Md Delwar

Hossain Bhuyan

Saeid Allahdadian

Michael Döhler

Yann Lecieux

Laurent Mevel

Franck Schoefs

Carlos Ventura

Transfer matrices-based output-only statistical damage localization and quantification on a frame structure

Damage localization in civil or mechanical structures is a subject of active development and research. In this paper, vibration measurements and a finite element model are used to locate loss of stiffness in a steel frame structure at the University of British Columbia. Damage localization is not very well developed so far and few methods show promising properties in practice. Among them, the Stochastic Dynamic Damage Locating Vector (SDDLV) approach has interesting capabilities. In this paper, it is compared to a sensitivity based approach developed by the authors. Both approaches have in common to be built on the estimated transfer matrix difference between reference and damaged states. Both methods are tested for localization and quantification on the Yellow Frame structure at the University of British Columbia.

INTRODUCTION

The detection, localization and quantification of damages based on measured vibration data are fundamental tasks for structural health monitoring (SHM) to allow an automated damage diagnosis [START_REF] Farrar | An introduction to structural health monitoring[END_REF]. We consider the case of output-only vibration measurements of a structure subject to ambient excitation. Damages are considered as changes in the structural stiffness.

The Stochastic Dynamic Damage Locating Vector (SDDLV) approach is a vibration based damage localization method using both finite element (FE) information of the structure in the reference state, and modal parameters estimated from measurements in both damage and reference states of the system [START_REF] Bernal | Load vectors for damage location in systems identified from operational loads[END_REF]. It is a theoretically sound method that evaluates the transfer matrix difference between reference and damaged states of a structure, without updating the FE model. In previous works, the deterministic SDDLV approach has been extended with a statistical framework, taking into account the uncer-tainties in the measurement data [START_REF] Döhler | Statistical decision making for damage localization with stochastic load vectors[END_REF]. Further robustness has been achieved through a statistical aggregation of results for different parameters of the method, notably lifting a constraint on the necessary number of sensors [START_REF] Marin | Robust statistical damage localization with stochastic load vectors[END_REF][START_REF] Bhuyan | Statistical damage localization with stochastic load vectors using multiple mode sets[END_REF]. In previous works, the SDDLV as an output-only method was only considered for damage localization, but not for damage quantification that required input information in related works [START_REF] Bernal | Damage localization and quantification from the image of changes in flexibility[END_REF]. In this paper, both damage localization and quantification are investigated based on the transfer matrix difference in a sensitivity-based approach [START_REF] Döhler | Fault detection, isolation and quantification from Gaussian residuals with application to structural damage diagnosis[END_REF] using output-only data. A damage-sensitive parameterization of the system is used, referring e.g. to the element stiffnesses in a finite element model. Through sensitivity analysis on the transfer function matrices with respect to this parameterization, damage is first localized by statistical tests, and then damage is quantified only for the damaged element by estimating the parameter change.

A similar sensitivity-based statistical framework has been successful for damage localization using a subspace residual, recently shown in PhD thesis [START_REF] Allahdadian | Robust Statistical Subspace-Based Damage Assessment[END_REF], where it has been applied in a real test case of a four-story steel frame structure, the Yellow Frame at the University of British Columbia [START_REF] Allahdadian | Damage localization of a real structure using the statistical subspace damage localization method[END_REF]. This case study was also subject of previous endeavors involving the deterministic version of the SDLV approach [START_REF] Turek | Development of a combined damage detection methodology[END_REF]. In the current paper, the proposed statistical approaches based on the SDDLV and on the transfer matrix sensitivities are demonstrated on the same structure, with the overall goal to obtain reliable damage assessment methods for real, complex structures.

MODELS

The behavior of a mechanical structure is assumed to be described by a linear timeinvariant (LTI) dynamic system

M Ẍ (t) + C Ẋ (t) + KX (t) = f (t) (1)
where M , C, K ∈ R d×d are the mass, damping and stiffness matrices, respectively, t indicates continuous time and X ∈ R d denotes the displacements at the d degrees of freedom (DOF) of the structure. The external force f (t) is not measurable and modeled as white noise. Let the dynamic system (1) be observed at r coordinates. Since f (t) is unmeasured, it can be substituted with a fictive force e(t) ∈ R r acting only in the measured coordinates and that regenerates the measured output. Furthermore, defining x = [X Ẋ ] T , this leads to the corresponding continuous-time state-space model

ẋ(t) = Ax(t) + Be(t) y(t) = Cx(t) + De(t) (2)
with state x ∈ R n , output y ∈ R r , state transition matrix A ∈ R n×n and output matrix C ∈ R r×n , where n = 2d is the system order and r is the number of outputs. The input influence and direct transmission matrices are B ∈ R n×r and D ∈ R r×r respectively.

DAMAGE LOCALIZATION AND QUANTIFICATION

Starting point of the SDDLV-based damage localization approach is the transfer matrix difference between reference and damaged states. The transfer function G(s) ∈ C r×r of the system at the sensor coordinates can be derived as [START_REF] Bernal | Load vectors for damage location in systems identified from operational loads[END_REF]

G(s) = R(s)D where R(s) = C(sI -A) -1 CA C † I 0 (3)
under the condition that the system order satisfies n ≤ 2r. Note that R(s) can be computed from output-only system identification, e.g. using subspace methods [START_REF] Marin | Robust statistical damage localization with stochastic load vectors[END_REF], while matrices B and D cannot be identified for the computation of G(s). The difference between the transfer matrices in both damaged (variables with tilde) and healthy states is δG(s) = G(s) -G(s). Based on this matrix, the subsequent damage assessment depends in particular on the sensor locations and positions and the value of s. Damage is considered as changes in the stiffness of system (1), while mass remains unchanged.

Localization with SDDLV Approach

In the SDDLV approach, a vector in the null space of the difference between the transfer matrices of the healthy and damaged systems is obtained. Since the matrices δG(s) and δR(s) T = R(s) T -R(s) T have the same null space due to D = D [2], the desired load vector v(s) ∈ C r can be obtained from the null space of δR(s) T using a singular value decomposition (SVD). Thus it can be estimated entirely on measurements in healthy and damaged states. It has been shown that when applying this load vector to the FE model of the healthy structure, then the resulting stress field S(s) is zero at the damaged element. The stress is stacked in a vector and yields a linear relationship

S(s) = L model (s)v(s) (4)
defined through the FE model of the healthy state. To decide if the estimated stress components are zero, the covariance of the stress Σ S has been estimated in [START_REF] Döhler | Statistical decision making for damage localization with stochastic load vectors[END_REF][START_REF] Marin | Robust statistical damage localization with stochastic load vectors[END_REF][START_REF] Bhuyan | Statistical damage localization with stochastic load vectors using multiple mode sets[END_REF]. Then, an appropriate hypothesis test is performed on each structural element t by selecting the respective stress components S t in S(s) as well as the respective covariance submatrix Σ t of Σ S , and computing the test statistic

χ 2 t = S T t Σ -1 t S t (5)
for each structural element t tested for damage. Since stress over damaged elements is zero in theory, potential damage is located in elements t corresponding to the lowest values of χ 2 t among all elements. Robustness of the localization is achieved by evaluating the stress in (4) for different Laplace variables s and for different mode sets when computing the null space vector v, and aggregating the results for each element in (5) [START_REF] Marin | Robust statistical damage localization with stochastic load vectors[END_REF][START_REF] Bhuyan | Statistical damage localization with stochastic load vectors using multiple mode sets[END_REF].

Localization and Quantification with Senstivity-Based Approach

The sensitivity-based approach evaluates a parameterized residual vector ζ whose mean is zero in the healthy state and that deviates from zero in the damaged state. The parameter vector θ ∈ R l describes the monitored system in the current state, with θ 0 in the reference system. It is convenient to choose θ as the collection of stiffness parameters of the elements of the structure, where θ 0 is obtained from a finite element model.

In [START_REF] Döhler | Fault detection, isolation and quantification from Gaussian residuals with application to structural damage diagnosis[END_REF], a statistical framework has been set up for Gaussian residual vectors parametrized by θ with the purpose to decide which parts of θ have changed (for damage localization) and then to estimate this change (for damage quantification). The Gaussian residual vector ζ ∈ R h is computed from the measurements of the system and needs to satisfy

ζ ∼ N (0, Σ) in reference state N (J δ, Σ) in damaged state, (6)
where δ = √ N (θ -θ 0 ) ∈ R l is an unknown change, N is the data length used for the computation of ζ, the sensitivity matrix J ∈ R h×l of the residual with respect to θ has full column rank and the residual covariance matrix Σ ∈ R h×h is positive definite.

Based on the transfer matrix in the previous section, such a residual is defined as

ζ def = √ N vec( R(s) T -R(s) T ) re , (7)
where (•) re denotes the stacked real-valued vector q re def = (q) (q) for any vector q with (•) and (•) denoting the real and imaginary parts, respectively, and vec(•) denotes the column stacking vectorization operator. Note that the mean J δ of the residual follows from a first-order Taylor approximation of ζ, based on

vec( R(s) T ) re ≈ vec(R(s) T ) re + J (θ -θ 0 ). ( 8
)
The covariance of the quantity vec( R(s) T -R(s) T ) re is derived in detail in [START_REF] Marin | Robust statistical damage localization with stochastic load vectors[END_REF], as well as the sensitivity of vec(R(s)) with respect to the modal parameters. To obtain the required sensitivity matrix J , the derivative of the modal parameters with respect to the structural parameters is needed in addition, which is described in detail in [START_REF] Heylen | Modal Analysis Theory and Testing[END_REF].

Note that the transfer matrix can be evaluated for several Laplace variables s i , i = 1, . . . , n s . Then, the residual can be defined as

ζ def =    vec( R(s 1 ) T -R(s 1 ) T ) re . . . vec( R(s ns ) T -R(s ns ) T ) re    . (9)
The respective sensitivity matrices J are stacked analogously, and the joint covariance is detailed in [START_REF] Marin | Robust statistical damage localization with stochastic load vectors[END_REF][START_REF] Bhuyan | Statistical damage localization with stochastic load vectors using multiple mode sets[END_REF].

DAMAGE LOCALIZATION TESTS

For damage localization it has to be decided which parts of vector δ are non-zero, i.e. which parts of the parameter vector are changed. The structural elements corresponding to the changed parameters are thus damaged. To this end, each component of δ will be tested one after another. Denote the component to be tested by δ t , and the remaining complementary components by δ C , such that

δ = δ t δ C . (10)
Then, δ t = 0 is tested against δ t =0. Following [START_REF] Turek | Development of a combined damage detection methodology[END_REF], the sensitivity matrix J and the Fisher information matrix F = J T Σ -1 J are analogously arranged as

J = J t J C , F = F t,t F t,C F C,t F C,C = J T t Σ -1 J t J T t Σ -1 J C J T C Σ -1 J t J T C Σ -1 J C . ( 11
)
Sensitivity tests: Assuming that δ C = 0 for testing δ t = 0 against δ t =0, the generalized likelihood ratio (GLR) test follows as

t sens = ζ T Σ -1 J T t (J T t Σ -1 J t ) -1 J T t Σ -1 ζ, (12)
which is called sensitivity test. The test statistic t sens is χ 2 distributed with non-centrality parameter δ T t F t,t δ t . For making decision about the damage location, the test variable is compared to a threshold.

Minmax tests: Instead of assuming the components of δ C = 0, δ t is tested using a residual blind to any change in δ C . This is achieved by a projection orthogonal to the space defined by δ C , also known as minmax test, as follows. Define the partial residuals

ζ t = J T t Σ -1 ζ, ζ C = J T C Σ -1 ζ, and the robust residual ζ * t = ζ t -F t,t F -1 C,C
ζ, whose mean is sensitive to changes δ t but not to δ C . Testing δ t = 0 against δ t =0 with the GLR test yields

t mm = ζ * t F * -1 t ζ * t , (13)
where

F * t = F t,t -F t,C F -1 C,C F C,t . The test statistic t mm is χ 2 distributed with non- centrality parameter δ T t F * t δ t .

ESTIMATORS FOR DAMAGE QUANTIFICATION

When damage is localized in the first step, the task for damage quantification is to estimate δ t for the damaged components in the second step. Then, the parameter change follows as θ -θ 0 = δ/ √ N . Sensitivity approach: An estimate of δ t can be derived from the residual vector ζ as

δsens t = (J T t Σ -1 J t ) -1 J T t Σ -1 ζ, ( 14
)
where δsens t ∼ N (δ t , F -1 t,t ) for the assumption δ C = 0. Minmax approach: An estimate of δ t based on minmax approach is

δmm t = F * -1 t ζ * t ∼ N (δ t , F * -1 t ). (15)

APPLICATION

Both localization approaches and the quantification have been demonstrated on a four-story steel frame structure, the Yellow Frame (Figure 1), at the University of British Columbia. The structure is described in detail in [START_REF] Allahdadian | Robust Statistical Subspace-Based Damage Assessment[END_REF][START_REF] Allahdadian | Damage localization of a real structure using the statistical subspace damage localization method[END_REF]. It is supposed to show a linear vibration behavior. Twelve sensors are located on the structure, three at the north, south and west side in each floor (Figure 1(b)). For damaged and undamaged states, acceleration data containing 219,900 and 550,000 samples, respectively, at a sampling frequency of 1000Hz were recorded. Damage is introduced by removing braces number 2 and 4 at the south side of the structure on the first floor.

After downsampling and decimation of the data by factor 5, seven well-estimated modes were obtained in the healthy and damaged states using subspace identification, together with their uncertainties. The identified frequencies are shown in Table I.

A finite element model of the structure is required to obtain L model for the stress computation in the SDDLV approach, and to obtain the sensitivity of the modal parameters with respect to the structural parameters for the computation of the sensitivity J [START_REF] Allahdadian | Robust Statistical Subspace-Based Damage Assessment[END_REF][START_REF] Allahdadian | Damage localization of a real structure using the statistical subspace damage localization method[END_REF]. For simplicity, damage is located only in the braces. Hence, stress is only computed in the braces in the SDDLV approach, and parameter θ is defined only as Young's modulus of each brace in the sensitivity approach. Altogether, 32 brace elements are tested.

In both approaches, the transfer matrix difference and its covariance are computed based on the identified seven modes for two different choices of Laplace variables s, comprising one or several variables s in the vicinity of the identified modes. Note that test values close to zero indicate potentially damaged elements in the SDDLV approach, and high values indicates the damaged elements in the sensitivity-based approaches.

SDDLV approach: localization

The computation of the stress and its covariance for the statistical evaluation in the χ 2 t -tests is carried out for two different sets of s-values, each in the vicinity of the respective identified poles. First, one s-value is chosen with s 1 = -1 + 76i, and second, two s-values are chosen as s 1 = -1 + 76i and s 2 = -1 + 30i for joint evaluation. To compare the ratios between the healthy and damaged elements, the computed values are normalized in the figures such that the smallest of the 32 values is 1.

The estimated stress and its statistical evaluation is shown in Figures 2(a)-2(d). It can be seen that it is impossible to locate the damage at elements 2 and 4 with the estimated stress alone in (a) and (c), while its statistical evaluation in (b) and (d) correctly indicates the damage (smallest values). The additional s-value in (d) leads to a slightly better contrast to the damaged elements than in (b).

Sensitivity-based approach: localization and quantification

The damage indicators are shown in Figures 3 for each tested element using the transfer-matrix based residuals. In all cases, it is seen that the test statistic at the damaged elements shows the highest value, correctly localizing the damage at braces 2 and 4. However, in Figures 3(a) and (b) it can be seen that the tests also react strongly at undamaged elements using one s-value. By choosing three s-values in Figures 3(c (d) it can be seen that the performance is much better. For the quantification of the damage extent, the values of δsens and δmm were estimated for the respective sensitivity and minmax approaches, and shown in Table II. Since the braces are removed, the stiffness loss should be 100%. The minmax approach gets closer to this value than the sensitivity approach, however the overall quantification accuracy is not entirely satisfying.

CONCLUDING REMARKS

In this paper, two transfer matrix based localization methods have been compared and achieved similar success on a real structure. The sensitivity based approach allows in addition the quantification of damage. The first quantification results are encouraging, however further understanding and work is needed to obtain more precise damage quantification estimates.

  (a) Photo of the Yellow Frame structure (south-east corner) (b) Scaled plan of each level and location of sensors

Figure 1 :

 1 Figure 1: Yellow Frame structure.

Figure 2 :

 2 Figure 2: SDDLV-based damage localization: stress computation and statistical evaluation at one s-value (-1 + 76i) and two s-values (-1 + 76i, -1 + 30i). Braces 2 and 4 are removed.

Figure 3 :

 3 Figure 3: Sensitivity-based damage localization: Sensitivity and minmax tests at one svalue (-1 + 76i) and three s-values (-1 + 10i, -1 + 250i, -1 + 280i). Braces 2 and 4 are removed.

TABLE I :

 I Identified frequencies (in Hz) of Yellow Frame.

	# healthy damaged
	1	8.11	5.79
	2	8.71	7.97
	3	15.5	14.0
	4	21.4	19.9
	5	23.8	23.7
	6	31.2	30.6
	7	46.5	46.2

TABLE II :

 II Quantification of damage extents (in %)

	# s-values	1		3
	Damaged brace	2	4	2	4
	δsens 48.2 48.7 70.9 70.0
	δmm 86.0 94.2 77.1 102.2

ACKNOWLEDGMENTS

Financial support from the Bretagne region is gratefully acknowledged.


In this paper the statistical subspace damage localization (SSDL) method is employed in localizing the damage in a real structure, namely the Yellow frame. The SSDL method is developed for real testing conditions and tested in two damage configurations. It was demonstrated that the SSDL method can localize the damage robustly in the Yellow frame for simple and multiple distinct damage scenarios using the analytical modal parameters. The method is described and its effectiveness is demonstrated.

INTRODUCTION

With the advent of new technologies, instrumentation of structures is becoming widespread. The data acquired from the instrumented structures can provide beneficial information on their structural conditions. However, efficient techniques and methods are necessary in processing this data in order to assess the functionality of the structure.

It is about five decades that vibration based health monitoring of structures is utilized to evaluate the conditions of the structures. At the beginning, it was basically focused over the offshore oil industries and aerospace engineering problems and now it is widespread throughout the civil and mechanical engineering communities as well. In this context, the damage can be identified as the change in: the boundary conditions of the structure, the geometrical/material properties of the composing elements and or the connectivity of these elements.

Damage identification methods are the main component of structural health monitoring which process the data in order to detect damages in the structure. Damage detection of the structures can be categorized into 4 groups based on their level of identification: I) identifying the damage existence, II) identifying the geometrical location of the damage, III) quantification of the severity of damage and IV) evaluation of the remaining service life of the structure. Most of the literature is concentrated on the level 1 to 3 of the detection on the controlled laboratory structures and some on-field tests.

Several extensive review papers can be found in literature on the topic of damage detection e.g. ( [START_REF] Fan | Vibration-based Damage Identification Methods: A Review and Comparative Study[END_REF] and [START_REF] Doebling | A Summary Review of Vibration-Based Damage Identification Methods[END_REF]). Among the damage identification methods, the statistical subspace damage detection method has a strong theoretical background which showed to be robust in practice for detecting the damage in S101 bridge [START_REF] Döhler | Structural health monitoring with statistical methods during progressive damage test of S101 Bridge[END_REF]. Several studies were also carried out on this technique for detecting the damage in [START_REF] Allahdadian | Investigation on the sensitivity of subspace based damage detection technique to damage and noise levels[END_REF]- [START_REF] Allahdadian | On the Influence of Sample Length and Measurement Noise on the Stochastic Subspace Damage Detection Technique[END_REF]. In this paper the statistical subspace damage localization (SSDL) approach [START_REF] Balmès | Statistical model-based damage localization: A combined subspace-based and substructuring approach[END_REF], [START_REF] Döhler | Fault detection, isolation and quantification from Gaussian residuals with application to structural damage diagnosis[END_REF] is employed in localizing the damage in a real structure, namely the Yellow frame. This method uses vibration measurements of the structure in a (healthy) reference state and in the damaged state, as well as a finite element (FE) model in the reference state. Damage is localized through statistical tests on the parameters of the model, avoiding the FE updating problem. Several methodologies and theories are developed and demonstrated in order to enable the practical implementation of this technique, which are shown in this paper.

First, the SSDL technique will be reviewed and then the methods and theories will be proposed. At the end the results of these method on the Yellow frame will be shown. Following that, the conclusions and discussions are presented.

STATISTICAL SUBSPACE DAMAGE LOCALIZATION TECHNIQUE

In this section, the theoretical background of the statistical subspace damage localization technique is introduced, mainly based on references [START_REF] Balmès | Statistical model-based damage localization: A combined subspace-based and substructuring approach[END_REF], [START_REF] Döhler | Fault detection, isolation and quantification from Gaussian residuals with application to structural damage diagnosis[END_REF].This technique can detect the damage in a structure by creating a subspace from measurement data in a reference (healthy) state. This subspace is based on the modes of the structure but without actual computation of them. This subspace is employed in a statistical comparison along with the data measured from the possibly damaged structure in assessing the condition of the structure. In this statistical comparison, sensitivities from an analytical model of the structure in the reference state are used, without computation of the modes from the test data.

Output-Only Covariance Based Subspace System Identification

The discrete-time state-space representation of the dynamic equilibrium equation of a model can be written by performing sampling with time step τ in step k as

1 k k k k k k x Fx w y Hx ε + = +   = +  (1)
in which n x ∈ R is representing the state of the system and the measured output is represented by r y ∈ R . Parameter n is the system order and r is the number of sensors. ε are generally assumed to be white noise for output-only system identification approaches. By defining the output covariance as ( )

T i k k i R y y - = Ε
, the block Hankel matrix

1, p q + H is composed as 1 2 2 3 1 1, 1 2
Hank( )

q p p q i p p p q R R R R R R R R R R + + + + +       = =         H ⋯ ⋯ ⋮ ⋮ ⋱ ⋮ ⋯ . ( 2
) Matrix 1, p q + H
contains information from the dynamic properties of the system and, hence, any change in the system will be reflected in a change in

H

to its value in the reference state or even simply to compute the cross product of it and its left null-space at the reference state 0 S . Then, if there is no change in the system 0 1, 0

T p q S + = H (3)
and if there is a change in the system 0 1, 0

T p q S + ≠ H
. Due to the stochastic nature of data, such a small change from zero is additionally corrupted by noise modeled as a random normally distributed variable, namely the residual function ζ , defined as 0 1, vec( )

T p q N S ζ + = H (4)
where N represents the number of samples used in the computation of

1, ˆp q + H estimated from 1 1 N T i k k k i N R y y = - = Σ
. By defining the system parameter θ as the current physical properties of the structure such as the stiffness of each element, and 0 θ the physical properties of the reference structure (undamaged), we have

0 N θ θ δθ = + (5) where 0 (
)

N θ δθ θ - =
is defined as the (unknown) normalized parameter change vector. Therefore, by defining two hypotheses based on the condition of the structure as 0 0 1 0 : unchanged system : changed system (damaged)

H H θ θ θ θ = ∴   ≠ ∴  (6)
the defined residual can be proved to be asymptotically normally distributed (for large

N) with 0 1 (0, ) under ( , ) under H J H ζ δθ Σ  →  Σ  N N (7)
in which J represents the asymptotic sensitivity of the residual with respect to the system parameterization, and Σ is the covariance matrix of the residual.

Parametric Hypothesis Test

In order to decide between hypotheses (6), a generalized likelihood ratio (GLR) test is employed [START_REF] Balmès | Statistical model-based damage localization: A combined subspace-based and substructuring approach[END_REF] which results in the following χ 2 -value, representing a damage index:

( )

1 2 1 1 1 T T T J J J J χ ζ ζ - - - - = Σ Σ Σ (8)
If this value exceeds a threshold it indicates the existence of damage in the system. In order to locate the damage two procedures, i.e. MinMax and sensitivity based approaches, can be employed. While in the sensitivity based approach, [START_REF] Balmès | Statistical model-based damage localization: A combined subspace-based and substructuring approach[END_REF]

T k k k k p F χ ζ ζ - = (9)
where * k F is a part of a Fisher-information-matrix for parameter k p while removing (projecting out) the effect of other parameters [START_REF] Döhler | Fault detection, isolation and quantification from Gaussian residuals with application to structural damage diagnosis[END_REF]. Similarly, the robust residual * k ζ is defined. The sensitivity approach should be less effective than the MinMax approach in case of a highly dependent parameterization. Herein, these two approaches employed in localizing the damage for the Yellow frame.

JACOBIAN COMPUTATION

The residual defined in the SSDL method is derived from operations on the measured data and has statistical background while the FE model is directly connected to the physical behaviour of the structure. A relation linking the data-based residual to the physical model is needed, which is the sensitivity of the residual with respect to the physical parameterization. This link is made by the derivative chain in [START_REF] Allahdadian | Practical Damage Identification of Structures Using the Statistical Subspace Damage Detection Technique[END_REF]. This will let us benefit from a physical model of the structure along with the measured data to localize the damage. Assume for simplicity that each element of the structure is uniquely related to one physical parameter k p . The chain rule is employed in connecting the residual function [START_REF] Allahdadian | On the Influence of Sample Length and Measurement Noise on the Stochastic Subspace Damage Detection Technique[END_REF] to the physical parameters k p as ( , ) ( , ) ( , , ) ( , ) ( , , ) ( ) ( , ) For the computation of ( ) k J p , there should be a decision made on the use of the estimated eigenstructure from the measured data versus the modal parameters from the analytical model, on each term in the computation of Jacobians. The parameters estimated from the measured data are assumed to be more precise when the identification is well performed. In general, the model updating on the analytical model is not necessary and not needed in the SSDL procedure unless the modal parameters are very different from the identified ones and the model is not a good representative of the dynamic behaviour of the structure. The important accuracy needed for the damage localization is in the sensitivity values of modal parameters with respect to physical parameters. Herein, two options are described and their performance will be investigated in the case study.

k f k f p J p J J J J λ ϕ µ ψ ξ µ ψ ξ λ ϕ ℘ ℘ = ( 10
Configuration 1: All terms in [START_REF] Allahdadian | Practical Damage Identification of Structures Using the Statistical Subspace Damage Detection Technique[END_REF] are evaluated from the analytical model. In this way, all the parameters are consistent and the Jacobian is purely computed from the analytical model. The modal parameters of the analytical model are used for the computation of ( , ) J λ ϕ , and only the Hankel matrix 1, p q + H and 0 S are computed from the measured data.

Configuration 2: The first, second and third terms of the Jacobian are evaluated from the eigenstructure estimated from the measured data and the last term is evaluated from the modal parameters of the analytical model as in [START_REF] Balmès | Statistical model-based damage localization: A combined subspace-based and substructuring approach[END_REF]. It should be noted that the modal parameters of the analytical model and the eigenstructure of the measured data need to be identified and adjusted in terms of scaling and order. Hence, one system identification is needed in the reference state prior to the localization of damage.

After the computation of Jacobians the vectors corresponding to each elements need to be clustered if they are close, using the Hierarchical Fisher-information-matrix-based clustering (HFC) approach [START_REF] Allahdadian | Practical Damage Identification of Structures Using the Statistical Subspace Damage Detection Technique[END_REF]. This clustering is based on the the statistical properties of the data as well as the modal behaviour of the elements that in turn is related to their geometrical and physical closeness and modal direction in the considered mode shapes.

SCALING OF MODE SHAPES

By using the second configuration described in previous subsection, there is a need in scaling the mode shapes obtained from the analytical model, i.e. ℘, and measured data, i.e. and ϕ ψ . This scaling can be performed in typical ways such as modal mass scaling or unit maximum member. However, usually the mass matrix of the analytical model is complicated to truncate and the mode shapes are obtained from limited number of degrees of freedoms on which a sensor is located. Therefore, the general scaling of these vectors is performed as follows. Since the scaling of the mode shapes is not matching,

j j α ϕ ℘ = ( 11
)
where α is a constant scalar. Thus, the adjusted mode shapes ℘ ɶ , can be evaluated as

H j j j j H j j ϕ ℘ ℘ = ℘ ℘ ℘ ɶ . ( 12
)

Coupled Mode Shapes Scaling

Coupled mode shapes happen mostly in symmetric or nearly symmetric structures which have sets of two close eigenvalues. In these cases, the identified mode shapes are mixed and result in a linear combination of the analytical mode shapes. In order to solve this issue, the scaling needs to be done by solving a double linear equation. ϕ , respectively. The linear combination is written as

1,2 1,2 C Γ = Φ where 1 2 1 1 1,2 1,2 3 4 2 2
, and

T T T T c c C c c ϕ ϕ     ℘   = Γ = Φ =       ℘       . ( 13
)
Therefore, we have † 1,2 1,2 C = Φ Γ and then the scaled and decoupled mode shapes

1 1,2 2 T T   ℘ Γ =   ℘   ɶ ɶ ɶ can be evaluated as † 1,2 1,2 1,2 1,2 Γ = Φ Γ Γ ɶ . ( 14
)

CASE STUDY: THE YELLOW FRAME

The Yellow frame is a modular 4 story, scaled (1/3) steel frame established in 2016 at the University of British Columbia (UBC), shown in Figure 1. Several damage scenarios are designed and tested by removal of braces of the structure to test the SSDL method. This structure is 3.6 m high and is composed of 2 spans in each direction with the total length of 2.5 m. Each floor of the structure is carrying dead loads applied to the structure by using 4 steel plates distributed on each level. Based on Figure 1.c, the considered damage scenarios are defined as Table 1.

The FE model of this structure is built and the sensitivity analysis of the mode shapes and natural frequencies with respect to each brace is computed using a finite difference approach. Subsequently, using the proposed scaling/decoupling approach, the coupled mode shapes can be perfectly decoupled; this is demonstrated in the following figure for mode shapes 1 and 2 which are among the coupled mode shapes. The sensitivities are used in composing the Jacobians either from configuration 1 or 2. Using the HFC approach the elements are clustered based on the Jacobian columns.

Finally the χ 2 -test is computed for each element (brace) using the sensitivity and MinMax approaches as illustrated in Figure 3. From these results it can be seen that the damage localization works best with Jacobian configuration 1 using the MinMax approach. With this setting, the test reacts well on the damaged elements in both scenarios S1 and S2, localizing the damage correctly. For the multiple distinct damage scenario S1, the MinMax approach using Jacobian configuration 1 seems to be the only robust damage localization solution. For the simpler case, where damage occurs only at a pair of neighboring elements, damage can also be localized by MinMax approach in configuration 2, whereas the sensitivity approach only works in configuration 1.

∈

  ℝ is the observation matrix. The state noise k w and output measurement noise k

  ) in which ( , ) λ ϕ and ( , ) µ ψ represent, in order, the discrete and continuous time eigenstructure. The modal parameters, i.e. natural frequencies, mode shapes and damping values, are denoted by ( , , ) f ξ ℘ . ( , ) J λ ϕ is the sensitivity of the residual to the eigenstructure ( , ) λ ϕ and J • is the consistent Jacobian of parameters • with respect to .

Let 1 ℘ and 2 ℘

 2 be the two coupled mode shapes corresponding to the analytical mode shapes 1 ϕ and 2

Figure 1 .Figure 2 .

 12 Figure 1. (a) The Yellow frame structure, (b) the schematic plan of the structure showing the location of sensors, (c) the numbering of the braces of the structure

Scenario S1 with configuration 1 Scenario S1 with configuration 2 Scenario S2 with configuration 1 Scenario S2 with configuration 2 Figure 3 .

 12123 Figure 3. SSDL method from sensitivity based and MinMax approaches for two configuration schemes of the Jacobian computation

DISCUSSION AND CONCLUSIONS

In this paper, the experimental data measured from the Yellow frame test was employed in validating the proposed methods allowing the SSDL approach to localize the damage in practice. It was shown that the SSDL method can localize the damage in this structure effectively for two damage scenarios.

Two configurations of the Jacobian matrix were investigated. It was illustrated that using the modes from the analytical model in composing the Jacobian matrix results in a more robust damage identification. The reasoning behind this choice relies to the following requirement: in order to compose the Jacobians from analytical and real data modal parameters, modes need to be matched from data and FE model; In that particular application, the number of identified parameters came out to be less than the analytical modal parameters (8 versus 10). This resulted in less precision in the tests using Jacobians derived from Configuration 2, where the estimates were used.

The MinMax and sensitivity based approach were used in localizing the damage. The MinMax approach requires a pre-clustering of the parameterization to be effective and the clustering obtained from HFC appeared to be an appropriate clustering scheme for the proposed method. Finally, it was observed that the MinMax test using Jacobians derived using Configuration 1 based on the analytical modes is the best choice to localize the damage for both simple and multiple distinct damages.