Discrete Variable Structural Optimization of Systems under Stochastic Earthquake Excitation - Structural Seismic Design Optimization and Earthquake Engineering - page 63

k

It is seen that, even if P

F j ({ }) is small, by choos-

y where { } δκ j is a vector

of constant, real valued coefficients. For samples
({ },{ }),

y = y + ∆ and { y { } { } }

ing m F j and F

, ({ }), , ..., = 1 1 appro-
priately, the conditional probabilities can still be
made sufficiently large, and therefore they can be
evaluated efficiently by simulation because the
failure events are more frequent. The intermediate
failure events are chosen adaptively using infor-
mation from simulated samples so that they cor-
respond to some specified values of conditional
failure probabilities (Au and Beck, 2001).

y k m

j k F j

θ ξ i = 1 , ..., m near the limit state sur-

i i s

face, that is, κ ({ },{ },{ }) ≈ 0 the perfor-

mance function is evaluated at n

y θ ξ

j i i

= × points q m

s s s

in the neighborhood of { } y k . That is, for each

sample ({ },{ }), θ i i q s designs are defined. These

points are generated as

{ }
{

η
η

{

lk

} { }

k

{ }

l

, 1 , ...,

y − y = ∆ y = R l = n

s

}

l

Sensitivity Estimation

(35)

As previously pointed out the approximation of
the reliability constraints requires the estimation
of the gradient of the transformed failure probabil-
ity functions h t . For the purpose of estimating
the gradients, it is assumed that the optimization
variables can have non-discrete values during the
solution process. Therefore, all design variables
are treated as continuous during the first step of
the optimization process (see Solution Scheme
section). The sensitivity of the transformed failure
probability functions with respect to the design
variables is estimated by an approach recently
introduced in (Valdebenito and Schuëller, 2011).
The approach is based on the approximate repre-
sentation of two different quantities. The first
approximation involves the performance function
κ j while the second includes the failure probabil-
ity function. Recall that the failure domain Ω F j
for a given design { y is defined as

where the components of the vector { } η l are in-

dependent, identically distributed standard Gauss-
ian random variables, and R is a user-defined
small positive number. This number defines the
radius of the hypersphere ({ } / { η η } ) R centered

l l

y k at the current design { } δκ j

are computed by least squares. To this end the
following set of equations is generated

. The coefficients { }

lk k

κ ({ y },{ },{ }) θ ξ = κ ({ },{ },{ }) y θ ξ

j i i j i i

{ }
{ }

η
η

+ { δκ }

T

R

l

i

l

l = + ( i q − × 1 ) m q , = 1 , ..., q , i = 1 , ..., m

s s

(36)

s

where the index l indicates a double loop in terms
of the indices i and q . Since the samples
({ },{ }), = 1 are chosen near the

limit state surface the approximate performance
function κ j is expected to be representative, on
the average, of the behavior of the failure domain
Ω F j in the vicinity of the current design { }

θ ξ i , ..., m

i i s

0
(33)

Ω F ({ }) y = {{ },{ } θ ξ | κ ({ },{ },{ }) y θ ξ ≤ }.

j

j

y k
(Jensen et al., 2009). Next, the failure domain
Ω F j for a given design { y can also be defined in
terms of the normalized demand function as

y k is the current design, the performance
function is approximated in the vicinity of the
current design as

If { }

∆
(34)

κ ({ },{ },{ }) y θ ξ = κ ({ },{ },{ }) y

k

θ ξ + { δκ } {

T

y }

j j j