Discrete Variable Structural Optimization of Systems under Stochastic Earthquake Excitation - Structural Seismic Design Optimization and Earthquake Engineering - page 66

Figure 3. Typical displacement-restoring force curves of the shear panels at the initial design. The first
and the second floor

interstory drift ratio reaches some critical level
for the first time. A threshold level value equal to
0.02 m is considered in this case, and the failure
events are defined as

with side constraints

K 9

∈ { . , 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 . , . , . , . , . , . , . , . , . }

10



{ . , 8 0 8 5 9 . , . , 0 9 5 10 0 10 5 11 0 11 5 12 0 N/m . , . , . , . , . , . }

F = max | δ ( ) | t > 0 02 . , i = 1 2 , (45) (48)

i t ∈ [ 0 , T ] i

K 9
10

∈ { . , 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 . , . , . , . , . , . , . , . , . }

where δ 1 ( t and δ 2 ( t are the relative displacements
of the first and second floor, respectively. The
target failure probability is taken equal to 10 − .
The optimization variables are the linear inter-
story stiffnesses K 1 and K 2 , with initial design
K 01



{ . , 8 0 8 5 9 . , . , 0 9 5 10 0 10 5 11 0 11 5 12 0 N/m . , . , . , . , . , . }

(49)

The problem is solved by using the sequential
approximate optimization approach previously
described. Subset simulation is used to estimate
the failure probabilities and their sensitivities. To
smooth the variability of the estimates, the aver-
age of failure probability estimates over five in-
dependent simulation runs is considered at each
design. The value of the parameter that controls
the curvature of the second-order terms in the
approximations is taken as χ = 0 . . It is noted

= 9 0 10 . ×

9

N/m and K 02 = 7 0 10 . ×

9

N/m.

The optimization problem is written as

Min f K K ( , ) = K + K (46)

1 2 1 2

subject to

P K K ( , ) ≤ P

*

, j = 1 2 , (47)

F 1 2 F

j j