Part 08 - Response Spectra
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9/22/2014 1 Earthquake Response of Linear Systems Ch 6 Chopra Examples of Ground motions Earthquake Response of Linear Systems El Centro ground motions Ground acceleration Ground velocity Ground displacement
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Response Spectra
Transcript of Part 08 - Response Spectra
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Earthquake Response of Linear Systems Ch 6 Chopra
Examples of
Ground motions
Earthquake Response of Linear Systems
El Centro ground motions
Ground acceleration
Ground velocity
Ground displacement
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Earthquake Response of Linear Systems
Earthquake Response of Linear Systems
u(t) = displacement of the mass relative to the ground
Internal forces, like moments and shears in the beams and columns,
are linearly related to the displacement function.
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Earthquake Response of Linear Systems
For a given ground motion, the deformation response, u(t), depends only on the
natural vibration period of the system and its damping ratio.
Observations:
1. Period of the response is close
to the natural period.
2. Among the 3 systems, the
longer the vibration period, the
greater the peak deformation.
3. Systems with more damping
respond less than lightly
damped systems.
4. For the systems with the same
period, the times to reach the
maxima and minima are similar.
Earthquake Response of Linear Systems
The equivalent static force, fs , is related to the displacement u(t) as:
fs (t) = k u(t)
Expressing k in terms of the mass m gives:
where
A(t) = pseudo-acceleration
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Earthquake Response of Linear Systems
Earthquake Response of Linear Systems
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Response Spectra
Definition:
A plot of the peak value of a response quantity as a function of the natural
vibration period Tn of the system, or a related parameter such as circular
frequency �n or cyclic frequency fn, due to a given forcing function is called
the response spectrum of that quantity .
Construction of Response Spectrum Response Spectra
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Response Spectrum
To construct the Response Spectrum for the pseudo-velocity, we can use the
Response Spectrum for the displacement, D. We just apply the relationship
Response Spectrum
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Response Spectrum
Response Spectra
To construct the Response Spectrum for the pseudo-acceleration, we can use the
Response Spectrum for the displacement, D. We just apply the relationship
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Response Spectra
One of the reasons is that each of the spectrum provides a physically meaningful quantity:
1. The deformation spectrum provides the peak deformation of a system.
2. The pseudo-velocity spectrum is related directly to the peak strain energy stored
in the system during the earthquake (see Eq. 6.6.2)
3. The pseudo-acceleration spectrum is related directly to the peak value of the
equivalent static force and base shear (see Eq. 6.6.4)
Response Spectra
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Response Spectra
V = �D log V = log � + logD
A = � V log A = log V + log � (b)
These are linear
relationships in
the log scale
or log D = log V – log � (a)
Thus, a four-way log plot as shown in Fig. 6.6.3 allows all three types of
spectra to be illustrated on a single graph.
From these relations, it is seen that when a plot is made with log V as the
ordinate and log � as the abcissa, Eq (a) is a straigh line with slope of +450
for a constant value of log D, and Eq (b) is a straight line with slope of -450
for a constant value of log A.
Response Spectra
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Response Spectra
Response Spectra
6.7 PEAK STUCTURAL RESPONSE FROM THE RESPONSE SPECTRUM
Response quantities of interest can be expressed in terms of D, V, A, and the
vibration characteristics of the vibrating system.
peak deformation:
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Response Spectra
Response Spectra
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Response Spectra
Response Spectra
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Response Spectra
(Study Ex 6.4 and 6.5)
Response Spectra
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Response Spectra
Response Spectra
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Response Spectra
Response Spectra
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Response Spectra
Response Spectra Effect of Damping
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Response Spectra
Ground motions during earthquakes show jagged response spectra. Even for
the same site, the response spectra for different earthquakes may have
peaks and valleys not necessarily at the same periods. (See Fig. 6.9.1)
It is not possible to predict the jagged response spectrum in all its detail for
a future earthquake. Thus, the design spectrum should consist of a set of
smooth curves or a series of straight lines with one curve for each level of
damping.
Fig. 6.9.1 Response Spectra for the
el Centro earthquake in 1940, 1956,
and 1968, at * = 2%
Response Spectra
The design spectrum should be representative of ground motions at the
site during past earthquakes. If no records are available, it should be based
on other sites under similar conditions. The factors that we try to match
include:
1. magnitude
2. distance of site from earthquake source
3. fault mechanism
4. geology of the travel path of seismic waves
5. local soil condition at the site
The design spectrum is based on statistical analysis of the response spectra
for a set of ground motions. For each value of the period Tn the probability
distribution, its mean value and its standard deviation are determined.
Connecting all the mean values gives the mean response spectrum.
Similarly, connecting all the mean-plus-one-standard-deviation gives the
mean-plus-one-standard-deviation response spectrum. (See Fig. 6.9.2)
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Response Spectra
Observe that the two
response spectra are
much smoother than
the response spectrum
for an individual
ground motion.
This smoothed
response spectrum
lends itself to
idealization by a series
of straight lines
Response Spectra
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Response Spectra
Response Spectra
Example
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Response Spectra
Response Spectra
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Response Spectra
Examples of Design Spectra
Response Spectra
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Response Spectra
Response Spectra
Relative velocity response spectrum = a response spectrum for the peak
velocity of an SDOF system obtained from ,- (t)
Pseudo-velocity response spectrum = a response spectrum for the peak
displacement multiplied by �n
Pseudo-velocity =
Relative Displacement Response
Relative Velocity Response
Total acceleration Response
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Response Spectra
Difference between
Relative Velocity and
Pseudo-Velocity
Response Spectra
Total acceleration Response
Pseudo-acceleration Response
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Response Spectra
