Geology Reference
In-Depth Information
Box 2.
2
N
∂
∂
u
N
∂
∂
u
1
2
i
i
∑
∑
j
j
u
(
ρ
)
=
u
+
(
ρ
−
ρ
0
)
+
(
ρ
−
ρ
0 2
)
(22)
j
i
j
0
0
i
i
2
0
i
i
ρ
=
ρ
ρ
ρ
=
ρ
ρ
ρ
=
ρ
i
i
i
i
i
i
i
=
1
i
=
1
i
i
M
≤
M M
≤
(24)
2.2.3 Plastic Rotation Constraint and
Sizing Constraint
y
u
y
=
where
M
y
can be found from Eq. (17) for a moment
hinge and Eq. (19) for an axial-moment hinge,
the corresponding value of
ρ
i
can be solved and
this value can then be taken as the instantaneous
upper bound value of
ρ
i
such that
ρ
On the one hand, by setting
M
M
Besides checking the inter-story drift responses
discussed above, local response quantities (i.e.,
sectional plastic rotation and strength of all mem-
bers) at the performance point also must not exceed
appropriate response limits. Therefore, the plastic
rotation,
θ
ph
, at the hth end of member
i
(where
the subscript
h
represents one end of a member
and
h
=1,2) should be constrained in the optimiza-
tion by
U
=
. On
ρ
i
the other hand, by setting
M
u
=
M
and assuming
=
1 1.
, the instantaneous lower bound
value of
ρ
i
can then be found such that
ρ
M
M
u
y
L
=
.
As a result, based on Eq. (24), the lower and up-
per bounds of
ρ
i
for each plastic hinge can be
instantaneously established during the OC itera-
tive resizing process.
ρ
i
≤
θ
θ
(23)
ph
p
where
θ
U
is the rotation limit of member i for a
specific performance level. Once the designer
determines the performance levels of the structure
(e.g., Immediate Occupancy, Life Safety, Collapse
Prevention), the corresponding limiting value of
θ
U
is then determined. In addition, in order to
reduce the practical building design problem to
a manageable size, the strength design of each
member is not considered explicitly as a design
constraint; rather, the strength-based steel rein-
forcement ratios in accordance with code speci-
fications are first calculated and these values are
then taken as the lower size bound for each mem-
ber in the inelastic seismic drift design optimiza-
tion.
It is found from Figure 2 that, in order to
maintain the relationship of
0≤ ≤
2.2.4 Design Optimization
Problem Formulation
Upon establishing the explicit inelastic drift for-
mulation, Eq. (22), the optimization problem of
minimizing the steel construction cost Eq.(10)
can be explicitly written in terms of the design
variable,
ρ
i
, as
N
i
∑
′
Minimize:
f
(
ρ
i
=
)
w
ρ
(25)
1
s
si
i
i
=
1
Subject to:
U
, the
internal moment,
M
, leading to the occurrence
of a plastic hinge must satisfy the following con-
dition:
θ
θ
N
N
1
1
2
i
i
∑
∑
≤
p
p
0
0 2
g
(
ρ
)
=
∆
u
+
α ρ
(
−
ρ
)
+
α ρ
(
−
ρ
)
1
j
i
p
j
0
1
i
i
i
2
i
i
i
ψ
h
ρ
=
ρ
i
i
j
j
i
=
1
i
=
1
(
j
=
1 2 3
,
,
N
j
, ...,
)
(26)
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