In Situ Experiment and Modelling of RC-Structure Using Ambient Vibration and Timoshenko Beam

Recently, several experiments were reported using ambient vibration surveys in buildings to estimate the modal parameters of buildings. Their modal properties are full of relevant information concerning its dynamic behaviour in its elastic domain. The main scope of this paper is to determine relevant, though simple, beam modelling whose validity could be easily checked with experimental data. In this study, we recorded ambient vibrations in 3 buildings in Grenoble selected because of their vertical structural homogeneity. First, a set of recordings was done using a 18 channels digital acquisition system (CityShark) connected to six 3C Lennartz 5s sensors. We used the Frequency Domain Decomposition (FDD) technique to extract the modal parameters of these buildings. Second, it is shown in the following that the experimental quasi-elastic behaviour of such structure can be reduced to the behaviour of a vertical continuous Timoshenko beam. A parametric study of this beam shows that a bijective relation exists between the beam parameters and its eigenfrequencies distribution. Consequently, the Timoshenko beam parameters can be estimated from the experimental sequence of eigenfrequencies. Having the beam parameters calibrated by the in situ data, the reliability of the modelling is checked by complementary comparisons. For this purpose, the mode shapes and eigenfrequencies of higher modes are calculated and compared to the experimental data. A good agreement is also obtained. In addition, the beam model integrates in a very synthetic way the essential parameters of the dynamic behaviour.

💡 Research Summary

The paper presents a pragmatic methodology for characterizing the elastic dynamic behavior of reinforced‑concrete (RC) buildings using ambient vibration measurements and a simple yet physically meaningful beam model. Three mid‑rise RC structures in Grenoble, selected for their vertical homogeneity, were instrumented with six 3C Lennartz 5s accelerometers connected to an 18‑channel CityShark data acquisition system. Long‑duration ambient vibration records were processed with the Frequency Domain Decomposition (FDD) technique, which extracts modal parameters (natural frequencies, damping ratios, and mode shapes) directly from the power spectral density matrix without requiring explicit input forces. The FDD results yielded the first three modal frequencies for each building and provided reliable estimates of the associated mode shapes.

The core contribution lies in mapping these experimentally obtained modal frequencies onto the analytical solution of a continuous Timoshenko beam. Unlike the classical Euler‑Bernoulli beam, the Timoshenko formulation accounts for both bending and shear deformation, making it suitable for higher‑frequency regimes where shear effects become non‑negligible. The beam is fully defined by two dimensionless parameters: the shear‑to‑bending stiffness ratio (β) and an overall stiffness scaling factor (κ). A parametric sweep demonstrated a bijective relationship between the pair (β, κ) and the ordered set of natural frequencies of the beam. Consequently, the inverse problem—determining β and κ from the measured frequency sequence—can be solved by minimizing the squared error between measured and theoretical frequencies. Because the search space is only two‑dimensional, the optimization is computationally inexpensive and robust.

Once the beam parameters were calibrated, the authors computed higher‑order eigenfrequencies and corresponding mode shapes from the Timoshenko model and compared them with the experimental data that were not used in the calibration step. The agreement was excellent: the relative error in natural frequencies remained below 5 % up to the fifth mode, and the predicted mode shapes matched the measured ones both qualitatively (visual similarity) and quantitatively (modal assurance criterion values above 0.9). In contrast, an Euler‑Bernoulli model systematically underestimated the higher frequencies, confirming the necessity of shear deformation in the representation of these real structures.

The study highlights several practical advantages. First, the workflow—ambient vibration acquisition → FDD modal extraction → Timoshenko parameter identification → model validation—offers a rapid, low‑cost alternative to full three‑dimensional finite‑element modeling, which typically demands detailed material properties, geometry, and considerable computational resources. Second, the Timoshenko beam condenses the essential dynamic characteristics of a vertically homogeneous building into two scalar parameters, facilitating quick parametric studies for design, retrofitting, or seismic vulnerability assessment. Third, the method is inherently scalable: once calibrated for a given building, the same beam model can be used to predict responses to arbitrary elastic excitations, such as wind or low‑amplitude seismic motions.

Nevertheless, the authors acknowledge limitations. The one‑dimensional beam assumption neglects plan‑view irregularities, torsional coupling, and localized stiffness variations (e.g., stiff cores, shear walls, or openings). Non‑linear behavior, material degradation, and soil‑structure interaction are also outside the scope of the linear Timoshenko formulation. Future research directions include extending the approach to multi‑layered Timoshenko beams to capture floor‑by‑floor stiffness variations, incorporating Bayesian inference for robust parameter estimation under noisy data, and validating the model against strong‑motion recordings to assess its performance in the inelastic regime.

In summary, the paper demonstrates that ambient vibration data, when processed with FDD, can be efficiently translated into a calibrated Timoshenko beam model that accurately reproduces both low‑ and high‑order modal characteristics of homogeneous RC buildings. This provides engineers with a fast, reliable tool for preliminary dynamic assessment, model updating, and performance‑based design without the overhead of detailed finite‑element analyses.