Optimal Performance- Based Seismic Design - Structural Seismic Design Optimization and Earthquake Engineering - page 195

the sensitivity procedures may be interpreted in
the context of nonlinear analysis procedure.

pattern. The load pattern can be different for
various levels of risks. It also can be different in
a single risk level for different directions. The
lateral earthquake load pattern can be either uni-
form, triangular, exponential, or etc.; FEMA356
recommends the exponential form. To be in the
safe side one may use more than one or two load
patterns. n δ , n θ and n s are number of stories,
number of sections that the plastic rotation has to
be controlled, and number of f sections for strength
check, respectively. δ and δ are inter-story dri ft
and its allowable upper limit. Similarly, θ and θ
are plastic rotation and its allowable upper bound.
F and S are internal force and strength respec-
tively. DI stands for damage index for any load
pattern and performance level; damage index
should be less than the pre-specified value DI p .
The constraint group C1 and C2 are the main
constraints in PBSD; they control the drift and
plastic rotation. The constraint group C3 are
strength constraints and for higher performance
levels, may be considered as secondary perfor-
mance constraints. The C 4 set of constraints are
not used so much. They are optional and stand
for limitations on damage index. One may adopt
different upper bounds for different performance
levels on a certain damage index. The C 5 group of
constraints are lower bound and upper bounds for
design variables. Obviously some other constraints
can be added to the optimization problem. For

OPTIMIZATION IN PERFORMANCE-
BASED DESIGN

To optimally design a structure, as was already
mentioned, the design problem should be ex-
pressed mathematically in terms of design vari-
ables in the framework of a standard optimization
problem, Eq.(15). To express design constraints
in its explicit form, Eq.(16) is employed. The
derivatives in this equation are obtained from
sensitivity analysis. Some examples of sensitiv-
ity analysis were discussed. In this section, the
general formulation of OPBSD is presented and
some, not all, published research works in the
field will be described.

General OPBSD Problem

For optimal Performance-based seismic design,
the general standard optimization problem can be
written in the form seen in Box 1.
In the above formulation, Z is the objective or
cost function. C 1 to C 5 are groups of constraints.
The superscript p relates to the performance
level and np is the number of performance levels.
If it is to design a structure for four performance
levels, np =4. The superscript lp denotes the load

Box 1.

= ( ) Minimize Z C X

lp p p

( ) ( )

Subject to C : δ − δ ≤ o s = 1 2 , , ... n ; p = 1,...,np ;( lp = 1,...,nlp)

1 s δ

( )

lp p p

(41)

C : θ − θ ≤ o r = 1 2 , , n

2 r r θ

( )

lp p p

C : F − S ≤ o i = 1 2 , , ... n

3 i i s

p

lp p

C : DI − DI ≤ o

4

( )

C : x

l

− ≤ x o x , − ≤ x

u

o j = 1 2 , , ... x n

5 j j j j