Geology Reference
In-Depth Information
Figure 2. Structural system with a passive energy
dissipation system under ground acceleration
g
( ( ), ( ))
δ
t u t
=
H
(
λ δ
( ))
t
×
i
i
λ
u t
( )
−
−
u
[
i
y
H u t
(
λ
( )
−
u
)
i
y
u
u
p
y
)
]
H u
(
−
λ
u t
( ))
+
H u t
(
λ
( )
−
u
p
i
i
p
(44)
, , ,
H
(·)
denotes the
Heaviside step function,
u
y
is a parameter speci-
fying the onset of yielding, and
k
d p
is the
maximum restoring force of the device. The val-
ues
u
p
=
1
(
i
−
1
)
1 2
where
λ
i
= −
i
=
6 0 10
3
.
×
−
m,
u
=
0 7
.
u
,
and
y
p
6 0 10.
N/m are used for each nonlinear
element. Because of the yielding, energy dissipa-
tion due to hysteresis is introduced in the struc-
tural response. For illustration purposes the be-
havior of the friction devices at the initial design
(
K
1
k
d
=
×
9
9
7 0 10
= ×
.
N/m)
is shown in Figure 3. In this figure typical dis-
placement-restoring force curves of the friction
devices at the first and second floor are presented.
The non-linear incursion of the devices is clear
from the figure. In general the overall effect of
the dissipation system is to decrease the response
of the structural system. The system is subjected
to a ground acceleration that is modeled as de-
scribed in a previous Section. Note that the dura-
tion of the excitation is 20 s and the sampling
interval is equal to 0.01 s (see Table (1)). There-
fore the number of random variables involved in
the characterization of the excitation is equal to
2000. This in turn implies that the estimation of
the failure probability for a given design represents
a high dimensional reliability problem.
=
9 0 10
.
×
N/m, and
K
2
random variable with a coefficient of variation of
40
%
. This high coefficient of variation accounts
for the considerable uncertainty in estimating
damping ratios in real structural systems.
For an improved earthquake resistance, the
model is reinforced with a passive energy dissipa-
tion system (shear panel) at each floor. The shear
panels follow the inter-story restoring force law
(
)
1
2
r t
( )
=
k
δ
( )
t
−
q t
( )
+
q t
( )
(42)
d
where
k
d
denotes the initial stiffness of the device,
δ
(
t
is the relative displacement between floors,
and
q t
2
( )
denote the plastic elongations
of the device. Using the auxiliary variable
u t
1
( )
and
q t
δ
1 2
the plastic elongations
are specified by the differential equations
( )
( )
t
q t
( )
q t
( ),
=
−
+
Discrete Optimization Problem
The objective function is defined in terms of an
initial cost which is assumed to be proportional
to the linear interstory stiffnesses. The reliability
constraints are given in terms of the interstory
drift ratios. Failure is assumed to occur when the
i
q t
( )
=
λ δ
( ) ( ( ), ( )),
t g
δ
t u t
i
=
1 2
,
(43)
i
i
where the nonlinear functions
g
i
are specified by
(Pradlwarter and Schuëller, 1993)
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