PSI - Issue 44
Guglielmo Amendola et al. / Procedia Structural Integrity 44 (2023) 1427–1434 Guglielmo Amendola, Giuseppe Carlo Marano/ Structural Integrity Procedia 00 (2022) 000 – 000
1429
3
=
( ) m x t m x t m x t m x t m x t m x t m x t c x t F t F t
d g m u t
7
6
5
4
3
2
1
1
1
d
d
d p
d p
d p
d p
d p
d d
a
p
m x t m x t m x t m x t m x t m x t F t F t
sp g m u t
6
5
4
3
2
1
1
2
sp
sp p
sp p
sp p
sp p
sp p
p
p
m x t F t F t
sa g m u t
8
1
2
sa
a
a
=
5 m x p t m x t m x t m x t m x t c x t c x t k x t F t m x t m x t m x t m x t c x t k x t c x t k x t = 5 5 4 5 3 5 2 5 1 5 5 5 5 2 p p p p p p p p p d d p p p p p
5 p g m u t
4 p g m u t
4 4
4 3
4 2
4 1
5 5
5 5
4 4
4 4
p p
p p
p p
p p
p p
p p
p p
p p
=
m x t m x t m x t c x
t k x t c x t k x t
3 p g m u t
3 3
3 2
3 1
4
4
4 4
3 3
3 3
p p
p p
p p
p
p
p p
p p
p p
=
m x t m x t c x t k x t c x t k x t
2 p g m u t
2 2
2 1
3 3
3 3
2 2
2 2
p p
p p
p p
p p
p p
p p
=
m x t c x t k x t c x t k x t
1 p g m u t
1 1
2 2
2 2
1 1
1 1
p p
p p
p p
p p
p p
(1 a,b,c,d,e,f,g,h) where m d , m sp , and m sa are respectively the masses of the deck and of the two isolation devices installed on the pier and on the abutment; m pi (i=1,..,4,5) is the i-th lumped mass of the pier segment; k pi and c pi (i=1,..,5) are the stiffness and viscous damping, assumed equal for each dof associated to the pier segments; t is the time instant; F ja (t) and F jp (t) are the reaction forces of the DCFP referred to the abutment and the pier, respectively, for the upper (j = 1) and lower sliding surface (j = 2). In particular, according to (Fenz and Constantinou, 2006), (Castaldo and Amendola, 2021a), (Constantinou, 2004), the reaction forces can be expressed as:
d m g
1
5
9 x
6 + + pi
( ) sgn x
F
x x x x
1
7
8
1 9 a
a
2
R
1
i
1
a
m
1
2 8 a x
8 sgn x
8 x
F
sa m g
d
(2 a,b,c,d)
2
a
2
R
2
a
d m g
1
1 7 p x
7 sgn x
7 x
F
1
p
2
R
1
p
m
1
2 6 p x
6 sgn x
6 x
F
sp m g
d
2
p
2
R
2
p
5
where , R 1 and R 2 are the upper and lower radius of curvature of the DCFP devices and ( ( )) j j x t (with j=1,2) is the sliding friction coefficient, estimated according to experimental investigation (Mokha et al., 1990), (Constantinou et al., 1990, 2007), with the following expression: ,max ,max ,min exp 1, 2 j j j j j j x f f f x for j (3) 9 6 7 8 x x x x 1 pi i x where f j,max and f j,min are the value of friction coefficient at high and near-zero sliding velocity respectively. Finally it is assumed α=30 and f j,max =3 f j,min according to (Mokha et al., 1990), (Constantinou et al., 1990, 2007). Then, in line with previous studies (Castaldo and Tubaldi, 2018), the system in Fig. (1) can be expressed in a non dimensional form, by means of mass ratios; the circular frequency of vibration of the isolated deck and of the i-th dof of the pier; the damping coefficient of the i-th dof of the pier, respectively as:
m
m
k
c
m
k
pi
sp
pi
pi
(4 a,b,c,d,e,f)
,
,
,
,
,
d
sa
comb
pi
sa
sp
pi
pi
2
m
m
m
m
m
m
d
d
d
d
pi
pi pi
In addition, according to (Castaldo and Tubaldi, 2018), the time scale τ=tωd can be introduced together with the seismic intensity scale factor a 0 , evaluated with the expression 0 ( ) ( ) g u t a , where ( ) is a non-dimensional function of time which describes the time history of the seismic event. Finally, the non-dimensional system of
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