Optimal Design of Nonlinear Viscous Dampers for Protection of Isolated Bridges - Structural Seismic Design Optimization and Earthquake Engineering - page 388

Table 1. Probability models for uncertain model parameters. In this table c.o.v=coefficient of variation,
ρ= correlation coefficient, μ= mean value

Gaussian
μ= 70 kN/mm
c.o.v = 10%

Correlated log-normal
median = 8%,
c.o.v = 20%, ρ = 50%

Log-normal
Median = 10km
c.o.v = 50%

ζ al
ζ ar

k p r

Truncated exponential in [6 8] with
parameter 0.9log e (10)

Log-normal
median = 3%
c.o.v = 25%

Correlated Gaussian
μ= 2500 kN/mm
c.o.v = 15%, ρ = 50%

M

k al
k ar

ζ p

v p Uniform in [0, π ]

Gaussian
μ= 0.3 c.o.v = 10%

Gaussian
μ= 0.1 c.o.v = 20%

μ p α p

x o
x ol
x or

Correlated log-normal
median = 10 cm
c.o.v = 15%, ρ = 70%

Gaussian
μ= 0.03m c.o.v=10%

Gaussian
μ= 2.0 c.o.v=15%

δ yp γ p

k cc
k cl
k cr

Independent log-normal
median given by 10
c.o.v = 40%,

Lognormal
Median Equation 21
c.o.v = 40%

m tl
m tr

Independent Exponential
Μ= 20 ton

f p

Correlated Gaussian
μ= 2500 kN/mm
c.o.v = 15%, ρ 50%

e cc
e cl
e cr

Independent Gaussian truncated in
[0 1]
μ= 0.6 c.o.v = 15%

Log-normal
Median Equation 22
c.o.v = 39%

k al
k ar

A v

The optimization problem is defined by Equa-
tion 24, in which the objective function is the
seismic risk given by Equation 23. The admis-
sible design space is defined in Box 7.where f dl
and f dr correspond to the damper force for the left
and right abutment, respectively.
A simplified design problem is also considered
where a d is set to 1, corresponding to a linear
viscous damper. This will allow for investiga-
tion of the performance improvement gained by
considering nonlinear, rather than linear, damper
implementation.
The bridge performance measure assumed in
this study addresses potential seismic damages
for all components of the bridge: the pier, the
abutments, and the deck. The failure criteria used
are: (i) the maximum pier shear V p , associated
with yielding and inelastic deformations for the
pier, (ii) the maximum displacement for the left
and right abutment z r and z l , respectively, associ-
ated with permanent deformations for the ground,
and (iii) the maximum velocity for impact between
the two spans v o or between each of the spans and
the left or right abutment v l and v r , respectively,
associated with the damages that occur during
pounding (Davis, 1992). The fragility related to

each of these quantities, i.e. the probability that
the response will exceed some acceptable perfor-
mance μ , is used for describing the system per-
formance. These fragilities are assumed to have
a lognormal distribution with median μ and coef-
ficient of variation β . The probabilistic description
for the fragility incorporates into the stochastic
analysis model prediction errors (Goulet et al.,
2007; Taflanidis & Beck, 2010). The seismic-risk
occurrence measure is the average of these fra-
gilities, see Box 8.where Φ g is the standard Gauss-
ian cumulative distribution function. The charac-
teristics for the median of the fragility curves are
μ p =2200 kN for the pier shear, μ z =6 mm for the
abutment displacement and μ v =10 cm/sec for the
impact velocity, whereas the coefficient of varia-
tion is set for all of them equal to β p = β z = β v =0.4.
Seismic risk is finally given by the integral in
Equation 1 which in this case corresponds to
probability of unacceptable performance for the
base-isolated bridge. The maximum value for
C ( φ ) is one while the minimum value for C ( φ ),
corresponding to best possible performance, is
zero.