PSI - Issue 44

Sandro Carbonari et al. / Procedia Structural Integrity 44 (2023) 27–34 Sandro Carbonari et al. / Structural Integrity Procedia 00 (2022) 000–000

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3

foundation level (node f ), the translation u p and φ p of the top of the pier (node p ), and the translation of the deck u d (node d ), which is connected to the pier cap through a spring of stiffness k , representative of the deck support devices. Masses of the deck m d , of the pier cap m c and of the foundation m f are lumped at the centroids of the relevant members, while the pier mass m p is assumed to be lumped at the pier ends. Furthermore, the foundation cap and the pier cap are assumed to be rigid while the pier is modelled as a Timoshenko beam. The geometric parameters of the system are reported in Fig. 1. The dynamic equilibrium equations of the system subjected to ambient accelerations or forced vibrations assumes the general form � � � ��� � � � � � � � � ��� � � � � � � � � ��� � � � � � � (1) where � � � � � � � � � (2) is the vector collecting displacements of the dof of the system, � , � and � are the mass, damping and stiffness matrices of the superstructure, respectively, while ��� , ��� and ��� are the mass, damping and stiffness matrices of the lumped system that accounts for the soil-foundation system compliance. The damping matrix of the superstructure is assumed to be proportional to both stiffness and mass, according to the Rayleigh approach, which provides � � � � (3) where coefficients α and β can be calibrated to get defined damping ratios in correspondence of two resonance frequencies of the structure. Finally, in equations (1), is the load influence vector. In case of forced vibration tests, determines where the known force is applied, while in case of ambient vibration tests, by assuming that the excitation only comes from the soil accelerations � � (i.e. � � � ), � � � � � � � � (4a,b) where ̅ � � 1 1 0 1 0 � is an influence vector determining the dofs where the ground acceleration acts. a m d c b d ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ � � � ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤

z

k x c x

h c h d

u d

m x I y

k

m c

f

u p

m p /2

p

 p

x

external node

h p

k ry c ry

z

h

rigid link

m p /2

k hx c hx

m f

h f

0

x

f

u f

hidden node

 f

H

m hx

Fig. 1. (a) Soil-foundation-pier system; (b) numerical model and (c) LPM proposed by Carbonari et al. (2018).