Geology Reference
In-Depth Information
L
U
ρ
≤ ≤
ρ
ρ
(
i
=
1 2 3
,
,
N
, ...,
)
(27)
analysis using commercially available soft-
ware such as SAP2000 (2000).
3. Read all the necessary input and output re-
sults of the analysis and establish the explicit
elastic design optimization formulation of
Eqs. (1) and (9).
4. Apply the recursive OC optimization algo-
rithm to resize the concrete element sizes.
5. Repeat Steps 2 and 4 for statically indeter-
minate structures until the concrete cost of
the structure between two successive design
cycles converges to be within certain accept-
able criteria (say, 0.5%).
6. After the elastic design optimization, fix the
optimal member sizes,
B
i
and
D
i
, in the
inelastic design optimization. Based on the
member size derived from the elastic opti-
mization, determine the minimum and
maximum size bounds of the steel reinforce-
ment ratios,
ρ
i
and
′
i
i
i
i
where
u
=
∂
∂
∆
∂
∆
θ
ρ
2
∑
j
ph
α
=
m
0
i
jh
1
0
0
ρ
ρ
=
ρ
∂
ρ
=
ρ
i
i
i
i
h
=
i
1
i
(28)
=
∂
2
∆
u
2
∂
∆
θ
2
∑
j
ph
α
=
m
0
i
jh
2
0
0
∂
ρ
2
ρ
=
ρ
∂
ρ
2
ρ
=
ρ
i
i
i
i
h
=
1
i
i
(29)
In Eq. (25),
′
w
si
is the cost coefficient for the
steel reinforcement,
ρ
i
;
ψ
j
p
is the allowable in-
elastic inter-story drift ratio. Eq. (26) defines the
set of seismic inelastic inter-story drift perfor-
mance constraints under specified earthquake
ground motions. Eq. (27) defines the sizing con-
straints for the steel reinforcement, where
ρ
i
L
and
ρ
i
U
correspond to the lower and upper size bounds
specified for the tensile steel reinforcement vari-
able,
ρ
i
, and they should be updated after each
nonlinear pushover analysis. For the sake of
simplicity, the compressive steel reinforcement,
′
ρ
i
, in accordance with
the strength-based code requirements.
7. Carry out the nonlinear pushover analysis
to determine the inelastic responses of the
structure at the performance point.
8. Read all the necessary input and output
results of the pushover analysis, establish
the lower and upper bounds of
ρ
i
for the
members with plastic hinges using Eq. (24)
and determine the values of the first-order
and second-order derivatives of inelastic
drift responses using Eqs (28) and (29).
9. Establish the explicit inelastic inter-story
drift constraints using a second-order
Taylor series approximation and formulate
the explicit inelastic design problem, Eqs
(25)-(29).
10. Apply the recursive OC optimization algo-
rithm to resize all steel reinforcement design
variables.
11. Repeat Steps 7 and 10 until the convergence
of the values of the steel cost objective func-
tion and the inelastic drift design constraints
is achieved.
ρ
i
, has been assumed to be simply related to
ρ
i
and therefore, it is not included in the explicit
optimization problem Eqs. (25)-(29).
2.3 Design Procedure
The overall design optimization procedure for
limiting lateral elastic and inelastic drifts of an
RC building structure is listed as follows:
1. Assume the initial member sizes and deter-
mine the design spectra corresponding to
minor and severe earthquakes.
2. In the first phase, i.e., the elastic design op-
timization, carry out the response spectrum
analysis of the structure subject to a minor
earthquake and conduct a static virtual load
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