Geology Reference
In-Depth Information
where
B
i
L
and
B
i
U
,
D
i
L
and
D
i
U
correspond to
the lower and upper size bounds specified for the
section width,
B
i
, and depth,
D
i
, respectively.
story drifts can be expressed using the general
and accurate complete quadratic combination
(CQC) method as
N
N
2.2 Second Phase - Inelastic
Design Optimization Problem
n
n
∑
=
1
∑
( )
n
( )
n
(
m
)
(
m
)
∆
u B D
(
,
)
=
ρ
(
u
−
u
)(
u
−
u
)
j
i
i
CQC
nm
j
j
−
1
j
j
−
1
n
m
=
1
(5)
2.2.1 Nonlinear Analysis Procedure
where
N
n
denotes the total number of modes
considered in the response spectrum analysis ;
u
j
n
−
1
In the newly developed performance-based seismic
design approach, nonlinear analysis procedures
become important in identifying the patterns and
levels of damage to assess a structure's inelastic
behaviour and to understand the modes of failure
of the structure during severe seismic events.
Pushover analysis is a simplified, static, nonlinear
procedure in which a predefined pattern of earth-
quake loads is applied incrementally to framework
structures until a plastic collapse mechanism is
reached. This analysis method generally adopts a
lumped-plasticity approach that tracks the spread
of inelasticity through the formation of nonlinear
plastic hinges at the frame element's ends during
the incremental loading process.
As graphically presented in Figure 1, the
nonlinear static analysis procedure requires de-
termination of three primary elements: capacity,
demand and performance. The capacity spectrum
can be obtained through the pushover analysis,
which is generally produced based on the first
mode response of the structure assuming that the
fundamental mode of vibration is the predominant
response of the structure. This pushover capacity
curve approximates how a structure behaves be-
yond the elastic limit under seismic loadings. The
demand spectrum curve is normally estimated by
reducing the standard elastic 5% damped design
spectrum by the spectral reduction method. The
intersection of the pushover capacity and demand
spectrum curves defines the “performance point”
as shown in Figure 1. At the performance point,
the resulting responses of the building should then
be checked using certain acceptability criteria. The
( )
,
u
j
−
( )
,
u
j
( )
are the respective nth and mth
modal elastic displacements at the (j-1)th and jth
floor levels;
ρ
nm
is a modal correlation coefficient
for the nth and mth modes, which can be obtained
from the following equation with the constant
damping ratio,
ξ
, as
3
2
8
2
(
1
)
ξ
+
β
β
ρ
=
nm
nm
(6)
nm
2
2
2
2
(
1
−
β
)
+
4
ξ β
(
1
+
β
)
nm
nm
nm
where
ω
β
=
≤1
(7)
n
ω
nm
m
Upon establishing the explicit formulation of
the elastic inter-story drift Eqs. (4) and (5), based
on the response spectrum analysis method, elas-
tic spectral inter-story drift constraints using the
CQC method can be written in terms of the design
variables,
B
i
and
D
i
, as
Subject to:
1
g B D
(
,
)
=
∆
u B D
(
,
)
≤
1
(
j
=
1 2
,
, ...,
N
)
j
i
i
j
i
i
j
e
ψ
h
j
j
(8)
Besides, the member sizing constraints can
be defined as,
L
U
L
U
B
≤
B
≤
B
;
D
≤
D
≤
D
i
i
i
i
i
i
(
i
=
1 2
,
, ...,
N
i
)
(9)
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