## Abstract

Building frequencies (fundamental and higher modes) are a critical parameter especially in the field of structural health monitoring mainly based on the stability of the structural dynamic parameters of individual building (frequencies, damping and modes shape). One of the most used methods to find out these parameters is based on the use on ambient vibration analysis. In this work, we study the fluctuations over a month period of the fundamental frequencies (transverse and longitudinal) of a 3.5-story RC-building made of 2 identical units connected by a structural joint. Time independent building frequencies is a strong assumption; as illustrated by our experiment showing that over an observation period of a month, building frequencies fluctuate of about 3.5 %. A clear correlation is found between the building frequency fluctuations and temperature variations, with a phase-shift interpreted as the characteristic time of heat diffusion within the walls. This allows: (1) determining the thermal diffusivity of the structure, (2) inferring its relative stiffness variations, and (3) showing that its Young modulus varies linearly with temperature.

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## Acknowledgments

We thank Djamel Boucherf (National Meteorological Center of Algeria) who provided the temperature records. We are grateful to Philippe Guéguen and Michel Bouchon for fruitful discussions. This study has been funded by CGS and IRD.

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## Appendix 1: Determination of the thermal diffusivity of the medium

### Appendix 1: Determination of the thermal diffusivity of the medium

In 1D, the Fourier law of thermal conduction reads

where \(\upalpha\) is the thermal diffusivity given by

where \(\lambda\) is the thermal conductivity, \(\rho\) the density and \(C_{p}\) the specific heat capacity.

A simple dimension analysis on Eq. (2) shows that the thermal diffusion time \(\tau_{D}\) is roughly given by

where \(L\) is a characteristic distance of the 1D structure. Equation (4) represents the diffusion time for heat to cross a structure of width \(L\). It is comparable to the phase lag \(\phi\) defined in the main text.

A more rigorous expression can be obtained using the Fourier transform of the temperature field \(T\)

Putting Eq. (5) into (2) yields the following dispersion equation

that can be injected back in Eq. (5) to give

where

Equation (7) shows that the temperature field is a superposition of spatially oscillating modes decaying with time with a characteristic time given by \(\tau \left( k \right)\) for mode \(k\). Considering that the dominating mode corresponds to the wavelength \(L\) where \(L\) is the width of the wall, the characteristic diffusion time \(\tau \left( L \right)\) is given by

When the width \(L\) of the wall and the phase lag \(\phi\) are known, we can use Eq. (9) using \(\tau \left( L \right)\) = \(\phi\) to estimate the thermal diffusivity \(\alpha\) of the building walls

### Implications for the variations of the elastic properties

To a first order, the resonance frequency of the building is given by

where \(K\) is the stiffness of the building and \(M\) its mass. The relative variations of frequency are given by

We assume that the variations of the building mass are negligible so that Eq. (12) reduces to

The relative variations in the elastic parameter \(K\) are hence given by

### Variations of the resonance frequency with temperature

Figure 5 show the longitudinal and transverse resonance frequency \(f\) as a function of temperature. A linear trend is observed suggesting that

where \(a\) and \(b\) are two parameters independent of temperature, at least in the temperature range considered in our experiment.

Combining Eq. (15) with (11) yields

where, again, the variations of mass with temperature are neglected (\(\frac{\partial M}{\partial T} = 0\)). Note that Eq. (15) predicts a quadratic dependence of the stiffness on temperature.

If the stiffness of the building K and the Young modulus \(E\) are proportional, then

where \(T_{0}\) is a reference temperature in the range considered in our experiment. We can combine Eq. (17) with (16) to get the quadratic expression

Considering the values of \(a\) and \(b\) given earlier and given that \(T\) varies from −0.5 to 26.4 °C, it is straightforward to show that Eq. (18) simplifies into

Equation (\({\text{A}}18\)) predicts a linear dependence of \(E\) with \(T\).

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Guillier, B., Chatelain, JL., Perfettini, H. *et al.* Building frequency fluctuations from continuous monitoring of ambient vibrations and their relationship to temperature variations.
*Bull Earthquake Eng* **14, **2213–2227 (2016). https://doi.org/10.1007/s10518-016-9901-z

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### Keywords

- Building frequencies
- Temperature
- Building health monitoring
- Diffusivity
- Young modulus