Sensitivity analysis of subspace-based damage indicators under changes in ambient excitation covariance, severity and location of damage

Engineering Structures 208 (2020) 110235

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Sensitivity analysis of subspace-based damage indicators under changes in ambient excitation covariance, severity and location of damage ⁎

T

⁎

Angelo Aloisio , Luca Di Battista , Rocco Alaggio, Massimo Fragiacomo Department of Civil, Construction-Architectural and Environmental Engineering, Università degli Studi dell'Aquila, Via G. Gronchi, 18, L'Aquila 67100, Abruzzo, Italy

A R T I C LE I N FO

A B S T R A C T

Keywords: Sensitivity analysis Damage detection tests Subspace-based algorithms Probabilistic structural safety assessment Structural dynamics

It is common practice in Structural Health Monitoring (SHM) to perform damage detection by detecting changes in subspace-based damage indicators. They are computed by comparing a reference state with the current (possibly damaged) state through a matrix based on the computation of residuals originated from the orthogonality defect of projected subspaces. In the last decades, several scalar indicators have been proposed based on the residual matrix. Still, a multivariate sensitivity analysis was not carried out to estimate the sensitivity of each indicator to the excitation changes. In the current paper, a covariance-based sensitivity analysis applied to a spatial truss structure is carried out to evaluate the sensitivity of eight damage indicators to ambient excitation covariance, damage severity, and damage location. The analysis aims at identifying the ranges of applicability of damage indicators within a simple statistical framework, possibly extendable to more general applications. It is argued whether any equivalent class of indicators can be obtained from the manipulation of the residual matrix. The role of damage location in the probabilistic assessment of damage is discussed: a methodological Bayesian approach, based on the results of the sensitivity analysis, is proposed for the development of a structural reliability analysis driven by damage indicators.

1. Introduction The problem of detecting faults, modelled as changes in the eigenstructure of a linear dynamical system, has been investigated following various approaches [1–10]. During the last two decades, there has been a growing interest in subspace-based linear system identification methods [11,12]. Many authors have attempted to design fault detection algorithms based on subspace identification methods [13,14,6]. Among damage detection tests, the so-called non-parametric damage detection tests do not require the computation of modal parameters. The methods considered in the current paper compare output data measured from the structure reference state with data from the possibly damaged state using a subspace-based residual function. However, while the modal parameters are not affected by a change in the ambient excitation statistics, damage detection tests, which skip the system identification step and handle the measured vibration data directly, may be influenced by changes of the excitation covariance [15]. Therefore, damage indicators should be robust to changes of the excitation covariance: the indicator would only depend on the occurrence of damage and not by the particular input. The problem of finding a damage test robust when changing the excitation covariance was addressed by [13,16–18] and recently by Döhler et al. [6], who derived a ⁎

robust indicator from a robust residual matrix. However, no rigorous demonstration of the indicator robustness, based on the use of a robust residual matrix, was given so far. Damage tests, based on a robust residual matrix, show their robustness when the damage location is fixed, like in Döhler et al. [15]; but what happens when the damage position changes? Is the metric of robust damage indicators significantly sensitive to damage location? It must be remarked that the issue of damage localization is not treated in this paper: the sole role of damage location is discussed. In particular, damage localization cannot be achieved in a merely data-driven approach. Two extreme and complementary paradigma face each other when dealing with damage indicators: a data-driven and a model-driven approach. The problem of detecting a structural modification arises from a data-driven approach: the indicator reveals a possible change in the eigenstructure of the dynamic system. However, unless a numerical model is developed or an extensive experimental programme is carried out (the latter economically inconvenient when dealing with complex structures), the modification of the indicator cannot be related to specific damage. Additionally, it would be improper to identify the notion of structural modification with that of structural damage, considering that modification does not always imply any “loss” or “harm” to the structure: a modification caused by temperature excursions cannot be

Corresponding author. E-mail addresses: [email protected] (A. Aloisio), [email protected] (R. Alaggio), [email protected] (M. Fragiacomo).

https://doi.org/10.1016/j.engstruct.2020.110235 Received 12 July 2019; Received in revised form 13 January 2020; Accepted 13 January 2020 0141-0296/ © 2020 Elsevier Ltd. All rights reserved.

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Fig. 1. Diachronic steps of a possible structural health monitoring program.

monitoring (SHM) [32,34,35,37,38]. It is usually divided into five subtasks of increasing difficulty [39]: damage detection (level 1), damage localization (level 2), identification of the damage type (level 3), quantification of the damage extent (level 4) and prediction of the remaining service life (level 5). The paper mainly deals with the first subtask and attempts to discuss the role of damage location for damage detection purposes.

considered, strictu sensu, damage to the structure. A structural modification can be identified with damage only when the indicator is not sensitive to the environmental factors and depends on the sole occurrence of damage. Nevertheless, no damage indicator, so far, can be set independent of all possible environmental factors. Even the robustness of the indicators to changes of the excitation covariance is a challenging task [6]. From simple damage detection tests to the more complex problem of the localization of damage, a numerical model would be necessary, at least for the estimate of correlations between damage severity and values of the indicator, given certain locations of damage, as later explained. In particular, in a diachronic perspective, a threestep procedure could be outlined in a structural health monitoring program based on damage indicators (see Fig. 1):

2.1. State-space representation State-space representation of output-only vibration-based structural monitoring corresponds to the following discrete time model

xk + 1 = Axk + vk yk = Cxk + wk

(i) Preliminary phase: model updating from acquired data and estimate of correlations between simulated damage scenarios and values of the indicator. (ii) Monitoring phase: structural monitoring and performance of damage tests; (iii) Assessment phase: probabilistic assessment of the structural reliability from the performed tests.

(1)

n ,

r

the outputs yk ∈ , the state transition matrix with the states xk ∈ A ∈ n × n and the observation matrix C ∈ r × n , where r is the number of sensors and n is the system order. The excitation vk is an unmeasured Gaussian white noise sequence with zero mean and constant covariance [0pt ] def

matrix Q = E(vk vkT ) = Qδ (k − k '), where E(·) denotes the expectation operator and wk is the measurement noise.

The first phase is objectively the most critical. The updated numerical model, “tuned” to the acquired data, is used for the simulation of damage scenarios [19–22]. The second phase involves the continuous acquirement of data, in a possibly automated acquisition pattern, their clustering and de-trend using other measured quantities (e.g. velocity of wind, temperature, humidity) and the performance of damage tests [23–26]. The third phase concerns the interpretation of damage tests. There are two alternative approaches: the first, based on the use of the updated model for the identification of damage severity and location, relying on optimization algorithms and other numerical methods extensively discussed in literature [18,27–33]; the second, considered in this paper, is based on the interpretation of damage tests using the correlations obtained from the updated model developed in the first phase. In the current paper, the unknown damage location and the indicator sensitivity to damage location are discussed for the formulation of a probabilistic assessment of damage. The possible adoption of other damage indicators, obtained by manipulating the residual matrix, is further investigated by addressing the issue of robustness to the excitation covariance. Hence, a variance-based sensitivity analysis is performed to assess the sensitivity of several damage indicators to damage severity, excitation covariance and damage location. The paper is organized as follows: In the first section, the background of subspace-based damage indicators is presented, and in the second section, the Sobol sensitivity analysis is introduced and referenced. In the third section, the case study, a spatial truss structure, is described. In the fourth and fifth section, the main findings of the sensitivity analysis are reported. In the last sections, comments and conclusions are given to provide an interpretative frame to the results. In particular, it is attempted to develop a more general elementary statistical framework for the interpretation of damage indicators for structural reliability analysis. It is questioned whether damage indicators can be regarded as possible estimators of damage severity when damage location and typology are known following a modeldriven approach.

2.2. Subspace residual Any damage diagnosis method requires the extraction of damagesensitive features from the measurement data of the monitored system [40,41]. This feature vector is generally defined in a way that it is approximately Gaussian distributed with zero mean in the reference state and non-zero mean in the damaged state, hence the designation residual vector [34]. Many residuals have been used in the literature: the subspace residual, the transfer matrix-based residual [42], residuals built on a nullspace based comparison of data Hankel matrices, [43], on the difference of output covariance Hankel matrices [44] or directly on the modal parameter differences [45]. In this paper, the subspace residual is tested, specifically the conventional and robust subspace residuals. Following the nomenclature from [15], the first residual will be addressed to as conventional, in comparison with the latter, robust to changing excitation. The residual matrix represents the orthonormality defect between the subspaces of responses due to noise effects mainly due to structural damages. In [5,18] a residual function was proposed to detect changes in the system's eigenstructure from measurements yk without actually identifying the eigenstructure in the possibly damaged state. The considered residual is associated with a covariance-driven output-only subspace identification algorithm. Let G = E(xk + 1 ykT ) be the cross-covariance between the states and the outputs, Ri = E(yk ykT− i ) = CAi − 1 G be the theoretical output covariances, and

Hp + 1, q

⎡ R1 ⎢ R2 = ⎢ ⎢⋮ R ⎢ ⎣ p+1

def

R2 R3 ⋮ Rp + 2

⋯ ⋯ ⋱ ⋯

Rq ⎤ Rq + 1 ⎥ def ⎥ = Hank (Ri ) ⋮ ⎥ Rp + q ⎥ ⎦

(2)

the theoretic block Hankel matrix. Using measured data (yk )k = 1, ⋯ , n , a consistent estimate Hp + 1, q is obtained from the empirical output covariances

2. Subspace-based damage detection

i = 1 R N

The diagnosis of damages is a fundamental task for structural health 2

N

∑ yk ykT−i k=1

(3)

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p + 1, q = Hank (R i ) H

(4)

Q Q −1 Id, r = (ξ N̂ )T Σ ξ̂ ⎜⎛ξ N̂ ⎞⎟ ⎝ ⎠

The residual function originally proposed by [5,18] compares the system undamaged or reference state with the damaged or current one. The considered residual matrix can be written as

c = S T Hp + 1, q R

which should be compared to a threshold. Σξ̂ and Σζ̂ are consistent Q Q estimates of the asymptotic covariance of ξ N̂ and ζ N̂ respectively. The computation of the asymptotic covariance matrix is a numerically critical issue, as it involves the inversion of big low-rank matrices. The numerically robust scheme, proposed by [15], is adopted for its computation.

(5)

1

where S is the left null space of the block Hankel matrix Hp + 1, q in the reference state and Hp + 1, q is the covariance block Hankel matrix in the current one. In practice, the excitation covariance Q may change between different measurement sessions of the system due to different environmental factors, while the excitation is still assumed to be stationary during one measurement. A change in the excitation covariance Q leads to a change in the cross-covariance between states and outputs G and thus in the Hankel matrix. Döhler et al. [15,43] proposed a new re1 be the matrix of sidual, which is robust to changing excitation. Let U 1 is a the left singular vectors obtained from an SVD of Hp + 1, q . As U matrix with orthonormal columns, it can be regarded as independent of the excitation Q: this property qualifies its use for a residual function that is robust to changes in the excitation covariance. Then, the residual matrix can be written as

r = S T U 1 R

(10)

2.2.2. Yan et al. damage detection test Yan et al. [43], clinging to the geometric interpretation of the residual matrix, as expression of a loss of orthonormality, proposed a r damage indicator given by the norm of matrixR

r ) σN2̂ = norm (R

(11)

where norm (.) is an operator giving the maximal singular value of a matrix. The indicators considered in this paper are:

(6)

c ) Iy, nr = norm (R

(12)

r ) Iy, r = norm (R

(13)

2.2.3. Other damage detection tests A set of two damage detection tests, based on the definition of both the robust and the conventional residual matrix (6), is proposed. In order to further assess the effectiveness of the trace operator over the damage test definition, the square matrix obtained self-projecting the residual matrix on itself is considered. Ia indicators can be written as:

As stated by [43], other residual matrices (see (5)) are not ideal candidates for structural damage detection since the amplitude of the residues, resulting from erroneous definition of the system order or the amplitude excitation, may mask the residues variation due to small structural damages. A set of damage indicators, obtained from the robust residual matrix, are detailed in next subsections.

c R cT ) Ia, nr = tr (R 2.2.1. Döheler et al. non-parametric damage detection tests The damage test2 is defined as non-parametric, as the system parameters do not need to be explicitly known in the reference state. Following a procedure used by Basseville et al. [5] and Fritzen et al. [47], the residual matrix (5), is rearranged in a vectorial form. The formulation adopted by [46] is followed3. Specifically two residual vector are considered, the first obtained from the conventional residual matrix (7) and the second from the robust one (8).

(14)

r R rT ) Ia, r = tr (R

(15)

The role of the covariance matrix (10) in the non-parametric Σ−ξ 1ζ , is a posteriori investigated in the last indicators: Q Q Il, nr = (ζ N̂ )T ⎛⎜ζ N̂ ⎞⎟ ⎝ ⎠

(16)

(17)

Q ζ N̂ =

c ) N vec (R

(7)

Q Q Il, r = (ξ N̂ )T ⎜⎛ξ N̂ ⎞⎟ ⎝ ⎠

Q ξ N̂ =

r ) N vec (R

(8)

3. Covariance-based sensitivity analysis

The index Q of the residual vectors indicates the excitation covariance of the system, for which the measured data (yk )k = 1, ⋯ , n are used for the computation. It is tested if this residual function is significantly different from zero. A non-parametric χ 2 -test to decide whether the residual vector is significantly different from zero or not, following the one reported by [46] (9) leads to Q Q −1 Id, nr = (ζ N̂ )T Σζ̂ ⎜⎛ζ N̂ ⎞⎟ ⎝ ⎠

χ 2 -test,

A sensitivity index is a number that gives quantitative information about the relative sensitivity of the model to a selected set of parameters. A set of global sensitivity indices is the group of Sobol indices [48]. The method proposed by Sobol decomposes the variance of the output of the model or system into fractions which can be attributed to inputs or sets of inputs. Suppose that the inputs are independently and uniformly distributed within the unit hypercube, ie I n = {x ≤ x i ≤ 1} . Assume that f (x) = f (x1, x2 , ...,x n ) is a real integrable function defined on the unit hypercube I n . Thus f (x) may be decomposed in the following way:

(9)

n

1

The system parameters in terms of eigenvalues and eigenvectors in the reference and current states are respectively θ0 and θ . [46] proposed a nonparametric version of the damage detection test, adopted in the current paper,where the system parameter θ0 does not need to be known explicitly in the reference state. Instead of using the null space S (θ0T ) on the parameterized observability matrix [5,18], an empirical (non-parametric) null space S is computed on an estimated block Hankel matrix from data in the reference state using e.g. an SVD. 2 Others [43] refer to damage detection tests as to damage indicators. 3 vec denotes the vectorization operator.

f (x ) = f0 +

n

∑ fi (xi) + ∑ fij (xi xj) + ...+f1,2, … , n (x1, x2, …, xn) i=1

i
(18)

where f0 is a constant and fi is a function of x i , fij a function of x i and x j , etc. A condition of this decomposition is that all the terms in the functional decomposition are orthogonal. Given the orthogonality the followings are obtained

f0 = 3

∫ f (x) d x In

(19)

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fi =

∫

f (x) d x

dx i − f0

The former assesses the indicator sensitivity to both variations of the excitation covariance and damage severity, the latter assesses the same indicator sensitivity to the excitation covariance and the severity and location of damage. The excitation covariance and the damage severity are randomly varied within specific ranges, in particular 0.0001 − 0.005 m and 0.01 − 0.2% respectively. Damage is simulated by a percentage decrease of No.5 stiffness in the first analysis (indicated in Fig. 2). An even random distribution is adopted for the selection of the members of the spatial truss in the second analysis to obtain a random damage location. The numerical simulations were very time consuming to achieve a satisfactory convergence of the sensitivity analysis.

(20)

I n−1

where dx dx i denotes the product of all the dxk except dx i , and

fij =

∫

f (x) d x

dx i dx j − f (x i ) − f (x j ) − f0

I n−2

(21)

In this way one can define inductively all fi1, ⋯ , is for any subset of variables x i1, ⋯, x is with indices 1 ≤ i1 < ⋯ < is . Furthermore, the following quantities are defined

D=

∫ f 2 (x) d x − f0

(22)

In

Di1, ⋯ , is =

∫ ⋯ ∫ fi2, ⋯, i Is

Is

1

s

(x i1, ⋯, x is ) dx i1⋯dx is

5. Two-parameter sensitivity analysis The results of the first analysis are reported in Table1 and Fig. 5. Table 1 is organized in horizontal sections for each indicator. In each section, the Sobol indices, the Total effect indices and their sum are shown. The sums are not exactly equal to one given the approximation and the statistical meaning of the indicators. The analysis decomposes the variance of each damage indicator into two fractions which can be attributed to the contributions of damage severity and excitation covariance. Concerning the first analysis, where member 5 is damaged, as reported in Table 1 and Fig. 5, the main findings are:

(23)

It has been established by Sobol that, under the assumption that x is uniformly distributed in In , the quantity Si1, ⋯ , is = Di1, ⋯ , is D is a measure of the global sensitivity of the function f (x) with respect to the group of variables. The Sobol indices are often called the “first-order sensitivity indices”, or “main effect indices”. However, when the number of variables is large, the first-order approach requires the evaluation of 2n − 1 indices, which can be too computationally demanding. For this reason, a different sensitivity index, known as the “Total-effect index” or “Total-order index”, SwT (24), is preferred [49].

• The ranking generated by the Sobol indices S

n

SwT = 1 −

∑ Si1, … , is ik ≠ j

(24)

The computation of Sobol integrals, is performed through a Monte Carlo simulation. Latin hypercube sampling is used for generating a near-random sample of parameter values from a multidimensional distribution to speed up the convergence [50]: in the first step, a matrix of N randomly sampled input combinations is built, each one made up of M components, where M is the number of model inputs. Both the Sobol index Sw and the total sensitivity index SwT are computed in this paper. While Sw measures the effect of varying a single parameter alone, SwT measures the contribution to the output variance of the selected parameter, including all variance caused by its interactions. In short, the more different is the ranking generated by the two indices, the more complex is the interaction between the parameters [51]. Information on parameter sensitivity is particularly valuable in system identification and system optimization. In the current paper, the functions adopted for the sensitivity analysis are the damage indicators themselves.

•

4. Application The damage indicators, shown in the previous sections, are tested against a FEM structural model, in order to show their performance under different excitation properties as well as under different severities and locations of damage. A spacial truss structure is analysed, with 60 DOF; the members have a uniform circular cross-section with internal radius Ri = 0.02m and thickness t = 0.001m ; the modulus of elasticity is E = 200000 MPa for all members; the specific weight is 2700 kg m3 . Output data are virtually acquired by a number of sensors in the vertical direction. The white noise excitation is generated in point A as a vertical yielding constraint, Fig. 2. The variability of the excitation condition is simulated by randomly changing the variance of the white noise input within given ranges. The truss model is simulated in SAP2000, while data are processed in MATLAB. By means of the SSI covariance-driven algorithm [52], the modal parameters of the FEM model, implemented in SAP2000, are compared with those from the Output-Only Identification, reported in Fig. 3. For each damage indicator, two sensitivity analyses are carried out.

• • 4

w closely matches those by the total sensitivity indices SwT , see Table 1. This result might suggest that interactions among the parameters are not significant over the specified range. This property is examined in the following section. The range of variation selected for global sensitivity analysis does not cover all applications: the sensitivity indices should be re-evaluated for each application, which may involve a range of variation not covered by the selected range. The use of a robust residual matrix Rr determines a significant reduction of the effects of the excitation covariance over the numerical value of all the indicators. The robustness does not depend on the sole use of a robust residual matrix: an odd choice of the indicator may disguise the expected robustness (see Ia, r ). The robustness does not only depend on the computation of the residual matrix regardless of its further manipulation: while the Iy, r , Id, r and Il, r are less sensitive to the excitation covariance, the Ia, r shows poor performance. Among the ones which have an excellent performance, the Id, r indicator is the most robust: it is a χ 2 -test where the asymptotic covariance matrix maximizes the effect of damage, which is made already significant by reducing the noise effects using the robust residuals. The Il, r indicator may provide evidence for the actual reasons why the Ia, r is an unsuccessfully robust predictor of damage, despite being obtained by manipulating the terms of the robust residual matrix. The Il, r indicator, which is deducted from the Id, r by setting the covariance matrix equal to the identity matrix, is the sum of the square of all members of the residual matrix. The satisfactory performance of this indicator, not far from Id, r , shows the importance of the residual matrix vectorization: it can be noticed by simple inspection that the residual matrix have elements very close to zero, possibly depending on the excitation covariance, and elements carrying the information of the orthogonality defect of the subspace projection. A successful indicator should avoid any matrix product of the residual matrix, like in Ia, r : the elements of the residual matrix which are very close to zero should not spoil the benefits of using a robust residual. The indicators based on the conventional residual matrix are susceptible to the effects of noise. Specifically the Id, nr could be considered the best indicator, since the χ 2 -test maximizes the effect of damage, while the Iy, nr , Ia, nr and Il, nr are strongly affected by variations of the covariance of the noise excitation. The indicators values corresponding to increasing severities of damage are reported in Fig. 4. The results are not general since they

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Fig. 2. (a) View of the FEM model of the spatial truss, where the virtual accelerometers (thick dots) and the member 5 are indicated; (b) Stabilization diagram of the SSI-COV algorithm.

correspond to stiffness reductions of No. 5 member, but the behaviour of the indicators is very different among each other. The indicators, obtained from the non-robust matrix in Fig. 3(a), (c), (e), (g), do not have a clear regression model for increasing values of damage. Specifically, given the dependence on the amplitude of the excitation, the indicators seem to detect damage for lower values of the noise covariance hardly: there is an increasing scatter of the indicator values for increasing severities of damage. Besides, except for Iy, r , the indicators obtained from the robust matrix have a clear increasing regression model. A first-order regression model is likely to provide a good fit of the Ia, r (Fig. 3(f)) and Il, r (Fig. 3(h)) indicators, while a higher-order one may be suitable for Id, r (Fig. 3(d)). The indicators obtained using all the robust residuals avoiding any cross product (i.e. Id, r and Il, r ) could be labelled as the best ones since they have a clear regression model with low dispersion due to the robustness to variations of the excitation covariance. It is interesting the role of the asymptotic covariance matrix in introducing a higher-order dependence of Id, r to the severity of the damage. Iy, r , as shown in the following section, depends on the damage severity, and it could exhibit a low-increasing regression model for other damaged members.

Table 1 Variance-based sensitivity analysis when changing of both the excitation covariance and the damage severity. Damage Index

Parameters

Sw

SwT

3[2]*Iy , nr

Noise Damage Sum Noise Damage Sum Noise Damage Sum Noise Damage Sum Noise Damage Sum Noise Damage Sum Noise Damage Sum Noise Damage Sum

0.7777 0.0426 0.8203 0.0795 0.6321 0.7116 0.5807 0.3435 0.9242 0.0191 0.9369 0.9561 0.8535 0.1862 1.0397 0.4586 0.4328 0.8914 0.8550 0.1561 1.0111 0.1291 0.8793 1.0084

0.9751 0.2783 1.2533 0.0898 0.8996 0.9894 0.7138 0.4374 1.1512 0.0965 1.0258 1.1223 0.8757 0.1494 1.0251 0.6231 0.7263 1.3494 0.9275 0.1213 1.0489 0.1632 0.8315 0.9947

3[2]*Iy, r

3[2]*Id , nr

3[2]*Id , r

3[2]*Ia , nr

3[2]*Ia , r

3[2]*Il , nr

3[2]*Il , r

6. Three-parameter sensitivity analysis The second analysis reported in Table 2 and Fig. 5, obtained by randomly changing the damage location among the 64 members of the spatial truss, highlights the significant influence of location over the

Fig. 3. Stable modal parameters of the FEM spatial truss obtained from the SSI covariance-driven algorithm [52]. 5

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Fig. 4. Graphic representation of the damage indicators, tested in the current analysis, for increasing values of damage.

indicator values. The main findings are reported:

general trend of the indicators for increasing severities of damage is reported for all members. Unlike from the previous analysis, the results are superposed: the different performance of the two classes of indicators, the ones from the conventional matrix and those from the robust, is not so evident like in Fig. 4. However, if the indicators are separately plotted for each damaged member, the dependence on the excitation covariance in non-robust indicators corrupt the possible regression model (indicator value-severity of damage), introducing an increasing dispersion of the results for more significant damages. Iy, r , like in Fig. 4(b), does not have increasing regression models for all damaged members, despite in some cases (e.g. No.40 member), a quadratic regression model is likely to transpire. In other cases, different regression models can be detected for all damaged members. The Il, r seems to exhibit the best performance: different linear fittings can be observed in Fig. 6(h). The effect of damages at different locations, which is here evidenced, can be estimated by considering modal parameters. A rigorous comparison between the effects over modal parameters and damage indicators is difficult to carry out unless modal parameters are somehow compressed into scalar values as well. However, this aspect is not the purpose of the current paper.

• Like in the previous analysis, the ranking generated by the Sobol

•

•

•

indices Sw closely matches that by the total sensitivity indices SwT , see Table 2. It suggests that interactions among the parameters are not significant over the specified range. Assuming a normal PDF distribution of the indicators given a specific damage location, the absence of considerable interactions between different variables means that the variance of the indicators is not markedly dependent on the damage location. All indicators are highly sensitive to location. In some, specifically Iy, nr Ia, nr and Il, r , obtained from the non-robust residual matrix, the sensitivity to changing of the excitation covariance prevails. However, when the robustness to the excitation covariance is somehow achieved, the location shows its non-negligible influence. The higher-order dependence of Id, nr to the severity of damage determines a higher sensitivity to location despite its non-robustness, if compared to the Il, nr indicator which is more dependent on the excitation covariance. The dependence of the indicators on the severity of damage is not uniform between all members of the spatial truss; Hence, the assessment of the damage severity may not be univocal when, given a particular value of the indicator, the location is unknown: damage indicators cannot be then used as deterministic predictors of the severity nor of the location of damage. The two covariance-based sensitivity analysis does not give information about the actual metric of each indicator, which is dependent on the particular structure under test [15]. In Fig. 6, the

7. Discussion on the probabilistic assessment of damage As evidenced in the introduction, a monitoring program aiming at detecting damages could be organized into three phases: Preliminary phase (i), Monitoring phase (ii) and Assessment phase (iii). In the first phase, the numerical model is updated to the first acquired results. The 6

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7.2.1. Probabilistic assessment of the severity of damage Theoretically, if damage tests are meant to give information about damage severity, damage indicators possibly independent on damage location could be reliable and almost deterministic predictors of damage severity: given a certain damage severity, different locations do not considerably affect the value of the damage indicator. However, the considered indicators do not have such quality, being all strongly affected by changes in the damage location. A certain indicator value may be related to the exceedance of several damage thresholds corresponding to different safety levels. The damage may occur in any part of the structure. In this paper, it is assumed that two possible damages will never be contemporary events: specifically, the exceedance of an indicator value could be related to the occurrence of a sole damage event among all possible ones. Indicating with Dj the generic j -th damage and n the number of possible damage scenarios, the Bayesian probability of the occurrence of an unknown damage D , given the indicator value I , reasonably assuming P (I ) = 1, writes

Table 2 Variance-based sensitivity analysis under changing of the excitation covariance, and severity and location of damage. Damage Index

Parameters

Sw

SwT

4[2]*Iy , nr

Noise Damage Location Sum Noise Damage Location Sum Noise Damage Location Sum Noise Damage Location Sum Noise Damage Location Sum Noise Damage Location Sum Noise Damage Location Sum Noise Damage Location Sum

0.8733 0.0662 0.1377 1.0772 0.1393 0.3330 0.6030 1.0753 0.1525 0.0412 0.9243 1.1180 0.0486 0.1266 0.5879 0.7631 0.6111 0.1707 0.1611 0.9429 0.1219 0.2193 0.6654 1.0066 0.5315 0.0904 0.3322 0.9440 0.1282 0.2507 0.6048 0.9836

0.8132 0.1773 0.0898 1.0803 0.0465 0.4543 0.7214 1.2222 0.2337 0.0862 0.7491 1.0690 0.0142 0.5558 0.8455 1.4156 0.7911 0.1044 0.4368 1.3323 0.2565 0.5365 0.9199 1.7129 0.5411 0.2505 0.2601 1.0516 0.0665 0.4372 0.7394 1.2431

4[2]*Iy , r

4[2]*Id, nr

4[2]*Id, r

4[2]*Ia, nr

4[2]*Ia, r

4[2]*Il, nr

4[2]*Il, r

n

P (D I ) =

∑ P (I Dj) P (Dj) I=1

(25)

Given the results reported in the previous sections, the likelihood P (I Dj ) can be approximated to a Gaussian distribution

P (I Dj ) ∝ e

−

I − I ̂(Dj ) σ2

(26)

As noticed in the sensitivity analysis, σ 2 could be assumed as constant when the variance of the indicator is not markedly affected by the location of the damage. I ̂(Dj ) is the correlation of the indicator to damage, estimated in a preliminary phase from the updated model. If the location of damage and its severity do not significantly interact in the ranges of interest, the regression function I ̂(Dj ) can be separately estimated from univariate parametric analysis, given a specific location by changing the severity of the damage. Consequently, the proposed Bayesian approach may be assimilated to a reliability analysis of a parallel system [53,54,55]. The accuracy of the assessment will depend on the discrepancies between the identified regression models: specifically, considering the limit case of having the same regressions for all damage scenarios, a certain indicator is univocally related to the intensity of damage by the unique regression model. This is confirmed by Eq.(25); In that case the P (D I ) → 1 if the coincident likelihoods P (I Dj ) → 1 when σ 2 → 0 and the actual level of damage D on the structure is assessed. The probabilistic treatment of damage tests is simulated for the spatial truss under test. The regression models representative of the correlation between damage indicators and intensity of damage are estimated from monovariate analysis, for all the 64 members of the truss. Given the regression models, it is possible to evaluate the joint probability function of Eq. (25), which is reported in Fig. 7(a). In this application, the sole Il, r indicator, which is normally distributed, is adopted, given its good performance. However, the Id, r indicator, which is χ 2 distributed, could be used for the same purposes: Thöns et al. [56] present a Bayesian updating procedure for the structural reliability assessment based on the χ 2 test of the conventional subspace residual. Given a certain damage indicator, in a real case application, it is possible to assess the probability of occurrence of a certain level of damage without knowing its actual location. For instance, assuming an arbitrary value of the indicator (the vertical dashed line in Fig. 7(a)) the PDF is shown in Fig. 7(b): it gives the probability of occurrence of a certain level of damage given the value of the indicator. For completeness, the corresponding CDF is shown in Fig. 7(c). The joint probability function in Fig. 7(a) is highly peculiar of the structure under test, showing several peaks. In particular, if the indicator were independent on location, the joint density function would

second phase concerns the continuous data analysis; the third phase deals with the assessment of damage from the results of the performed damage tests. The third phase consists of three different tasks having increasing complexity: damage detection (i), assessment of the severity of damage (ii) and localization of damage (iii).

7.1. Damage detection The first task is fairly simple as it reduces to the observation of the exceedance of a certain threshold, any of all previously defined. However, this task is not exhaustive when the damage location can change and consequently so does the associated damage thresholds: the exceedance of a certain threshold should be considered critical or not depending on the actual location of the damage. In order to overcome the problem of “false alarms”, the second task should be tackled. If damage tests are meant to be exploited for the sole assessment of the existence of damage, without concerning the damage severity, the sensitivity to damage location is not influent.

7.2. Severity of damage It does not matter if different indicators show different sensitivity to location, provided that a distinct regression model is estimated for each damage typology and location for rising severities of damage. The indicators obtained from the conventional residual matrix and those obtained by picking the maximum singular value of the same matrix (Iy, r and Iy, nr ) may not be suitable for this purpose since they do not possess a distinct regression model. The correlations between indicator and severity of damage may lead to a probabilistic assessment of the severity of the damage: a simple probabilistic approach is developed in the next paragraphs. 7

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Fig. 5. Bar plot of the results of the sensitivity analysis; in (a) and (b), the Sobol indices and total effect indices of the two-parameter analysis; in (c) (d), the Sobol indices and total effect indices of the three-parameter analysis.

In order to show a methodological application of the proposed procedure, the reliability analysis driven by subspace-based damage indicators is carried out for the spacial truss under test. The failure mechanisms corresponding to the exceeding of the axial forces in each member of the truss is considered. The load condition reported in Fig. 8(a) ( f = 100N ) is assumed for the computation of the axial forces in each member, as well as the tensile strength σy = 200 MPa . If the resistance of each member (R = σy A , where A is the undamaged crosssection area) and the axial forces S are Normal distributed with variance σR = 15N and σS = 200N respectively, the safety margin M is also Normal Distributed with mean ( μ = μR − μs ) and variance

be a Gaussian function translated along the regression curve and its sections would be Gaussian. In that case, the uncertainty of the estimation would be related to the intrinsic variance of the indicator. The efficiency of the proposed method may increase if more complex residuals, like the ones reported by Allahdadian et al. [34], are adopted. 7.2.2. Damage indicators and probabilistic reliability analysis If the probability of occurrence of certain damage can be related to failure mechanisms, the damage indicators could be effectively used for probabilistic reliability analysis. The structural reliability could be assessed through the probability of failure if models are established for resistances R and loads S. In that case, the probability of failure reads

PF =P (g ((R − S ) < 0)

(σM =

(27)

P (Fj ) = Φ(−β )

where g is the limit state function corresponding to the considered failure mechanism. Following the Bayesian approach outlined in the previous paragraph, the probability of failure given the exceedance of a certain damage indicator P (F I > I )̂ reads:

P (F |I 〉 I )̂ =

I=1

(28)

where Fj is the j − th failure mechanism. The (28) is easily computable if the P (Fj ) is assessed from a reliability analysis. The likelihood P (I Fj ) , as reported in Eq.(26), can be estimated if the damage indicators corresponding to the considered failure mechanisms are previously estimated IF̂ = I ̂(DF , j ) , where DF , j is the damage associated to the failure mechanism.

P (I Dj ) ∝ e

−

7.3. Localization of damage The issue of localization is a complex task. A reliable assessment of localization cannot derive from a merely data-driven probabilistic assessment of damage tests, and numerical models should be adopted. More complex algorithms should be adopted, as noticed from the extensive scientific literature on this topic [18,60,21,61,62,63,64]. Noise

I − IF̂ σ2

(30)

where β = μM σM is the reliability index. The likelihoods are then estimated and the results are shown in Fig. 8(b) and (c) [57,58,59]. In conclusion, even without dealing with the issue of localization, it would be possible to estimate the probability of structural failure, given the exceeding of a certain damage indicator, assuming regression models of the same damage indicators with increasing damage for a determined location.

n

∑ P (I > I ̂ Fj) P (Fj)

σR2 + σS2 ); then the probability of failure is the P (Fj ) of Eq.(28).

(29) 8

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Fig. 6. Graphic representation of the damage indicators, tested in the current analysis, for increasing values of damage.

carried out. The aim is to understand possible limits in using these indicators as predictors of damage and their eventual ranges of applicability. The analysis was carried out on a spatial truss model and cannot rigorously be extended to any structure. The main results and considerations are listed herein after:

robustness is significantly evidenced by using the robust residual, which, however, is sub-optimal for localization purposes. More complex residuals [34] should be implemented for more effective localization methods. 8. Conclusions

• The use of a robust residual matrix is not the sole warranty of the

robustness of a scalar damage indicator obtained by manipulating the terms of the residual matrix. The use of a robust residual matrix does not diminish the sensitivity of the indicators to the excitation covariance unless cross products between all terms of the residual matrix are avoided. This goal could be achieved by vectorizing the residual matrix [46]: many terms of the residual matrix are close to zero and highly influenced by the excitation covariance and other

Damage detection tests are the most widely accepted synthetic indicators of the occurrence of structural damage. They are routinely used in damage detection of complex systems. Subspace-based damage indicators may represent an effective tool for SHM applications, since they condense the structural status into a single scalar value. In this paper, a variance-based sensitivity analysis of several damage indicators to damage severity and location and excitation covariance is

Fig. 7. (a) Probability Joint density function of the indicator and level of damage; (b) section cut of (a) in correspondence of the vertical dashed line; (c) CDF of (b). 9

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Fig. 8. (a) Load condition for the probabilistic reliability analysis; (b) Probability of failure of the structure given the exceedance of a certain damage indicator; (c) Reliability of the structure given the exeedance of a certain damage indicator.

• • •

possible variables (sampling frequency, time shift of the covariancedriven SSI algorithm etc.), others store the residual matrix information to be processed. All indicators are highly sensitive to the location of damage, although such dependency is hidden by the variations of the excitation covariance in the indicators derived from the conventional residual matrix. The indicators, which could be considered as the best candidates for real-case applications, are the Id, r and Il, r : they are robust to change in the excitation covariance and can be described by a distinct regression law (indicator-level of damage), given a specific damage location, obtained from univariate analysis. If the indicators are meant to be used for the sole assessment of damage occurrence, any damage test could be theoretically a good candidate, despite the ones based on the robust residual have the best performance. If the same indicators are meant to give information about damage severity, no deterministic assessment can be carried out. However, a Bayesian probabilistic assessment of damage could be developed if the indicators possess distinct regression models (indicator-level of damage) for the considered damage scenarios.

[11] Van Overschee P, De Moor B. N4sid: Subspace algorithms for the identification of combined deterministic-stochastic systems. Automatica 1994;30(1):75–93. [12] Van Overschee P, De Moor B. Subspace algorithms for the stochastic identification problem. Automatica 1993;29(3):649–60. [13] Benveniste A, Basseville M, Moustakides G. The asymptotic local approach to change detection and model validation. IEEE Trans Autom Control 1987;32(7):583–92. [14] Zhang Q, Basseville M, Benveniste A. Early warning of slight changes in systems. Automatica 1994;30(1):95–113. [15] Döhler M, Mevel L, Hille F. Subspace-based damage detection under changes in the ambient excitation statistics. Mech Syst Sig Process 2014;45(1):207–24. [16] Moustakides GV, Benveniste A. Detecting changes in the ar parameters of a nonstationary arma process. Stochastics: Int J Probab Stochastic Processes 1986;16(1–2):137–55. [17] Vanlanduit S, Parloo E, Cauberghe B, Guillaume P, Verboven P. A robust singular value decomposition for damage detection under changing operating conditions and structural uncertainties. J Sound Vib 2005;284(3–5):1033–50. [18] Basseville M, Mevel L, Goursat M. Statistical model-based damage detection and localization: subspace-based residuals and damage-to-noise sensitivity ratios. J Sound Vib 2004;275(3–5):769–94. [19] Weng J-H, Loh C-H, Yang JN. Experimental study of damage detection by datadriven subspace identification and finite-element model updating. J Struct Eng 2009;135(12):1533–44. [20] Jafarkhani R, Masri SF. Finite element model updating using evolutionary strategy for damage detection. Comput-Aided Civ Infrastruct Eng 2011;26(3):207–24. [21] Yan A-M, De Boe P, Golinval J-C. Structural damage diagnosis by kalman model based on stochastic subspace identification. Struct Health Monit 2004;3(2):103–19. [22] Aloisio A, Alaggio R, Kohler J, Fragiacomo M. Extension of generalized bouc-wen hysteresis modeling of wood joints and structural systems. J Eng Mech 2020:146. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001722. [23] Hu W-H, Moutinho C, Caetano E, Magalhães F, Cunha A. Continuous dynamic monitoring of a lively footbridge for serviceability assessment and damage detection. Mech Syst Sig Process 2012;33:38–55. [24] Magalhaes F, Cunha A, Caetano E. Online automatic identification of the modal parameters of a long span arch bridge. Mech Syst Sig Process 2009;23(2):316–29. [25] Cabboi A, Magalhães F, Gentile C, Cunha Á. Automated modal identification and tracking: application to an iron arch bridge. Struct Control Health Monit 2017;24(1):e1854. [26] Hu W-H, Cunha Á, Caetano E, Magalhães F, Moutinho C. Labview toolkits for output-only modal identification and long-term dynamic structural monitoring. Struct Infrastruct Eng 2010;6(5):557–74. [27] Lim TW, Kashangaki TA. Structural damage detection of space truss structures using best achievable eigenvectors. AIAA J 1994;32(5):1049–57. [28] Teughels A, De Roeck G, Suykens JA. Global optimization by coupled local minimizers and its application to fe model updating. Comput Struct 2003;81(24–25):2337–51. [29] Jaishi B, Ren W-X. Finite element model updating based on eigenvalue and strain energy residuals using multiobjective optimisation technique. Mech Syst Sig Process 2007;21(5):2295–317. [30] Shabbir F, Omenzetter P. Particle swarm optimization with sequential niche technique for dynamic finite element model updating. Comput-Aided Civ Infrastruct Eng 2015;30(5):359–75. [31] Fritzen C-P, Jennewein D, Kiefer T. Damage detection based on model updating methods. Mech Syst Sig Process 1998;12(1):163–86. [32] Aloisio A, Di Battista L, Alaggio R, Fragiacomo M. Analysis of the forced dynamics of a masonry facade by means of input-output techniques and a linear regression model. In: COMPDYN, 2019, 7th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering; 2019. [33] Galeota D, Antonacci E, Aloisio A, Alaggio R. The S. Maria di Collemaggio basilica: from the vulnerability assessment to the first results of SHM. In: Proceedings of the XVII National Conference ANIDIS, 13–17 September 2019, Ascoli Piceno, Italy, vol. 17, ANIDIS; 2019. [34] Allahdadian S, Döhler M, Ventura C, Mevel L. Towards robust statistical damage localization via model-based sensitivity clustering. Mech Syst Sig Process 2019;134:106341. [35] Aloisio A, Alaggio R, Fragiacomo M. Dynamic identification of a masonry façade from seismic response data based on an elementary ordinary least squares

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.engstruct.2020.110235. References [1] Willsky AS. A survey of design methods for failure detection in dynamic systems. Automatica 1976;12(6):601–11. [2] Frank PM. Fault diagnosis in dynamic systems using analytical and knowledgebased redundancy: a survey and some new results. Automatic 1990:26(3):459–474. [3] Patton R, Frank P. R. (eds.) clark. fault diagnosis in dynamic systems: theory and application. englewood cliff; 1989. [4] Basseville M, Nikiforov IV, et al. Detection of abrupt changes: theory and application, vol. 104. Prentice Hall Englewood Cliffs; 1993. [5] Basseville M, Abdelghani M, Benveniste A. Subspace-based fault detection algorithms for vibration monitoring. Automatica 2000;36(1):101–9. [6] Döhler M, Mevel L. Subspace-based fault detection robust to changes in the noise covariances. Automatica 2013;49(9):2734–43. [7] Gertler JJ. Survey of model-based failure detection and isolation in complex plants. IEEE Control Syst Mag 1988;8(6):3–11. [8] Isermann R. Process fault detection based on modeling and estimation methods survey. Automatic 1984;20(4):387–404. [9] Alaggio R, Bedon C, Benedettini F, Dilena M, Morassi A. Ambient vibration testing and structural identification of a cable-stayed bridge. Proc IOMAC 2015;15:p. 6th. [10] Pantoli L, Muttillo M, Ferri G, Stornelli V, Alaggio R, Vettori D, et al. Electronic system for structural and environmental building monitoring. Convegno Nazionale Sensori 2018;481–488 [Springer].

10

Engineering Structures 208 (2020) 110235

A. Aloisio, et al.

[37]

[38]

[39] [40] [41]

[42]

[43] [44]

[45] [46]

[47] [48] [49]

[50] McKay MD, Beckman RJ, Conover WJ. Comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 1979;21(2):239–45. [51] Ma F, Zhang H, Bockstedte A, Foliente GC, Paevere P. Parameter analysis of the differential model of hysteresis. J Appl Mech 2004;71(3):342–9. [52] Peeters B, De Roeck G. Reference-based stochastic subspace identification for output-only modal analysis. Mech Syst Sig Process 1999;13(6):855–78. [53] Köhler J. Reliability of timber structures. No. 301, vdf Hochschulverlag AG; 2007. [54] Singh C, Billinton R. System reliability, modelling and evaluation, vol. 769. Hutchinson London; 1977. [55] Melchers RE, Beck AT. Structural reliability analysis and prediction. John Wiley & Sons; 2018. [56] Thöns S, Döhler M, Long L. On damage detection system information for structural systems. Struct Eng Int 2018;28(3):255–68. [57] Bjerager P. On computation methods for structural reliability analysis. Struct Saf 1990;9(2):79–96. [58] Madsen HO, Krenk S, Lind NC. Methods of structural safety. Courier Corporation 2006. [59] Shinozuka M. Basic analysis of structural safety. J Struct Eng 1983;109(3):721–40. [60] Balmes É, Basseville M, Mevel L, Nasser H, Zhou W. Statistical model-based damage localization: a combined subspace-based and substructuring approach. Struct Control Health Monit: Official J Int Assoc Struct Control Monit Eur Assoc Control Struct 2008;15(6):857–75. [61] Kopsaftopoulos F, Fassois S. A functional model based statistical time series method for vibration based damage detection, localization, and magnitude estimation. Mech Syst Sig Process 2013;39(1–2):143–61. [62] Gautier G, Mencik J-M, Serra R. A finite element-based subspace fitting approach for structure identification and damage localization. Mech Syst Sig Process 2015;58:143–59. [63] Chao S-H, Loh C-H, Tseng M-H. Structural damage assessment using output-only measurement: localization and quantification. J Intell Mater Syst Struct 2014;25(9):1097–106. [64] Mevel L, Hermans L, Van der Auweraer H. Application of a subspace-based fault detection method to industrial structures. Mech Syst Sig Process 1999;13(6):823–38.

approach. Eng Struct 2019;197:109415. https://doi.org/10.1016/j.engstruct.2019. 109415. Aloisio A, Di Battista L, Alaggio R, Antonacci E, Fragiacomo M. Assessment of structural interventions using bayesian updating and subspace-based fault detection methods: the case study of S. Maria di Collemaggio basilica, L’aquila, Italy. Struct Infrastruct Eng 2020. Antonacci E, Aloisio A, Galeota D, Alaggio R, Fragiacomo M. The S. Maria di Collemaggio basilica: from vulnerability assessment to first results of SHM. J Archit Eng 2020. [In press]. Farrar CR, Worden K. An introduction to structural health monitoring. Philos Trans Royal Soc A: Math Phys Eng Sci 2006;365(1851):303–15. Döhler M, Mevel L. Modular subspace-based system identification from multi-setup measurements. IEEE Trans Autom Control 2012;57(11):2951–6. Döhler M, Mevel L, Zhang Q. Fault detection, isolation and quantification from gaussian residuals with application to structural damage diagnosis. Ann Rev Control 2016;42:244–56. Bhuyan MDH, Viefhues E, Döhler M, Lecieux Y, Mevel L, Hille F, Schoefs F. Outputonly subspace and transfer matrix-based damage localization and quantification; 2017. Yan A-M, Golinval J-C. Null subspace-based damage detection of structures using vibration measurements. Mech Syst Sig Process 2006;20(3):611–26. Gres S, Ulriksen MD, Döhler M, Johansen RJ, Andersen P, Damkilde L, et al. Statistical methods for damage detection applied to civil structures. Procedia Eng 2017;199:1919–24. Parloo E, Guillaume P, Van Overmeire M. Damage assessment using mode shape sensitivities. Mech Syst Sig Process 2003;17(3):499–518. Balmès E, Basseville M, Bourquin F, Mevel L, Nasser H, Treyssède F. Merging sensor data from multiple temperature scenarios for vibration monitoring of civil structures. Struct Health Monit 2008;7(2):129–42. Fritzen C, Mengelkamp G, Guemes A. Elimination of temperature effects on damage detection within a smart structure concept. Struct Health Monit 15–17, 2003. Sobol IM. Sensitivity estimates for nonlinear mathematical models. Math Model Computat Exp 1993;1(4):407–14. Homma T, Saltelli A. Importance measures in global sensitivity analysis of nonlinear models. Reliab Eng Syst Saf 1996;52(1):1–17.

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