Geology Reference
In-Depth Information
Table 1. Parameters for the stochastic ground acceleration model
Parameter
Numerical Value
Parameter
Numerical Value
σ r (km)
r (km)
20.0
9.0
b
U
10 20
1.8
ρ s (gm/cc)
β s (km/s)
2.8
3.5
1/ 2
R Φ
V
0.55
R 0 (km)
F
2.0
1.0
T (s)
t (s)
20.0
0.01
M min
M max
6.0
8.0
SEQUENTIAL OPTIMIZATION
wise (Atkinson and Silva, 2000). The modification
of seismic waves by local conditions, site effect
G ( ), is expressed by the multiplication of a
diminution function D ( ) and an amplification
function A ( ). The diminution function accounts
for the path-independent loss of high frequency
in the ground motions and can be accounted for
a simple filter of the form D f
The solution of the reliability-based optimiza-
tion problem given by Equations (1, 2 and 3)
is obtained by transforming it into a sequence
of sub-optimization problems having a simple
explicit algebraic structure. Thus, the strategy
is to construct successive approximate analyti-
cal sub-problems. To this end, the objective and
the constraint functions are represented by using
approximate functions dependent on the design
variables. In particular, a hybrid form of linear,
reciprocal and quadratic approximations is con-
sidered in the present formulation (Haftka and
Gürdal, 1992). For the purpose of constructing the
approximations all design variables are assumed
to be continuous.
= 0 03 π (An-
derson and Hough, 1984). The amplification
function A ( ) is based on empirical curves given
in (Boore et al., 1997) for generic rock sites. An
average constant value equal to 2.0 is considered.
Finally, the filter that controls the type of ground
motion I ( ) is chosen as I f
( )
e
.
f
= 2 π for ground
acceleration. The particular values of the different
parameters of the stochastic ground acceleration
model considered in this contribution are given
in Table 1. For illustration purposes Figure 1
shows the envelope function, the ground motion
spectrum and a corresponding sample of ground
motion for a nominal distance r = 20 km, and
moment magnitude M= 7.0. For a detailed discus-
sion of this point-source model the reader is re-
ferred to (Atkinson and Silva, 2000; Boore, 2003).
( )
(
)
f
First and Second-Order
Approximations
Let p y
({ }) be a generic function involved in the
optimization problem, i.e. the objective or con-
straint functions, and y { } a point in the feasible
design space. The function p y
({ }) is first ap-
proximated about the point { }
y 0
by using a hybrid
 
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