## 5.1 Modal analysis of the operational condition buildings

The singular value decomposition from FDD applied to theHBB8 dataset 3 is given on Fig. 3. Black bell curves are defined by applying the Modal Assurance Criteria (MAC > 70%) to select physical modes (see Michel et al. 2010 for an explanation). Three modes are clearly identified at frequencies Fr(DirX) = 1.88Hz, Fr(DirY) = 2.46Hz and Fr(DirZ) = 2.76Hz, which correspond to bending in the E and N axis, and torsion around the Z axis, respectively (Fig. 4). Translation mode frequency values are different from the empirical model estimate for HBB8 (i.e., 1.64Hz), with differences that may reflect the variability within the population of nominally identical building or the natural wandering of the modal parameters. For example, OMA results correspond to resonance frequency values close to the examples given in Fig. 2b, corresponding to given weather conditions. Higher modes are also identified at 5.78 Hz, 7.06 Hz and 7.62 Hz in DirX, DirY and DirZ respectively. The frequency ratios between the first and the second horizontal modes are close to 3 (i.e., 5.78/1.88 = 3.1 in DirX and 7.06/2.46 = 2.87 in DirY), corresponding to shear beam-like building behavior (e.g., Michel and Gueguen, 2018).

Furthermore, PSD applied to synchronized 10-minutes windows recorded in the concrete stairwell and at HB02 (DirX and DirY) is displayed on Fig. 5. The three first modes are clearly identified on both HBB8 and stairwell structures, corresponding to the HBB8 modal frequencies obtained by OMA. This reflects the dynamic coupling between the external stairwell and the building HBB8, which calls into question the effectiveness of the seismic joints of adjacent structures and the transmission of energy through the foundations at this level of vibrations. Above 3Hz (Fig. 5), the PSD is different, corresponding to the specific modes of the stairwell or coupling with the adjacent structure HBB5 (Fig. 1a). In conclusion, the identified modes with OMA correspond to HBB8 and these frequencies will be considered for dataset 2 (i.e., SHM phase).

## 5.2 Variation of modal parameters during the construction phase

The HBB8 modal model is subject to variations over time. Using datasets 1 and 2, the modal parameter variations are assessed during the construction phase, the residents’ moving-in phase, and the operating phase of the building. These variations result from modifications to the structure during the construction phase, occupancy and weather conditions. During the monitoring phase, only the modal parameters (frequency and damping) from one station (HB02) are considered because of their similitude (Fig. 2b).

Figure 6 shows the variations of the resonance frequencies during the construction and operating phases (dataset 1 and 2). The first value on March 11, 2021 corresponds to the installation of the battens in the roof structures. The resonance frequencies are Fr(DirX) = 2.56 Hz, Fr(DirY) = 3.22 Hz and Fr(DirZ) = 4.35 Hz. Two month later (i.e., the May 25, 2021–75 days), a drop to 2.18 Hz (-15%), 2.83 Hz (-12%) and 3.44 Hz (-21%) is observed in X, Y and Z direction, respectively, due to the end of the structural elements and exterior carpentry.

After May 25th 2021 and the end of structural elements work, the stiffness is approximately assumed constant. The total weight load, given by the designer is 900 tons. At the first order, the equivalent SDOF resonance frequency f for both translation (DirX and DirY) and rotation (DirZ) are proportional to stiffness k and mass m as follows:

$$\begin{array}{c}f=\frac{1}{2\pi }\sqrt{\frac{k}{m}} \#\left(3\right)\end{array}$$

The stiffness after the end of the structural elements work is approximately 1.69 108 N/m, 2.85 108 N/m and 4.30 108 N/m in DirX, DirY, DirZ, respectively. Stiffness in DirZ (4.30 108 N/m) is close to the stiffness in DirX + DirY (1.69 108 N/m + 2.85 108 N/m = 4.54 108 N/m) which is consistent with shear beam theory. After May 25th 2021, Fr drop to 2.18 Hz, 2.83 Hz and 3.48 Hz for DirX, DirY, DirZ respectively that corresponds to the installation of plasterboards ended on June 3, 2021 (2.16 Hz, 2.77 Hz, 3.46 Hz for DirX, DirY, DirZ respectively). This phase can be associated with the increase of mass only, as follows:

$$\begin{array}{c}\varDelta m=\frac{k}{4{\pi }^{2}}\varDelta \frac{1}{f} \#\left(4\right)\end{array}$$

that corresponds to Δm = 17 ton, Δm = 39 ton, Δm = 10 ton. The difference of mass variation between the three modes can be explained by the modal mass distribution which is not taken into account in the equivalent SDOF model.

After June 3rd 2021 until March 2022, Fr drop to 1.91 Hz, 2.50 Hz, and 2.86 Hz for DirX, DirY, DirZ respectively that corresponds to the finishing works. This drop is associated with the increase of mass of Δm = 244 ton, Δm = 209 ton and Δm = 426 ton for DirX, DirY, DirZ, respectively. The designer estimated the sum of imposed load (Q) and snow load (S) to Q + S = 2116 kN. S is estimated using the Eurocode1: S = 383kN (without load combination coefficients). The mass of the equivalent imposed load Q is estimated around 177 tons which is lower than the result obtained by the drop of frequency by finishing works, showing that finishing work can contribute to stiffness.

Construction phase is almost finished on the 1st January of 2022, i.e. the starting date of the semi-permanent network dataset considered in this study. Figure 7 shows the overall variations of the three first modes (frequency and damping values) and the weather condition over the period January 2022 to June 2022. Missing data close to the end of February is due to an electricity shut-down due to works, not supplied by the local batteries installed at each station. Several windows with remarkable behavior are observed.

First, from 1st January to end of February 2022, the construction phase is ending. The three resonance frequencies are decreasing linearly, with a general coefficient of 7 10− 4 Hz/day for mode DirX, 1.5 10− 3 Hz/day for mode DirY and 2.2 10− 3 Hz/day for mode DirZ (Fig. 8-left). In our case, the variation corresponds to the mass variation of the buildings, with values lower than during the construction phase (2021/03/11 to 2021/05/25) for which the coefficients corresponded to 5 10− 3, 5 10− 3 and 10− 2 Hz/day for DirX, DirY and DirZ, respectively. Over this period, the damping values did not vary linearly. The mean damping values are 1.76% for DirY, 1.95% for DirX, and 2.02% for DirZ with coefficient of variation (COV) of 11.48%, 13.83% and 13.66%, respectively. Damping c in such system is usually approximated by viscous damping ξ, as follow:

$$\begin{array}{c}\xi =\frac{c}{2\sqrt{km}} \#\left(5\right)\end{array}$$

However, in Fig. 7, damping values does not vary like the frequency as 1/m1/2. This may be explained by the various source of damping (i.e. damping material, structural damping, friction between connectors, etc.) which are not all viscous and thus not dependent on 1/m1/2. Damping is quite complex and further researches are needed to confirm this assumption.

## 5.3 Variation of modal parameters during the resident move-in phase

Since March 2022, residents have been started to move in. This corresponds to a linear decrease of the three resonance frequencies (Fig. 8-right). The variations correspond to -5 10− 3 Hz/day for DirX, DirY and DirZ, associated to the mass of inhabitants and furniture. At the end of the finishing work, the frequency values are 1.92 Hz, 2.50 Hz and 2.86 Hz for DirX, DirY and DirZ respectively. On April 1st 2022, all residents are assumed installed. The frequency values drop to 1.85 Hz, 2.45 Hz and 2.77 Hz for DirX, DirY and DirZ respectively. These variations correspond to a variation of mass Δm = 87 ton, Δm = 56 ton, Δm = 87 ton for DirX, DirY and DirZ respectively. The imposed load per unit surface (q = Q/A) of furniture assumed uniformly distributed along the 8 stories of surface A = 351m² represents q = 0.30kN/m², q = 0.20kN/m² and q = 0.30kN/m² for DirX, DirY and DirZ respectively. According to Eurocode 5, the maximum imposed load per surface unit for timber floor is 1.5kN/m², consistent with the calculated values.

## 5.4 Variation of modal parameters during the operational phase

After the movie-in phase, the operational phase has started. Figure 9 shows a zoom on this period, since the time the weather station has been installed. First of all, there is an opposition between humidity and temperature. These two parameters therefore appear to be particularly anti-correlated, and their respective impact on structure dynamics will be difficult to distinguish. In general, diurnal and seasonal variations are also closely linked to weather parameters. Frequencies are characterized by daily oscillations (day/night alternation in phase with temperature and humidity variations) and longer-period variations linked to specific weather episode (e.g. around mid-May 2022). In terms of damping, diurnal and seasonal variations are visible, but with a priori less amplitude than for frequencies.

More specifically (Fig. 9), we note a significant variation in weather parameters at the end of March/beginning of April, the effects of which can be seen on frequencies and, more moderately, on damping. These effects do not carry over in the same way in all directions, with the Y direction in particular being more reactive to this episode. This may be interpreted as a consequence of the presence of the external staircase and the adjacent HBB5 building, which protect HBB8 against the wind in this direction. This effect can also be seen in Fig. 9b on the coefficients of variation (COV) averaged per hour, with COV DirY (mean hourly COV values equal to 0.9%) being around 2 times (0.5% and 0.6%) greater in this direction, reflecting greater sensitivity to external loading. On the other hand, the COV of DirX and DirZ are identical and very stable over time, while their frequency values fluctuate according to the time of the day. In particular, DirX and DirZ are particularly in phase, with a decrease in frequency during the day, in contrast to DirY. These variations adjust to variations in humidity, with frequency decreasing as humidity decreases. At the material scale, an increase of moisture content in wood leads to a decrease in stiffness and an increase of mass of the timber structure, which lead to a decrease of natural frequencies.

At the building scale, this frequency/humidity anti-correlation has also been observed by Alarcon et al., 2023 with a similar behavior on a 6-storey light-frame timber building. They also proposed some explanations, with swelling which could have an impact on tightening in assemblies or with delaying between ambient moisture and timber moisture. Others counter-intuitive phenomena have been observed in concrete building by Mikael et al. (2013) who suggested that the expansion of concrete or cladding in relation to sun exposure could close pre-existing cracks, changing its stiffness. Furthermore, the effects humidity on soil-structure interaction had already been shown for classical material buildings (e.g. Clinton et al., 2006; Todorovska, 2009; Guéguen et al., 2017), which will be confirmed in a second phase by the exploitation of SHM data over a longer time series and the use of the HB04 sensor located at the base of the building. To the contrary of Alarcon et al. (2023), frequencies COV are of the same order of magnitude than in reinforced concrete buildings but this issue should be improved thanks to longer monitoring analysis.

Concerning damping, COV are found to be 10 times higher than frequency COV, as already reported on concrete structures by Mikael et al. (2013) and Nayeri et al. (2008). Hourly variations in damping are highly correlated with wind speed, which on average increases in the middle of the day (12:00–20:00). Few first-order similarities are observed with temperature and humidity. This means that damping is highly sensitive to wind speed, associated with dynamic loading. This can be interpreted by the activation of heterogeneities (e.g., cracks, links between structural elements, etc.), which dissipate vibration energy via attenuation mechanisms even at low load levels. This interpretation was advanced on examples of more heterogeneous structures (concrete in particular) by Brossault et al. (2018) based on the fluctuation-dissipation theorem (e.g., Kubo, 1966). As a general result of statistical thermodynamics, the latter quantifies the relationship between fluctuations in a system (loading) and the system's response to applied loadings (dissipation). In other words, Brossault et al (2018) showed that at low levels of loading (i.e., when the system's stiffness remains in linear elasticity), the entropy of the system (e.g., a building or beam-like building with different amount of heterogeneities) was characteristic of the relationship between loading and attenuation/dissipation. The consequence is an increase of damping for a constant stiffness values at low level of loading. This needs to be confirmed for timber buildings with continuous data over longer durations.

These observations are confirmed in Fig. 10, which summarizes the frequency or damping relationships with weather conditions. On the one hand, assuming a SDOF system, frequency and damping are anti-correlated. Also, air temperature does not seem to directly control the building response, with frequency and, to a lesser extent, damping variations highly nonlinear with temperature. On the other hand, we see that humidity essentially controls the frequency variations of the DirX, DirY and DirZ modes with an increase in frequency with humidity. To the opposite, a high correlation with wind speed, both in frequency and damping, is observed: the frequency decreases and the damping increases when the wind increases. We therefore see that even if the temperature does not seem to have an impact on the modal parameters at first order but for DirY. This difference could be explained by the presence of the concrete stairwell in this direction or by the zinc cladding and the sun exposure, as suggested by Mikael et al. (2013). It is also important to remind that the weather conditions produce crossed effects, correlation does not mean causality. Using longer data, it will be necessary to practically analyze and quantify the coupled effects between weather parameters on the influence of modal parameters. However, a key findings is even if the loading levels remain low, nonlinear elasticity (NLE) is observed, similar to what is observed in the laboratory (e.g., Ostrovsky and Johnson, 2001) or on concrete buildings (Guéguen et al., 2016). This NLE behavior is a function of the levels of heterogeneities and it will then be interesting to quantify the NLE according to the level of loading or deformation of the timber structure.

## 5.5 Peak acceleration level for comfort

Finally, peak acceleration at the top floor is measured to compare with norm ISO 10137 on serviceability of buildings with vibrations (International Organization for Standardization, 2007), defining the peak threshold in acceleration by frequency for tall buildings. The lowest value is set to 0.04 m/s2 for frequencies between 1 and 2 Hz. Figure 11 shows mean and maximal hourly values of top peak acceleration in DirX and DirY during the operating phase. Same mean, maximal and COV values per hour in both direction are observed with increasing values for the day time (04:00 to 21:00) in relation with human activity. The maximum value is 0.0013 m/s2, almost 30 times lower than the serviceability threshold from ISO 10137 (International Organization for Standardization, 2007) over this period characterized by no high wind speed values. For this phase, serviceability of the building is then complied. However, thanks to the long term SHM phase (the semi-permanent instrumentation is still running and the data are available online), a specific long-term serviceability analysis could be done considering specific wind speed event, moderate earthquakes that struck the region of Grenoble and traffic-induced loading, compared to design and rules for timer buildings.