EACS 2012 – 5th European Conference on Structural Control
Genoa, Italy – 18-20 June 2012 Paper No. #008
Structural Reliability Updating with Stochastic Subspace Damage Detection Information Sebastian THÖNS* BAM Federal Institute for Materials Research and Testing Unter den Eichen 87, 12205 Berlin, Germany [email protected]
Michael DÖHLER Inria Rennes – Bretagne Atlantique Campus de Beaulieu, 35042 Rennes, France [email protected]
ABSTRACT Damage detection algorithms as a part of Structural Health Monitoring (SHM) are widely applied in research and industry and have shown their capabilities to efficiently detect structural damages. These algorithms usually compare a model from a safe reference state of a structure to vibration data from a possibly damaged state. For such a comparison, special properties of real vibration data introduce uncertainties, such as low signal-to-noise ratios, non-stationary or nonwhite ambient excitation, non-linear behavior and many more. Recently, statistical damage detection algorithms based on stochastic subspace identification have been proposed that take into account the uncertainties in the data. Building upon the uncertainty modeling, the next step in the view of the authors is to utilize damage detection algorithm information in the context of the structural reliability theory. Therefore, this paper introduces an approach for the updating of the structural reliability with damage detection algorithm information. Two steps are described namely the determination of a probability of detection (PoD) distribution function for damage detection algorithms accounting for the relevant uncertainties and the concept of Bayesian updating of the structural reliability. The introduced approaches are applied in generic examples. In this way the potential of the utilization of damage detection system information for more reliable structural systems are demonstrated. Keywords: Stochastic subspace, structural reliability, Bayesian updating, probability of detection
1 INTRODUCTION In order to detect changes in the modal parameters linked to the structural parameters, often eigenstructure identification results are used and evaluated for changes, e.g. , but their automatic estimation and matching from measurements of different states of the structure might require an extensive preprocessing step. In recent years a subspace-based output-only damage detection method was developed -, where a χ²-type test is used to analyze changes in the dynamic response of the system, instead of identifying the modal parameters. The parameters of the test are obtained from vibration data recorded in the reference state, while the value of a χ²-test computed on data from the possibly damaged state is compared to a threshold to decide if damage occurred. 1 *
The method has been successfully applied to several laboratory and real applications from aerospace as well as mechanical and civil engineering, e.g. in -. Despite the outlined developments of damage detection procedures, they are rarely applied in conjunction with the structural reliability theory . However, the idea to employ damage detection algorithms for a reliability evaluation of structures was e.g. formulated by Yao and Natke in 1994 . Yao and Natke based the condition assessment of a structural system entirely on the system identification approaches for damage detection and not on the structural reliability theory nor on the Bayesian probability. The starting point of this paper is the perspective of structural reliability and its Bayesian basis. Thus this paper suggests Bayesian updating of the structural reliability with damage detection system information on the basis of an approach known for inspection systems. The paper starts with a summary of the statistical subspace based damage detection in Section 2. Here, the basic procedure and the mathematical basis are described. Section 3 introduces the approach of utilizing damage detection system information in the context of structural reliability. The approach builds upon a well known concept utilizing Bayesian updating. Section 4 introduces a concept for the determination of the probability of detection (PoD) for damage detection algorithms and outlines the relevant uncertainties which should be taken into account. The application of the introduced approaches is performed in Section 5. The paper closes with the conclusions and an outlook for further research. 2 STATISTICAL SUBSPACE BASED DAMAGE DETECTION 2.1 Models and Parameters The behaviour of a mechanical system is assumed to be described by a stationary linear dynamical system MZɺɺ(t ) + CZ (t ) + KZ (t ) = v(t ), Y (t ) = LZ (t ) ,
where t denotes continuous time, M, C and K are the mass, damping and stiffness matrices, highdimensional vector Z collects the displacements of the degrees of freedom of the structure, the nonmeasured external force v modelled as non-stationary Gaussian white noise, the measurements are collected in the vector Y and matrix L indicates the sensor locations. The eigenstructure of (1) with the modes µ and mode shapes ϕµ is a solution of
det( µ 2 M + µ C + K ) = 0, ( µ 2 M + µ C + K )φµ = 0, ϕ µ = Lφµ . Sampling model (1) at some rate 1/τ yields the discrete model in state-space form X k +1 = FX k + Vk +1 ,
Yk = HX k + Wk ,
whose eigenstructure is given by det( F − λ I ) = 0, ( F − λ I )φλ = 0, ϕλ = H φλ .
Then, the eigenstructure of the continuous system (Equation (1)) is related to the eigenstructure of the discrete system (Equation (2)) by eτµ = λ , ϕ µ = ϕλ . The collection of modes and mode shapes (λ,ϕλ) is a canonical parameterization of system (Equation (2)) and considered as the system parameter θ Λ
θ = , vec Φ
where Λ is the vector whose elements are the eigenvalues λ and Φ is the matrix whose columns are the mode shapes ϕλ. 2.2 Statistical Subspace-based Covariance-Driven Damage Detection 2.2.1
Stochastic Subspace Identification
To obtain the system parameter θ from measurements (Yk)k=1,…,N, the covariance-driven output-only subspace identification algorithm , can be used. From the data, a block Hankel matrix H is filled with the correlation lags Ri = E(YkYk-iT) of the output data
R1 R2 H= ⋮ R p +1
R2 R3 ⋮ Rp+2
⋯ ⋯ ⋱
Rq Rq +1 , R p + q
k =i +1
T k k −i
It possesses the factorization property H = OC
into observability matrix H HF O= ⋮ p HF and controllability matrix C, where O is obtained from H by an SVD and truncation at the desired model order
H = (U1
∆ U0 ) 1
T 1/ 2 V , O = U1∆1 . ∆0
The matrices H and F can be recovered from the observability matrix O to finally obtain the eigenstructure (λ,φλ) of the system (Equation (2)) and system parameter θ in Equations (3)-(4). 2.2.2
Subspace-based Damage Detection
In order to detect changes in the system parameter θ, the statistical subspace-based damaged detection from - is recalled, where a residual function for damage detection is associated with the subspace identification from above. This function compares the system parameter θ0 of a ˆ as in (5) computed on new data corresponding to an reference state with a block Hankel matrix H unknown, possibly damaged state, without actually computing the system parameter θ in the unknown state. Let S be the left null space of the block Hankel matrix (5), which can be obtained from S = U0 in (Equation (6)). Then, the characteristic property of the system in the reference state writes 3
ˆ = 0 if θ =θ0. ST H For checking whether new data agree with the reference state corresponding to θ0, the residual function
ˆ) ζ N = N vec ( S T H
is introduced in ,, where N is the number of samples on which H is computed. It was shown that this residual function is asymptotically Gaussian for large N with zero mean in the reference state and non-zero mean in the damaged state. Let Σ = lim N (ζ N ζ NT ) be the residual covariance matrix. Then, a change in the system parameter θ corresponds to a change in the mean value of the residual function and manifests itself in the variable
χ N2 = ζ NT Σˆ −1ζ N ,
where Σˆ is a consistent estimate of Σ. This variable is used as a damage indicator. It is compared to a threshold t to decide if the measured data agree with the reference state or if the system is damaged. Note that Equation (8) is the empirical version of the damage detection test , where the Jacobian of the residual ζN is not taken into account. 2.2.3
Probability Distribution of the Damage Indicator The damage indicator variable χ N2 is an asymptotically χ 2 -distributed variable with d
degrees of freedom , where d is the dimension of residual ζ N (asymptotic because only a limited number of data samples is used in ζ and Σˆ ). This variable is compared to a threshold to decide N
between reference state and damaged state. In the damaged case, the variable χ N2 is asymptotically non-central χ 2 -distributed with noncentrality parameter δ T Σ −1δ , where δ is the change in ζ N due to damage. This means that the asymptotic distribution of the damage indicator variable has different parameters with every change in the structure. Its asymptotic expected values and variances are: reference state: E( χ 2 ) = d , V ( χ 2 ) = 2d damaged state: E( χ 2 ) = d + δ T Σ −1δ , V ( χ 2 ) = 2d + 4δ T Σ −1δ .
As the dimension d of residual ζ N is usually quite high, the (non-central) χ 2 -distribution can be approximated by a normal distribution.
3 UPDATING OF THE STRUCTURAL RELIABLITY APPLYING DAMAGE DETECTION INFORMATION Structural risks are characterized by low probabilities of failure but high consequences. The structural reliability is characterised by very low failure rates and various failure mechanisms necessitating an explicit modelling of loading and resistances for the individual failure mechanisms. The structural reliability theory in its present form is developed since the 1960s and constitutes a well developed discipline and framework (e.g. , ). 4
The starting point of the consideration is the concept of the updating the structural reliability with inspection system information. The concept was developed in the 1970s and is mentioned e.g. by Tang . Madsen, Skjong et al. introduced this concept within the framework of structural reliablity theory in 1987 . It is widely used in science and in the offshore industry (e.g. ). The concept uses the inspection outcome (i.e. the indication I or no indication event I ) and information about the inspection system namely the probability of detection given a damage (PoD). The inspection information of no indication I at a certain point of time can be utilised to update the probability of a damage P ( D ) of a component (with the damage event D ) calculating the probability of damage given no indication of the inspection system P ( D | I ) applying Bayesian updating (Equation (10)). P(D | I ) =
P ( I | D) P ( D)
For inspection technologies, the PoD is defined as the integral of the probability density of the signal s containing noise, given a damage of size d , from a predefined threshold t to infinity (Equation (11)). The threshold t is set up to produce a low probability of false alarm (PFA), which is defined as the integral of the noise from threshold t to infinity, and to facilitate at the same time a reasonable probability of detection for small damages (e.g. , ). A PoD curve is derived for various damage sizes leading to a continuous distribution which can be interpreted as a cumulative probability distribution function. ∞
P ( I | D ) = PoD = ∫ f ( s | d ) ds
The concept of Bayesian updating of the structural reliability can be applied to updating of structural systems being monitored with a damage detection system consisting of a measurement system and a damage detection (stochastic subspace) algorithm. Equation (10) can be rewritten using I DD ,T denoting no indication by the damage detection system in the period T of the application of the system (Equation (12)). P ( DT | I DD ,T ) =
P ( I DD ,T | D ) P ( D )
P ( I DD ,T )
Equation (12) can be transformed to Equation (13) by the definition of conditional probability. Here, the limit state equation for the damage is denoted with g and the limit state equation for calculating the probability of no indication with g I DD ,T . P ( D | I DD ,T ) =
P ( D ∩ I DD ,T ) P ( I DD ,T )
P g < 0 ∩ g I DD ,T < 0
P g I DD ,T < 0
The limit state function for determining the probability of no indication can be derived with the approaches in  (Equation (14)) with a standard Normal distributed random variable u and with Φ −1 denoting the inverse cumulative normal distribution.
g I DD ,T = Φ −1 F ( I DD ,T | d ) − u
The cumulative probability distribution of indication given a damage F ( I DD ,T | d ) is defined as the probability of detection curve (PoD curve). The quantification of this probability distribution 5
is usually done on an experimental basis with certain artificial defects and/or damaged components (see e.g. ) which is described in the section to follow. 4 APPROACH FOR THE DETERMINATION OF THE PROBABILITY OF DETECTION The determination of a PoD curve for inspection systems involves typically (1) selecting representative structures with statistically significant number of known defects, (2) subjecting the damaged structural specimens an inspection both in the laboratory and in field installations with a representative sample of inspection technicians and inspection conditions, and (3) analyzing the results of the inspection in terms of damage detection probabilities as a function of damage size (see e.g. ). The uncertainties accounted for with this procedure can be summarized and classified in measurement uncertainties of the sensor and amplifier system, in uncertainties of the damage detection algorithm and in human errors. It thus is suggested to determine the PoD curve for the stochastic subspace algorithm (described in Section 2) building upon the outlined procedure. For the determination of the PoD distribution for the stochastic subspace algorithm two steps are necessary consisting of the determination of a threshold t to differentiate between a damaged state and an undamaged state and the determination of the PoD for the individual damage sizes. The approach here is to replace the signal (for inspection technologies, see Section 3) with the damage indicator of the stochastic subspace algorithm. A low PFA, which can be interpreted as the probability of indication by the damage detection algorithm given no damage P ( I DD | D) , is accomplished by determining the threshold t to achieve a high probability of no indication given no damage P ( I DD | D ) = 1 − P ( I DD | D ) = α which can be chosen e.g. as α = 0.99. The following procedure is suggested for the determination of the threshold t: • •
Record several data sets from the reference, i.e. the healthy state of the system; Compute S and Σ on one or on several data sets from the reference state (see Section 2);
Compute the values of χ N 2 in Equation (8) on the remaining data sets of the reference state
to empirically determine their mean and variance, assuming their distribution is approximated by a normal distribution fχref in the reference state; Set up the threshold t, such that t
P ( I DD | D ) =
f χ ref ( χ N 2 | D ) d χ N 2 = α
in the reference state. Given the threshold t, the PoD , i.e. the probability of indication given a damage P ( I DD | d ) , can be calculated by integrating the probability density of the indicator variable χ N 2 from the threshold to infinity (Equation (16)). ∞
P ( I DD | d ) = ∫ f ( χ N 2 | d ) d χ N 2
Calculating P ( I DD | d ) for different damage sizes d leads to a cumulative distribution of the
PoD which can be written as F ( I DD ,T | d ) . The structural reliability can then be updated with the continuous information of a damage detection system, i.e. the probability of a damage given no indication of the damage detection system can be calculated applying Bayesian updating (Equations (13) and (14)). 6
5 EXEMPLARY APPLICATION OF THE DEVELOPED APPROACHES The steps outlined in the previous Sections 3 and 4 are exemplarily applied comprising the determination of the threshold t, the calculation of the PoD (Section 5.1) and the application of these results to a generic structural reliability example (Section 5.2). The first two steps namely the determination of the threshold t and the calculation of the PoD can be understood as a part of the procedure for the PoD determination outlined at the beginning of section 4. Here, this procedure is replaced by a generic example constituting a mass-spring model with six degrees of freedom. With this example it is exemplarily accounted for the measurement uncertainties of the sensor and amplifier system (with white noise) and the uncertainties of the stochastic subspace algorithm. A Lognormal distributed random variable E is introduced to account for the human errors. The third step is the application of the determined PoD curve to a generic structural reliability example. Here, it is shown how the probability of component damage can be updated with the information of no indication of a damage detection system. 5.1 Distributions of χ N2 and Determination of the Probability of Detection First, the probability density functions of the damage indicator variable χ N2 are estimated in a simulation study for the reference state and several damage scenarios of a structure. The simulated structure is a mass-spring model of six degrees of freedom (Figure 1). Using this model, 1000 sets of output-only data were generated to obtain the necessary measurements in each structural state, each data set containing 50,000 samples, with white noise as input. Output noise was added on the data, such that the resulting signals have a signal-to-noise ratio of 10 dB. Damage was simulated by reducing the stiffness of spring 2 in several steps.
Figure 1 – Mass-spring model for simulations.
Figure 2 – Empirical probability density functions (scaled) of χ N2 for different structural states. 7
The first data set of the reference state was used to compute the parameters S and Σˆ of the damage indicator variable χ N2 . From the remaining data sets, the values of χ N2 were computed in (7)-(8) for different structural states. They are plotted in a histogram in Figure 2 for the reference state and three different damage states, where the stiffness of spring 2 was reduced by 1%, 2% and 3%, respectively. From these simulations, the mean and standard deviation of χ N2 can be determined for the different structural states, see Table 1. Table 1 – Empirical mean and standard deviation of χ N2 for different structural states. Standard Stiffness Mean 2 reduction (%) deviation of χ N2 of χ N 0 1379 168.8 0.4 1410 175.8 0.8 1550 195.4 1.2 1805 224.8 1.6 2171 252.3 2 2667 333.6 2.4 3280 376.4 2.8 4005 493.3 3.2 4890 599.1 3.6 5861 707.8 From the distribution in the reference state (with the stiffness reduction 0), the threshold t is determined, such that Equation (10) is fulfilled for some probability α and the (approximately normal distributed) damage indicator variable χ N2 . For α = 0.99, the threshold t is determined as t = 1772. Using this threshold, the probability of detection is determined from Equation (16), which is presented in Figure 3.
Figure 3 – Probability of detection for damage indicator χ N2 with α = 0.99 for a reduction of stiffness in spring 2.
For the determination of the PoD the uncertainties associated with the sensor and amplifier system and the uncertainties of the stochastic subspace algorithm were considered. To account for human errors a Lognormal distributed random variable E with a mean of 1.0 and a standard deviation of 0.3 is introduced and is multiplied to the damage indicator variable (Equation (17)). 2 χ app = E χ N2
The steps, namely the threshold determination to fulfill α = 0.99 and the calculation of the probability of detection with Equation (16) is repeated. The threshold t is determined as t = 2743. The probability of detection function is now more flat as the human errors lead to an increase of the uncertainties (see Figure 3 and 4).
2 Figure 4 – Probability of detection for damage indicator χ app with α = 0.99 for a reduction of stiffness in spring 2.
5.2 Structural Reliability Updating The introduced concept for the utilisation of damage detection information (Section 3) in the framework of the structural reliability theory is applied to an example. A generic limit state function for the damage event D referring to a specific component of a structure, a specific damage mechanism and to a certain point of the service life is assumed for simplicity. The limit state function is written as the stiffness resistance R minus the stiffness reduction K (Equation (18)). The stiffness reduction can be detected with the damage detection system with different probabilities according to the PoD (Figure 4) derived in the previous section. g = R−K
The distribution of the stiffness resistance is modelled Lognormal distributed with a mean of 0.3 and a standard deviation of 0.15. The distribution of the stiffness reduction is also assumed Lognormal distributed with a mean of 0.03 and a standard deviation of 0.02. The probabilistic model is summarised in Table 2.
Table 2: Probabilistic model for the example Random variable Stiffness resistance Stiffness reduction
LN: Lognormal distribution
With the limit state Equation (18) and Equation (19), a reliability index without the information of the damage detection system β D of 3.09 is calculated.
β D = −Φ −1 ( P ( D ) )
Now it is assumed that the structural component is monitored with a damage detection system and that the PoD determined in Section 5.1 (Figure 4) applies. Further it is assumed that in the monitoring period T the damage detection algorithm delivers a continuous no indication information I DD ,T . Applying Equations (12) to (14) and (20), the reliability index given no indication β D|I DD ,T of 4.31 is calculated.
β D| I
= −Φ −1 P ( D | I DD ,T )
The reliability index is considerable higher when the information of the damage detection system are utilized. The information include the probability of detection which is determined to account for the uncertainties associated with the application of a damage detection algorithm (Section 4) and the information of no indication of a damage. It is assumed in the example that the damage detection algorithm can detect the stiffness reduction with a relatively high probability (see Figure 4). Clearly, the reliability given no indication is dependent on the PoD distribution and changes as the PoD curve changes. 6 CONCLUSIONS AND OUTLOOK A damage detection algorithm such as stochastic subspace approach delivers continuous information about the condition of a structure. Recently, statistical damage detection algorithms based on stochastic subspace identification have been proposed that take into account uncertainties in the data which can arise due to e.g. low signal-to-noise ratios, non-stationary or non-white ambient excitation and non-linear behavior. The information of a stochastic subspace damage detection algorithm can be utilised for the updating of the structural reliability similar to inspection information and constitutes the basic idea of the paper. The approach here is to replace the signal for inspection technologies with the damage indicator of the stochastic subspace algorithm. The updating of the structural reliability necessitates that all relevant uncertainties are considered and modelled. The uncertainties are outlined and a procedure to account for the uncertainties for updating the structural reliability is suggested based on the determination of a PoD distribution. The application of the developed approaches for the determination of the PoD and the updating of the structural reliability is exemplarily shown. It is demonstrated that the structural reliability of a component can be significantly increased with the no indication information of a damage detection system. The updated structural reliability applies to the period the damage detection system is operated.
The task for further research on the utilization of damage detection information in the structural reliability theory has explicitly to address the specific limit states and failure mechanisms and the time dependency of these. Furthermore, a procedure for determining the PoD distribution of damage detection algorithms including the uncertainties due to human factors should be developed. ACKNOWLEDGEMENTS The financial support of the project Integrated European Risk Reduction System (IRIS, CP-IP 213968-2) is gratefully acknowledged.
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