PSI - Issue 44

Diego Gino et al. / Procedia Structural Integrity 44 (2023) 1435–1442 Diego Gino et al./ Structural Integrity Procedia 00 (2022) 000 – 000

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3

2. Response of bridges isolated with FPS In line to Jangid (2008), Kunde and Jangid (2009), the structural behaviour of an isolated three-span continuous deck bridge (Figure 1), is reproduced in this work adopting the following modelling strategy: 5 dof relates to the lumped masses of the RC pier and 1 dof to the mass of the stiff RC deck. As introduced before, the RC abutment is considered as infinitely rigid. The dynamic equilibrium equations which control the behaviour of a bridge isolated with FPS subjected to seismic action according to configuration of Figure 1 described by the expressions of Eq.(1).

a

b

Fig. 1. (a) Modelling of the bridge of FPS devices by means of 6 dof model; (b) force-displacement response of the FPS on the RC pier.

 

 

 

 

 

 

 

 

 

+

+

=

( )

d d m u t

 m u t m u t m u t m u t m u t c u t    

b f t

a f t

d g m u t





5

4

3

2

1

d p

d p

d p

d p

d p

d d

       

       

   

   

 

 

 

 

 

 

=

 m u t m u t m u t m u t m u t c u t c u t k u t f t    

5 p g m u t

 

   

 



5 5

5 4

5 3

5 2

5 1

5 5

5 5

p p

p p

p p

p p

p p

d d

p p

p p

b

 

 

 

 

 

=

 u t c u t k u t c u t k u t  

4 p g m u t

 m u t m u t m u t m



1

5 5

5 5

4 4

4 4

4 4

4 3

4 2

4

p

p p

p p

p p

p p

p p

p p

p p

p

(1)

 

 

 

 

 

 

=

   m u t m u t m u t c u t k u t c u t k u t m u t m u t c u t k u t c u t k u t          = p p p p p p p p p p p p p p       3 3 3 2 3 1 4 4 4 4 3 3 3 3 2 p g m u t p p p p p p p p p p p p   2 2 2 1 3 3 3 3 2 2 2 2

3 p g m u t



  m u t c u   1 1 2 p p  p p 2

 

 

 

 

 =

 t k u t c u t k u t  

1 p g m u t

2 2

1 1

1 1

p p

p p

p p

The term u d represents the displacement in horizontal direction of the deck relative to the pier, u pi is the displacement of the i th (i:1-5) lumped mass of the pier with respect to the i th -1 dof, m d and m pi respectively the mass of the deck and of the i th (i:1-5) lumped mass of the pier, c d is the constant value of viscous damping of the deck, k pi and c pi are the stiffness of the pier and related viscous damping constant of the i th (i:1-5) dof, t is the time, f p (t) and f a (t) are the reactions of the FP isolators on the pier and on the abutment evaluated as:

  u m g u

  sgn d

 



d d m gu t

 

+ p d d

(2a)

p f t



2

2

R

  

      d

  

1 2

d m g

5

5

5

  

  

  

  

  

  







 

+

( )

( )

sgn ( ) u t

a f t

d u t

u

u t

u m g

u

(2b)











pi

a d

pi

d

pi

2

R

1

1

1

i

i

i







where g is the acceleration of gravity, R denotes the FPS radius of curvature,      u t is the friction coefficient of the FP device on the abutment (a) or of the isolator on the pier (p), which depends on the sliding velocity and, finally, sgn(∙) is the sign function. The variation of      u t can be reproduced according to the results of Mokha et al. (1990) and Constantinou et al. (1990) as also performed by Castaldo and Ripani (2017). By means the division of the Eq.(1)-(2) by the value of the deck mass m d , the related dimensionless equations can be evaluated according to following parameters: mass ratio of the i th dof of the pier /   pi pi d m m ; damping ratio of the deck with isolation and of