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# The fluctuation-dissipation theorem used as a proxy for damping variations in real engineering structures

Marc-Antoine Brossault, Philippe Roux, Philippe Guéguen

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Marc-Antoine Brossault, Philippe Roux, Philippe Guéguen. The fluctuation-dissipation theorem used as a proxy for damping variations in real engineering structures. Engineering Structures, 2018, 167, pp.65-73. 10.1016/j.engstruct.2018.04.012. hal-04322171

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https://hal.science/hal-04322171v1

Submitted on 25 Jul 2025

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The fluctuation-dissipation theorem used as a proxy for damping variations in real engineering structures

Marc-Antoine Brossault, Philippe Roux, Philippe Guéguen

ISTerre, Université de Grenoble Alpes, CNRS/IFSTTAR

**Abstract**

The damping ratio and the resonance frequency represent relevant health indicators in structural health monitoring, and they are key parameters in the design of buildings. The combination of two modal parameters (i.e., damping, frequency) is reported to vary in time due to changes in environmental conditions, and with damage to the structure and dynamic loading. Amplitude dependence of the damping and frequency parameters has also been reported for strong loading, such as strong winds. However, none of these reasons can explain the much higher fluctuations of the damping values compared to the resonance frequency in the amplitude range of natural ambient vibrations. The objective here is to investigate if any amplitude dependence in this lower range of loading can explain the observed differences in the scattering of these two modal parameters. To do so, the first approach consists of a study of vertical clamp-free beams made out of different materials. Secondly, the observations and interpretations obtained are scaled up to buildings under continuous monitoring. Finally, we consider the fluctuation-dissipation theorem to explain these uncommon results.

**Keywords:** damping ratio; fluctuation-dissipation; laboratory tests; civil engineering structure; Grenoble City Hall; Factor Building; Prefecture Nice
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# I. Introduction

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Since the 1990s, the use of ambient vibrations to monitor civil engineering structures has been driven by both the increasing demand for safety assessment and the technical progress made in the acquisition of continuous data. Vibrations recorded in a structure provide access to temporal monitoring of various health indicators, including the percentage of critical damping ζ, termed the damping or damping ratio herein, and the modal frequency f. When associated to a one-dimensional, single-degree-of-freedom like structure, these two modal parameters are directly related to the stiffness of the structure (i.e., ω²=k/m, ζ=0.5 c/[km]⁰.⁵, where ω=2πf is pulsation, c is viscous damping, k is stiffness, m is mass). They are very sensitive to damage [1][2], to variations in the dynamic loading that produce nonlinearities [3][4][5], to boundary conditions[6], and to weather conditions, such as air temperature and humidity [5][7]. Consequently, in view of these nonrepresentative changes in the health of a structure, these two modal parameters of damping and frequency are often considered not to be sufficiently effective to be used in structural health monitoring.

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As well as their use for structural health monitoring purposes, damping and frequency are also the most critical parameters in the structural design to prevent earthquake damage, as the force that is applied to a civil engineering structure is related to a corresponding coefficient that depends on its period and damping. However, under weak motion, the fluctuations of damping that have been observed during continuous monitoring (e.g., [7][8][9]) has indeed been reported to be very high, and especially to be stronger than the resonance frequency, even though damping and frequency have direct first-order relationships to the stiffness. In the amplitude range of ambient vibrations recorded in civil engineering structures, the fluctuations of the damping over time are more than one order of magnitude higher than those of the resonance frequency [7][8]. The dependency of the damping on the loading amplitude applied to the structure and observed in previous studies [9][10][11] is a possible source of this overall strong variability. As discussed by Jeary [10], in between two loading thresholds below and above which damping can be considered as constant, the damping is reported to increase significantly. Tamura [9] showed for instance that for acceleration of 1 ×10⁻² m/s² to 4.5 ×10⁻² m/s², damping increases from 1.5% to ~2.5%. However, in the present case (i.e., using ambient vibrations), the range of strain considered is very weak, and we assume it would correspond to the zero-amplitude plateau of the Jeary model [10]. Moreover, Modena et al. [12] showed that cracks in structures that are
not detectable by visual inspection might promote larger variations in damping than in the
resonance frequency, which confirms the high sensitivity of damping to the presence of
cracks.

However, at the thermodynamic equilibrium of a system, the fluctuation-dissipation theorem relates the spontaneous fluctuations of a system and its response to a small external perturbation [14][15]. Combining the formulations proposed by Kubo [14] and Campillo and Roux [13] for the Ward identity applied to the noise correlation theorem leads to the formulation of a proportional relation between the dissipation (or damping) in a system and the intensity of the external loading. This theoretical development is made for weak forcing parameters for which all deformations are considered to remain in the elastic domain, and consequently, the frequency should remain constant.

The objective of the present study was to experimentally verify the dependency of
these two modal parameters (i.e., damping, frequency) with respect to the external loading
amplitude. This paper is structured as follows. Section II presents the experimental set-up
conducted at the laboratory scale on vertical clamp-free beams; i.e., a reduced-scale
equivalent model of structure-like high-rise building, behaving like a continuous beam. The
basic interpretation of the dynamic response of the beam is also considered. Section III
describes the processing applied to the data to assess the continuous damping and modal
frequency values, according to the loading amplitude. The results of the laboratory
experiment are presented in section IV. Then, data from real-scale buildings are investigated
in section V, to confirm the application of the fluctuation-dissipation theory to civil
engineering structures. Finally, the interpretation of the main results is discussed in section
VI, with the conclusions arrived at in the framework of the fluctuation-dissipation theory.

## II. Experimental set-up and data

Figure 1 shows the experimental set-up used for this study. Continuous recordings of
vibrations are performed at the top of three different vertical clamp-free beams: a granite and
a limestone beam, with sections of 2 cm × 5 cm, and a Plexiglas beam with a section of 1 cm
× 5 cm. The beams are fixed on a large limestone block with epoxy glue. Two accelerometers
are used (4518-003; Brüel & Kjaer), one at the base and one at the top of each beam. They
have a sensitivity of 100 mV/g, and a sampling frequency of 5 kHz. The acquisition is
through a signal conditioner (Type 2694-A; Brüel & Kjaer), which enables pre-amplification
of the data with a gain of 10 dB. An acquisition card (USB-6259 BNC; National Instruments) connects the conditioner to the computer. The beam is loaded through a continuous air jet applied at the top. As a first order approximation, this very low, random and stationary flow reproduces the physical characteristics of the ambient vibrations recorded in real-scale structures having a modal response equivalent to continuous beam model. This was confirmed by comparison with numerical modeling [16] and modal analysis [17]. The same experimental protocol was used, avoiding the disturbance on the tip of the beam and allowing a correct modal analysis of the beam.

Figure 2a shows 3 hours of continuous recording of the acceleration at the top of the granite beam. The recordings are actually quasi-continuous, with their gathering into 1-h files to allow the acquisition program to transfer the data to the storage computer system. As a result of the mechanical variation of the air compressor, the amplitude of the air jet applied at the top of the beam varies, with a period of 40 min. This period of variation is long enough compared to the characteristic period T of the resonances of the beams; i.e., related to their dynamic responses above 0.1 s. The small value of the beam resonance frequencies with respect to the time characteristic of the flow (40 minutes) allows the loading to be considered as quasi-stationary, as assumed for ambient vibrations recorded for real-scale buildings.

Fourier transforms of 1-h-long vibrations recorded at the top of each beam are shown in Figure 2b. For these clamp-free conditions, we can define the dynamic response of each beam using the Euler-Bernoulli continuous beam model [18], as given in Equation (1):

$$ \frac{\partial^2}{\partial x^2} EI(x) \frac{\partial^2 u(x,t)}{\partial x^2} + m(x) \frac{\partial^2 u(x,t)}{\partial t^2} = 0 \quad (1) $$

where $u(x,t)$ is the horizontal time displacement along the beam (x), $m(x)$ is the mass per unit length along the direction x, $EI(x)$ is the Young's modulus, and $I(x)$ is the moment of inertia. Solving Equation (1) for a homogeneous beam model provides the modal frequencies of Equation (2):

$$ f_n = \frac{(\alpha_n H)^2}{2\pi} \sqrt{\frac{EI}{mH^4}} \quad (2) $$

where H is the length of the beam, and the $\alpha_n H$ coefficients are a discrete series defined by:
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$$ \alpha_1 H = 1.8751 $$

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$$ \alpha_2 H = 4.6941 $$

(3)

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$$ \alpha_3 H = 7.8548 $$

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$$ \alpha_n H \approx (2n - 1) \times \frac{\pi}{2} $$

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The solution of Equation (1) for the vibration motion of the free-clamped beam leads to the expression of the mode shapes as in Equation (4):

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$$ \varphi_n(x) = \cosh \alpha_n x - \cos \alpha_n x - \frac{\cosh \alpha_n H + \cos \alpha_n H}{\sinh \alpha_n H + \sin \alpha_n H} (\sinh \alpha_n x - \sin \alpha_n x) \quad (4) $$

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where *ch* and *sh* are the hyperbolic cosines and sines, respectively. In Figure 2b, the frequency peaks observed for each beam correspond to their theoretical Euler-Bernoulli modal frequencies (Eq. 2), as previously confirmed in equivalent laboratory experiments [16]. Considering the ratio of the resonance frequencies, all of the beams experience pure bending deformation. This is confirmed by the mode shapes computed using a peak-picking method [19] with 10-min-long recording of the ambient vibrations on 16 accelerometers located along the granite beam. Figure 3 compares the experimental to the theoretical modal shapes of the granite beam (Eq. 4). As well as showing high visual agreement, the modal assurance criteria values [20] calculated for each mode and given in Figure 3 are >0.9; i.e., higher than the widely used match threshold of 0.8 [3][20]. Comparison of the experimental and theoretical results allows the conclusion that the observed flexural frequencies correspond to the vibration modes of the beam and the fixed-base condition is respected.

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### III. Processing

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The continuous acceleration recorded at the top of each beam is processed using the random decrement technique, to track the damping and frequency variations over time. This random decrement technique was first proposed in the late 1960s by Cole [21], and then further studied especially by Vandiver et al. [22], who thus provided details of the theory and operability. This method relies on the concept that at each time *t*, the dynamic response of a system to a stationary and stochastic loading is composed of the response to the initial triggering conditions that correspond to the free-oscillating response, and the response to the

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random loadings [6]. In the present case, these initial conditions are selected to be the local extrema, which provides efficient conditions for the random decrement technique [23]. Due to the stationary nature of ambient vibrations, the averaging of a large number of time segments with identical initial triggering conditions provides the free-oscillating part of the response. This average is known as the random decrement signature, and considering the stationary of the loading, it is then assumed to be at the first order equivalent to the impulse response of the system [24][25]. A simple formula for the random decrement signature (RDS) is found in Guéguen et al. [6], as in Equation (5):

$$RDS(\tau) = \frac{1}{N} \sum_{i=1}^{N} s(t_i + \tau) \qquad (5)$$

where N is the number of windows that respect the chosen initial conditions, s is the recording signal, tᵢ is the time to verify the initial triggering conditions, and τ is the time index in the random decrement signature, which varies from zero to the window duration. The damping ratio ξ and the circular frequency ω are estimated from the random decrement signature using a fit with an exponential decay that corresponds to the impulse response (free oscillation) of the structure [18], according to Equation (6):

$$x(\tau) = Xe^{-\xi\omega\tau}\cos(\omega\sqrt{1-\xi^2}\tau) \qquad (6)$$

The exponential fit is employed rather than the classical logarithmic decrement method in order to simultaneously invert the frequency and the damping, following the model given in Eq. 6. Temporal monitoring of both damping and frequency are obtained mode by mode. The raw data are initially band-pass filtered around each modal frequency, using a fourth-order Butterworth filter. The two cutting frequencies are equal to the central frequency, as plus and minus 10%. In the present study, the random decrement technique is applied to the filtered data, with a sliding time window of 1,000× the estimated period of the mode, and a 50% overlap. The duration of the time windows to obtain each random decrement signature is set to τ, which is equal to 10 periods [16]. The damping and frequency values are then flagged to the central time of each time window. Simultaneously, the amplitude of the filtered signal is associated to each window, which is computed as the mean value of the envelope of the accelerometric time history.
The output of this processing provides the damping, frequency, and acceleration vectors for each mode. Due to the 40-min cycle period of the air compressor, these three parameters are finally band-pass filtered between 20 min and 1 h, to focus the data on the dynamics of interest.

IV. Results

Figure 4 shows the damping and frequency values with respect to the loading amplitude for the four first modes for the granite beam. To match the energy unit, the loading is computed as the square of the aforementioned average acceleration. The damping and frequency versus the squared acceleration are shown in Figure 4 as a color scale that is proportional to the density of the points measurement. A clear variation in damping with respect to the energy of the loading is seen, whereas the resonance frequency remains constant on the same range as the energy.

The damping increases from ~0.5% to ~1.0% for the first mode, from ~0.4% to ~0.6% for the second, from ~0.6% to ~0.8% for the third, and from ~0.35% to ~0.5% for the fourth, whereas the resonance frequencies are stable around the average values (μ) of 14.2 Hz, 89.9 Hz, 248.8 Hz, and 492.2 Hz for modes 1 to 4, respectively. For information, the order of the magnitude of the deformation that corresponds to the acceleration is between 10⁻⁷ m and 10⁻⁸ m; i.e., below the classical yield deformation of beam-like structure. Herein, we do not discuss the contribution of each source of damping, the aerodynamic damping being able to contribute significantly in the percentage of critical damping, in the same manner as for real-case buildings. In all cases, the increasing trend of the percentage of critical damping (ξ) is fitted with a linear relationship. This linear fit allows extrapolation toward null acceleration. This particular value of the damping will be referred to in the following as the ‘zero-amplitude’ damping, which corresponds to the intrinsic damping of the structure. From our measurements, the intrinsic damping ratios are 0.42%, 0.30%, 0.41%, and 0.31% for modes 1 to 4, respectively; i.e. they decrease with increasing mode number.

Moreover, it can be noted that the standard deviation (σ) is stable whatever the loading amplitude, with values of 7.90% ×10⁻², 2.17% ×10⁻², 1.83% ×10⁻², and 8.68% ×10⁻³ for the damping ratios, and 7.33 ×10⁻³ Hz, 1.42 ×10⁻² Hz, 2.74 ×10⁻² Hz, and 3.29 ×10⁻² Hz for the resonance frequency of each mode. The stability of σ confirms that no bias is introduced in the measurement in the laboratory experiment set-up. On real-scale structures,
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Mikael et al. [7] reported a coefficient of variation (σ/μ) for the damping that was one or two orders of magnitude higher than for the frequency. In the present study, the average fluctuation coefficients for the damping ratio are 10.50%, 4.62%, 2.74%, and 2.08%, respectively, and for the resonance frequency, they are 5.15 × 10⁻²%, 1.58 × 10⁻²%, 1.10 × 10⁻²%, and 6.68 × 10⁻³%, respectively, and are thus up to three orders of magnitude higher for the damping than for the frequency. Unlike studies on real structures, there are no variations in the other external factors that are known to influence the modal parameters, such as air temperature [8], and the fluctuation coefficients in these data only reflect the measurement accuracy of each parameter, relative to the loading amplitude.

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Additional damping measurements were performed for the granite beam. These consist of measuring the free-oscillation of the beam under transient forcing, using two methods. First, an impulse force is applied to the limestone block beam support, and second, a 10-s-long conditioning signal emitted via a piezoelectric transducer is applied at the base of the beam. These two methods are comparable to the pull-out test and the shaker test, respectively, which are two classical experiments that are applied to real buildings for damping and frequency measurements. The deformations generated in the beam are ~10⁻⁷ m and 10⁻⁵m. In these cases, the damping values are 0.41% and 0.44% for the second mode, 0.47% and 0.44% for the third mode, and 0.40% and 0.38% for the fourth mode, from the first and second experiments, respectively. These values are higher than the zero-amplitude damping obtained from the linear trends observed in Figure 4. Indeed, the damping obtained with the two methods aforementioned (i.e., transient external impulse forcing) corresponds to the upper limits; i.e., including all of the sources of attenuation (e.g., aerodynamic, intrinsic, radiative, scattering).

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Figure 5 shows analog results for the third mode of the limestone and Plexiglas beams, same observation having been done for other modes. As for the granite beam, the resonance frequency remains constant whatever the loading amplitude. These data show stable σ, with coefficients of variation of 4.68% and 0.16% for the damping and frequency, respectively, for the Plexiglas beam, and 6.39% × 10⁻³ and 2.08% for the limestone beam, respectively. A clear linear dependency of the damping with respect to the squared acceleration is observed for the limestone beam, which is characterized by an increasing linear trend with a slope of 1.11 and a zero-forcing damping of 0.31%. On the other hand, the Plexiglas beam does not show any variations with acceleration.

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The main differences between the Plexiglas beam and the granite and limestone beams are the density and size of the heterogeneities in the material. These range from purely

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homogeneous materials in the case of Plexiglas, to a fine-grained one for limestone, and macro-grains and some cracks for granite. TenCate et al. [26] discussed the experimental measurements of the elastic properties on diverse samples submitted to weak deformation. They showed that heterogeneities in the rock samples were responsible for the appearance of the nonlinear elastic response; i.e. variations in the elastic properties of the system under weak deformation. To explain this particular behavior, Johnson and Sutin [27] proposed a mechanism that involved grains and their associated bond system. In the present case, the deformation is weak and the origin of the damping variation for the granite and limestone beams with increasing loading might be explained by an equivalent dependency on heterogeneities, compared to the homogeneous Plexiglas beam. This interpretation will be discussed in section VI.

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## V. Application to real-scale buildings

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Three well-studied real buildings shown on Figure 6 are also considered in this study. The first is the Factor Building (FB) of the University of California, Los Angeles, which is located on the Los Angeles campus [29]. This is a 17-story (plus two basement levels) steel special moment resisting frame building supported by concrete bell caissons and spread footings. The floor area increases by approximately 14% for floors 10 to 16. This building is permanently instrumented with 72 3C accelerometers that are continuously recording at 100 Hz [29]. The horizontal channels are oriented North-South and East-West. The output range scale is ±4 g, which is suitable for a seismically active area such as California, while being sensitive enough for recording ambient vibrations [6][29]. A view of the FB is given in Figure 6a, together with the Fourier transform of a 1-h-long recording of the ambient vibrations at the top of the building. The peaks located ~0.6 Hz and ~1.85 Hz are the first and second bending modes in the North-South direction, and the peak at ~0.8 Hz corresponds to a torsion mode [30], although this is not considered further in the present study.

The second building is Grenoble City Hall (GCH) [3][31]. This is a 13-story reinforced concrete tower with a rectangular plan section (44 x 13 m) and external glass frontage fixed to a steel framework. Between the third and twelfth floors, the inter-storey height is regular (3.2m) and higher for the first (4.68 m) and second storey (8m). Above the second storey, a pre-stressed slab of 23m span is supported by two inner cores. The structural strength system

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combines the shear walls of the inner cores with RC frames bearing the RC floors. The foundation system consists of deep piles, anchored in an underlying stiff layer of sand and gravel. The instrumentation of this building, which is part of the French Permanent Accelerometric Network (RAP) [32], includes six 3C accelerometers, at the bottom and the top of the structure. The sampling frequency is 125 Hz and the output range scale is ±1 g, which is sensitive for the recording of ambient vibrations [3]. The focus is set here on the longitudinal component of one of the top stations. Figure 6b shows the fundamental frequency around 1.2 Hz associated to the first bending mode, and the first torsion mode around 1.45 Hz [3], which is not considered further in the present study.

Finally, the third building is the Prefecture Tower in Nice (NPT) [33]. This consists of twin towers that are separated by a 10-cm joint. Due to their similarity, only the eastern, 22-story building is instrumented. The resistance of the structure is composed of reinforced concrete shear walls in inner core containing lift ans stairs, that support RC slabs. An external glass frontage is present and inter-storey thin columns placed to limit deformation. These columns are laid on a box girder at the first floor. The foundation of the building consist on deep piles foundation, not rigidly connected to the pile cap. Twenty-four accelerometric sensors are positioned from the bottom to the top of the structure, with the instrumental features as for the GCH array (i.e., the RAP technical specifications). Figure 6c shows an elevation view of the NPT building and the Fourier transform of a 1-h-long recording in the longitudinal direction on the top floor. The peak at around 1.2 Hz corresponds to the first bending mode, and the peak around 1.6 Hz to the first torsion mode [33].

Considering the first bending mode of these three buildings, the same methodology was applied to extract the continuous resonance damping and frequency values as in the laboratory experiment set-up. The results here are shown in Figure 7, in the same way as in Figures 4 and 5. All of these buildings show similar behaviors for both the damping ratio and the frequency. The damping ratio increases significantly: 91.2% (for FB), 12% (for GCH), and 13.2% (for NPT). In contrast, the frequency decreases slightly over the range of excitation, with the loading squared acceleration of about 0.32% (FB), 0.34% (GCH), and 0.33% (NPT). The intrinsic damping ratios are calculated by linear extrapolation toward the zero-forcing squared acceleration value: the intrinsic damping is 1.38% (FB), 1.03% (GCH), and 0.76% (NPT). As observed by Michel et al. [31] for the GCH building, the slight frequency decrease observed in the real structures might be due to their complex properties, including the units assembled with joints, but might also be due to the soil-structure
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interaction that might soften the building response in the presence of a soft soil condition, even under weak deformation.

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The fluctuation coefficients computed at a given squared acceleration loading are stable over the entire range for the damping ratio, at 14.33% (FB), 7.28% (GCH), and 6.74% (NPT), and for the frequency, at 0.502% (FB), 0.137% (GCH), and 0.118% (NPT). This stability indicates that no bias is caused by the natural ambient vibration, as in the case of the laboratory experiments. The fluctuation coefficients are much higher for the damping ratio than the frequency, as observed by Nayeri et al. [8] and Mikael et al. [7] for the FB and the GCH building, respectively. These two studies indicated fluctuation coefficients of around 70% (FB) and 12% (GCH) for the damping ratio, which are five-fold and two-fold larger, respectively, than these measurements in the present study. These differences are due to the differences in the calculation of the fluctuation coefficient between the present study and Nayeri et al. [8] and Mikael et al. [7]. This was computed over the whole monitoring period by Nayeri et al. [8] and Mikael et al. [7], and at constant amplitude of loading in the present study. These differences thus highlight the impact of taking into account the amplitude dependency of the damping in the uncertainty evaluation.

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# VI Discussion

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The experimental data presented in this study indicate clear dependency on the amplitude of the damping ratios, both in the laboratory experiments and in the real civil engineering structures, even if the experimental condition are not the same but very similar with respect to the system dynamics (continuous beam-like structure) and loading (equivalent ambient vibration). This dependency has been reported in previous studies performed on structures under strong shaking [9][10][11][34][35][36]. For example, Jeary [10] proposed an explanation that involved the mobilization of larger and more abundant cracks with increasing excitation amplitude. Assuming a constant mass, the opening of these discontinuities implies alterations in the stiffness, which impact in the opposite way on the modal damping and frequency. The Tamura [9] observations are also consistent with the theory proposed by Jeary [10]. Depending on the acceleration, three different behaviors were seen: (1) a 'weak-forcing' plateau with a threshold of $10^{-2}$ m/s$^2$ top acceleration, below which no cracks are opened and no variations in the modal parameters are seen; (2) a decrease in the damping simultaneous with an increase in the frequency, for acceleration up to $4.5 \times 10^{-2}$ m/s$^2$;

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(3) a plateau at higher acceleration without variations in the damping and frequency. However, during the second phase, the relative variations in the damping and frequency were very different, with shifts of ~66% (increase) and ~2.5% (decrease), respectively, as a consequence of exceeding the yield deformation.

Although the trends for increased damping and decreased frequency are similar between the present study and Tamura [9], the loading amplitudes are totally different. In the present study, the maximum top acceleration in the buildings is around $10^{-4}$ m/s² with a corresponding deformation of about $10^{-6}$ m for all of the buildings, which corresponds to the ‘weak-forcing’ plateau of the Jeary theory, where no variations in damping and frequency are expected. The same also applies to the laboratory experiments, with deformation below $10^{-7}$ m. However, in both cases (i.e., laboratory, real buildings) damping increases significantly, whereas there are no significant variations in frequency, except for the Plexiglas beam.

One interpretation can be considered through the fluctuation-dissipation theorem applied to the laboratory and building cases. The explanation that we propose here relies on mathematical developments derived from the Helmholtz equation and their association to the fluctuation-dissipation theorem. Note that this approach can be further generalized to the bi-harmonic flexural wave equation for beams (Eq. 1). In a volume V, consider two fixed points 1 and 2 and a varying position x, the Ward identity states [13] as in Equation (7):

$$Im(G_{12}(\omega)) = \frac{2\omega\kappa}{c} \int_V G_{1x}G_{2x}^* dV \quad (7)$$

where $G_{12}$ is the Green's function between positions $r_1$ and $r_2$, c is the wave velocity, and $\kappa$ is the attenuation. In the same system, the ambient noise correlation is expressed as given in Equation (8):

$$C_{12}(\omega) = \sigma_v^2 \int_V G_{1x}G_{2x}^* dV \quad (8)$$

where $C_{12}$ is the two-point correlation function, and $\sigma_v^2$ is the intensity of the ambient vibrations in the system. By combining Equations (7) and (8) and choosing points 1 and 2 to coincide, we obtain the following Equation (9):

$$\frac{2\omega\kappa}{c} C(\omega) = \sigma_v^2 Im(G(\omega)) \quad (9)$$
where Im(G) is the imaginary part of the Green's function. In principle, the attenuation of Green's function (or elastic waves) depends on the density of local heterogeneities and the nature of defects present in a complex system. The fluctuation-dissipation relationship proposed by Kubo [14] becomes as shown in Equation (10) at the classical limit, or in other words, without taking into account the quantum fluctuations [15]:

$$ \mathrm{Im}(G(\omega)) = \frac{\omega}{2k_B T} C(\omega) \qquad (10) $$

where k_B is the Boltzmann constant, and T is the effective temperature of the solid at equilibrium. k_B is introduced as the consequence of the thermodynamic equilibrium state that results from the balance between the energy loading of the structure under continuous noise excitation and the required damping that prevents the structure from collapsing. k_B characterizes the entropy in thermodynamics, that can be interpreted as the measure of the microscopic disorder in a solid. Generally, the response of a solid to a stress can be divided into elastic and anelastic part. The latter contains information about temperature, stress and the type of local heterogeneities. The nature and the density of microscale defects are, in turn, controlled by temperature, pressure and stress. For that reason, at the thermodynamic equilibrium (k_B T=constant) the attenuation (or dissipation) is controlled by the small-scale heterogeneities in the solid and the continuous noise excitation applied to the structure.

Finally, if the temperature remains constant, Equations (9) and (10) lead to a proportional relationship between the system attenuation and the intensity of the ambient vibrations, as in Equation (11):

$$ \kappa = \frac{c}{4k_B T} \sigma_V^2 \qquad (11) $$

Equation (11) was established for linear systems that experience linear elastic deformations. In these examples given here, the damping variations are consistent with the increasing loading intensity, as described by the fluctuation-dissipation approach. The slopes between the damping ratio and the intensity of loading are proportional to $\frac{c}{4k_B T}$ according to Equation (11). However, the Plexiglas beam, which is the most representative example of linear systems, did not show the expected dependency between the loading and the
attenuation. We interpret this according to the presence of multi-scale heterogeneities in the two rocky beams compared to the crystal-like structure of the Plexiglas beam. The heterogeneities (e.g., cracks or/and grains) in the granite and limestone rocks are sources of nonlinearity, which manifests themselves with different physical observations, such as slow dynamics at higher deformation levels [26][27][28], which is characterized by slow recovery of the elastic properties of the materials after shaking. In this case, a variety of recovery processes is operative [37][38]. Assuming that the relaxation process is associated to the loading intensity (or activation) E, the recovery process depends on the relaxation time τ, which can be expressed according to Arrhenius' Law [39][40][41], as given by Equation (12):

$$ \tau = \tau_0 e^{\frac{E}{k_B T}} \qquad (12) $$

where τ₀ is a relaxation constant that is associated to the intrinsic material properties. The relaxation time τ is directly proportional to the level of heterogeneities in the system. By comparing Equations (11) and (12), we can conclude that for a given temperature, the recovery process and the fluctuation-dissipation are correlated, which thus explains the fluctuation-dissipation process observed in the rock beams and the real buildings, but not that observed in the Plexiglas beam. Therefore, under weak deformation, the increase in the damping reflects the presence of heterogeneities in the system.

442

## VII Conclusions

444

We have shown here that variations in elastic responses of civil engineering structures and Euler-Bernoulli-like beams can be explained by the fluctuation-dissipation theory. This is illustrated here through their modal parameters, as the damping and frequency variations to the weak external loading. We interpret this result as a connection between the damping and the rate of heterogeneities of the system concerned. We demonstrated that under weak deformation, the modal damping increases with the loading intensity while the modal frequency remains constant. Surprisingly, these observations can be performed on both free-clamped beams at the laboratory scale and real-scale buildings, which confirm the invariant scale of this phenomenon. For a homogeneous material such as the Plexiglas beam, the fluctuation-dissipation relationship is not observed at all. Although this observation requires more investigations, it already highlights the need to correct the damping measurements in structures from the effects of the loading intensity at the time of the measurements, to reduce

446

447

448

449

450

451

452

453

454

455

456
457 its assessment uncertainty.

458 Data-driven structural health monitoring (SHM) methods are based on detecting
459 changes relative to initial conditions, and their efficiency depends on the level of knowledge
460 of prior conditions. In fact, for civil engineering structures, the most challenging issue is that
461 the modal frequencies, whose measurements are considered to be the most cost-effective
462 solution for data-driven SHM methods, is subject to changes caused not only by damage, but
463 also by environmental and operating conditions (traffic, wind, humidity, solar radiation and,
464 most importantly, temperature). It can be imagined that the damping variation is proportional
465 to the total number of heterogeneities in the system, with consequences for the monitoring of
466 the real state of existing buildings (i.e., SHM). The present study suggests that the sensitivity
467 of the damping variations is correlated to the presence and density of heterogeneities in the
468 material constituting the civil engineering structures. If this hypothesis can be confirmed by
469 other experimental data, this might provide a new monitoring method for structural health
470 monitoring using ambient vibrations.

471

472

473 Acknowledgments

474 The GCH and NPT data were provided by the French Accelerometric Network (RAP-RESIF
475 (1995): RESIF-RAP doi:10.15778/RESIF.RA http://data.datacite.org/10.15778/RESIF.RA
476 (last accessed: 20/02/2017). The RESIF network is a national Research Infrastructure that is
477 recognized as such by the French Ministry of Higher Education and Research. The RESIF
478 network is additionally supported by a public grant overseen by the French National Research
479 Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-11-
480 EQPX-0040), and the French Ministry of Ecology, sustainable development. The FB
481 instrumentation was funded by USGS and data were downloaded from www.iris.edu. This
482 study was sponsored by the Urban Seismology project at the Institute of Earth Science
483 ISTerre of the University of Grenoble-Alpes, and by a grant from Labex OSUG@2020
484 (Investissements d’avenir, ANR10-LABX56). M.A. Brossault is funded by the Auvergne-
485 Rhône-Alpes Region.

486
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594 **Figure Captions**

595

596

597 FIG. 1. View of the experimental set-up at the laboratory scale, showing the three beams (as indicated), the air jet that excites the beams, the accelerometers at the top and the base of each beam, and the associated conditioning amplifier.

600

601 FIG. 2. (a) Three-hour-long signal recorded at the top of the granite beam. The white strips at 1 h and 2 h correspond to breaks in the recording due to the data transfer and storage. (b) Fourier transforms of 1-h-long accelerometric signals recorded at the top of each of the three beams (as indicated). The theoretical frequencies of the first modes (i=1 to 5) computed using the Euler-Bernoulli beam model for each beam are shown as vertical dashed lines.

606

607 FIG. 3. Experimental (red) and theoretical (black) modal shapes computed using the Euler-Bernoulli beam model for the five first modes of the granite beam. MAC: modal assurance criteria value.

610

611 FIG. 4. Frequency (a) and damping ratio (b) *versus* the squared modal acceleration for modes 1 to 4 for the granite beam. The colors represent the densities of the measurement points (dark blue, 0; dark red, 1). The x and y axes have different scales according to the average damping and frequency values of each mode, and different squared modal acceleration.

615

616 FIG. 5. Frequency (a) and damping ratio (b) *versus* the squared acceleration of the third mode of the limestone beam and the Plexiglas beam. The x and y axes have different scales according to the average damping and frequency values of each mode, and different squared modal acceleration.

620

621 FIG. 6. View of the real buildings included here, and Fourier transform of 1-h-long ambient vibrations recorded at the top floor in the longitudinal direction. (a) Factor Building on the University of California, Los Angeles campus. (b) Grenoble City Hall (France). (c) Prefecture Tower in Nice (France).

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FIG. 7. Frequency (a) and damping ratio (b) versus the squared acceleration of the fundamental mode. Top row: Factor Building on the University of California, Los Angeles campus. Middle row: Grenoble City Hall (France). Botton row: Prefecture Tower in Nice (France). The color scale represents the density of the measurement points (dark blue, 0; dark red, 1). The x and y scales are different between the panels. The black dashed lines highlight the linear fits.
<figure><img src="image_3.png" /><figcaption>Figure 1</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 2</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 3</figcaption></figure>
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Figure 4</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 5</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 6</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 7</figcaption></figure>