Civil Engineering Reference
In-Depth Information
nonlinear demand spectrum for appropriate target level of ductility can be
determined as follows, given a value of
A
u
:
if
TT
<
c
[
()
−
()
]
2
TA
2
β
0
AT
g
μ
()
()
=
c
el
nl
if
AAT
≥
⇒
SD
μ
∗
u
nl
nl
()
(
)
2
2
μ
−
1
4
π
AAT
nl
[
()
−
]
2
TA
2
β
0
1
A
g
μ
()
<<
()
⇒
()
=
c
el
u
if
AT
A
AT
SD
μ
∗
nl
c
u
nl
nl
(
)
2
2
μ
−
4
π
A
u
[
()
]
2
gT
2
β
πμ
A
A
0
()
()
=
c
el
if
AAT
≤
⇒
SD
μ
u
nl
c
nl
2
4
u
[13.5]
if
TT
≥
c
[
()
]
()
2
2
gT
β
πμ
A
AT
0
()
()
=
c
el
if
A
≥
AT
⇒
SD
μ
u
nl
nl
2
4
el
[
]
2
()
gT
2
β
πμ
A
A
0
()
()
=
c
el
if
AAT
<
⇒
SD
μ
u
nl
nl
4
2
u
where two different formulations are provided for values of ultimate capac-
ity
A
u
greater or smaller than the nonlinear acceleration
A
nl
(
T
c
); period
T
c
identifi es the transition from constant acceleration to constant velocity
section of the elastic spectrum. In Equation (13.5),
SD
nl
is the nonlinear
spectral displacement, as function of the chosen target ductility
is the
acceleration amplifi cation factor calculated as the ratio of the elastic
maximum spectral acceleration
A
el
(
T
)
and the peak ground acceleration
A
el
(0)
for
T
μ
;
β
0;
A
nl
(
T
) is the non-linear spectral acceleration for the value
of natural period
T
that defi nes the elastic branch of the capacity curve;
g
is the gravity acceleration. Note that in Equation (13.5)
A
el
(
T
)
, A
nl
(
T
) and
A
u
are dimensionless quantities, expressed as a proportion of
g.
In Fig. 13.6 the performance points (PP) for the limit state of signifi cant
damage (SD) for each building in the Nocera Umbra, Italy, case study are
compared with the response spectrum for 475 year return period (or 10%
probability of exceedance in 50 years). For the nonlinear spectrum obtained
with the N2 method approach a ductility
=
3 has been chosen in agree-
ment with experimental evidence. It can be seen that there is a signifi cant
scatter of performance and although the representative LS points on the
mean curves for the three types of behaviour are all beyond the intersection
with the spectrum, there is a consistent number of PPs under the curve,
equally defi cient in terms of acceleration and displacement, especially for
the out of plane behaviour. Such outliers should not be overlooked as they
usually point to inherent construction defi ciency in a regional context,
reducing the seismic resilience of a community.
μ
=
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