Geotechnical Earthquake

Engineering

by

Dr. Deepankar Choudhury

Professor

Department of Civil Engineering

IIT Bombay, Powai, Mumbai 400 076, India.

Email: dc@civil.iitb.ac.in

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)

IIT Bombay, DC 15

Bracketed duration is the measure of time between the first and

last exceedence of a threshold acceleration 0.05 g.

Duration

Ref: Kramer (1996)

IIT Bombay, DC 16

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)