Civil Engineering Reference
In-Depth Information
Table 3.10
Equivalent damping ratios for single-degree of freedom systems (
after
Borzi
et al.
, 1998 ).
μ
= 2
μ
= 3
μ
= 4
μ
= 6
EPP
8%
13%
16%
19%
HHS,
K
3
= 0
11%
15%
18%
20%
HHS,
K
3
= 10%
K
y
10%
13%
15%
17%
HHS,
K
3
= − 20%
K
y
13%
20%
28%
—
HHS,
K
3
= − 30%
K
y
15%
27%
—
—
Key
: EPP = elastic - perfectly plastic model; HHS = hysteretic hardening - softening model.
Ky and K
3
are the secant and post-yield stiffness of the primary curve of the HHS model.
The equivalent damping values given in Table 3.10 are recommended for use with the elastic response
spectra given before, for periods of up to 3.0 seconds. Nil entries in Table 3.10 indicate that structures
with highly degrading response (
K
3
= −20% and − 30%
K
y
) would not have ductility capacity of 4 or
more. If a more refi ned value of ductility-damping transformation is sought, the relationships given
by Borzi
et al.
(1998) as a function of magnitude, distance, soil condition and period should be
consulted.
Problem 3.5
Draw the displacement spectrum with ductility
μ
= 4 on a site at a distance of 15 km from the source,
on stiff soil and subjected to an earthquake of magnitude
M
= 6.5. What options are available to
the designer if it is deemed necessary to decrease the displacement demand imposed on the structure
below the value implied by the above spectrum, if the fundamental periods of vibration are about
1.7 and 3.0 seconds? For the inelastic displacement spectrum, assume a viscous damping of
0.5%.
3.4.3.2 Spectra from Ground - Motion Parameters
By plotting the response spectra of an ensemble of earthquake records, normalized by the relevant
ground-motion parameter, Newmark and Hall (1969) derived statistical values of the amplifi cation
factors for acceleration, velocity and displacement. These amplifi cation factors, expressed as ratios
between peak ground parameter and peak response of the system, are provided in Table 3.11 .
To establish the elastic response spectrum for the full range of periods, use is made of a four- way
log paper. The resulting spectrum is referred to as a 'tripartite plot', since it includes the three spectral
forms, and is shown in Figure 3.14. This is made possible by the simple relations between spectral
acceleration, velocity and displacement given in equation (3.18). Indeed, the tripartite plot is based
upon the following relationships:
log
S
=
log
ω
+
log
S
d
(3.20.1)
v
log
S
=
log
ω
−
log
S
(3.20.2)
v
a
Equations (3.20.1) and (3.20.2) indicate that the logarithm of the spectral velocity is linearly related
to the logarithm of the natural frequency
ω
, provided the spectral displacement or acceleration remains
constant. The slope is either +1 or
1 depending on whether the displacement or the acceleration is
assumed as a constant. Thus, to draw the elastic spectrum for use, for example in spectral analysis
illustrated in Section 4.5.1.1, the following steps are required:
−
