Module – 1
Introduction to 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 - 4
D. Choudhury, IIT Bombay, India
Fig. Location, year and number of fatalities (in parenthesis)
for earthquakes in India during 1800 – 2001
(Modified
after
Bilham and
Gaur, 2000)
D. Choudhury, IIT Bombay, India
Earthquake Distribution during 1800 to 2007, reference USGS
D. Choudhury, IIT Bombay, India
Seismic Zonation Map of India as per IS 1893: 2002, Part 1
Zone PGA
II 0.10g
III 0.20g
IV 0.25g
V 0.40g
D. Choudhury, IIT Bombay, India
Module – 2
Basics of Vibration
Theory
20
D. Choudhury, IIT Bombay, India
Reference:
NPTEL Video Course on
Soil Dynamics
Module – 2
by Prof. Deepankar Choudhury,
IIT Bombay, Powai, Mumbai, India.
21
Dynamic loads :
1. Earthquake load,
2. Wind load,
3. Moving load,
4. Guide way unevenness,
5. Machine induced load,
6. Blast load,
7. Impact load etc.
Vibration
D. Choudhury, IIT Bombay, India
Degrees of Freedom (DOF) o No of independent co-ordinates (displacements) required to define the
displaced position of all the masses relative to their all the position is
defined as degrees of freedom.
o Generally in Dynamics, mass property dictates the DOF whereas in
Statics , the stiffness property dictates the DOF
Examples
ln ( ) 4.141 0.868 1.09ln[ 0.0606exp(0.7 )]PHA g M R M
D. Choudhury, IIT Bombay, India
26
Simple Vibrating System (SDOF system)
Mass-Spring-Damper (MSD) System
m Kinetic Energy
k Potential Energy
c Dissipation
D’Allembart’s principle
For any object in motion, the externally applied forces, inertial force and
forces of resistance form a system of forces in equilibrium.
ln ( ) 4.141 0.868 1.09ln[ 0.0606exp(0.7 )]PHA g M R M
D. Choudhury, IIT Bombay, India
27
Linear Model for Equation of Motion
Governing Equation of Motion
Units MLT
system
FLT system SI unit
m M F/LT-2 kg
k MT-2 F/L N/m
c MT-1 F/LT-1 N-s/m
2
2. . . ( )d u du
m c k u p tdt dt
( )mu cu ku p t
D. Choudhury, IIT Bombay, India
28
Type of vibrations
Vibration
Free Vibration
[p(t) = 0)]
Forced Vibration
[p(t) = 0)]
Undampe
d (c = 0)
Damped
(c = 0)
Undampe
d (c = 0)
Damped
(c = 0)
Periodic Aperiodic
Transient (t tf) Steady state (t )
D. Choudhury, IIT Bombay, India
29
SDOF system
Free Vibration
1. Undamped Free Vibration
ln ( ) 4.141 0.868 1.09ln[ 0.0606exp(0.7 )]PHA g M R M
The structure is disturbed from its
static equilibrium and then vibrates
without any applied forces.
The equation of motion is:
The solution is: n nu(t) A cos( t) Bsin( t)
n k m (rad/s) natural circular frequency
A and B are determined by the initial conditions
D. Choudhury, IIT Bombay, India
30
ln ( ) 4.141 0.868 1.09ln[ 0.0606exp(0.7 )]PHA g M R M
t 0 o o
t 0 o o n
u u u A
u u u B
which can be written as nu(t) Csin( t )
2 2 o n oo o n
u uC u (u ) cos sin
C C
natural period n
n
2T (s)
πnatural frequency n
n
n
1f (Hz)
T 2π
D. Choudhury, IIT Bombay, India
31
Equation of motion: Earthquake excitation
ln ( ) 4.141 0.868 1.09ln[ 0.0606exp(0.7 )]PHA g M R M
S
D
t
I
f ku
f cu
f mu
0tmu cu ku
D. Choudhury, IIT Bombay, India
33 Prof. Deepankar Choudhury, Department of Civil Engineering, IIT Bombay, Mumbai, India
Forced Vibration: Response to Step Excitation
Now,
0 0
f(t) ( )
= 1, t>t
= 0, t<t
= 1/2, t=t
of motion
( )
(0) ,
(0)
a
a
a
a
u t t
Equation
mx cx kx Fu t
Initial conditions x x x x
34 Prof. Deepankar Choudhury, Department of Civil Engineering, IIT Bombay, Mumbai, India
Response to Step Excitation
(0) (0) 0x x
2 0
0
2
0
2
2
( )
= ( cos sin )
Using the initial conditions,
( ) 1 cos sin1
n
n
n n
t
D D
n
t
D D
Fx x x
m
x t CF PI
Fe A t B t
m
Fx t e t t
k
1 Prof. Deepankar Choudhury, Department of Civil Engineering, IIT Bombay, Mumbai, India
Forced Vibration due to Arbitrary excitation (Duhamel’s Integral)
0
0
0 0
0 00
( ) ( ) . ( - ) ( - ). ( )
, ( ) ( ). ( )
( )
= ( cos sin ) ( ). ( )
conditions, (0) , (0)
( )= ( cos
i
s n
n
n
t
t
t
D D
t nD D
d
dx t f d h t h t f d
So x t h t f d
x t CF PI
e A t B t h t f d
Initial x x x x
x xx t e x t
0
0
) ( ). ( )
1, ( ) .sin
, (0) 0, (0) 0
( ) ( ). ( ) Duhamel's Integral
n
t
t
D
d
t
t h t f d
where h t e tm
If x x
x t h t f d
D. Choudhury, IIT Bombay, India
Example Problem
1
D. Choudhury, IIT Bombay, India
For the system shown in Figure, mention (with reasoning) the
number of degrees of freedom for the system for a small
oscillation. Derive the governing equation of motion (from first
principle). Consider, mass of the linkage AB and other
connectors are negligible. Calculate the natural frequency and
natural period of vibration for the system if k1 = k2 = 90 N/m
and m1 = m2 = 10 kg. And c1 = c2 = 6 N-s/m. Estimate the
damped frequency, damped period, damping ratio of the
system.
2
IIT Bombay, DC 22
Module – 3
Engineering Seismology
IIT Bombay, DC 23
Seismology is the branch of Geophysics concerned
with the study and analysis of Earthquakes and the
science of energy propagation through the Earth's
crust.
Engineering Seismology is concerned with the
solution of engineering problems connected with the
Earthquakes. Seismology is extremely important
because:
•Study of earthquakes gives us important clues
about the earth’s interior
•Understanding earthquakes allows us to minimize
the damage and loss of life
What is Earthquake ?
• An earthquake is the result of a sudden release of energy in the Earth's
crust that creates seismic waves. Earthquake is also known as a quake,
tremor or temblor.
IIT Bombay, DC 25
What is an Earthquake?
An earthquake is the vibration of Earth
produced by the rapid release of accumulated
energy in elastically strained rocks
Energy released radiates in all directions from its
source, the focus
Energy propagates in the form of seismic waves
Sensitive instruments around the world record
the event
Focus and Epicenter of Earthquake
IIT Bombay, DC 27
What causes an Earthquake?
Movement of Tectonic Plates
Earth is divided into sections called Tectonic
plates that float on the fluid-like interior of the
Earth. Earthquakes are usually caused by
sudden movement of earth plates
Rupture of rocks along a fault
Faults are localized areas of weakness in the
surface of the Earth,sometimes the plate
boundary itself
Why an earthquake occurs?
• The earth's crust (the outer layer of the planet) is made up of several
pieces, called plates.
• The plates under the oceans are called oceanic plates and the rest
are continental plates
Earthquakes usually
occur where two plates
are running into each
other or sliding past
each other.
An image of the world's plates and their boundaries.
IIT Bombay, DC 29
Where do Earthquakes occur?
Plate boundaries
Faults
IIT Bombay, DC 30
Release of Accumulated energy
IIT Bombay, DC 31
The point within Earth
where faulting begins is
the focus, or hypocenter
The point directly above
the focus on the surface
is the epicenter
The Focus and Epicenter of an Earthquake