Optimal Seismic Performance- Based Design of Reinforced Concrete Buildings - Structural Seismic Design Optimization and Earthquake Engineering - page 220

3.3 Explicit Multi-Objective Problem
Formulation

earthquakes. Five classes of structural damage
(namely negligible, slight, moderate, severe and
complete damage) of a building are given based
on the maximum inter-story drift value in accor-
dance with the Chinese code for seismic design
of buildings (GB50011-2001). Due to the fact that
the classification of the five different degrees of
damage is discrete in nature and the set of inter-
story drift responses of a building is continuous,
the fuzzy-decision theory is employed to best
estimate the damage loss with specified proba-
bilities of occurrence of different levels of earth-
quakes. For simplicity, τ is used to represent the
inter-story drift index such that τ =

Upon obtaining the explicit nonlinear objective
function (i.e., the initial structure cost f 1 in Eq.
(11) and the damage loss f 2 in Eq. (32)), and the
explicit inelastic drift formulation in Eq. (22), the
multi-objective design optimization problem can
be written in terms of the design variables ρ i as
Minimize:

ρ i = {

}

F ( ) f , f (33)

1 2

∆ u R h ,
where ∆ u is the inter-story drift; h is the story
height; R is equal to 500 for concrete frames and
1000 for concrete shear-wall structures and frame-
wall structures. Finally, the total structural damage
loss f 2 presented in Eq. (31) can be explicitly
expressed in terms of the variable τ (

×

Subject to:

N N

1 1

2

i i

∑ ∑

≤

g ( ρ ) = ∆ u + α ρ ( − ρ

0

) + α ρ ( − ρ

0 2

) 1

j i j

0

1 i i i 2 i i i

h ψ

p ρ = ρ

i i

i = 1 i = 1

j j

( j = 1 2 , , ..., N j ) (34)

∆ u as )

N L U

N r
∑

ρ ≤ ≤ ρ ρ ( i = 1 2 3 , , N , ..., ) (35)

j

( )

∑ τ (

i i i i

f = f P × a ∆ u ) + a (32)

2 1 r 1 j 2

r = 1 j = 1

Eqs. (33) defines the life-cycle cost function
F , which consists of the construction cost f 1 and
the damage loss f 2 ; Eq. (34) defines the inelastic
inter-story drift constraint at the structural perfor-
mance point for a specified ground motion; Eq.
(35) defines the lower and upper size bounds
specified for the design variables, ρ i ; α 1 and α 2
are given in Eqs.(28)-(29).

where a 1 and a 2 are the coefficients of an ex-
pected failure loss, which depend on not only
damage levels but also building classes and details
are presented in Li (1998). For example, assum-
ing that the importance of a frame building belongs
to Class B, the values of a 1 and a 2 can be derived
as shown in Table 1. Details of derivation can be
found in Zou (2002) and Zou et al (2007).

Table 1. The coefficients of expected failure loss for a Class B building

τ < 0 5 . 0 5 . ≤ < τ 1 5 . 1 5 . ≤ < τ 3 ≤ < τ ≤ < τ τ ≥ 10

3 7 7 10

a 1

0.000 0.080 0.800 1.350 4.770 0.000

a 2

0.02 -0.020 -1.100 -2.750 -26.667 21.000