Civil Engineering Reference
In-Depth Information
analytical model, the mass-proportional damping is not included (
a
0
=
0) because the mass-
proportional damping, if applied to the structure in the analysis, would act as external dampers
which do not physically exist in base-supported structures such as structures under earthquake
action. The damping proportional to the converged stiffness at each time step is applied to
model the energy dissipation arising from story deformations. The coefficient
a
1
is given by
Equation (9.47).
2
ζ
a
1
=
(9.47)
(
ω
1
+
ω
2
)
where
ω
2
are the natural frequencies associated with the
first and second modes at the initial condition.
The damping ratios were determined based on the different damage levels of the structures.
The recommended damping values for reinforced concrete structures at different stress levels,
given by Chopra (2005), are 3-5% for reinforced concrete with considerable cracking; and
7-10% at the stress level at or just below the yield point. However, the values are roughly esti-
mated for reinforced concrete structures and are not specified for different types of structures,
such as frames or shear walls. Since the damping in structures is determined by both material
damping and system damping, a specific damping value for reinforced concrete shear walls
should be used for wall-related structures. Farrar and Baker (1995) conducted experiments to
measure the damping ratio of low-rise reinforced concrete shear walls. They found that the
damping ratio was 1-2% for undamaged low-rise shear walls. With an increasing damage level
of a wall, from moderate to severe, the measured damping ratio correspondingly increased
from 2 to 8 %. The measured damping ratio of the wall at final failure was as high as 22%. In
the seismic tests of reinforced concrete shear walls performed by Ile and Reynouard (2000),
the damping ratio was found to increase from slightly more than 1% at the first test run to
approximately 4% before the last test run at failure.
In this topic, the damping ratios for reinforced concrete shear walls found by Farrar and
Baker (1995) and Ile and Reynouard (2000) have been adopted. In general, a damping ratio
of 2% is used when the structure is in the elastic stage and only minor cracking is observed.
A damping ratio of 4% is used for the case that the steel in the wall yields and considerable
cracking is observed. If minor crushing of concrete is observed, a damping ratio of 8% is used.
When severe crushing of the concrete is observed and the structure has failed, high damping
ratios of 15 and 25% are used, respectively, in order to consider the damaged condition of
the specimen.
In the case under earthquake excitations, Equation (9.46) is replaced by Equation (9.48).
ζ
is the damping ratio and
ω
1
and
⎧
⎨
⎩
⎫
⎬
m
1
y
g
m
2
y
g
m
3
y
g
{
F
} =−
(9.48)
⎭
where
y
g
is the ground acceleration.
