PSI - Issue 64

Marco Martino Rosso et al. / Procedia Structural Integrity 64 (2024) 507–514 Author name / Structural Integrity Procedia 00 (2019) 000–000

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3. Numerical vibration response simulation of a monitored reinforced concrete existing building case study

Case study herein under investigation is a RC existing building located in Northern Italy. As illustrated in Fig. 4, the building is four-story, regular in plan and height, characterized by an inter-story height of 3.20 m (4.00 m for the first floor), with plan dimensions of 24.90 m x 13.90 m. RC columns have constant cross section of 55 x 30 cm for the entire elevation of the building. Beams’ cross-sections at 1st floor are 55x30 cm, whereas at 2nd and 3rd floors are 50x30 cm, and at last 4 th level are 40x30 cm. As illustrated in Fig. 5, due to its planar and elevation regularity, the building behavior is shear-type diaphragmatic, thus allowing to describe its dynamics with a 3D lumped mass multiple degrees of freedom (MDOF) system. Therefore, a point mass is concentrated in the centre of the mass of each floor, and it is associated to three generalized DOFs each, viz. two translations in X ( u ) and Y ( v ) directions respectively, and a rotation ( θ ) around the vertical axis. Indeed, this building is described by 12 DOFs, i.e. 4 translations for each principal direction and 4 rotations. The mass and stiffness matrices of this RC building have been determined, and the damping matrix was computed by assuming a constant damping ratio equal to 2% for every mode. 12 natural frequencies (1.23 Hz, 1.28 Hz, 2.01 Hz, 3.53 Hz, 3.69 Hz, 5.63 Hz, 6.00 Hz, 6.10 Hz, 7.34 Hz, 8.22 Hz, 9.54 Hz, and 12.95 Hz) have been found by solving the eigenvalue problem for this 3D lumped mass system. Afterward, in order to simulate a SHM system collecting ambient vibration responses, the following stochastic state space (SSS) representation was formulated (see Rainieri and Fabbrocino, 2014), i.e. accounting for a random white noise dynamic excitation applied at the ground level DOFs in the process noise term  :          (1)       (2) Eq. (1) are called state equations, whereas Eq. (2) are the observation equations. The symbol x (t) indicates the state vector which encompasses the state variables, i.e. the displacements and velocities of the system, whereas  (t+dt) denotes the first derivative with respect to the time of the next-state prediction, and the symbol A indicates the state transition matrix. The symbol y (t) refers to the acceleration vibration responses, observed at any time instant due to the random dynamic excitation of the process noise term w (t). The acceleration responses were further contaminated by an additive zero-mean Gaussian white noise called measurement noise process v (t). Eventually, the symbol C denotes the output influence matrix. The SSS model numerically simulated a monitoring system at 200 Hz which recorded 1 hour of acceleration ambient vibration response sampled using one biaxial accelerometer placed on every floor of this building (8 acceleration histories encoded in y (t)).

Fig. 4. Shear-type existing RC frame case study: plan view and lateral sectional view.