Alaa Hamze

Philippe Gueguen
email: philippe.gueguen@ujf-grenoble.fr

Philippe Roux

Laurent Baillet

DAMAGE DETECTION AND LOCALISATION USING MODE-BASED METHOD AND PERTURBATION THEORY

Keywords: perturbation, detection, localisation

In this paper, the detection and the localization of a local perturbation are assessed by analysing the frequency changes only (fundamental mode and overtones). After describing the method used herein applied to the bending beam and based on the perturbation theory, experimental application to a 1D plexiglas beam is shown using frequency and modal analysis technique. The damage is considered as a local perturbation of Young's modulus. Finally, the localisation of damage is done using classical modal-based methods and perturbation theory. The frequency values are caught by the Random Decrement Technique applied to the time history vibrations for one sensor at the free extremity of the beam. Detection and localization are successful, even for small and transient changes of the structure properties.

INTRODUCTION

During the last two decades, a significant amount of research has been conducted in the area of nondestructive damage evaluation (NDE) via changes in the dynamic modal responses of a structure. Rytter [START_REF] Rytter | Vibration based inspection of civil engineering structures[END_REF] classified the NDE methods into 4 levels following their objectives: level 1 is the detection of damage, level 2 corresponds to the identification of the damage location, level 3 is the quantification of the damage severity, and level 4 is the prognosis of the integrity of the structure. The first level LV1 is to detect if changes has occurred. The analysis of the damage related to the variation of the fundamental frequency of the buildings was a common practice in earthquake engineering. As supported by Farrar et al. [START_REF] Farrar | Vibration-based structural damage identification[END_REF], frequencies are certainly the modal parameter the most sensitive to changes, especially because the loss of stiffness directly impacts the frequency values. Data were collected and processed for understanding the variations of the modal parameters during and after earthquakes, related with the shaking level and the damage observed [START_REF] Celebi | Dynamic characteristics of five tall buildings during strong and low-amplitude motions[END_REF] [4] [START_REF] Dunand | Utilisation du bruit de fond pour l'analyse des dommages des bâtiments de Boumerdes suite au séisme du 21 mai 2003[END_REF] [6] [START_REF] Michel | Time-Frequency Analysis of Small Frequency Variations in Civil Engineering Structures Under Weak and Strong Motions Using a Reassignment Method[END_REF]. The second level (LV2) is to simultaneously detect and determine the location of damage. For example, variation of frequencies may only reflect a global change of the system properties and it is not often sufficient to locate within the structure the origin of the perturbation. For that reason, damage detection methods were developed based on the mode shape analysis. The experimental assessment of mode shapes is sometimes insufficiently accurate to detect and locate small variations, when compared to the sensitivity of the modal frequency analysis. The third level (LV3) is to detect, locate and estimate the severity of damage. Few applications of LV3 are available in practice. However, the estimate of the severity may contribute significantly to the action of the decision-makers in case of emergency after an extreme event.

In most cases, structural damage detection is based on a comparison between response from a "pre-damage" state and a "post-damage" state. Several methods have been described in the literature to identify and locate damages, most of them based on comparisons between a damaged and an undamaged condition and efficiency for severe damage. For Structural Health Monitoring application using vibration techniques, these methods focused on mode shape derivatives as indications of damage, such as the mode flexibility method [START_REF] Pandey | Damage detection in structures using changes in flexibility[END_REF], the curvature flexibility method [START_REF] Zhang | The damage indices for constructed facilities[END_REF], the mode shape curvature method [START_REF] Pandey | Damage detection from changes in curvature mode shapes[END_REF] or a combination of these methods. The experimental assessment of mode shapes is sometimes insufficiently accurate to detect and locate small variations, compared to the sensitivity of the modal frequency analysis. Some example in the literature concluded on the efficiency of these methods. Nevertheless, this last procedure requires recordings at several places of the building for defining the mode shapes and the frequencies of the structure, both modal parameters being used to identify the origin of the variations observed. For example, variation of frequencies may only reflect a global change of the system properties and it may not be sufficient to locate within the structure the origin of changing.

The objective of this study is to propose a natural-frequency-based method for structural damage detection and localization through perturbation theory, as associated with a high-resolution deconvolution method to treat the inverse problem. The second objective is to apply this method to experimental conditions, when ambient vibrations excite a structure. After presenting the theoretical approach for the one-dimensional (1-D) clamped-free beam with bending and the detection of weak changes in elastic properties based on perturbation theory, experimental tests are then performed, considering different perturbations applied to the beam, which are localized through a linear inversion approach. An adaptive technique is then further applied that optimizes the perturbation localization.

THEORETICAL APPROACH FOR 1D BENDING BEAM

Neglecting shear deformation, the 1-D equation governing the transverse vibration of a beam of length L is the Bernoulli-Euler equation:

! 2 !x 2 EI x ( ) ! 2 ! x,t ( ) !x 2 " # $ $ % & ' ' + m x ( ) ! 2 ! x,t ( ) !t 2 = 0 (1)
where is the spatial-temporal beam deformation at position x and time t, m(x) is the mass per unit length, and EI(x) is the product of the elastic Young's modulus and the inertial moment at each point on the beam. Since the goal of this approach is to evaluate, to first order, the change in frequency when a slight perturbation is made to the elastic properties of the beam, we may assume that the stiffness coefficient EI(x) is perturbed at position x such that:

EI x ( ) ! EI ! x ( ) = EI x ( ) + !EI x ( ) (2)
This perturbation naturally induces a relative change in the modal deformation ! n ( x ) and the frequency of the beam and the perturbed system now satisfies the modified wave equation:

! 2 !x 2 EI ! x ( ) ! 2 ! n ! x ( ) !x 2 " # $ $ % & ' ' (! n ! 2 m x ( ) ! n ! x ( ) = 0 (3)
In perturbation theory for linear operators, the change in the modal deformation is classically projected over the set of the unperturbed modes that lead us to consider the perturbation as:

!" n = 1 2" n ! EI x ( ) ! 2 ! n x ( ) !x 2 " # $ $ % & ' ' 0 L ( 2 dx (4)
This equation can be used to invert for the stiffness perturbation !EI x ( ) at each point x on the beam from the measurement of the frequency change observed for each mode n. In practice, both ! x,t ( )

! n ! n ! n ( x ) !" n ! n
and are obtained either (1) from a numerical simulation using a model of the beam for a set of modes or (2) from the data acquired on the beam by one or a set of sensors.

In a second step, Eq. ( 5) is used to invert for the stiffness perturbation at each point x on the beam from the measurement of the frequency change !" n observed for each mode n. In this inversion process, we assume that and are known for the unperturbed problem. The inversion algorithm is built from the discretization of the integral in Eq. ( 4) over a finite number of segments of height Δx on the beam. Linear inversion processes as the Fréchet kernel associated to mode n and a perturbation Δx located at x k on the beam is used. From the linear discretization, the inversion problem can be rewritten using a matrix formulation involving p modes along the beam:

!" 1 !" 2 ... !" p ! " # # # # # $ % & & & & & = G !EI x 1 ( ) !EI x 2 ( ) ... !EI x M ( ) ! " # # # # # # $ (5)
where the kernel matrix G is such that:

G = 1 2! 1 L M ! 2 " 1 x 1 ( ) !x 2 " # $ $ % & ' ' 2 ... 1 2! 1 L M ! 2 " 1 x M ( ) !x 2 " # $ $ % & ' ' 2 ... ... ... 1 2! p L M ! 2 " p x 1 ( ) !x 2 " # $ $ % & ' ' 2 ... 1 2! p L M ! 2 " p x M ( ) !x 2 " # $ $ % & ' ' 2 ( ) * * * * * * * * * + , - - - - - - - - - (6)
Note that following the same methodology the perturbation theory can be applied to the case of a 1-D shear beam. From Eq. ( 6), the solution of the inverse problem requires the estimation of G -1 . Assuming Gaussian-distributed uncertainties, the estimated inverse of G is obtained as in Tarantola [START_REF] Tarantola | Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation[END_REF]:

G !1 ! = (G T C d !1 G + r *C m ) !1 G T C d !1 (7)
where the a-priori model covariance matrix C m resembles a diagonal matrix, but with Gaussian decrease outside the diagonal. The spatial correlation length l is assumed to be stationary along the beam, and should be of the order of the smallest wavelength associated with the highest-order mode taken into account in the inversion. The data covariance matrix C d is diagonal, with diagonal elements that correspond to the uncertainty on the measured frequency fluctuations . From the estimate of G !1 ! , we then obtain an estimation of the stiffness perturbation as:

!EI ! x 1 ( ) !EI ! x 2 ( ) ... !EI ! x M ( ) ! " # # # # # # $ = G '1 ! !" 0 !" 1 ... !" p ! " # # # # # $ ( 8
)

DESCRIPTION OF THE EXPERIMENT

The structure used in this study is made of a continuous Plexiglas beam anchored at the bottom and free at the extremity (i.e. clamped-free beam). Figure 1 shows the experimental setup used to study the bending vibration in the (xy) plane in a clamped-free configuration. The data acquisition is

! n ( x ) !EI x ( ) ! n ! n ( x )
!" n performed through 29 accelerometers powered by two conditioning amplifiers. The accelerometers are mono axial (Bruel and Kjaer, Type 4344) with 1.45 g of mass and a frequency band between 1-20 kHz. All the sensors were oriented in the horizontal direction, on the widest face of the beam in order to record the beam vibration in the y direction. The amplifiers are connected to a data acquisition unit by an RS232 port that allows to define and to control some parameters of the acquisition such as the gain of each accelerometer. A maximum gain of 40 dB was selected to amplify the signal recorded by each sensor. The beam vibration was gathered by these 29 accelerometers and transmitted directly to the computer through the data acquisition unit. The 29 accelerometers were spread along the height of the beam with a 3.5 cm interval. The beam was continuously excited by air jet assuming to ba a white noise low-amplitude excitation. Data was collected by windows of 10 seconds at a sampling frequency of 5000 Hz (Fig. 2). The transfer of the data to the data acquisition unit requires about 20 seconds, that is to say the recording obtained for the experiments was a not totally continuous recording. The local perturbation was produced by 30 seconds of heating the continuous Plexiglas beam in one positions corresponding to the position A, B and C in Fig. 1. A first analysis of the beam response can be easily obtained by computing the Fourier spectrum (Fast Fourier Transform) at the top or at each accelerometer. Figure 3 shows the variation of the Fourier amplitude along the height of the beam. The theoretical behaviour of continuous beam is observed, with nodes and anti-nodes corresponding to the shape of the modes. The frequencies ratio obtained by the experimental tests f 2 /f 1 =6.3, f 3 /f 1 =18.44 and f 4 /f 1 =35.79, let us assume in the rest of the document an experimental bending beam behaviour.

RESULTS

The variation of the frequency values are first tracked by the Random Decrement Technique [START_REF] Cole | On-line failure detection and damping measurement of aerospace structures by random decrement signatures[END_REF][13] [START_REF] Asmussen | Theory of the vector random decrement technique[END_REF] and applied to the recordings provided by the sensor located at the top of the beam. In this case, RDT is computed mode by mode, using a sliding window of 110 seconds with 10% of overlapping. For each mode, the signal is filtered by a band-pass filtered using a second-order Butterworth filter, centred on ±10 % of the frequency. The length of the sliding window, the bandwidth of the filter and number of periods are chosen so as to reduce the error to estimate the frequency.

Figure 4 shows the variations of the frequency obtained by RDT for modes 4 to 9 using data recorded at the top. As shown by Mikael et al. [START_REF] Mikael | Long-term frequency and damping wandering in buildings analysed using the Random Decrement Technique (RDT)[END_REF], the RDT method provides a very accurate estimate of the frequency of the system, detecting clearly the local perturbation effects. Figure 5 gives the experimental mode shapes extracted using the FDD method applied to the experimental data. The modal analysis method used is the Frequency Domain Decomposition, a non-parametric frequency domain method [START_REF] Brincker | Modal identification of output-only systems using frequency domain decomposition[END_REF]. More details and theoretical background of this method can be found in [START_REF] Brincker | Modal identification of output-only systems using frequency domain decomposition[END_REF]. FDD was applied to the data recorded by the 29 accelerometers located along the beam. Two analyses were done, considering the undamaged and the damage states for detecting and localizing the damage using the Uniform Load Surface method proposed by [START_REF] Zhang | The damage indices for constructed facilities[END_REF]. This method combines certain aspects of the MSC method [START_REF] Pandey | Damage detection from changes in curvature mode shapes[END_REF] and flexibility method [START_REF] Pandey | Damage detection in structures using changes in flexibility[END_REF] to develop the change in uniform load surface curvature (ULS). The basic idea is that a localized loss of stiffness will produce a curvature increase at the same location.

The flexibility matrices can be approximated by the modal parameters of damaged and undamaged beam and at the end of the process the curvature change at location i is evaluated as follows:

!F i '' = F i ''* { } " F i '' { } i=1 N # (9)
where F i ''* { } and F i '' { } are the damaged and undamaged curvature of the uniform load surface at the i th degree of freedom, respectively. !F i '' represents the absolute curvature change, and N represent the number of the degree of freedom (or identified number of mode shapes). The elements of ∆! ! !!

that have comparatively large values correspond to damage location. Note that the absolute change in curvature is first evaluated for each unit load flexibility shape and then summed. Considering damaged and undamaged states, we can apply the ULS method for detecting and localizing the damage. We observe Fig. 6 the ULS estimate for the A, B and C positions of the damage. The curve represents the value of the ULS function along the beam. In this case, the localization of the damage position is bad, showing some information for A localization, a lot of fluctuation along the beam for case B and a ghost of the damage not defined at the good position for the case C. Because of the small and transient nature of the change, mode shapes used for the ULS method are not modified enough for being accounted for the application of the ULS.

The second method used for detecting the damage is based on the perturbation theory was presented previously. Only one recording at the top of the beam is required and it is not necessary to define the mode shapes. Moreover, for small and fast changes, the frequencies react immediately while the mode shapes remain unchanged or at least with changes that could not be detected. Figure 7 shows the three steps of the localization process based only on the variation of the modal frequencies obtained with the RDT method. The three steps of the localization method are described for the three cases of damage, i.e. at the positions A, B and C. Because of the theoretical model of the beam (bending beam) and the bad resolution of the first modes, the inversion scheme has difficulties to distinguish the real to the ghost localization. Similar difficulties are also observed for the case B, considering modes 1 to 9, the position of the damage being localized close to the symmetry axis of the beam, and then of the modes. For cases A and B, even if the symmetrical ghost of localization is not completely removed, the amplitude of the inversion permits us to discriminate the position of the damage, which could be more difficult in case of blind analysis. For the last case, position C, the method is very efficient for localizing the changes, considering modes 1 to 9. Even after the second step, because of the distance of the damage to the symmetry axis of the beam, the localization is quite effective.

CONCLUSIONS

This study has discussed the non-invasive evaluation of damage via changes in the dynamic modal response of a structure. The basic idea is that the modification of the stiffness, mass or energy dissipation characteristics of a system can alter its dynamic response. Unlike conventional methods of detection, localization and quantification of changes, this study is devoted to the development of a novel approach of localization based only on variations of the modal frequencies tracked using a time-domain method (the RDT) and on perturbation theory. A new inversion method is proposed, based on the sensitivity kernel approach. Finally, to be placed under equivalent conditions as the experiment, the beam is excited with equivalent white noise and its motion is recorded at the top with time-varying localization of the perturbation.

The effectiveness and robustness of the method is tested in a configuration that is equivalent to that used experimentally. The instantaneous variations in the frequencies caused by the change in the position of the perturbation along the beam are tracked by applying the RDT to the top motion. Even for small frequency, this new approach improves the accuracy of the damage localization, which suggests temporal and geometrical monitoring of the perturbation. In conclusion, the combination of the linear inversion, the high-resolution deconvolution, and the RDT provide effective tools to detect and localize damage in a beam-type structure.

Figure 1 .

 1 Figure 1. Experimental bed test for detection and localization of changes used in this work (a) Clamped-free beam set up tested in the laboratory. (b) Sketch diagram with geometric dimensions and accelerometer layout

Figure 2 .

 2 Figure 2. Example of 10 seconds window recorded at the top of the beam.

Figure 3 .

 3 Figure 3. Amplitude of the Fourier spectra computed at the 29 sensors and representing the shapes of the 9 first modes of the undamaged beam. The black line is the numerical mode shapes.

Figure 4 .

 4 Figure 4. Time variation of the normalized frequencies of modes 4 to 9 computed by RDT using sensor located at the top of the beam (free condition) for experiment E1. A, B and C mark the time when the heat flow was applied for 30 seconds at positions A,B and C respectively.

Figure 5 .

 5 Figure 5. Mode shapes extracted by the FDD method for undamaged and damaged beam at positions A,B and C.

Figure 6 .

 6 Figure 6. Localization of the damage at position A, B and C using the ULS method and considering modes 2 to 5 for experiment E1.

Figure 7 .

 7 Figure 7. Results of the damage localization method applied to the experimental data for the three cases of perturbation (a: A; b: B; c: C).

& & & & & &

& & & & &

EWSHM 2014 -Nantes, France

ACKNOWLEDGMENTS

This work was supported by the French Research National Agency (ANR) under the RiskNat program (project URBASIS n°ANR-09-RISK-009). The authors thank the "Region Rhône Alpes" for financial support.


A Billon

J.-M Henault

M Quiertant

F Taillade

A Khadour

R.-P Martin

K Benzarti

QUANTITATIVE STRAIN MEASUREMENTS WITH DISTRIBUTED FIBER OPTIC SYSTEMS: QUALIFICATION OF A SENSING CABLE BONDED TO THE SURFACE OF A CONCRETE STRUCTURE

Keywords: Truly distributed fiber optic sensors, surface-mounted sensors, strain monitoring, crack detection, civil engineering structures

Distributed Fiber Optic Systems (DOFS) are an emerging and innovative technology that allows long-range and continuous strain/temperature monitoring with a high resolution. However, strain profiles measured in the optical fiber may differ from actual strain in the structure, due to the shear transfer through the intermediate material layers between the optical fiber and the host material (i.e., in the protective coating of the sensing cable and in the adhesive). Therefore, fiber optic sensors need to be qualified in order to provide accurate quantitative strain measurements. This study presents a methodology for the qualification of a DOFS. It is proposed to establish a numerical modeling of the system, in which the mechanical parameters are calibrated from experiments. A specific surface-mounted sensing cable connected to an Optical Frequency Reflectometry Domain (OFDR) interrogator is considered as case study. We found that (i) tensile and pull-out tests are able to provide full information about materials and interfaces of our modeling; (ii) the calibrated model allows us to compute strain profiles along the optical fiber, (iii) which proved to be consistent with strain profiles measured on a cracked concrete beam during a 4-points bending test.

INTRODUCTION

Context

Structural Health Monitoring (SHM) is an active field of research that aims at assessing the structural behavior and performance of civil infrastructures through the monitoring of keyparameters at various time scales. SHM can provide effective indexes to predict the residual lifespan of the instrumented structure and can help practitioners and infrastructures owners to optimize repair operations and reduce maintenance costs.

Reinforced Concrete (RC) structures are usually monitored with either short-gauge sensors like Vibrating Wire Gauges (VWG) or long-gauge sensors like Linear Variable Differential Transducers (LVDT). More recently, fiber optic sensors have proven to be a promising alternative to traditional sensors [START_REF] Glisic | Fibre optic methods for structural health monitoring[END_REF]. They are light, small, insensitive to electromagnetic waves, and they do not corrode. Thanks to the low attenuation of light propagating into the fiber, they can perform measurements over tens of kilometers. Finally, as the fiber is both the transmission medium and the transducer, a high degree of multiplexing is possible and installation cost of cable wires is reduced.

Mechanical transfer function (MTF) of DOFS

Distributed fiber optic sensors consist of an interrogation unit paired with a standard optical fiber. Measurement is based on the analysis of a backscattered light signal that allows continuous monitoring of strain and temperature along the fiber with a user-tunable spatial resolution. Various technologies are available, but the present study focuses on the use of Optical Frequency Domain Reflectometry (OFDR) for Rayleigh scattering. OFDR interrogation provides strain measurements with a centimeter scale spatial resolution, but the distance range is limited to a hundred meters. Optical fibers are either surface-mounted or embedded into civil engineering structures [START_REF] Henault | Quantitative strain measurement and crack detection in RC structures using a truly distributed fiber optic sensing system[END_REF]. For the monitoring of RC structures, fibers are usually fastened to steel reinforcing bars before pouring of the concrete. Hence, such embedded sensors are only intended for instrumenting new structures, unlike surface-mounted fibers which can be installed at the surface of any existing structure.

To protect the fiber from hazardous installation processes and environmental conditions, it is inserted in a cable with one or more protective layers. Cable geometries and designs are versatile and can be rather complex. Several fibers are often packaged together into a cable. For their attachment to the surface of a structure, sensing cables can be mechanically anchored or stuck with a suitable adhesive.

From a mechanical point of view, strain of the structure surface is transferred to the optical fiber through the protective and adhesive layers, which mainly deforms under shear stress. Thanks to this shear mechanism, the stress level is lower in the fiber than in the structure, so that the fiber is protected against high stress concentrations arising from cracks. But, since the strain measured in the fiber optic sensors can be different from the actual strain in the structure, measurements in the fiber are not directly quantitative.

In the case of a linear stress transfer (through interfaces and materials), the relationship between the actual strain of the structure and the strain measured in the fiber can be written as follows:

𝜀 𝐹𝑂 𝑥 = 𝑀𝑇𝐹⨂𝜀 𝑆 (𝑥) (1)
where 𝑥 is the position along the fiber, 𝜀 𝐹𝑂 the strain in the optical fiber and 𝜀 𝑆 the strain at the surface of the structure (both in m.m -1 ), while ⨂ is the convolution operator. The 𝑀𝑇𝐹 is called the Mechanical Transfer Function. According to the previous equation, the 𝑀𝑇𝐹 is equal to the strain profile along the fiber when the strain of the structure is a Dirac delta function, i.e., an infinitely narrow crack:

𝜀 𝐹𝑂,δ 𝑥 = 𝑀𝑇𝐹⨂𝛿 𝑥 = 𝑀𝑇𝐹(𝑥) (2)
In other words, the 𝑀𝑇𝐹 is the impulse response of the system formed by the fiber optic sensor (FOS) bonded to the structure. For cracks with a finite opening 𝜔, small enough with respect to the system dimensions, Equation (2) leads to:

𝜀 𝐹𝑂,𝜔 𝑥 = 𝜔 • 𝑀𝑇𝐹(𝑥) (3)
This 𝑀𝑇𝐹 is of paramount importance to obtain accurate measurements and to evaluate crack openings. Therefore, several analytical studies based on Volkersen's theory [START_REF] Volkersen | Die Nietkraftverteilung in zugbeanspruchten Nietverbindungen mit konstanten Laschenquerschnitten[END_REF] were undertaken by different authors to build a model for a FOS hosted by a concrete structure subjected to various boundary conditions. For instance, Ansari and Libo [START_REF] Ansari | Mechanics of bond and interface shear transfer in optical fiber sensors[END_REF] have studied an optical fiber with a protective coating embedded into a concrete substrate; D. Li et al. [START_REF] Li | Mechanical property and strain transferring mechanism in optical fiber sensors[END_REF] examined the behavior of a multi-layered axisymmetric cable. W.Y. Li et al. [START_REF] Li | Investigation of strain transmission of surface-bonded FBGs used as strain sensors[END_REF] investigated the case of a surface-mounted fiber with a bonding layer only while Her and Huang [START_REF] Her | Effect of coating on the strain transfer of optical fiber sensors[END_REF] added a protective coating. Finally, Feng et al. [START_REF] Feng | Theoretical and experimental investigations into crack detection with BOTDR-Distributed Fiber Optic Sensors[END_REF] developed an axisymmetric model for a FOS bridging a crack in the concrete substrate, which was also subjected to a constant deformation. They considered both the presence of protective and adhesive layers. These studies are all based on the following assumptions: all the materials present linear elastic behaviors, a perfect bond is assumed at all interfaces, shear and transverse deformations of both the fiber and the concrete adherent are neglected, while a pure shear deformation is considered for the intermediate layers. In the present study, the same assumptions are made, which will be then validated by the experimental tests.

From the results in [START_REF] Feng | Theoretical and experimental investigations into crack detection with BOTDR-Distributed Fiber Optic Sensors[END_REF], the theoretical 𝑀𝑇𝐹 under the stated assumptions can be deduced:

𝑀𝑇𝐹(𝑥) = 𝑘 2 𝑒 -𝑘|𝑥| (4)
with

𝑘 2 = 2 𝐸 𝑓 𝑟 𝑓 2 1 𝐺 𝑐 ln 𝑟 𝑐 𝑟 𝑓 + 1 𝐺 𝑎 ln 𝑟 𝑎 𝑟 𝑐 (5)
where the subscripts f, c and a respectively represent the quantities related to the fiber core, coating and adhesive. 𝐸 𝑓 is the Young's modulus of the silica fiber, 𝐺 𝑐 and 𝐺 𝑎 are the shear modulii, and 𝑟 𝑓 , 𝑟 𝑐 , 𝑟 𝑎 the radii. 𝑘 is called the shear-lag parameter and depends on the mechanical properties of the considered system (cable and host structure). It is thus a distinctive feature of this system. 𝑘 is also inversely proportional to the effective bond length of the system 𝐿 𝜏 :

𝑀𝑇𝐹 𝑥 d𝑥 = 0.97 𝑀𝑇𝐹 𝑥 d𝑥 +∞ -∞ ⟺ 𝐿 𝜏 = - ln (0.03) 𝑘 𝐿 𝜏 -𝐿 𝜏 (6)
𝐿 𝜏 corresponds to the shortest interfacial length necessary to transfer 97% of the applied load [START_REF] Yuan | Improved theoretical solutions of FRP-to-concrete interfaces[END_REF]. Actually, cables may not be axisymmetric and materials/interfaces may exhibit more complex behaviors. Thus, it is often necessary to go through a numerical modeling of the system in order to determine the stress transfer mechanism. However, this implies that all mechanical properties of the cable constituents and all interface laws are known, but this is rarely the case in practice.

A general methodology for determining the 𝑀𝑇𝐹 of any cable was introduced in a previous work and applied to a specific embedded cable [START_REF] Henault | Analysis of the strain transfer mechanism between a truly distributed optical fiber sensor and the surrounding medium[END_REF]. In the present study, tensile and pull-out tests were carried out to assess relevant properties of the materials and interfaces, and subsequently, to determine domains of linear elastic behavior and perfect bonding. A Finite Element (FE) modeling was then used to interpret experimental results. Then, a global FE model of the cable hosted by the structure was built, whose parameters were identified by fitting experimental results. Finally, the global model was used to establish the strain profile along the optical fiber induced by a surface crack opening, thus leading to the numerical determination of the cable's 𝑀𝑇𝐹.

In a previous study, such methodology has proven to be efficient for an embedded cable [START_REF] Henault | Mesures réparties de déformation par fibre optique[END_REF]. The present paper aims at demonstrating its effectiveness when applied to a surface-mounted cable. In the following, the methodology is carried out step by step, and results are then compared to experimental evidences collected during a 4-points bending test in a last section.

DETERMINATION OF THE MTF

Presentation of the cable

The FutureNeuro TM cable developed by Neubrex Co. [12], and manufactured by Fujikura Ltd., consists of two single-mode optical fibers inserted in a foamed adhesive tape designed to be stuck to the surface of a RC structure, and coated with a protective layer (Figure 1.A). Thus there is no need for an additional binding agent. The two fibers, made of silica glass, are protected by a primary coating with an outer diameter of 250 µm and are assembled together within a polymer matrix. The whole set is called the cable core (Figure 1.B). Total cable width and thickness are about 8 mm and 0.6 mm, respectively. This cable is intended for strain sensing of a host surface only. It works nominally over temperature range from -20°C to 60°C and can be stuck to various substrates, but the present study is concerned with concrete structures only. where two fibers and their primary coatings are assembled into a polymer matrix

Mechanical testing

Mechanical tests aim at providing all relevant mechanical properties that will be used in the global FE model of the cable stuck to the concrete structure.

Experimental devices

Mechanical tests were carried-out with an Instron 5960 universal testing machine of loading capacity 50 kN. For the present experimental campaign, the apparatus was equipped with a 100 N load cell and suitable grip devices. Tensile load was transmitted to samples attached to upper and lower grips. All tests were performed with a 1 mm/min displacement rate of the upper crosshead. An Advanced Video Extensometer (AVE) was used for accurate displacement and strain measurements. It consists of a high-resolution digital camera and real-time image processing software, which tracks the distance between two markers drawn at the surface of the specimens.

The Rayleigh interrogator was the Optical Backscatter Reflectometer OBR-4600 commercialized by Luna Technologies. It makes it possible to measure strain profiles along the fiber with a maximum resolution of 1 µm/m and 0.1°C. A wavelength sweep from 1545.87 nm to 1588.63 nm was used. The tunable spatial resolution was set at 1 cm, for measurements over a 70 m-long range.

Tests were performed at a controlled room temperature of 20°C ± 1°C. The experimental setup is shown in Figure 2.

Tensile tests

Tensile tests were performed on samples of optical fiber with primary coating after they were stripped out of the cable. To avoid squeezing or damaging of the fiber inside the grips, each extremity of the sample was first stuck with superglue (ethyl cyanoacrylate glue with approximate Young's modulus after curing of 1 GPa [START_REF] Ferreira | Stress analysis of lap joints involving natural fibre reinforced interface layer[END_REF]) to a metallic holder, which was then itself clamped in the grip (Figure 2.B and C). Data were collected by a computer that displayed real-time load-strain curves, as shown in Figure 3.A. The Young's modulus of the silica fiber 𝐸 𝑓 was calculated from the average slope of the linear curves, and was found equal to 77 GPa. A typical OBR strain profiles is reported in Figure 3.B. It exhibits exponential shapes on its two sides accounting for the stress transfer between the fiber and the holder through shear strain of intermediate layers, and are in agreement with the abovementioned analytical models. The shear-lag parameter of the corresponding system (optical fiber and coating perfectly stuck to the holder) depends on the shear modulus of the coating 𝐺 𝑐 and on the Young's modulus of the silica fiber 𝐸 𝑓 . Therefore we can calculate 𝐺 𝑐 = 0.46 MPa.

A failure of the primary coating around the silica fiber systematically occurred near the holder for all tested specimens, as shown in Figure 2.C. Such a failure is more likely caused by test conditions (flexure due to misalignment of the holders, damage of the primary coating caused by the holder or micro-damages of the coating during stripping and sample preparation), and do not correspond to an inherent failure of the material which is expected at a higher load level.

Pull-out tests

Pull-out tests are commonly used to evaluate the interface behavior between two materials of a bonded assembly [START_REF] Ferrier | Influence of the properties of externally bonded CFRP on the shear behavior of concrete/composite adhesive joints[END_REF]. In our case, both core-adhesive (Figure 2.C) and adhesive-concrete (Figure 2.B) interfaces were tested. The boundaries of the elastic domain were determined from the load-strain curves: the damage threshold was identified by a change of slope at the end of the linear domain. Again, failure of the tested specimens was consistently due to polymeric matrix and primary coating breaking at the exit point of the holder. Two shear-lag parameters could be calculated from the exponential parts of the OBR strain profiles (Figure 3.B). They depend on the shear modulii of a material equivalent to the primary coating-polymeric matrix assembly 𝐺 𝑒𝑞 and of the adhesive 𝐺 𝑎 . Together with a FE modeling of the experiments, these shear lag parameters enabled us to calculate 𝐺 𝑒𝑞 = 0.97 MPa and 𝐺 𝑎 = 60 kPa. Moreover, a comparison between the FE modeling (assuming a perfect bond for all interfaces) and test results showed a fair agreement, suggesting that interfaces are really perfectly bonded. This endorsed the working assumptions.

Global FE model and computation of the 𝑴𝑻𝑭

Thanks to the previous mechanical tests, a global 3D FE model of the cable stuck to the surface of a concrete prism could be calibrated. Numerical modeling was implemented using the commercial software COMSOL Multiphysics® with the structural mechanics module. The model was based on a simplified geometry of the sensing cable, whose section is depicted in Figure 4. Symmetries of the cable made it possible to divide by four the volume to be meshed, but not to reduce the representative structure to a 2D modeling. Tetrahedral elements were used to build the mesh, and a high density of elements was considered inside and around the optical fiber. To avoid singularities in the mesh pattern, an homogeneous equivalent material accounting for both the polymeric matrix and the primary coating had to be considered. A 2D-section of the mesh pattern is shown in Figure 5. The Poisson ratios 𝜈 of the materials were arbitrary fixed, and the Young's modulii were chosen in accordance with the previous experimental results. Perfect bond were considered for all interfaces, i.e. the stiffness was set at a very high value (10 14 Pa.m -1 ). In these conditions, linear and elastic equations of mechanics ruled the behavior of the structure.

In order to compute a strain profile in the fiber which is (in first approximation) proportional to the 𝑀𝑇𝐹, boundary conditions were chosen to simulate the opening of a crack in the concrete medium, in agreement with Equation (3). In practice, a pre-existing crack is inserted in the middle of the concrete prism, which is consequently separated into two blocks that are moved away from each other through an imposed displacement. The resulting strain profile shown in Figure 6.B exhibits an exponential shape, and as a consequence, so does the 𝑀𝑇𝐹. The predictions of analytical models and Equation (4) are thus verified, which means that our real system and the theoretical simplified models behave in a close manner. The shear-lag parameter of the 𝑀𝑇𝐹 was identified by fitting the profile with Equation (4), providing a value of 𝑘 = 12 m -1 . Therefore the effective bond length was: 𝐿 𝜏 = 29 cm ± 3 cm

The uncertainty value of the transfer length was calculated by evaluating experimental uncertainties and propagating them through Equation [START_REF] Li | Mechanical property and strain transferring mechanism in optical fiber sensors[END_REF].

As pointed out before, the observed failure modes were not due to material failure, but rather to experimental testing conditions. With the help of numerical modeling, it is possible to convert the average failure load into a stress threshold: the study is valid, i.e. elastic and linear assumptions are verified, until the shear stress in the polymer matrix/primary coating set reaches a value of 0.2 MPa, or equivalently until the crack reaches a width of 3 mm. However, this is a conservative limit and the domain of validity of the sensing cable is probably much broader.

In order to validate the previous methodology for the determination of the cable FTM, a confrontation with experimental evidences is needed.

EXPERIMENTAL VALIDATION OF THE METHODOLOGY

In this section, numerical results are compared with experimental data collected from a 4-points bending test conducted on a plain concrete beam equipped with the sensing cable. The experimental setup is shown Figure 6.A. Following the application of the bending load, a single failure was initiated on the tensile face and at the center of the beam, where LVDT sensors were also placed. As this crack propagated through the beam, the strain profile measured by the sensing chain (the cable paired with an OBR) was once again proportional to the 𝑀𝑇𝐹. According to Equation (3), the 𝑀𝑇𝐹 could be found by dividing the strain profile by the crack opening value. This crack opening displacement corresponds to the area under the strain profile, since the 𝑀𝑇𝐹 is normalized. Figure 6.B displays both the above modeled 𝑀𝑇𝐹 and the experimentally determined 𝑀𝑇𝐹 experimental. The two curves are remarkably similar. Besides, the experimental effective bond length can be calculated by fitting the experimental profile using Equation (4): a value 𝐿 𝜏 = 25 cm was found while the modeled value was 𝐿 𝜏 = 29 cm, suggesting a good agreement among the two methods.

CONCLUSION

A fiber optic cable paired with a Rayleigh OFDR interrogation unit makes it possible to perform continuous strain monitoring with a centimeter scale spatial resolution for the SHM of very large infrastructures. A generic methodology was introduced in order to determine the relationship between the strain profile measured along the optical fiber and the actual strain of the monitored structure. Such a methodology is based on mechanical testing and modeling and has proved to be fully efficient for a specific surface-mounted cable. In addition, those results were consistent with experimental data obtained from a four-point bending test conducted on a plain concrete beam with a single crack.

Figure 1 :

 1 Figure 1: A -View of the cross-section of the cable stuck to the concrete surface. B -Zoom of the cable core,where two fibers and their primary coatings are assembled into a polymer matrix

Figure 2 :

 2 Figure 2: A -Experimental setup. B -A sample for the adhesive-concrete pull-out test. C -Failure of the polymer matrix and primary coating during a core-adhesive pull-out test

Figure 3 :

 3 Figure 3: A -AVE load-strain curve for a core-adhesive pull-out test. B -Corresponding OBR strain profiles.

Figure 4 :

 4 Figure 4: Simplified geometry of the model: A -Total section of the cable. B -Zoom of the core structure.

Figure 5 :

 5 Figure 5: 2D-section of the 3D mesh used for FE computing: A -total section; B -zoom in the sensing cable

Figure 6 :

 6 Figure 6: A -4-points bending test on a plain concrete beam with a single crack formed in the central part. B -Comparison between computed and measured MTFs, showing a good agreement

EWSHM 2014 -Nantes, France


Philippe Gueguen
email: philippe.gueguen@ujf-grenoble.fr

Mickael Langlais

Philippe Roux

Jakub Schinkmann

Isabelle Douste- Bacqué

FREQUENCY AND DAMPING WANDERING IN EXISTING BUILDINGS USING THE RANDOM DECREMENT TECHNIQUE

Keywords: Frequency, damping, temperature, buildings

With the increasing use of permanent, continuous and real-time networks, ambient vibrations can provide a simple tool for the identification of dynamic building parameters. This study is focused on the long-term variation of frequency and damping in several buildings, using the Random Decrement Technique (RDT). RDT provides a fast, robust and accurate long-term analysis and improves the reliability of frequency and damping measurements for structural health monitoring. This reveals particularly useful information in finding out precisely how far changes in modal parameters can be related to changes in physical properties. This paper highlights the reversible changes of the structure's dynamic parameters, correlated with external forces, such as temperature and exposure to the sun. Contrasting behaviours are observed, including correlation and anti-correlation with temperature variations.

INTRODUCTION

Since the design forces in structures are frequency and damping dependent (based on the seismic coefficient C(T,ξ) where T is the period of the building and ξ is the damping ratio), these two parameters are the subject of special attention and focus of many research activities. Most losses produced by earthquakes throughout the world are due to deficient seismic behaviour in existing buildings in spite of improvements made to seismic codes [START_REF] Spence | Comparing loss estimation with observed damage: a study of the 1999 Kocaeli earthquake in Turkey[END_REF]. A critical step in seismic risk assessment is therefore to be able to predict the expected damage for a given earthquake in existing structures. Spence et al. [START_REF] Spence | Comparing loss estimation with observed damage: a study of the 1999 Kocaeli earthquake in Turkey[END_REF] asserted that the adjustment of structural models should assume a large set of unknown parameters influencing the response of existing buildings and introducing a large range of errors and epistemic uncertainties for the establishment of fragility curves, generally due to the lack of structural plans, aging and structural design. Knowing frequency and damping can then reduce the range of errors and epistemic uncertainties for representing the vulnerability as fragility curves [START_REF] Michel | Seismic vulnerability assessment to slight damage based on experimental modal parameters[END_REF].

New instrumentations and new signal processing methods provide information on ageing effects or after extreme events. The basic idea is that any modifications of the stiffness of a system alter its dynamic response [3] [4]. Variations in these modal parameters can result from a change in the boundary conditions (e.g. fixed-or flexible-base structure), mechanical properties (e.g. reinforcement or retrofitting) or the elastic properties of the material (e.g. Young's modulus). The causes may also be related to non-linear responses of the buildings, transient variations having been observed during seismic excitation due to the non-linear response of the soil-structure boundary [5] [6] or to the closing/opening process of pre-existing cracks within the elements of the reinforced concrete buildings [START_REF] Clinton | The observed wander of the natural frequencies in a structure[END_REF] [START_REF] Michel | Time-frequency analysis of small frequency variations in civil engineering structures under weak and strong motions using a reassignment method[END_REF]. Finally, these variations can also be long-term, reversible and slight, as recently observed by Clinton et al. [START_REF] Clinton | The observed wander of the natural frequencies in a structure[END_REF] and Todorovska and Al Rjoub [START_REF] Todorovska | Effects of rainfall on soil-structure system frequency: example based on poroelasticity and a comparison with full-scale measurements[END_REF], often correlated to the temporal variations of the atmospheric conditions (temperature, humidity, etc.) Most previous studies conducted in civil engineering structures (e.g., [START_REF] Clinton | The observed wander of the natural frequencies in a structure[END_REF][10] [START_REF] Hua | Modelling of temperature-frequency correlation using combined principal component analysis and support vector regression technique[END_REF] [START_REF] Nayeri | A novel approach for the structural identification and monitoring of a full-scale 17-story building based on ambient vibration measurements[END_REF]) have shown that temperature is the most significant cause of variability of modal frequencies.

The main purpose of this paper is to analyse the long-term variations of the frequency and damping of one building. After presenting the Random Decrement Technique (RDT) applied to the data in section 2, the experiment is applied to the data in section 3 and the variations in frequency and damping are discussed in section 4. Concluding remarks are presented in section 5.

THE RANDOM DECREMENT TECHNIQUE

An effective solution to track frequency and damping variations over time is to apply the random decrement technique, RDT. By stacking a large number of windows with identical initial conditions, ambient vibrations remain stationary and the impulse response of the structure is revealed. Vandiver et al. [START_REF] Vandiver | A mathematical basis for the random decrement vibration signature analysis technique[END_REF] and Asmussen et al. [START_REF] Asmussen | Statistical theory of the vector random decrement technique[END_REF] provide details on the theory of RDT and its mathematical formulation that can be simplified by:

RDT (! ) = 1 N s(t i + ! ) i=1 N ! ( 1
)
where N is the number of windows with fixed initial conditions, s is the ambient vibration window of duration τ, and ti is the time verifying the initial conditions. The choice of initial conditions is a key point in ensuring the stability of the Random Decrement signature. The null displacement and positive velocity conditions proposed by Cole [START_REF] Cole | On-line failure detection and damping measurement of aerospace structures by random decrement signatures[END_REF] and verified by Asmussen et al. [START_REF] Asmussen | Statistical theory of the vector random decrement technique[END_REF] were used in the present work. The number of windows N is also critical to obtain a stable and relevant damping estimation. In our case, we optimised the parameters of the RDT by testing the τ and N values for providing the smallest dispersion. Before the RDT processing, a fourth-order band pass filter was applied to the raw data, centred on the expected fundamental frequency with a 10% frequency band. The RDT signature of the mode is exponentially damped and its period is computed by averaging the time laps between two upward zero crossing points.

Since the first real-scale structure instrumentations at the beginning of the 60's throughout the world, the quality of the sensor and the accessibility of the data have increased a lot, providing a large number of real data for testing as those provided in real-time and continuous recording by permanent networks. With this new source of data, the RDT is certainly the most appropriate solution for monitoring the health of the structure, with a good cost/quality ratio. For example, Mikael et al. [START_REF] Mikael | Long-term frequency and damping wandering in buildings analysed using the Random Decrement Technique (RDT)[END_REF] tested the RDT in two buildings monitoring by the French Accelerometric Network and they concluded in the ability for the RDT to detect very small variations that finally could be related to the integrity of the structure.

DESCRIPTION OF THE EXPERIMENT

The building of the present study was constructed before 1970 in the Grenoble city (France), resting on a huge sedimentary basin made of soft sediments and providing strong seismic site effects. The number of storeys is 7. They are located at the corner of an urban block (Fig. 1). The building is designed in RC shear wall elements. The building is instrumented at the 7th floor with a lowsensitivity/high dynamics 24bit seismic station associated to a CMG5 (Güralp) 3 components accelerometer. The sensor was aligned with the main orientation of the building, along the longitudinal (L) and transverse (T) directions. This sensor has a flat response in the 0.5 -40 Hz frequency band. The sampling frequency is set to 200 Hz. The GPS is continuously on, in order to provide the best time correction to the records and to be able to compare them with external information (e.g. temperature). The instrumentation started in March 2009, and ended in May 2011, i.e. two years of recording are available. Fourier transforms of one-hour ambient vibrations window show a clear frequency peak, corresponding to the fundamental frequency of the building in the L and T directions, with values of 2.6 Hz and 3.4 Hz, respectively. For two years, the variations of the frequency in the two directions are displayed in Fig. 2. The color scale corresponds to the air temperature recorded by a meteorological station located 2 km far from the building, in the Grenoble basin. As mentioned by Mikael et al. [START_REF] Mikael | Long-term frequency and damping wandering in buildings analysed using the Random Decrement Technique (RDT)[END_REF], two scales of variation are usually observed: daily variation as function of the day and night variation of temperature, and the seasonal variation related with the winter-summer alternation. In Fig. 2, a time window of 120 hours length smooths the variation of the frequency, i.e. only the seasonal variations are displayed. First, we observe that the RDT allows a very accurate assessment of the frequency, providing information on very small and fast shift of frequency. Same correlation are observed in the L and T direction, that led us confirm the physical origin of the wandering observed for two years, with alternate variation between hot and cold periods.

RESULTS

Frequency

With the RDT method, the damping coefficient of the building can be computed and we can compare the damping versus frequency variation. Under the single-degree-of-freedom system (SDOF) assumption, the fundamental frequency is related to stiffness K and mass M, through the equation:

! = K M (2)
where ω=2π.f is the resonant circular frequency of the building. To first order, we assume that the mass M remains constant. The frequency variations are then related to the variation of stiffness K, K depending on the properties of the building (e.g., Young's modulus, height, design of the building, etc.), but also on the cracks opening and the boundary conditions (e.g., fixed-or flexible-base building).

In order to investigate the effect of the temperature on the building frequency, we focused on one direction (longitudinal) in the following. The correlation between the temperature and the frequency is displayed Figure 3, analysing also the two phases of variation: the « cooling » and the « heating » phases corresponding to the decrease and the increase of the frequency and the temperature, respectively. No differences between these two phases are clearly pointed by this figure. At this time scale (seasonal variations), the response of the structure to the external forcing is about the same whatever the cooling or the heating periods. However, looking at the faster variations, i.e. related to the daily time scale, we note (Fig. 4) that the cooling and the heating gradients are different for the temperature and the frequency variation. In fact, while the increase (heating) of the frequency and the temperature has the same rate, the decrease of the frequency (cooling) is faster than the frequency rate let us suppose inertial effect of the temperature on the structure. This is clearly observed in Figure 5 that shows the rate of the frequency decreases or increases closely related with phases of heating and/or cooling, but also with the summer-winter period.

Damping

Even though the damping coefficient is a critical parameter for assessment design, it is not usually easy to determine the damping coefficient in experimental conditions and its physical origin in most practical systems are seldom fully understood. The free-oscillating response of the SDOF, as controlled by the e -ξωt function, is thus proportional to the frequency ω and the critical damping coefficient ξ. The critical damping coefficient ξ is expressed as follows:

! = c 2 K.M (3)
where c represents the energy-loss mechanism, and K and M the stiffness and mass of the SDOF respectively. The physical origin of the damping coefficient in existing buildings remains complex, mixing material damping, radiative damping at the soil-structure boundary and certainly frequencydependent damping related to the scattering of waves in the structure. We observe Figure 6 the seasonal (smoothed) variation of the damping with the temperature. The damping varies also with the temperature but this variation seems to be correlated or/and anticorrelated with the temperature according to the value of the temperature and the season. While we observed previously a strong correlation between temperature and frequency, this is not the case for the damping, even if the main parameter controlling damping and frequency is the stiffness (Eqs. 2 and 3).

If we look at the link between the frequency and the damping as function of the air temperature, the relationship is strongly non-linear. Mikael et al. [START_REF] Mikael | Long-term frequency and damping wandering in buildings analysed using the Random Decrement Technique (RDT)[END_REF] have already observed this nonlinear response of frequency and damping and this observation confirms the lack of knowledge on the damping phenomena observed in the structure.

CONCLUSIONS

The natural wandering of the frequency and damping that control the seismic response of a building during an earthquake have been analysed under ambient vibrations. The use of RDT has shown its efficiency for measuring very small variations of damping and frequency for the long-term monitoring of buildings. In our study, the main parameter controlling the fluctuations is the outside temperature. This was confirmed by the analysis performed by Mikael et al. [START_REF] Mikael | Long-term frequency and damping wandering in buildings analysed using the Random Decrement Technique (RDT)[END_REF] that showed different frequency versus temperature trend, based on independent measurements and for independent buildings. Conclusion is that these variations are of physical origin. In our case, we observe two different behaviours depending on the time-scale considered: daily or seasonal.

While almost all previous studies showed a positive correlation between frequency and temperature, no definitive conclusions can be done from this analysis since non-linear relationship have been observed.

Further works must be done for improving the analysis of this building, testing additional information for explaining the variation observed. This information can be related to the water table level that may wander and change the boundary condition of the structure (e.g., soil-structure interaction effect) or to the winter-summer differences of behaviour of the building with the internal and domestic heat flow.

Figure 1 :

 1 Figure 1: Description of the building tested

Figure 2 .

 2 Figure 2. Variation of the L (blue line) and T (green line) fundamental frequencies for two years, displayed with the variation of the air temperature (min °C: -10 bluemax °C: +40 orange).

Figure 3 .

 3 Figure 3. Variation of the frequency (blue line) and temperature (black line) for two years and differences of the gradient of temperature and frequency during the heating and cooling phases. (Color scale -min °C: -10 blue -max °C: +40 orange).

Figure 4 .

 4 Figure 4. Rate of daily variations of the frequency and the temperature for one week.

Figure 5 .

 5 Figure 5. Comparison between the gradient of the frequency and temperature variation during the increase (heating) and decrease (cooling) phases. The color plots correspond to the values of the temperature (blue: cold, red: hot).

Figure 6 .

 6 Figure 6. Variation of the L fundamental frequencies (thick blue line) and damping (thin black line) for two years, displayed with the variation of the air temperature (min °C: -10 blue -max °C: +40 orange).

EWSHM 2014 -Nantes, France

ACKNOWLEDGMENTS

This work was supported by the French Research National Agency (ANR) under the RiskNat program (project URBASIS n°ANR-09-RISK-009).