.0% (mode shapes) 0 0 31.2 0.5 5.0 78.1 1.0 10.0 100.0 2.0 15.0 75.4 5 40 25.2
4
5
6
43.2 63.6 47.2 48.4
3.0 77.0 35.2 52.3
55.1 36.1 47.0 48.0
60.2 8.2 45.0 39.0 20.3
64.2 99.0 74.6 79.8 18.1
62.0 7.0 41.2 22.1 11.3
49.2 10.1 43.2 4.2 32.1
24.3 18.5 17.3 22.9 11.2
27.0 43.8 3.4 77.0 23.2
52.1 98.3 96.0 46.8 50.7
60.3 40.5 17.4 40.8 46.1
37.4 6.0 4.1 5.2 32.1
(d) Uncertainties in training data = 2.0% (frequencies) and 15.0% (mode shapes) 0 0 28.1 46.9 0.5 5.0 35.4 28.1 1.0 10.0 76.8 24.1 2.0 15.0 92.1 4.0 5 40 37.4 4.0
27.4 35.2 46.1 5.8 33.4
25.1 59.1 73.9 100 23.4
2.1 54.1 29.2 11.1 31.9
43.0 3.2 32.1 3.3 15.7
(a) Sensor location.
(b) Experimental setup. Fig. 9. Experimental setup and sensor location.
5. Experimental example A two-span reinforced concrete slab with dimensions of 6400 mm × 800 mm × 100 mm was tested. The concrete grade
was 32 and the mass density was 2.55 × 103 kg/m3 . The slab rested on wooden planks placed over three steel UB sections. Two point loads were applied to each span using hydraulic jacks as shown in Fig. 9(a) to induce damage. For the dynamic
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Fig. 10. Segment on the slab.
(a) Crack pattern at level 1.
(b) Crack pattern at level 2.
(c) Crack pattern at level 3. Fig. 11. Crack patterns for different damage level.
test, a 5.4 kg impact hammer and nine sensors (accelerometers) were used. The sensors were placed at 27 points on the slab. Fig. 9(b) illustrates the sensor and impact locations. In this study only the vertical bending mode shapes of the middle slab are taken as the input of the ANN models. Three different load levels were applied by the jacks to produce three damage states. The load levels were (i) 18 kN at the left span (level 1); (ii) 18 kN each at both the left and right span (level 2); and (iii) 32 kN each at both the left and right span (level 3). At levels 2
and 3, the loads at left span were applied first, followed by at the right span. For the purpose of damage detection, the slab is divided into seven segments, as shown in Fig. 10. The crack patterns observed during the experiment for the three damage states are shown in Fig. 11(a)–(c). At level 1, some cracks are clearly seen at the middle of the left span, while no cracks observed in another segment. When the load at level 2 was applied, the cracks in the left span seemed to be more obvious, and a group of small cracks appeared at the bottom of right span. There were
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Fig. 12. ANN prediction without considering uncertainties. Table 7 Frequencies of the slab at different load level Undamaged (Analytical) 1st mode 2nd mode
18.222 28.576
Measurement Undamaged
Level 1
Level 2
Level 3
17.810 25.458
16.910 24.109
15.707 22.275
11.878 13.126
also some cracks noticed at the top of the slab at the middle support. At level 3, more cracks appeared in the left and right span, while the existing cracks were getting more severe. Vibration tests were performed before damage and after each load increment. The tests were carried out without the loads on the slab. DIAMOND software, which runs on MATLAB platform, was used to post-process the data using peak-picking method to obtain the natural frequencies and mode shapes. Only the first two modes were clearly identified from the test. The measured modal frequencies at different damage states are given in Table 7. It is clearly seen that the frequencies reduced at every load level. 5.1. Damage identification First, damage identification is performed using ANN model trained using deterministic data. Modal analysis using FEM are conducted to generate training data. The RC slab is modelled with shell elements with 189 nodes and 156 elements. The supports are idealized as simply supported. The slab is divided into seven segments and every element within the same segment is assumed to have the same material properties. The material properties used are: E = 3.3 × 1010 N/mm2 , ρ = 2.45 × 103 kg/m3 , v = 0.2. The undamaged analytical frequencies for the first two modes are also listed in Table 7. For training, 2000 damage cases are used. Frequencies and mode shapes for the first two modes and the E values for every segment are used as the input and output respectively. The best ANN model obtained has 82 hidden neurons. For
damage identification, the experimental data recorded at the three damage levels are introduced to the trained ANN model. Fig. 12 shows the ANN output for those three damage states. The results show that the deterministic ANN model is unable to provide accurate predictions for those three damage states. For all levels, the damages are incorrectly identified at segments 1 and 7, while underestimated at segment 5. Second, using the procedure described earlier, the damage identification is performed using the statistical ANN model with the same ANN architecture as for the deterministic ANN. By assuming that the noise levels of the measured modal data and FE model are 2% (frequencies) and 15% (mode shapes), the PDEs of the slab are obtained. Fig. 13 illustrated the PDEs for those three damage levels. It is seen that the damages identified using the proposed statistical method are closer to the observed patterns than with the deterministic method. The PDEs at level 1 show that the highest value is obtained at segment 2 which is the observed damage location, while the PDEs for the other segments are low. The result at damage level 2 shows that the damages at right span and at the middle support are not identified. This may be due to the damage that occurred in this span at this load level being insignificant. However, at level 3 better agreement is obtained, and the damage in both spans is identified with high confidence. Generally these results demonstrate that the statistical approach provides more reliable prediction of damage occurrence than the common deterministic ANN model by taking into consideration the uncertainties present in the real data when performing the training. 6. Conclusion This study presented a statistical ANN method that accounts for the inevitable FE modelling error and measurement noise for structural condition identification. Rosenblueth’s point estimation method is used to derive the statistical ANN model and to identify the structural condition. Both the modelling error
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Fig. 13. Probability of damages existence for slab using statistical ANN.
and measurement noise are assumed to have normal distribution and zero means. The accuracy of the statistical approach was proved using Monte Carlo simulation. Using this method, the probability of damage existence can be estimated. The numerical and experimental results demonstrated that, compared with the normal ANN approach, the statistical ANN approach gives more reliable identification of structural damage. Acknowledgements The authors would like to acknowledge the partial financial support from the Australian Research Council and Main Roads Western Australia under grant number LP0453783. Financial support from the Universiti Teknologi Malaysia for the first author to pursue a Ph.D. study in Australia is also gratefully acknowledged. References [1] Cawley P, Adams RD. The location of defects in structures from measurements of natural frequencies. Journal of Strain Analysis 1979; 14(2):49–57. [2] Wu X, Ghaboussi J, Garrett JH. Use of neural networks in detection of structural damage. Computers & Structures 1992;42(4):649–59. [3] Pandey PC, Barai SV. Multilayer perceptron in damage detection of bridge structures. Computers & Structures 1995;54(4):597–608. [4] Zhao J, Ivan JN, DeWolf JT. Structural damage detection using artificial neural network. Journal of Infrastructure Systems 1998;4(3):93–101. [5] Zapico JL, Worden K, Molina FJ. Vibration-based damage assessment in steel frames using neural networks. Journal of Smart Materials and Structures 2001;10:553–9.
[6] Yeung WT, Smith JW. Damage detection in bridges using neural networks for pattern recognition of vibration signatures. Engineering Structures 2005;27(5):685–98. [7] Sahin M, Shenoi RA. Quantification and localisation of damage in beamlike structures by using artificial neural networks with experimental validation. Engineering Structures 2003;25(14):1785–802. [8] Yun C-B, Bahng EY. Substructural identification using neural networks. Computers & Structures 2000;77(1):41–52. [9] Lee JJ, et al. Neural network-based damage detection for bridges considering errors in baseline finite elements models. Journal of Sound and Vibration 2005;280(3–5):555–78. [10] Matsuoka K. Noise injection into inputs in back-propagation learning. IEEE Transaction of System, Man and Cybernatics 1992;22(3):436–40. [11] Papadopoulos L, Garcia E. Structural damage identification: A probabilistic approach. AIAA Journal 1998;39(11):2137–45. [12] Rosenblueth E. Point estimates for probability moments. Proceedings of the National Acedemy of Science 1975;72(10):3812–4. [13] Bakhary N, Hao H, Deeks A. Vibration based damage detection using artificial neural network with consideration of uncertainties. In: Proceeding of structural faults and repair, structural faults + repair 2006. Edinburgh: Engineering Technics Press; 2006. [14] Prechelt L. Early stopping — but when? In: Orr GB, Muller OR, editors. Neural networks: Tricks of the trade. Berlin (Germany): Springer-Verlag Telos; 1999. p. 57–69. [15] Xia Y, et al. Damage identification of structures with uncertain frequency and mode shape data. Earthquake Engineering and Structural Dynamics 2002;31(5):1053–66. [16] Helton JC, Davis FJ. Latin hypercube sampling and the propagation of uncertainty in analyses of complex system. Reliability Engineering and System Safety 2003;81:23–69. [17] Maren AJ, Jones D, Franklin S. Configuring and optimizing the backpropagation network. In: Maren A, Harston C, Pap R, editors. Handbook of neural computing applications. San Diego (CA): Academic press; 1990. [18] Bishop CM. Neural network for pattern recognition. Oxford: Oxford University Press; 1995.
Damage detection using artificial neural network with consideration of uncertainties Norhisham Bakhary ∗ , Hong Hao, Andrew J. Deeks School of Civil and Resource Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia Received 27 October 2006; received in revised form 8 January 2007; accepted 8 January 2007 Available online 12 March 2007
Abstract Artificial Neural Networks (ANN) have received increasing attention for use in detecting damage in structures based on vibration modal parameters. However, uncertainties existing in the finite element model used and the measured vibration data may lead to false or unreliable output result from such networks. In this study, a statistical approach is proposed to take into account the effect of uncertainties in developing an ANN model. By applying Rosenblueth’s point estimate method verified by Monte Carlo simulation, the statistics of the stiffness parameters are estimated. The probability of damage existence (PDE) is then calculated based on the probability density function of the existence of undamaged and damaged states. The developed approach is applied to detect simulated damage in a numerical steel portal frame model and also in a laboratory tested concrete slab. The effects of using different severity levels and noise levels on the damage detection results are discussed. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Damage detection; Neural networks; Uncertainties; Rosenblueth’s point estimate; Random noise; Modal data
1. Introduction Various structural health monitoring techniques have been researched in order to obtain a reliable and efficient approach to increase the safety of civil engineering structures. Many different techniques have been proposed and investigated ranging from application of electrical impedance techniques to structural dynamics approaches. Among these techniques, structural dynamics approaches have been extensively explored by many researchers, as they can provide information on unforeseen potential failure mechanisms. Since the earliest work by Cawley and Adams [1], dynamics parameters such as natural frequencies, and mode shapes have been widely applied for damage detection, as the modal parameters are functions of structural properties. This implies that any degradation of the structural properties results in changes of the modal parameters. Artificial Neural Networks (ANN) have been utilized by many researchers to identify damage location and severity from various types of input and output variables, as they provide an efficient tool for pattern recognition. Wu et al. [2] explored the use of an ANN to detect member damage in a 3-storey frame. ∗ Corresponding author. Tel.: +61 8 6488 3072; fax: +61 8 6488 1044.
E-mail address: [email protected] (N. Bakhary). c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.01.013
Pandey and Barai [3] provided a more detailed treatment of ANN architecture in their study identifying damage in a 21bar bridge truss. Zhao et al. [4] applied a counter-propagation neural network to detect damage and support movement in a continuous beam. Zapico [5] developed a procedure for damage assessment of steel structures based on ANN. Most of the studies concluded that ANNs are capable of providing correct damage identification, especially when the structural damage and the associated changes in vibration properties are simulated numerically and are error free. However, in practice uncertainties in the FE model parameters and modelling errors are inevitable. The existence of modelling error in the FE model due to the inaccuracy of physical parameters, nonideal boundary conditions, finite element discretization and nonlinear structural properties may result in the vibration parameters generated from such a FE model not exactly representing the relationship between the modal parameters and the damage parameters of the real structure. On the other hand, the existence of measurement error in the measured data that is normally used as testing data in an ANN model is also unavoidable. Since the efficiency of an ANN prediction relies on the accuracy of both components, the existence of these uncertainties may result in false and inaccurate ANN predictions. Therefore, the impact of uncertainties on the
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reliability of ANN models for structural damage detection needs to be analysed. Several studies have considered uncertainty in application of ANN [6–9], and the noise injection learning method proposed by Matsuoka [10] has been a popular method of considering the uncertainties in ANN applications. Through this method, certain amounts of noise are applied to the training data to consider either modelling error or measurement error. However, studies that consider both errors in the FE model and noises in measurement data are quite limited. The objective of this paper is to study the influence of uncertainty on damage identification using a combination of frequency and mode shape as the input variables. To consider the uncertainties in the FE modelling and the measurement data, an approach introduced by Papadopoulos and Garcia [11] is applied. Using this method, the probability of damage existence (PDE) can be estimated by comparing the probability distribution of the undamaged and damaged models. To consider the effect of FE modelling error, a statistical ANN model is trained with vibration data generated from the FE model, but smeared with random variations. To include the effect of noise in the measurement data, the testing data used as input to the statistical ANN model for damage identification are also smeared with random noises. The probability moments of the undamaged and damaged states of the structural parameters are estimated using the point estimation method and verified by Monte Carlo simulation. The Monte Carlo simulation data is also used to determine the type of probability distribution function of the structural parameters of both the undamaged and damaged states. The PDEs are determined from the probability distribution for each structural member. A numerical steel frame model and a laboratory tested two-span reinforced concrete slab are used to demonstrate the developed procedure. Some parametric calculations are also performed to investigate the influences of using different noise levels and damage severities on the damage identification results. In this study, only random errors are considered. The systematic errors which may also exist, especially in FE model are not considered in the present study. 2. Theoretical background In this study, the uncertainties in the FE model and the measurement data are assumed to be normally distributed independent random variables with zero means and specified coefficients of variation. This implies that the parameters with uncertainties are equal to true values plus the random variations. Thus, the frequencies, mode shapes and stiffness parameters for training and testing are derived as below: λi = λi0 + λi0 X λi = λi0 (1 + X λi )
(1a)
λˆ i = λˆ i0 + λˆ i0 X λi = λˆ i0 (1 + X λi )
(1b)
φi = φi0 + φi0 X φi = φi0 (1 + X φi )
(1c)
φˆ i = φˆ i0 + φˆ i0 X φi α j = α 0j + α 0j X α j
(1d)
= φˆ i0 (1 + X φi ) = α 0j (1 + X α j )
(1e)
Table 1 Training functions and testing variables used in point estimation method Model
Training function
Testing variable Input
Output
1
α j++ = fn(λi0 + σλi , φi0 + σφi )
λˆ i0 + σλi , φˆ i0 + σφi
αˆ ++
2
α j−− = fn(λi0 − σλi , φi0 − σφi ) α j+− = fn(λi0 + σλi , φi0 − σφi ) α j−+ = fn(λi0 − σλi , φi0 + σφi )
λˆ i0 − σλi , φˆ i0 − σφi
αˆ −−
λˆ i0 + σλi , φˆ i0 − σφi
αˆ +−
λˆ i0 − σλi , φˆ i0 + σφi
αˆ −+
3 4
where λi , φi and λˆ i , φˆ i are the ith frequencies and mode shapes for training and testing, respectively, and α j is a parameter related to the stiffness of the jth segment. The change of stiffness parameters is described by the reduction of Young’s modulus (E values). Stiffness Reduction Factor (SRF) defined as the ratio of E value change to the initial E value is used as the damage indicator. Superscript ‘0’ represents the corresponding mean value and X λi , X φi , X αi are the zero mean normally distributed random variables with specific COV in frequencies, mode shapes and stiffness parameters, which are assumed in this study to be the same for both training and testing data. In this study, both the training and testing data sets contain randomly varying errors or noises. Because of the randomly varying training data set, the trained ANN model will be statistical with probability distribution parameters such as mean and standard deviation for each variable. Moreover, since the testing data set is random, the ANN prediction of the condition of each variable, in this case the Young’s modulus of each structural member, is also statistical. The most direct method to obtain the probability parameters is Monte Carlo simulation. However, Monte Carlo simulation requires high computational effort and is very time consuming. Another approach is to attempt a theoretical solution, but it is very difficult to obtain a closed form solution of probability distributions for such a multi degree of freedom dynamic system, and derivation of such a solution is beyond the scope of this study. In this study the probability moments of the undamaged and damaged states of the structural parameters are approximated using the point estimation method introduced by Rosenblueth [12]. By using this method, the mean values E(α) and standard deviation σ (α) of each variable can be calculated based on the upper limit (αi+ ) and the lower limit (αi− ) of the function corresponding to the ith variable. The upper and lower limit can be obtained by using mean plus one standard deviation and mean minus one standard deviation of each random variable in training and testing of the ANN model. Since two variables are used (frequency and mode shape) in this study, two upper limits (α++ , α−+ ) and two lower limit (α−− , α+− ) need to be obtained. Thus, four ANN models are developed by considering the mean values and standard deviations (σ ) of the random noises applied to each variable. The training functions and the input and output variables for testing are listed in Table 1. The relationship between input and output for ANN model is denoted by fn(·). The expectation (mean value, µα ) and standard deviation (σα ) of α can then be approximately calculated by:
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Fig. 1. Probability density functions for α j and α 0j and probability of damage j existence, Pd .
µα = E(α) =
1 (αˆ ++ + αˆ −− + αˆ +− + αˆ −+ ) 4 1
σα = [E(α 2 ) − [E(α)]2 ] 2
(2) (3)
where E(α 2 ) is calculated using Eq. (2) with α 2 terms substituted for the α terms. Using this approach substantially reduces the computational time in deriving the statistical mean and standard deviation of each stiffness parameter for structural condition monitoring. The PDE can be estimated from statistical distributions of the stiffness parameters of the undamaged and damaged models. For example, if the stiffness parameter (α j ) of the undamaged segment j is normally distributed with mean E(α j ) and standard deviation σ (α j ), the probability density function can be obtained as illustrated in Fig. 1, where L α j is the lower bound of the healthy parameter. In this study, the confidence level is set to 95%, thus the lower bound is L α j = E(α j ) − 1.645σ (α j ), which indicates that there is a probability of 95% that the healthy stiffness parameter falls in the range of [E(α j )−1.645σ (α j ), ∞]. Similarly, for the stiffness parameter of segment j in the damaged state (α 0j ), the distribution is again assumed as normal with mean E(α 0j ) and standard deviation σ (α 0j ), and the corresponding probability density function is also plotted in Fig. 1. The PDE is defined as the probability of α 0j not being within the 95% confidence healthy interval. Thus the PDE of segment j is j
Pd = 1 − prob(L α j ≤ xα 0 ≤ ∞) = prob(−∞ ≤ xα 0 ≤ L α j ).
(4)
PDE is a value between 0 and 1, and if the PDE of a segment is close to 1, then it is most likely the element is damaged. If the PDE is close to 0, damage existing in the element is very unlikely [13]. It should be noted that the stiffness parameters of the undamaged and damaged state have normal distributions because the random variations in Eq. (1) are assumed as zero mean normally distributed random variables. This will be proved later.
Fig. 2. Finite element model of the frame.
The input data consist of natural frequencies and mode shapes, and the output layers consists of Young’s modulus (E values) to represent the stiffness parameter. The change of the stiffness parameter or the damage severity for each segment are denoted by a Stiffness Reduction Ratio (SRF), defined as SRF = 1 −
E0 E
(5)
where E is the Young’s modulus in the intact state and E 0 is that at the damage level of interest. In most ANN applications for damage detection, the training data are obtained from FE analysis, which involved generating large number of damage cases based on an initial baseline FE model. Once the ANN model is well-trained, the testing data are then applied to the ANN model to obtain the locations and severities of any damages. In most of the previous studies, both training and testing data are assumed to be free from modelling and measurement error. In practice, however, modelling error and measurement noise are inevitable. According to Xia [15], the inaccuracy due to modelling and measurement error can be overcome by taking into account the uncertainties through a statistical method. In this study, modelling error and measurement noise are assumed to be normally distributed with zero means and specific variance. The noise is applied in terms of coefficient of variations (COV). The statistical properties of E value for each segment are obtained by using Rosenblueth’s point estimation method verified by Monte Carlo simulation. A Kolmorogov–Smirnov goodness-offit test (K–S test) is then applied to verify the distribution type of E values. This is followed by calculation of the PDE of E values for each segment.
3. Methodology 4. Numerical example A multilayer perceptron with Levenberg–Marquardt backpropagation algorithm is utilized to train the ANN model in this study. Sigmoid functions are employed as nonlinear activation functions for all layers. To reduce the effect of overfitting, the early-stopping method [14] is applied during the training.
To demonstrate the proposed method, the single span steel portal frame shown in Fig. 2 is used as an example. The cross section of beam is 40.50 × 6.0 mm2 , and column is 50.50 × 6.0 mm2 . The span length and height of the frame are
N. Bakhary et al. / Engineering Structures 29 (2007) 2806–2815 Table 2 E values for scenario 1 and scenario 2 Segment
1
2
3
4
5
6
Scenario 1 Scenario 2
0.4× E 0.4× E
1.0× E 1.0× E
1.0× E 0.3× E
0.2× E 1.0× E
1.0× E 0.4× E
1.0 × E 0.3 × E
Table 3 First three frequencies values for scenario 1 and scenario 2
Mode 1 Mode 2 Mode 3
Undamaged
Scenario 1
Scenario 2
4.628 16.112 20.649
3.9373 12.567 16.491
3.530 11.269 14.891
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the actual values. The results show that the ANN model is able to provide good output, which indicates that the ANN model is reliable in detecting damage from noise-free data. To investigate the reliability of the ANN model in predicting structural damage with noisy data, 2% and 15% random noise are applied respectively to the frequencies and mode shapes of the testing data. Applying these errors implies that the measured data is no longer exact. These data are then introduced to the trained ANN model. Fig. 5 shows the ANN output. The figure shows that for scenario 1, the false damage identification occurs at segments 2 and 3, while the stiffness of segment 6 is overpredicted. The same situation occurred in scenario 2, where the ANN model falsely identifies damage at segment 2, overestimates damage at segments 1 and 3, but underestimates at segment 6. This indicates that the common ANN model trained with simulated vibration parameters from finite element model cannot give reliable structural damage prediction if the errors exist. 4.1. Statistical artificial neural network
Fig. 3. First three mode shapes for undamaged, scenario 1 and scenario 2.
both 1000 mm. Rigid connections are applied between the beam and the columns, and the supports are assumed as clamped. 11 2 The material properties used are: E = 2.1 × 10 N/m , ρ = 7.67 × 103 , v = 0.2. The frame is modelled with 10 elements in each member. To reduce the complexity of ANN training, the frame is divided to six segments as shown in the figure. Each segment consists of five elements. Modal analysis is conducted using the FE model to generate input and output data to train the ANN models. Two damage scenarios are generated to assess the ANN prediction performance. Scenario 1 consists of damage in two segments of the frame (1 & 4), and scenario 2 consists of damage in four segments (1, 3, 5 & 6). The damage is imposed by reducing the E values of each corresponding segment. Table 2 shows the E values for scenario 1 and scenario 2. The frequencies and mode shapes of the first three modes are shown in Table 3 and Fig. 3. To train the ANN model, 1200 data sets are generated from the FE model based on the Latin hypercube sampling method [16]. To apply the early-stopping method, the data are divided into three parts in a ratio of 2:1:1. A trial and error method based on Kalmorogov and Lippmann’s approach [17] was utilized to attain the best ANN topology. To reduce the ‘curse of dimensionality’ as discussed by Bishop [18], only nine mode shape points and frequencies for the first three modes were used as the input parameters. The output parameters are Young’s modulus (E values) of every segment. The numbers of neurons in the input and output layers are the same as the respective number of input and output variables. The best ANN model is obtained with 97 hidden neurons. The trained ANN model is then assessed by introducing the modal parameters of the two damage scenarios mentioned above. Fig. 4 shows the predicted E values in comparison with
As demonstrated above, noise in measured data may lead to unreliable and false prediction of structural damage. To consider the uncertainties in the FE model and in the measured data, normally distributed random noises with zero means and specific variance are added to training data. The noise levels of the frequencies and mode shapes are also assumed to be 2% and 15% respectively. These data are then used to train the ANN model. A statistical ANN model is developed using the Rosenblueth’s point estimation method. To verify the reliability of the Rosenblueth’s point estimation method, the mean values and standard deviations of every segment in the undamaged state are obtained by using both the point estimation method and Monte Carlo simulation. In Monte Carlo simulation, randomly distributed training data and testing data are generated and used to predict structural damage. The training and testing process is repeated until the mean values and standard deviations of E values converge. The Monte Carlo simulations converge after about 125 iterations. Fig. 6 shows the Monte Carlo simulation result. The Monte Carlo simulation data are also used to determine the distribution type of the E value. A Kolmorogov–Smirnov goodness-of-fit test (K–S test) is applied to verify the accuracy of the selected distribution type of E values. The results show that the E values have normal type of characteristic. Fig. 7 illustrates the cumulative distribution function (CDF) for segment 3 as compared with the theoretical one. Based on the goodness-of-fit test, a normal distribution hypothesis of the E values for all segments is accepted with a confidence level of 95%. Fig. 8 shows the mean values and COV of E values obtained from both methods. It is observed that both methods provide quite similar results, indicating the point estimation method is reliable. Using the trained ANN model with 2% and 15% random errors (COV) in frequencies and mode shapes, and the testing data with the same level of noise, the mean values and standard deviations of structural stiffness parameters corresponding to the two damage scenarios are estimated. From the normally
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Fig. 4. ANN prediction for scenario 1 and scenario 2 compared to the actual value using noise-free input data.
Fig. 5. ANN prediction for scenario 1 and scenario 2 compared to the actual value using noisy input data.
Fig. 6. Monte Carlo simulation result. Table 4 Probability of damage existence (%) for every segment of scenario 1 and scenario 2
Scenario 1 Scenario 2
Fig. 7. K–S test result for segment 3.
Segment 1 2
3
4
5
6
96.0 99.7
3.0 100.0
99.8 13.5
0.00 99.7
0.00 99.9
38.5 1.0
distributed probability density function of the damaged and undamaged states, the PDEs can be calculated. The PDEs for scenario 1 and scenario 2 are listed in Table 4. From Table 4, it is observed that in scenario 1, the PDEs of segments 1 and 4 are very high and the PDEs of the other segments are low. This indicates that it is very likely damage exists in segments 1 and 4 only. For scenario 2, the highest
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Fig. 8. Mean values and coefficients of variation (COV) of E values in undamaged state. Table 5 Probability of damage existence (%) for different damage severities Severity level (%) −25 −10 −7 −5 −2
Segment 1
2
3
4
5
6
100.0 89.1 62.6 38.0 10.0
14.3 3.5 3.2 3.0 2.7
0.0 4.5 3.9 3.5 2.9
100.0 71.2 47.1 29.3 8.9
5.8 8.9 6.5 5.1 3.4
0.0 1.5 1.8 2.0 2.3
PDE occurred at segments 1, 3, 5 and 6, which are also the true damage locations. These results show that using the proposed statistical ANN model, the damages are detected with high confidence and undamaged segments are less likely to be falsely identified, as compared to the common ANN model. However, further investigation needs to be conducted in order to see the sensitivity of different uncertainty levels to the result, so does the damage severities. 4.2. Parametric study More detailed studies are carried out in this part to investigate the sensitivity of the proposed method to different damage levels and different uncertainties levels. To assess the reliability of the proposed method at different damage levels, an analysis is performed by varying the damage level, while the uncertainty level remains unchanged. Damage is introduced to segments 1 and 4 and results generated using the FE model with SRF equal to −25%, −10%, −7%, −5% and −2%. The uncertainty level is constant at 1% for frequencies, and 10% for mode shapes. By using the same method as described above, the PDE of each segment is calculated. The PDEs are listed in Table 5. It is observed that the highest PDEs occurred at the correct damage location (segments 1 and 4) for all cases. The results also show the highest PDEs are gained with a −25% damage level and the lowest are with a −2% damage. The results indicate that when the damage is severer, it can be confidently identified. However, when the damage is not significant, the damage cannot be detected confidently, although the damaged segments always have higher PDEs than the undamaged segments. The reason is that when the damage levels are small, the changes of modal parameters are not so apparent, and the influences of modelling and measurement noises are comparatively more significant.
In order to investigate the effect of different uncertainty levels on the proposed damage identification method, four ANN models are developed. Each model is trained using the training data that is generated from the same baseline FE model but smeared with different levels of uncertainties in frequencies and mode shapes respectively. These levels are (1) 0.0% and 0.0%, (2) 0.5% and 5.0% (3)1.0% and 10.0% and (4) 2.0% and 15.0%. The trained models are then tested with testing data which are also smeared with different levels of uncertainties. In this case, −25% damage levels are applied to segments 1 and 4. Table 6(a)–(d) illustrate the results in terms of PDE of different combinations of uncertainties in training and testing data. Table 6(a) lists the PDEs of each segment when the ANN model is trained with deterministic data. It is evident that the ANN model trained with deterministic data performed poorly when tested with noisy data. For example, the highest PDEs are observed at segments 1 and 6 when the network is tested with 0.5% (frequencies) and 5% (mode shapes) uncertainties. The ANN model gets more unreliable when the testing data contain higher noise. Table 6(b) shows the results obtained when the ANN model was trained with noisy data with COV 0.5% for frequencies and 5% for mode shapes. From the results, it is observed that in most cases the highest PDEs are obtained at segments 1 and 4, which indicates that the damage is correctly identified. The results show that the highest PDEs are obtained when the testing data also has noise with COV 0.5% for frequencies and 5% for mode shapes, as might be expected. When the measured data contains a significantly higher noise level than the training data, i.e. 5% in frequency and 40% in mode shape, the method, nevertheless, fails to give correct identification of damage locations. The same situation occurs when higher uncertainties are applied to the training data. Table 6(c) and (d) show the results when the training data are smeared with 1% and 10%, and 2% and 15% noise in frequencies and mode shapes respectively. In most cases the damage can be detected correctly. The highest confidence levels are obtained when the noise level in the training and testing data are the same. It is also clear that the PDEs values decreased when the difference between the uncertainty levels in training and testing data increased. This may indicate that if the uncertainties in either the training data or the testing data are larger than each other, the true information is submerged in noise, and thus the actual damage location cannot be identified accurately.
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Table 6 Probability of damage existence (%) for different combination of uncertainties in training and testing data Uncertainties level in testing data Frequencies (%) Mode shapes (%)
Segment 1
2
3
(a) Uncertainties in training data = 0.0% (frequencies) and 0.0% (mode shapes) 0.5 5.0 91.0 1.0 10.0 88.3 2.0 15.0 32.3 5 40 2.4
10.1 43.5 81.0 13.2
47.0 42.3 3.6 0.16
(b) Uncertainties in training data = 0.5% (frequencies) and 5.0% (mode shapes) 0 0 60.5 0.5 5.0 99.9 1.0 10.0 98.6 2.0 15.0 45.3 5 40 12.2
31.0 3.2 11.9 53.3 49.2
(c) Uncertainties in training data = 1.0% (frequencies) and 10.0% (mode shapes) 0 0 31.2 0.5 5.0 78.1 1.0 10.0 100.0 2.0 15.0 75.4 5 40 25.2
4
5
6
43.2 63.6 47.2 48.4
3.0 77.0 35.2 52.3
55.1 36.1 47.0 48.0
60.2 8.2 45.0 39.0 20.3
64.2 99.0 74.6 79.8 18.1
62.0 7.0 41.2 22.1 11.3
49.2 10.1 43.2 4.2 32.1
24.3 18.5 17.3 22.9 11.2
27.0 43.8 3.4 77.0 23.2
52.1 98.3 96.0 46.8 50.7
60.3 40.5 17.4 40.8 46.1
37.4 6.0 4.1 5.2 32.1
(d) Uncertainties in training data = 2.0% (frequencies) and 15.0% (mode shapes) 0 0 28.1 46.9 0.5 5.0 35.4 28.1 1.0 10.0 76.8 24.1 2.0 15.0 92.1 4.0 5 40 37.4 4.0
27.4 35.2 46.1 5.8 33.4
25.1 59.1 73.9 100 23.4
2.1 54.1 29.2 11.1 31.9
43.0 3.2 32.1 3.3 15.7
(a) Sensor location.
(b) Experimental setup. Fig. 9. Experimental setup and sensor location.
5. Experimental example A two-span reinforced concrete slab with dimensions of 6400 mm × 800 mm × 100 mm was tested. The concrete grade
was 32 and the mass density was 2.55 × 103 kg/m3 . The slab rested on wooden planks placed over three steel UB sections. Two point loads were applied to each span using hydraulic jacks as shown in Fig. 9(a) to induce damage. For the dynamic
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Fig. 10. Segment on the slab.
(a) Crack pattern at level 1.
(b) Crack pattern at level 2.
(c) Crack pattern at level 3. Fig. 11. Crack patterns for different damage level.
test, a 5.4 kg impact hammer and nine sensors (accelerometers) were used. The sensors were placed at 27 points on the slab. Fig. 9(b) illustrates the sensor and impact locations. In this study only the vertical bending mode shapes of the middle slab are taken as the input of the ANN models. Three different load levels were applied by the jacks to produce three damage states. The load levels were (i) 18 kN at the left span (level 1); (ii) 18 kN each at both the left and right span (level 2); and (iii) 32 kN each at both the left and right span (level 3). At levels 2
and 3, the loads at left span were applied first, followed by at the right span. For the purpose of damage detection, the slab is divided into seven segments, as shown in Fig. 10. The crack patterns observed during the experiment for the three damage states are shown in Fig. 11(a)–(c). At level 1, some cracks are clearly seen at the middle of the left span, while no cracks observed in another segment. When the load at level 2 was applied, the cracks in the left span seemed to be more obvious, and a group of small cracks appeared at the bottom of right span. There were
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Fig. 12. ANN prediction without considering uncertainties. Table 7 Frequencies of the slab at different load level Undamaged (Analytical) 1st mode 2nd mode
18.222 28.576
Measurement Undamaged
Level 1
Level 2
Level 3
17.810 25.458
16.910 24.109
15.707 22.275
11.878 13.126
also some cracks noticed at the top of the slab at the middle support. At level 3, more cracks appeared in the left and right span, while the existing cracks were getting more severe. Vibration tests were performed before damage and after each load increment. The tests were carried out without the loads on the slab. DIAMOND software, which runs on MATLAB platform, was used to post-process the data using peak-picking method to obtain the natural frequencies and mode shapes. Only the first two modes were clearly identified from the test. The measured modal frequencies at different damage states are given in Table 7. It is clearly seen that the frequencies reduced at every load level. 5.1. Damage identification First, damage identification is performed using ANN model trained using deterministic data. Modal analysis using FEM are conducted to generate training data. The RC slab is modelled with shell elements with 189 nodes and 156 elements. The supports are idealized as simply supported. The slab is divided into seven segments and every element within the same segment is assumed to have the same material properties. The material properties used are: E = 3.3 × 1010 N/mm2 , ρ = 2.45 × 103 kg/m3 , v = 0.2. The undamaged analytical frequencies for the first two modes are also listed in Table 7. For training, 2000 damage cases are used. Frequencies and mode shapes for the first two modes and the E values for every segment are used as the input and output respectively. The best ANN model obtained has 82 hidden neurons. For
damage identification, the experimental data recorded at the three damage levels are introduced to the trained ANN model. Fig. 12 shows the ANN output for those three damage states. The results show that the deterministic ANN model is unable to provide accurate predictions for those three damage states. For all levels, the damages are incorrectly identified at segments 1 and 7, while underestimated at segment 5. Second, using the procedure described earlier, the damage identification is performed using the statistical ANN model with the same ANN architecture as for the deterministic ANN. By assuming that the noise levels of the measured modal data and FE model are 2% (frequencies) and 15% (mode shapes), the PDEs of the slab are obtained. Fig. 13 illustrated the PDEs for those three damage levels. It is seen that the damages identified using the proposed statistical method are closer to the observed patterns than with the deterministic method. The PDEs at level 1 show that the highest value is obtained at segment 2 which is the observed damage location, while the PDEs for the other segments are low. The result at damage level 2 shows that the damages at right span and at the middle support are not identified. This may be due to the damage that occurred in this span at this load level being insignificant. However, at level 3 better agreement is obtained, and the damage in both spans is identified with high confidence. Generally these results demonstrate that the statistical approach provides more reliable prediction of damage occurrence than the common deterministic ANN model by taking into consideration the uncertainties present in the real data when performing the training. 6. Conclusion This study presented a statistical ANN method that accounts for the inevitable FE modelling error and measurement noise for structural condition identification. Rosenblueth’s point estimation method is used to derive the statistical ANN model and to identify the structural condition. Both the modelling error
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Fig. 13. Probability of damages existence for slab using statistical ANN.
and measurement noise are assumed to have normal distribution and zero means. The accuracy of the statistical approach was proved using Monte Carlo simulation. Using this method, the probability of damage existence can be estimated. The numerical and experimental results demonstrated that, compared with the normal ANN approach, the statistical ANN approach gives more reliable identification of structural damage. Acknowledgements The authors would like to acknowledge the partial financial support from the Australian Research Council and Main Roads Western Australia under grant number LP0453783. Financial support from the Universiti Teknologi Malaysia for the first author to pursue a Ph.D. study in Australia is also gratefully acknowledged. References [1] Cawley P, Adams RD. The location of defects in structures from measurements of natural frequencies. Journal of Strain Analysis 1979; 14(2):49–57. [2] Wu X, Ghaboussi J, Garrett JH. Use of neural networks in detection of structural damage. Computers & Structures 1992;42(4):649–59. [3] Pandey PC, Barai SV. Multilayer perceptron in damage detection of bridge structures. Computers & Structures 1995;54(4):597–608. [4] Zhao J, Ivan JN, DeWolf JT. Structural damage detection using artificial neural network. Journal of Infrastructure Systems 1998;4(3):93–101. [5] Zapico JL, Worden K, Molina FJ. Vibration-based damage assessment in steel frames using neural networks. Journal of Smart Materials and Structures 2001;10:553–9.
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