Geology Reference
In-Depth Information
Box 2.
σ
σ
log
(
1
/
R
)
1
1
2
log (1/
R
)
2
0
(19)
=
(
)
−
−
(
)
g
( )
u
ν β
,
u
min
=
Y
exp
β σ
/
−
min
≤
Y
T
π
T
Y
details may be found in Lutes, & Sarkani, (1997);
Datta, (2010).
The limit state corresponds to the first excur-
sion of a structural response
Y
(
stochastic dynamic response of structure under
earthquake load modelled as stationary stochastic
process. However, application of the proposed
RDO presented in this chapter is not restricted to
such stochastic random process only and extension
to non-stationary earthquake model will be straight
forward. However, this will involve time depen-
dent response statistics evaluations and subse-
quently to deal with time dependent performance
function in the optimization procedure.
,
t
can be ex-
pressed as, G [Y(u, t)] > 0, where G is a function.
For a stationary stochastic process, the reliability
of structure can be written as,
)
R t
(
,
u
)
=
P G
{ [ (
Y u
,
t
)]
≥
0
|
}.
(17)
t T
<
The optimum DVs must satisfy one or more
probabilistic constraint(s) consisting of limiting
the failure probability for a given value of reli-
ability,
R
min
. For a specified threshold barrier (
β
), the stochastic constraint becomes,
Stochastic Dynamic
Response Approximation
It can be noted that in a typical SSO procedure,
the safety measures are required to be evaluated
several times to obtain an optimal design. The
evaluations of safety measures for every change
of the DVs require several evaluations of the
dynamic responses of a structural system. The
dynamic responses as described by Eq. (9) are
implicit function of u and normally obtained by
numerical methods like the finite element proce-
dure. Furthermore, the formulations described in
the previous section assume that all the DVs and
DPs are deterministic. If the effects of uncer-
tainty are considered in the analysis,
D
{
}
R
(
β
, T,
u
)
=
P G
[ (
Y u
,
t
)
≥
0
]
=
exp
(
−
ν β
(
,
u
)
T
)
≥
R
t T
<
min
(18)
Using Eqs. (15) and (16), the constraint can
be finally expressed in Box 2.
Finally, the SSO problem under stationary
earthquake model can be expressed in Box 3.
In the above,
f
( )
u
is the objective function
of the optimization problem. Usually the weight
of the structure or some important stochastic re-
sponse measure is considered as the objective
function. The SSO presented here is based on the
ω
,
( )
and thereby the FRF vector will also involve
uncertainty. Therefore, the analysis will require
u
Box 3.
{ }
find
x
to minimize:
f
( )
u
(20)
σ
σ
log
( /
1
R
m
)
1
1
2
log
( /
1
R
)
2
Y
−
(
)
such that :
g
( )
u
=
ν β
( , )
u
−
in
=
exp
β σ
/
−
min
≤
0
.
Y
T
π
T
Y
Search WWH ::
Custom Search