PSI - Issue 64

Dario De Domenico et al. / Procedia Structural Integrity 64 (2024) 784–790 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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that reduce from 1.7 m at the intermediate piers to 1.3 m at the abutments and are connected via transverse diaphragms and a 0.16 m thick reinforced concrete (RC) slab. It is worth noting that the most recurrent case consists of a deck of 10.0 m width with 5 girders. Further, the decks are connected to the abutments by means of Mesnager hinges. The reinforcement of each girder consists of 6Ø10 longitudinal bars both in the top flange and central web and 13Ø12 in the bottom flange, Ø12/15 cm transverse web reinforcement and parabolic prestressed cables made of cold-drawn Ø7 steel wires. Each girder of the lateral spans has 4 cables, 3 of which consist of 32 wires anchored at the half-joint whereas the remaining one of 42 wires is anchored at the abutment. Similarly, in the central span, 3 cables of 32 wires, anchored at the half-joints, are adopted. The RC slab is reinforced with Ø10/25 cm transverse bars for the entire slab width, 4Ø14 longitudinal bars in the areas above the girders, and 6Ø8 longitudinal bars elsewhere. Finally, the dapped ends (0.70 x 0.90 x 0.75 m) are reinforced with 2Ø24 hangers, 4Ø24 diagonal bars, 6Ø24 longitudinal bars and stirrups Ø14/15 cm. 3. Modal identification methodology The identification of modal parameters is carried out by exploiting the approach proposed by Mazzeo et al. (2023) which allows processing signals recorded under free vibration tests. Notably, this method relies on the Variational Mode Decomposition – VMD (Dragomiretskiy et al. 2014) to concurrently isolate all the significant K modal components (also called Intrinsic Mode Functions – IMFs) and the corresponding center frequencies from a given signal, through the solution of the following constrained variational problem: (1) where v k is the k th IMF and ω k the corresponding k th circular frequency, t  is the gradient operator, ‖∙‖ 2 the L 2 – norm operator, δ ( t ) the Dirac distribution and j the imaginary unit. The actual solution is obtained by evaluating the saddle point of the following augmented Lagrangian function: (2) where α is a quadratic penalty factor, λ is the Lagrangian multiplier and 〈∙〉 is the L 2 – inner product. In the context of the considered identification methodology, the central frequencies extracted via the VMD from the analyzed signal are assumed as the sought modal frequencies of the investigated structure. The modal damping ratios are therefore evaluated exploiting an area ratio-based approach (Mazzeo et al. 2023). Specifically, considering that the k th IMF has 2 1 k N zero-crossing points and, thus, 2 k N areas enclosed between the response function and the time axis, the modal damping ratio is calculated as follows: ( ) ( ) ( ) t ( ) v t e ( ) ( ) v t = 1 v t 1 2 , , , , min   1 2 1 s.t. k j t  K v t K K K t k k v t k k j t   − = =                  +    ( ) t ( ) v t e ( ) + − ( ) ( ) ( ) 2 2 1 2 1 1 2 ,  k j t  K K K t k k v t k v t k k k j L v t v t t   =    − = = =           +  + −  

1 1 2 / 

k 

(3)

2

k k N A

where A k is the area ratio for the kth IMF defined as

N

S

k

, i k

1

ln

i

A

(4)

k

2

N

S

k

, i k

1

i N

k