Optimal Seismic Performance- Based Design of Reinforced Concrete Buildings - Structural Seismic Design Optimization and Earthquake Engineering - page 214

be invariant in the optimization under the presup-
position that adequate shear strength is provided
for all members.
The total construction cost, f 1 , of a reinforced
concrete framework, which consists of the concrete
cost f 1 given in Eq. (1) and the steel reinforcement
cost f 1 shown in Eq. (10), can be expressed as

In order to facilitate a numerical solution of
the drift design problem, it is necessary that the
implicit story drift constraint Eq. (12) be expressed
explicitly in terms of the design variables, ρ i and
′

ρ i . Before the drift formulation can be discussed,
three assumptions must be made. The first is that
all the inelastic deformation is assumed to occur
at the plastic hinges, which are located at the ends
of each frame member and members are fully
elastic between the plastic hinges. Secondly, the
plastic hinges are assumed to be frictionless and
have zero length. The third assumption is that
beam-column joints are much stronger than any
adjacent framing components so that the joint
region may be modelled as a stiff or rigid zone.
Based on the internal element forces and mo-
ments of the structure obtained from the pushover
analysis at the performance point, the principle
of virtual work can be employed to express the
pushover displacement. The pushover story dis-
placement, u j , at the performance point includes
the virtual work, u j mem , , produced by the struc-
tural members and the virtual work, u j hinge

N N i

i

∑ ∑ 1

′ = + =

+ ′

′

f B D ( , ,

ρ ρ

, ) f f w B D L + w L ( ρ L ρ i )

1 i i i 1 c 1 s ci i i i si si i si

i = 1 i =

(11)

It has been previously shown that, in the elas-
tic design optimization, the concrete material cost
of the structure with respect to the width, B i , and
depth, D i , is minimized and is kept to the mini-
mum value in the inelastic design optimization.
Under the condition that the concrete element
dimensions are fixed to their minimum values,
the steel material cost is to be minimized in the
inelastic design stage.
In seismic performance-based design, it is
necessary to check the “capacity” of a structure
against the “demand” of an earthquake. At the
performance point, where capacity equals demand
as shown in Figure 1, the resulting responses
of the building should then be checked using
certain acceptability criteria. In this chapter, the
inelastic drift responses at the performance point
of a building, generated by a severe earthquake
demand, are to be checked against appropriate
limits corresponding to a given performance ob-
jective. The inelastic interstory drift constraint at
the performance point is defined as below.

,

,

generated by the plastic hinges. That is,

u = u + u (13)

j j memb , j hinge ,

in which

Li

N i

F f
EA

F f
GA

F f
GA

M m
GI

M m
EI

M m

= 1

∫

X Xj
X

Y Yj
Y

Z Zj
Z

X Xj Y Yj Z Zj

u = + + + + + dx

j memb ,

EI

i X Y Z

o

i

(14)

N i

2

∆ u
h

u − u

∑ ∑

0

u = m θ (15)

j j j − 1

= ≤ ψ ( j = 1 2 , , ..., N )

j hinge , pjh ph

j j

h

i = 1 h = 1

i

j j

(12)

Considering rectangular concrete elements
with width ( B i ) and depth ( D i ) and expressing
the cross sectional properties in terms of B i and
D i , the displacement, u j mem , , in Eq. (14) can be
simplified in terms of B i and D i , as shown in

where u j and u j −1 are the respective displacement
of two adjacent j and j-1 floor levels; ψ j is the
specified inelastic inter-story drift ratio limit for
the j th story in the inelastic design phase.