Geology Reference
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model is the well-known stationary white noise
process, whose correlation function is Delta Dirac
and associated PSD function is constant at each
frequency. However, it cannot sufficiently describe
the spectral behaviour of many real stochastic
dynamic loads. To improve the spectral charac-
terization of ground motion, a second order filter
is introduced to colour the white noise, known as
the Kanai-Tajimi model (Tajimi, 1960). This model
has been widely applied in the random vibration
analysis of structures as it provides a simple way
to describe the ground motion characterized by
a single dominant frequency. However, actual
earthquake recorded data show a non-stationary
nature both in the amplitude and frequency
contents and a more generalized non-stationary
model obtained by enveloping the stationary input
stochastic process should be used for more refined
analysis. The Kanai-Tajimi PSD function used in
the present study is represented by,
∞
∫
( )
=
j
(
)
λ
u
ω
S
ω
,
u
d
ω
,
j
YY
−∞
( )
( )
σ
( )
u
=
λ
u
and
σ
( )
u
=
λ
u
Y
o
2
Y
(13)
The reliability of the structure based on the
first passage failure criterion for double barrier
problem can be obtained as,
T
( )
=
∫
r T
r
0
exp
−
h t dt
( )
(14)
0
where,
r
0
is the survival probability at time, t=0
and
h
( )
is the hazard function and T is the dura-
tion of the ground motion. Following the Poisson's
assumption of rare and independent threshold
crossings events, h(t) can be replaced with un-
conditional threshold crossing rate,
ν
+
which can
be expressed as:
(
4
2
2
2
4
)
ξ ω ω
+
ω
( )
=
g
g
g
S
ω
S
(11)
u u
0
2
2 2
2
2
2
(
ω
−
ω
)
+
4
ξ ω ω
g g
g
g
g
σ
σ
1
2
1
2
β
σ
( )
=
+
2
ν
β
Y
−
(
)
(15)
Y
π
Y
Y
where,
ω
g
and
ξ
g
denote the natural frequency and
damping ratio of the soil layer, respectively. The
PSD of the white noise process at bed rock,
S
0
can
be related to the peak acceleration, (
x
g max
In the above,
β
is the first time bi-lateral dis-
placement crossing barrier. If the failure is due to
double symmetric threshold crossing (as consid-
ered herein), the threshold crossing rate can be
given by,
) by,
,
2
0 0707
1
.
ξ
x
g
g max
,
S
=
(12)
(
)
0
2
ω
+
4
ξ
( )
=
( )
+
2
Y
g
g
ν β
ν
β
(16)
The applicability of the above is limited to
the assumptions that the structure can recover
immediately after suffering failure (threshold
crossing) and such failures arrive independently,
that is, they constitute a Poisson process. More
The spectral moments (
λ
j
) and the associated
root mean square (RMS) values (
σ
Y
and
σ
Y
) of
the responses useful for reliability evaluation can
be evaluated as,
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