Geology Reference
In-Depth Information
Theoretical Formulation
avoid singularity. An exponential form of weight
function has been adopted in the present nu-
merical study. Eventually, a weight matrix W(x)
can be constructed by using the weighting func-
tion in the diagonal terms as:
It is well-known that the sensitivity information
are useful to a designer as it provides a measure
of performance deviations in a design associated
with an increase or decrease of the respective vari-
ables. To reduce the number of significant random
variables before in reliability evaluation process
the use of importance measure as given below is
quite common (Bjerager, & Krenk, 1989; Haldar,
& Mahadevan, 2000; Gupta, & Manohar, 2004):
w x x
( -
)
0
0
0
1
0
w x x
( -
)
..
0
2
W
( )=
x
..
..
..
..
..
..
..
..
0
0
..
w x x
n
( -
)
(24)
2
2
=
∂
∂
G
u
N
∂
∂
G
u
∑
I
(26)
(ii)
By minimizing the least-squares estimators
L x
k
=
(ii)
1
k
y
( )
, the coefficients
a
( )
can be obtained by
the matrix operation as below,
where, G is the failure surface defining the safe
and unsafe regions and N is the total number of
random variables in a generic structural reliability
analysis problem. The applicability of the above
form of importance measure was studied by
Gupta, & Manohar (2004) through Monte Carlo
Simulation (MCS) study. Based on the entries of
Ii's, the uncertain variables can be grouped into
'important' and 'unimportant' variables. The un-
certain variables for which the failure surface is
more sensitive are identified as dominant variables.
This is hinged on the fact that all the gradients
are not equally important in the expression of
the failure surface in a typical reliability analysis
problem. This intuitively indicates that all the
gradients do not have equal importance in the
expression of the dispersion of the performance
function as defined by Eq. (2). Hence, it is ap-
parent that the importance measure should also
play a role to indicate the measure of robustness
of the performance.
It can be readily realized from Eq. (2) that
the dispersion
∆
(ii)
depends on two factors: (i)
the amplitude of uncertainty in the DVs and DPs
represented by the corresponding dispersion,
∆
u
(ii)
' s
and (ii) the gradients of the performance
−
1
=
a
( )
x
x W x
T
( )
x
x W y
T
( )
x
(25)
It is important to note here that the coefficients
a
(
x
are the function of the location
x
, where the
approximation is sought. Thus, the procedure to
calculate
a
(
x
is a local approximation and “mov-
ing” processes performs a global approximation
throughout the whole design domain.
The Proposed RDO APPROACH
As mentioned earlier, the present chapter deals
with an efficient RDO procedure for structures
subjected to stochastic earthquake load and char-
acterized by UBB type DVs and DPs. A heuristic
algorithm is proposed here which allows the use
of importance factor obtained using the respec-
tive sensitivity information of the performance
function and constraints. In the following sub-
sections, the related theoretical developments,
implementation of the algorithm and numerical
study are presented.
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