Geology Reference
In-Depth Information
The objective function is defined in terms of
initial, construction, repair, or downtime costs.
In the context of reliability-based optimization
of structural systems under stochastic excitation
a reliability constraint can be written as
formance functions can also be considered in the
above formulation.
STRUCTURAL MODEL
*
h
({ })
y
=
P
({ })
y
−
P
≤
0
(4)
A quite general class of dynamical systems under
ground acceleration can be cast into the follow-
ing form
j
F
F
j
j
F
j
({ })
is the probability of occurrence
of the failure event
F
j
evaluated at the design
{ }
where
P
y
y
[
M x t
]{ ( )}
+
[
C x t
]{ ( )}
+
[
K x t
]{ ( )}
=
(6)
, and
P
F
j
*
is the corresponding target failure prob-
ability. The failure probability function
P
−
[
M g x t
]{ } ( )
−
{ ({ ( )},
f
x t
{ ( )})}
z t
g
F
j
({ })
evaluated at the design
{
y
can be expressed in
terms of the multidimensional probability integral
y
where
{ ( )}
x t
denotes the displacement vector of
dimension
n
,
{ ( )}
x t
the acceleration vector,
{ ({ ( )},{ ( )})}
x t
the velocity vector,
{ ( )}
f
x t
z t
the
=
∫
Ω
P
({ })
y
q
({ }) ({ }) { } { }
θ
p
ξ
d
ξ
d
θ
vector of non-linear restoring forces,
{ ( )}
z t
the
vector of a set of variables which describes the
state of the nonlinear components, and
x t
F
j
({ })
y
F
j
(5)
g
( )
the
[ ]
,
[ ]
, and
[ ]
describe the mass, damping, stiffness, respec-
tively. The vector
{
g
couples the base excitation
to the degrees of freedom of the structure. All
these matrices are assumed to be constant with
respect to time. This characterization of the non-
linear model is particularly well suited for cases
where most of the components of the structural
system remain linear, and only a small part behaves
in a nonlinear manner. For a more general case,
the formulation can still be applied at the ex-
penses of more computational efforts due to pos-
sibly necessary updating of the damping and
stiffness matrices of the system with respect to
time. In general, the matrices involved in the
equation of motion depend on the vector of design
variables and uncertain system parameters and
therefore the solution is also a function of these
quantities. The evolution of the set of variables
{ ( )}
ground acceleration. The matrices
M
where
Ω
F
j
({ })
is the failure domain correspond-
ing to the failure event
F
j
evaluated at the design
{
y
. The vectors
{ },
y
θ
θ
i
,
i
=
1
, ...,
n
, and
u
=
1
represent the vector of uncer-
tain structural parameters and random variables
that specify the stochastic excitation, respec-
tively. The uncertain structural parameters
{ }
{ },
ξ
ξ
i
,
i
, ...,
n
T
θ
are modeled using a prescribed probability den-
sity function
q
({ }
θ
while the random variables
{
ξ
are characterized by a probability density
function
p
({ }
ξ
. The failure probability function
P
F
j
({ })
accounts for the uncertainty in the system
parameters as well as the uncertainties in the
excitation. It is noted that for structural systems
under stochastic excitation the multidimensional
integral (5) involves in general a large number of
uncertain parameters (in the order of thousands).
Therefore the reliability estimation for a given
design constitutes a high dimensional problem
which is extremely demanding from a numerical
point of view. Finally, it is noted that constraint
functions defined in terms of deterministic per-
y
z t
is described by a first-order non-linear
differential equation
{ ( )}
z t
=
{ ({ ( )},{ ( )},{ ( )})}
h x t
x t
z t
(7)
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