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into this model (Taflanidis & Beck, 2009a). All
these forces are discussed in more detail next.
The hysteretic behavior of each isolator is
modeled by a Bouc-Wen model which is described
by the following basic differential equation
(Taflanidis, 2009):
; if
; if
x
−
x
<
x
f
=
sl
sr
o
β
c
(
)
(
)
x
k
x
−
x
−
x
+
c
x
−
x
−
x
≥
x
c
sl
sr
o
cc
sl
sr
o
cc
sl
sr
(8)
where
k
cc
is the contact stiffness,
c
cc
the non-linear
damper coefficient and
β
c
the contact exponent,
taken here with the nominal value for Hertz type
of impact, i.e.
β
c
=1.5 (Muthukumar & DesRoches,
2006). The damper coefficient may be expressed
in terms of the ratio of relative velocities of the
pounding bodies before and after the contact,
called coefficient of restitution
e
cc
, as (Muthuku-
mar & DesRoches, 2006):
2
δ
z
=
α
x
−
z
(
γ
sgn(
x z
)
+
β
)
x
(5)
yi
is
b
is
b
is
b
where
x
b
is the displacement of the isolator, which
corresponds to the relative displacement of the
span relative to its support;
z
is a dimensionless
hysteretic state-variable that is constrained by val-
ues ±1;
δ
yi
is the yield displacement; and
α
is
,
β
is
, and
γ
is
are dimensionless quantities that characterize
the properties of the hysteretic behavior. Typical
values for these parameters are used here, taken
as
α
is
=1,
β
is
=0.1, and
γ
is
=0.9 (Taflanidis, 2009).
The isolator forces
f
i
may be then described based
on the state-variable
z
and the relative isolator
displacement
x
b
. For friction-pendulum isolators
and lead-rubber bearings, these forces are given
respectively, by (Taflanidis, 2009):
β
x
−
x
−
x
c
sl
sr
o
0 75
.
(
1
2
)
c
=
k
−
e
(9)
cc
cc
cc
v
con
where
v
con
is the relative velocity at the initiation
of contact. The contact stiffness is a function of
the elastic properties and the geometry of the
colliding bodies. For elastic contact between two
isotropic spheres with radius
R
1
and
R
2
the follow-
ing relationship holds (Werner, 1960)
f
=
k x
+
µ
N z
(6)
R R
4
i
p b
t
k
c
=
1
2
(10)
3
π δ
(
+
δ
)
R
+
R
1
2
1
2
f
k x
k
k
z
=
+
(
−
)
δ
(7)
i
p b
e
p
yi
where
δ
i
,
i
=1,2 is a material parameter for the ith
body given by
where
k
p
is the post yield stiffness,
N
t
the average
normal force at the bearing,
μ
is the coefficient
of friction,
k
e
the pre yield stiffness. For the left
span
x
b
is given by
x
b
=
x
sl
-
x
al
and
x
b
=
x
sl
-
x
p
for the
isolators supporting it to the abutment and the pier,
respectively. For the right span the corresponding
quantities are
x
b
=
x
sr
-
x
ar
, and
x
b
=
x
sr
-
x
p
, respectively.
The force due to pounding between the adjacent
spans is modelled as a single-sided Hertz contact
force with an additional damper that incorporates
in the analysis the energy dissipated during the
contact (Muthukumar & DesRoches, 2006)
v
E
2
1
−
δ
=
i
(11)
i
π
i
with
v
i
and
E
i
representing its Poisson's ratio
and modulus of elasticity, respectively. To cal-
culate(10) each of the spans can be roughly ap-
proximated as a sphere with radius (Muthukumar
& DesRoches, 2006)
m
3
4
R
=
πρ
i
(12)
3
i
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