Geology Reference
In-Depth Information
With this target displacement, the internal
forces and deformation of structural components
are obtained and checked to the acceptance criteria.
From this comparison, the performance level of
structure is known. The checking process is simi-
lar to linear analysis, but with different criteria;
i.e., it is similar in the sense that it depends on
the properties of structural component whether
it is deformation-controlled or force-controlled
component.
Having the capacity curve converted to capac-
ity spectrum curve it remains to draw the demand
response spectrum in the same coordinate. Noting
that response spectrum as specified by a design
code is a curve specified in S a -T coordinates, and
we need demand spectrum in S a -S d coordinates,
the horizontal axis should be transformed to S d
simply noting that S d =T 2 /4π 2 S a g. The outcome of
this conversion is an acceleration-displacement
response spectrum (ADRS) curve.
As building yields in response to seismic de-
mand, it dissipates energy with hysteretic damping;
that means its damping increases. With an increase
in damping, the demand spectrum reduces. i.e.,
the ADRS curve is not unique, and it depends on
the effective damping property of the structure.
According to ATC 40, the effective equivalent
viscose damping, is obtained from β eq 0 +0.05
in which 0.05 is the inherent viscose damping of
the structure that is considered to be constant. β 0
is the hysteretic damping represented as viscose-
equivalent hysteresis damping and is calculated
from: β 0 =E D /(4πE S0 ) in which E D is the damping
dissipated energy and E S0 is the maximum strain
energy. According to ATC 40 (section 8.2.2),
E D and E S0 are calculated based on performance
point that is the intersection of the ADRS curve
and the reduced demand spectrum. This implies
that the performance point is obtained through an
iterative process.
There are several techniques for finding this
intersection. Here, we explain one of them. Since
the intersection of demand and capacity spectra,
(the target/performance point) is not known a
priori, an initial target point is assumed on the
capacity spectrum curve. Then with the infor-
mation of this point and the capacity spectrum
curve, the corresponding viscose-equivalent
hysteresis damping β 0 and β eq are obtained and
the corresponding reduced demand spectrum is
constructed and intersected to capacity spectrum
curve to produce a new performance/target point.
If the obtained target point is close enough to the
assumed one, the performance point is found;
The Capacity Spectrum Method
In the capacity spectrum method that is recom-
mended by ATC40, the performance point that is
alternatively called the target point is obtained by
intersecting the capacity spectrum and demand
spectrum. The capacity spectrum is indeed the
capacity curve converted from force-displacement
ordinates to spectral acceleration-spectral dis-
placement ordinates. This is done by converting
the base-shear to S a and displacement to S d . The
following formulae apply.
V W
S
a =
(10)
α 1
roof
S
=
(11)
d
PF
(
ϕ
)
1
roof
,
1
2
n
m
ϕ
i
i
1
1
i
=
1
α
=
(12)
1
W
n
2
m
(
ϕ
)
i
i
1
i
=
1
n
m
ϕ
i
i
1
PF
=
i
=
1
(13)
1
n
2
m
(
)
i
i
1
i
=
1
 
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