PSI - Issue 44

Riccardo Martini et al. / Procedia Structural Integrity 44 (2023) 657–664 Riccardo Martini et al. / Structural Integrity Procedia 00 (2022) 000–000 3 where ̈ 1 and ̈ 2 are the vertical accelerations of the truck and bridge, respectively, 1 and 2 the vertical displacements. In order to obtain a non-dimensional solutions of the eigenvalue problem of the 2-DoF system, the following two non-dimensional parameters are introduced: � = 2 1 ⁄ (2) � = 2 1 ⁄ (3) where � is the ratio between the bridge and the truck masses and � is the ratio of the bridge and truck fundamental circular frequencies. Introducing the eigenvalue problem, in which is the vector of modal displacements and the natural circular frequency of the system, the following system is obtained: �� 1 − 1 − 1 1 + � � 2 � − 2 12 � 1 0 0 � �� � 1 2 1 � = (4) where 1 and 2 are the modal displacements of the truck and the bridge, respectively. Eq. 4 has non-trivial solutions if: �� 1 − 1 − 1 1 + � � 2 � − 2 12 � 1 0 0 � �� = 0 (5) Solving Eq. 5, the non-negative non-dimensional circular frequencies of the 2-DoF system ( and ) can be calculated as: � 2 � , = 1 � � 1+ � ( 1+ � 2 )± � �1+ � ( 1+ � 2 ) � 2 −4 � 2 � 2 2 � (6) Eq. 6 expresses the relationships between the two circular frequencies of the system ( and ) and the non dimensional parameters, previously introduced, ( � and � ) that can be easily determined during the static proof test procedures. The mass ratio � can be easily calculated by computing the bridge mass (that can be evaluated with good accuracy from the design documents) and by weighting trucks before the load test (as routinely done for static proof load tests). The frequency ratio � requires the estimation of the first vertical frequency of the unloaded bridge, by performing an OMA of the unloaded bridge based on the accelerations recordings of AVT, and the vertical frequency of the trucks, that can be determined in the same manner or obtained from the literature (Gara et al. (2020)). In order to investigate the BTI and to provide a specific graphical tool to support the dynamic identification procedure, a parametric investigation is performed considering the variability of the � ratio. The latter is identified taking into account many issues, such as the most common deck typology for new bridges (steel-concrete composite and the precast RC decks characterised by I-shaped or box girders), the number of the loading truck lines, the number of trucks in a single line, and the ratio between the length of the loading lines and the span length. For the estimation of the bridge mass, the self-weight of the structural and non-structural components (i.e. kerbs, barriers and road pavement) are considered, depending on the bridge typology. The above analysis determined the range of variability of the ratio � , which goes from the lower value of 0.5 to the higher value of 10. By fixing the value of � (i.e. � = 0.5, 1, 2, 3, 4, 5 and 10) and by solving Eq. 6 as function of the frequencies ratio � , that can be assumed to vary in a reasonable range (0.5-5), a graphical representation of the coupled system solutions is obtained for each value of � (continuous blue lines), as shown in Fig. 1. The dashed red line represents the natural frequency of the unloaded bridge ( / 2 =1). 659