Optimal Design of Nonlinear Viscous Dampers for Protection of Isolated Bridges - Structural Seismic Design Optimization and Earthquake Engineering - page 386

ILLUSTRATIVE EXAMPLE to period of 0.079 sec for the left and 0.089 sec

for the right abutment. Both damping coefficients
C al and C ar are selected based on modal damping
assumption. The damping ratios for each abutment
ζ al , ζ ar are treated as correlated uncertain variables
following a log-normal distribution with median
8%, coefficient of variation 20% and correlation
coefficient 50%. The high correlation in the abut-
ment model parameter characteristics is attributed
to the common soil properties.
For the left and right span of the bridge the
self-weight of the deck is taken as 1000 and 1200
ton respectively. Vehicle traffic is modeled as
additional loads m tl and m tr for the left and right
span, respectively, that follow independent expo-
nential distributions with mean value 20 ton. The
isolators connecting each span to its supports are
lead-rubber bearings modeled by Equation 7. All
isolators have same properties; post-yield stiffness
k e =3.0 kN/mm, pre-yield stiffness k p =30.0 kN/mm,
and yield displacement δ yi =2.5cm. These choices
correspond to a natural period of 2.57 sec and 2.81
sec for the right and left span, respectively, based
on the post-yield stiffness and no vehicle traffic.
The respective gap dimensions x o , x ol , x or , whose
potential variability is influenced by common
weather conditions, are modeled as correlated
log-normal variables with median 10 cm, coef-
ficient of variation 20%, and large correlation
coefficient, 70%.
The contact stiffnesses k cc , k cl , k cr for the Hertz
impact forces between the spans or between each
span and the neighboring abutment are taken as
independent log-normal variables with median
value given by 10 and large coefficient of variation
40%, again reflecting our limited knowledge. For
calculating the median value of k c , Poisson's ratio
is taken as 0.15, the modulus of elasticity as 30
GPa and the density 2.4 ton/m 3 . The equivalent
sphere radius 12 for each span is calculated by
considering their total mass, including vehicle traf-
fic. The coefficients for restitution for the energy
dissipated during contact for each span are mod-
eled as independent, truncated in [0 1] Gaussian

Models and Uncertainty Description

The design of nonlinear viscous dampers for a
two-span seismically isolated concrete bridge is
considered as illustrative example. The model
described previously is adopted for the bridge and
probability distributions are established, based
on our available knowledge, for all parameters
that are considered as uncertain. All probability
models are described here in terms of the mean
(or median) value, characterizing most probable
model, and the coefficient of variation, charac-
terizing the potential spread for each parameter.
For variables whose values are expected to be
correlated, such dependency is expressed through
their correlation coefficient.
The length of the span of the bridges is equal
to 30 and 34 m, respectively. The mass of the pier
is taken as m p =100 ton. For the pier restoring force
f p as illustrated in Figure 3, the initial stiffness k p ,
post-yield stiffness coefficient α p , over-strength
factor μ p and yield displacement δ yp , have mean val-
ues 70 kΝ/mm, 10%, 30% and 0.04 m, respectively.
All these parameters are treated as independent
Gaussian variables with coefficient of variation
10%. For the nominal, i.e. most probable, model
this assumption corresponds to period of 0.237
sec based on the initial stiffness. The damping
coefficient C p is selected based on modal damping
assumption, using the initial period of the pier. The
damping ratio ζ p is treated as uncertain variable
following a log-normal distribution with median
3% and coefficient of variation 25%.
The mass of the left and right abutments are
taken, respectively, as m al =400 ton, m ar =500 ton.
For the right and left abutment restoring forces f ar ,
f al the stiffness's k ar and k al , respectively, are mod-
elled as correlated Gaussian variables with mean
value 2500 kN/mm (Saadeghvaziri & Yazdani-
Motlagh, 2008), coefficient of variation 15%, and
correlation coefficient 50%. For the nominal, i.e.
most probable, model this assumption corresponds