geotechnical earthquake engineering - nptel
TRANSCRIPT
Geotechnical Earthquake
Engineering
by
Dr. Deepankar Choudhury
Professor
Department of Civil Engineering
IIT Bombay, Powai, Mumbai 400 076, India.
Email: [email protected]
URL: http://www.civil.iitb.ac.in/~dc/
Lecture – 14
IIT Bombay, DC 2
Module – 4
Strong Ground Motion
IIT Bombay, DC 3
Amplitude Parameters
From the time histories of acceleration, velocity and displacement
are obtained by integrating the acceleration records. All other
amplitude parameters are calculated from these time histories.
Ref: Kramer (1996)
IIT Bombay, DC 4
Peak Acceleration
Most commonly used measure of amplitude of a ground motion is the
Peak horizontal acceleration (PHA). It is the absolute maximum value
obtained from accelerogram.
Maximum resultant PHA is the vector sum of two orthogonal
components. Estimation of PHA is most important for any design. PHA
and MMI relationship (Trifunac and Brady, 1975) are often used.
PVA is not that important and PVA = (2/3)PHA is commonly assumed
for design (Newmark and Hall, 1982).
Peak acceleration data with frequency content/duration of earthquake is
important. Because for e.g. 0.5g PHA may not cause significant damage
to structures if earthquake duration is very small.
Ref: Kramer (1996)
IIT Bombay, DC 5
Peak Acceleration
Proposed relationships between PHA & MMI (Trifunac & Brady, 1975).
IIT Bombay, DC 6
Peak Velocity and Displacement
Peak horizontal velocity (PHV) is also used to characterize ground
motion. PHV is better than PHA for intermediate frequencies as velocity
is less sensitive to higher frequency.
For above reason, many times PHV may provide better indication for
damage than PHA. PHV and MMI relationship (Trifunac and Brady,
1975) are also used.
Peak displacements are associated with low frequency components of
earthquake motion. Hence signaling and filtering error of data is common
and hence not recommended for practical uses over PHA or PHV.
Ref: Kramer (1996)
IIT Bombay, DC 7
Other Amplitude Parameters Sustained Maximum Acceleration: The absolute values of
highest accelerations that sustained for 3 and 5 cycles in
acceleration time history are defined as 3-cycle sustained and 5-
cycle sustained accelerations respectively.
Effective Design Acceleration: The acceleration which is
effective in causing structural damage. This depends on size of
loaded area, weight, damping and stiffness properties of structure
and its location with respect to epicenter.
Kennedy (1980) proposed EDA as 25% higher than 3-cycle PHA
recorded in filtered time history.
Benjamin and Associates (1988) proposed EDA as the PHA after
filtering out accelerations above 8-9 Hz.
Ref: Kramer (1996)
IIT Bombay, DC 8
Frequency Content Parameters
The frequency content of an earthquake history is often
described using Fourier Spectra, Power spectra and
response spectra.
Ground Motion Spectra - Fourier Spectra
A periodic function (for which an earthquake history is an
approximation) can be written as
where cn and n are the amplitude and phase angle
respectively of the nth harmonic in the Fourier series.
)sin()(1
0 nnn
n tcctx
Ref: Kramer (1996)
IIT Bombay, DC 9
Frequency Content Parameters
The Fourier amplitude spectrum is a plot of cn versus n
Shows how the amplitude of the motion varies with
frequency.
Expresses the frequency content of a motion
The Fourier phase spectrum is a plot of n versus n
Phase angles control the times at which the peaks of
harmonic motion occur.
Fourier phase spectrum is influenced by the variation of
ground motion with time.
Ref: Kramer (1996)
IIT Bombay, DC 10
Fourier Amplitude Spectrum
fc fmax
Frequency (log)
Fou
rier
Am
pli
tud
e (l
og)
The Fourier amplitude spectra of actual
earthquakes are often plotted on
logarithmic scales, so that their
characteristic shapes can be clearly
distinguished from the smoothed curves.
Two frequencies that mark the range of
frequencies for largest Fourier
acceleration amplitude are corner
frequency (fc) and cutoff frequency (fmax)
fc is a very important parameter because it is inversely proportional to
the cube root of seismic moment, thus indicating that large earthquakes
produce greater low-frequency motions.
Ref: Kramer (1996)
IIT Bombay, DC 11
Frequency Content Parameters
Power Spectra
The power spectrum is a plot of G( ) versus n . The power
spectrum density (PSD) function is defined by the following
equation and is closely related to the Fourier amplitude
spectrum:
where G( ) is the PSD, Td is the duration of the ground
motion, and cn is the amplitude of the nth harmonic in the
Fourier series. PSD function is used to characterize an
earthquake history as a random process.
2
dT π
1)(
ncG
Ref: Kramer (1996)
A response spectrum is used to provide the most descriptive representation of the influence of a given earthquake on a structure or machine.
Response spectra are widely used in earthquake engineering.
A response spectrum is a plot of the maximum response amplitude (displacement, velocity or acceleration) versus time period of a system to a given component of ground motion.
Using the response spectrum, peak response of buildings to earthquakes can be assessed and their natural frequency can be determined.
Response Spectra
12
Frequency Content Parameters
Ref: Kramer (1996)
IIT Bombay, DC 13
Response Spectra
Response spectra are widely
used in earthquake
engineering. The response
spectrum describes the
maximum response of a
SDOF oscillator to a
particular input motion as a
function of frequency and
damping ratio. The response
spectra from two sites (one
rock and the other soil) are
shown in figure.
Frequency Content Parameters
Ref: Kramer (1996)
IIT Bombay, DC 14
Duration of an earthquake is very important parameter that influences the
amount of damage due to earthquake. A strong motion of very high
amplitude of short duration may not cause as much damage to a structure
as a motion with moderate amplitude with long duration can cause. This
is because the ground motion with long duration causes more load
reversals, which is important in the degradation of stiffness of the
structures and in building up pore pressures in loose saturated soils.
Duration represents the time required for the release of accumulated
strain energy along a fault, thus increases with increase in magnitude of
earthquake.
Relative duration does not depend on the peak values. It is the time
interval between the points at which 5% and 95% of the total energy has
been recorded.
Duration
Ref: Kramer (1996)
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Bracketed duration is the measure of time between the first and
last exceedence of a threshold acceleration 0.05 g.
Duration
Ref: Kramer (1996)
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Other Spectral Parameters
RMS acceleration : This is the parameter that includes the
effects of amplitude and frequency, defined as
dT
0
2
d
dttaT
1rms
a
Where a(t) is the acceleration over the time domain and Td is
the duration of strong motion
AI - The Arias Intensity is a measure of the total energy at the
recording station and is proportional to the sum of the squared
acceleration. It is defined as
dttag
AI2
02
Ref: Kramer (1996)
IIT Bombay, DC 17
SI - The Spectrum Intensity is defined as the integral of the pseudo-
Spectral velocity curve (also known as the velocity response spectrum),
integrated between periods of 0.1 - 2.5 seconds. These quantities are
motivated by the need to examine the response of structures to ground
motion, as many structures have fundamental periods between 0.1 and
2.5 sec. The SI can be calculated for any structural damping ratio.
Dominant frequency of ground motion (Fd) is defined as the
frequency corresponding to the peak value in the amplitude spectrum.
Thus, Fd indicates the frequency for which the ground motion has the
most energy. The amplitude spectrum has to be smoothed before
determining Fd.
Other Spectral Parameters
Ref: Kramer (1996)
IIT Bombay, DC 18
Predominant Period (Tp): Period of
vibration corresponding to the maximum
value of the Fourier amplitude spectrum.
This parameter represents the frequency
content of the motion. The predominant
period for two different ground motions with
different frequency contents can be same,
making the estimation of frequency content
crude.
Bandwidth (BW) - of the dominant
frequency; measured where the amplitude
falls to 0.707 (1 /sq. root 2) of the amplitude
of the dominant frequency. Again, this is
based on a smoothed amplitude spectrum.
Other Spectral Parameters
Tp
Period
Fou
rier
Am
pli
tud
e
GM1
GM2
Tp is same for the two
ground motions, though
the frequency content is
different
Ref: Kramer (1996)
IIT Bombay, DC 19
Central Frequency: Power spectral density function can be used to
estimate statistical properties of ground motion. The nth spectral
moment and central frequency ( ) is given by,
Central frequency is used to calculate theoretical median peak
acceleration as follows,
Shape Factor – It indicates the dispersion of the power spectral
density function about the central frequency,
Other Spectral Parameters
n
dGn
n
0
)(n
dGn
n
0
)(
n
dGn
n
0
)(0
2
28.2ln2 0max
dTu
20
2
11It lies between 0 and 1, higher value
indicates larger bandwidth.
IIT Bombay, DC 20
Other Spectral Parameters
vmax/amax ratio: It is related to the frequency content of the motion.
For SHM with period T, vmax/amax = T/2 .
Seed and Idriss (1982) proposed average values of vmax/amax for
different sites within 50 km of source.
Rock – 0.056 sec., Stiff soils (<200ft) – 0.112 sec., Deep stiff soil
(>200ft) – 0.138 sec.
Ref: Kramer (1996)