Geology Reference
In-Depth Information
Box 1.
1
6
{
z
(
j
+
1
)
(
t
+
∆
t
)}
=
{ ( )}
z t
+
[{
k
(
j
+
1
)
}
+
2
{
k
(
j
+
1
)
}
+
2
{
k
(
j
+
1
)
}
+
{
k
j
4
(
+
1
)
}]
(10)
1
2
3
where
{
h
is a non-linear vector function that
characterizes the behavior of the non-linear com-
ponents. From the last equation it is seen that the
set of variables
{ ( )}
Equation (8). The iteration starting with solving
Equation. (8) needs to be repeated until the norm
of the vector
{ (
+∆
is in two consecutive
iterations sufficiently close. Numerical experience
shows that in general only few iterations are re-
quired within each time interval [
t t
z t
t
)}
z t
is a function of the displace-
x t
, that is,
ments
{ ( )}
x t
and velocities
{ ( )}
+∆
].The
solution of the equation for the evolution of the
set of variables
{
z
is obtained by a modified
Runge-Kutta method of fourth order. The solution
at time
t t
{ ({ ( )},{ ( )})}.
z x t
x t
Numerical Integration
Since the differential equation that satisfy the
variables
{ ( )}
+∆
is written in Box 1, 2, and 3.
For actual implementation, the characterization
of the vector of non-linear restoring forces is
modeled in local component specific coordinates
(local displacements and velocities) with a mini-
mal number of variables. Therefore, the relation-
ships given by Equations (11) and (12) are
evaluated in local displacements and velocities
increasing in this manner the efficiency of the
above numerical integration scheme.
z t
is non-linear in terms of the re-
sponse
{ ( )}
x t
, Equations. (6) and (7) must be
solved in an iterative manner. Equation (6) is
solved first by any suitable step-by step integra-
tion scheme, leading from the solution at time
t
at the one at time
t t
+∆ ,
that is
[
M x t
]{ (
t
)}
[
C x t
]{ (
t
)}
[
K x t
]{ (
t
)}
+
∆
+
+
∆
+
+
∆
=
−
[
M g x t
]{ } (
+
∆
)
t
−
{ ({ (
f
x t
+
∆
t
)},{ (
z t
+
∆
t
)})}
g
(8)
It is seen that in order to compute the solution
at time
t t
EARTHQUAKE EXCITATION MODEL
+∆
, provided that the solution at time
t
is known, the value of
{ (·)
z
is required at time
t t
The ground acceleration is modeled as a non-
stationary stochastic process. In particular, a
point-source model characterized by the moment
magnitude
M
and epicentral distance
r
is con-
sidered here (Atkinson and Silva, 2000; Boore,
2003). The model is a simple, yet powerful, means
for simulating ground motions and it has been
successfully applied in the context of earthquake
engineering. The time-history of the ground ac-
celeration for a given magnitude
M
and epicen-
tral distance
r
is obtained by modulating a white
noise sequence by an envelope function and sub-
sequently by a ground motion spectrum through
the following steps:
+∆
. To this end, at the beginning of the it-
eration within the time interval
[ ,
t t t
+∆
it is
],
assumed that (
j
=0
)
{
z
( )
j
({ (
x t
t
)},{ (
x t
t
)})}
{ ({ ( )},{ ( )})}
z x t
x t
+
∆
+
∆
=
(9)
The solution of Equation. (8), gives the re-
sponses
{
j
+
1
∆
.
Then, the nonlinear differential Equation (7) can
be integrated to obtain new estimates for
{ ({ (
(
j
+
1
)
(
)
x
(
t
+
∆
and
{
t
)}
x
(
t
t
)}
z x t
+
∆
t
)},{ (
x t
+
∆
t
)})},
and the right
hand side
{ ({ (
f
x t
+
∆
t
)},{ (
z t
+
∆
t
)})}
in
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