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where {0} is a zero vector.
In order to solve the Equation (5), it is written
in state space as given in the following:
of the first two modes are assumed to be about 2%
of the critical value (Pourzeynali & Zarif, 2008).
Furthermore, it should be noted that all ten
vibrational modes of the building are considered
in the analysis, from which the first 5 modal
frequencies are given as: 1.46, 3.86, 6.27, 8.69,
and 10.84 Hz.
The base isolated building is analyzed under
action of the accelerogrames mentioned earlier,
and the results are compared with those of fixed
base building. In this step, the structural parameters
of the base isolation system are taken from the
references (Matsagar & Jangid, 2003; Matsagar
& Jangid, 2004) as the initial values, given in the
following (Pourzeynali & Zarif, 2008):
{ }
=
Z
{ }
+
{ }
A Z
B P
(10)
1
1
where {
Z
} is the state vector; and
A
1
,
B
1
and
{ }
are, respectively, the state matrix, input matrix,
and input vector given in the following:
{ }
=
{ }
X
X
(11a)
{ }
I
M K M C
0
A
=
−
m
=
m m
/
=
1 0
.
,
k
=
k
/
k
=
0 10
.
,
−
1
−
1
1
−
0
b
1
0
b
1
ξ
=
c
/ ( *
2
m
*
ω
)
=
0 10
.
,
ω
=
k m
/
(11b)
b
b
s
b
b
b
s
0
{ }
= −
{ } ( )
B
=
and P
R x
t
I
g
1
where
ξ
b
, is the damping ratio of base isolation
system;
m
1
and
k
1
are the mass and stiffness of
the building first story;
m
0
and
k
0
are the mass and
stiffness ratios of the base isolation system to that
of the building first story;
ω
b
is the natural circu-
lar frequency of the base isolation system; and
m
s
is given in Equation (9).
By considering the above values, equations of
motion of the building are solved and the results
are compared for both fixed base, and base isolated
building supported on isolators with linear behav-
ior in Tables 2 and 3, respectively (Pourzeynali
& Zarif, 2008). It should be mentioned that, for
brevity, only the results of the four most impor-
tant earthquakes (e. g., Kobe, El Centro, Loma
Prieta, and Northridge Earthquakes), as well as
the ensemble average values of the responses for
18 reference earthquakes are shown in the tables.
As it is seen from the tables, displacements of
the isolated building significantly are reduced. In
this case, in average 43.74% reduction is obtained
on building top story horizontal displacement
where
I
is an identity matrix.
The same procedure is performed for solving
the Equation (8). Damping matrix of the building
is also calculated using the well known Rayleigh
method.
Numerical Analysis
It is well known that the base isolation systems
are more effective for low buildings with low
dominant vibrational periods, but in this study,
their effect is studied on medium-height build-
ings. For this purpose, a 2-D realistic ten-story
steel building frame, located in Mashhad, Iran,
is selected. For this 2-D idealized frame, the
mass and stiffness matrices are calculated us-
ing matrix analysis procedure. Damping matrix
of the building is also calculated using the well
known Rayleigh method, and for calculating the
proportionality coefficients, modal damping ratios
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