Hi everyone. My name is Arnaud Montabert. I'm working in the Laboratory of Geology from ENS (Ecole Normale Supérieure). My research work is focused on archeoseismic studies, which is at the interface between seismology, earthquake engineering, and building archaeology. I'm going to present a part of my PhD thesis in collaboration with IRSN (Institut de Radioprotection et de Sûreté Nucléaire), CEREMA (Centre d'études et d'expertise sur les risques, l'environnement, la mobilité et l'aménagement), and the University of Siena. A great of part of historic buildings consists on monumental masonry structures. As a cultural heritage computational analysis of historical buildings is important to better predict and assess their structural behavior. But defining a realistic model of historic buildings is a very challenging task. Today, I'm gonna show you the process I use to validate a digital model. Let's consider a medieval church. The validation process is based on a very simple principle. The dynamic behavior of any physical system like the church we're looking at, can be decomposed into a discrete set of specific motion called: modes of vibration. A mode is described by three modal parameters: the natural frequency, the mode shape, and the modal damping. Let's explain in a few words what is model updating. Let's assume we get some experimental data like three natural frequencies of the church, f1, f2, and f3. Using a numerical model of my church, I can easily compute the three eigenfrequencies: f1, f2, and f3. This allows me to compare my church model to the actual data by computing an error function, defined as the discrepancy between numerical frequencies and frequencies directly measured in the church. If the error is too large, then I propose a second model by updating material properties and boundary conditions. I repeat this process as long as the error is too large. At the end of the process, I finally select the best model that allows me to obtain the minimum difference between numerical model parameters and the experimental ones. To achieve a good model updating. I'm sure you understand how important is the quality of experimental modal parameters. I'm gonna show you today how to measure them directly in the field. To identify the modal parameters, we're going to record the vibration of the church. Recordings are done by using seismometers like the ones used in classical seismology. Seismometer are located at different points in the church, for example, at the top of the bell tower or at the top of the nave. There are two main methods, experimental modal analysis and operational modal analysis. In experimental modal analysis, an input is emitted at the bottom of the church. The response of the structure to this artificial stress is then recorded by seismometers. You can imagine that this method is difficult to apply, especially in the context of a historic building because you cannot produce any damage. But don't panic, there is another way. In operational modal analysis, tremor is supposed to be sufficient to stress the main vibration modes of the building. It is then sufficient to place the seismometers in the structure in order to record the response of the church to this micro tremor. The frequency contents of micro tremor is rich. Let's explain the main sources. Indeed, noise originates by coastal waves generated by global marine phenomena or global meteorology or local meteorology like the winds, but also in the anthropogenic noise at higher frequencies covering the frequency range of interest for buildings. The earthquakes themselves generates energy through the other frequency bands. We shall use operational modal analysis to identify the modal parameters of our church. A very important question is, where to locate your seismometers. Indeed their location have a crucial impact on the results. Preliminary numerical models can be used to identify the main modes of vibration of the structure, and thus to identify the best location of the seismometers. In the case of the first bending modes in the x direction or in y direction, a minimum of two sensors is required to record the modal displacements. For torsion mode, three sensors for the same level can be used to record the building response. Let's do a little practice and see what's going on in the field. Here is a measurement survey we carried out in Italy, in a medieval church. Some parts of the historical building are very difficult to access. However, we need to be able to install sensors in the strategic locations we have identified just before. We therefore use an aerial platform to transport the equipment and the operator at the height of more than 10 meters. Our objective is to be able to install a sensor as close as possible to the wall of the nave. Because we work in risky condition, the first step is to secure the sensor and especially the cable, that allow its connection and to avoid any fall of the seismometer. For this, we use traps. By securing the cable in this way, we also avoid any vibration that could pollute recordings. The second step is to position the sensor on the beam. All the seismometer are then oriented in the same direction here towards the North. Then we can hook up the seismometer. This connection cable connects the seismometer to a data logger. The data logger ensures the recording of all the seismometer and their synchronization. Finally, the seismometer is adjusted horizontally using a level bubble positioned on top of the sensor and the screws located under the seismometer. Now we are ready to record. We installed in this way several dozen sensors throughout the church. But to simplify this presentation, we are interested in the measurements that we made only in the bell tower. In particular, we're interested in three sensors arranged vertically: 1, 2, 3. The seismometer record in all three directions of space. Let's analyze now records using the Enhanced Frequency Domain Decomposition technique. In a specific frequency range, the cross spectral density matrix between the different seismometers are computed to analyze the correlation between the different records. We repeat the process for the different frequency ranges. In the second step, a singular value decomposition of the previous power spectral density matrices allow us to extract eigenvalues for each frequency range and their associated eigenvectors. Let's display, for example, the evolution of the first two eigenvalues corresponding to the two lines of our power spectral density matrix. The peaks correspond to the natural frequencies of the bell tower. Each peak is associated with an eigenvector corresponding to the mode shape. In our case, the first frequency corresponds to the first bending modes in the x direction. The second frequency corresponds to the first bending mode of the bell tower in the y direction. Finally, the third frequency correspond to the torsional mode. Only one modal characteristic remains to be determined, modal damping. To do this, we select each of the spectral peaks, and then we compute the impulse response by using the inverse Fourier transform. The logarithmic decrement method is used to identify the modal damping that best reproduces the experimental data. This experimental process therefore allows us to identify the real modal characteristics of our church. Three natural frequencies, three mode shapes, and three modal dampings. This fundamental step will now allow us to calibrate our numerical model, which will be ready to carry out reliable numerical simulation.
