PSI - Issue 44

Elena Miceli et al. / Procedia Structural Integrity 44 (2023) 1419–1426 Elena Miceli et al. / Structural Integrity Procedia 00 (2022) 000 – 000

1422

4

 

   

   

0 u g a



2   2 d g

1 2

d   g

 

 

 

 

 

 

 

 

  d 

p d

+

+

+

+

+

+

2

sgn

               

 





5

4

3

2

1

d

p

p

p

p

p

d

d

d

2

(4a)





 

  

2   

1 2

g u

  

  

5

5





d    

 

    pi

5

+

=

(

) sgn 

( ) 

u











d

d

a d

pi

pi

2  g

2

a

1

i



1

1

i

i





0

d 

 

 

 

 

 

      d d

 

 

2

              





5

5

4

3

2

1

p

p

p

p

p

p

(4b)

g

 

   



2 p 

g

p 

p d  

2   2 d g

1 2

 

 

 

  d      = 5 ( ) p

5

5

2

sgn



5 5        5 5 p p p p

 





5

p

d

2  g

2

a

g 

 

0

2 p 

2 p 

p 

p 

(4c)

 

 

 

 

       5 5 2 p p p

 

 

 

5

4

5

4

( )               =

 

 

2

           





4

4

3

2

1

5

4

4 4

5

5

4

4

4

p

p

p

p

p

p

p p

p

p

p

p

p

2  g

2  g

g 

g 

2 p 

2 p 

p 

p 

(4d)

 

 

 

       4 4 2 p p p

 

 

 

4

3

4

3

( )               =

         

 

2







3

3

2

1

4

3

3 3

4

4

3

3

3

p

p

p

p

p

p p

p

p

p

p

p

2  g

2  g

g 

g 

2 p 

2 p 

p 

p 

(4e)

          

       3 3 2 p p p

 

 

 

3

2

3

2

( )               =

 

2







2

2

1

3

2

2 2

3

3

2

2

2

p

p

p

p

p p

p

p

p

p

p

2  g

2  g

g 

g 

2 p 

2 p 

p 

p 

(4f)

       1 1 2 p p p

 

 

 

 

2

1

2

1

2 2                        = 1 1 1 2 2 1 1 1 2 2 2 ( ) p p p p p p p p p p

2

g

g

g

g

2

0 pi pi d u a    are the nondimensional expressions for the displacement of the deck and

2 d d d u a    and 0

where

d d k m

d  

pi pi k m

and

are the respective circular vibration frequencies,

of the pier respectively,

pi  

2 d d d c m    and pi d m m   is the mass ratio of the i th lumped mass (equal for all the lumped masses). Then, the expression (4) is in terms of the following non dimensional parameters:   g     d 2 c m pi pi pi pi    are the respective damping factors and

p

d

,

,

,

,

,

p              g  d p d p  d  pi      

(5)

a

d 

0

g

  from the velocity, its expression

More precisely, to avoid the dependency of the nondimensional parameter

*   

max 0 / f g a

is substituted by . All the previous considerations are valid for the case of a single-column bent viaduct (Fig. 1a). The only difference is that the term F a (t) is not present in both equations (1) and (4) since the abutment is not modelled. 3. Parametric analysis In the following, the assumptions and considerations regarding the parametric analysis (valid, without distinction, for the two models) are presented. Starting with the structural properties, the pier period T p varies from 0.10s to 0.20s, the damping ratios are assumed equal to 0% d d      and 5% p p      , the deck period T d varies from 2s to 4s, the mass ratio varies in the range 0.1, 0.15, 0.2 and the normalized friction coefficient *   ranges between 0 and 2. For what concerns the seismic input, both the structures are subjected to 30 different seismic natural ground motions, selected from 19 different seismic events, coming from PEER, ITACA and ISESD databases. The main characteristics varies in the following ranges: the magnitude from 6.3 to 7.5, the peak ground acceleration from 0.13 to 0.82 g. the source-to-site distance from 13 km to 98 km. Regarding the seismic intensity 0 a , also indicated as intensity measure in line with the Performance Based Earthquake Engineering (PBEE) approach Aslani and Miranda (2005), Porter (2003), is herein equal to the spectral pseudo-acceleration   , A d d S T  . In line with Ryan and Chopra