Civil Engineering Reference
In-Depth Information
Figure 9.12
Single-column model representation of a shear building
9.5.2 A Three-degrees-of-freedom Building
Actually, we can further idealize the shear building as a single column (Figure 9.12a). In each
of the two representations (Figures 9.11 and 9.12) for a viscously damped shear building, the
stiffness and damping coefficients shown between any two consecutive masses are the forces
required to produce relative unit displacements and velocities, respectively, of the two adjacent
floor levels.
From the free body diagrams in Figure 9.11(b) or Figure 9.12(b), we obtain
m
1
y
1
+
c
1
y
1
+
k
1
y
1
−
c
2
(
y
2
−
y
1
)
−
k
2
(
y
2
−
y
1
)
=
F
1
(
t
)
m
2
y
2
+
c
2
(
y
2
−
y
1
)
+
k
2
(
y
2
−
y
1
)
−
c
3
(
y
3
−
y
2
)
−
k
3
(
y
3
−
y
2
)
=
F
2
(
t
)
(9.38)
m
3
y
3
+
c
3
(
y
3
−
y
2
)
+
k
3
(
y
3
−
y
2
)
=
F
3
(
t
)
In matrix notation
[
M
]
{
y
} +
[
C
]
{
y
} +
[
K
]
{
y
} = {
F
(
t
)
}
(9.39)
where
⎡
⎤
m
1
00
⎢
⎢
⎣
⎥
⎥
⎦
[
M
]
=
0
m
2
0
(9.40)
00
m
3
