Module 1 Introduction to 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 - 4
Fig. Location, year and number of fatalities (in parenthesis) for earthquakes in India during 1800 2001 (Modified after Bilham and Gaur, 2000)
Earthquake Distribution during 1800 to 2007, reference USGS
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
Module 2 Basics of Vibration Theory 20
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
ln PHA( g) 4.141 0.868M 1.09ln[ R 0.0606exp(0.7 M)] 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 PHA( g) 4.141 0.868M 1.09ln[ R 0.0606exp(0.7 M)] 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. 26
Linear Model for Equation of Motion Governing Equation of Motion 2 d u du m. c. k. u p( t) 2 dt dt mu cu ku p() t 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 27
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 28 Aperiodic Transient (t t f ) Steady state (t )
ln PHA( g) 4.141 0.868M 1.09ln[ R 0.0606exp(0.7 M)] SDOF system Free Vibration 1. Undamped Free Vibration The structure is disturbed from its static equilibrium and then vibrates without any applied forces. The equation of motion is: The solution is: u(t) A cos( nt) Bsin( nt) n km (rad/s) natural circular frequency A and B are determined by the initial conditions 29
ln PHA( g) 4.141 0.868M 1.09ln[ R 0.0606exp(0.7 M)] u u u A t 0 o o u u u B t 0 o o n which can be written as u(t) C sin( nt ) u C u (u ) cos sin C 2 2 o n o o o n u C natural period 2π T (s) n n natural frequency 1 n f n (Hz) Tn 2π 30
ln PHA( g) 4.141 0.868M 1.09ln[ R 0.0606exp(0.7 M)] Equation of motion: Earthquake excitation f f S D ku cu t mu cu ku 0 f I mu t 31
Forced Vibration: Response to Step Excitation Now, f(t) u( t t ) = 1, t>t = 0, t<t = 1/2, t=t Equation of motion mx cx kx Fu() t a a a a Initial conditions x(0) x, x(0) x 0 0 Prof. Deepankar Choudhury, Department of Civil Engineering, IIT Bombay, Mumbai, India 33
Response to Step Excitation x(0) x(0) 0 x 2 x x x() t CF PI n F 2 0 n m nt = e ( Acos t Bsin t) Using the initial conditions, D D m F 0 2 n F0 nt x( t) 1 e cos Dt sin 2 Dt k 1 Prof. Deepankar Choudhury, Department of Civil Engineering, IIT Bombay, Mumbai, India 34
Forced Vibration due to Arbitrary excitation (Duhamel s Integral) dx( t) f ( ) d. h( t - ) h( t - ). f ( ) d So, x( t) h( t ). f ( ) d x() t CF PI t 0 nt = e ( Acos t Bsin t) h( t ). f ( ) d Initial conditions, x(0) x, x(0) x x t e x t 0 0 0 nt 0 n 0 ( )= ( 0 cos D i d 0 D x D x 1 nt where, h( t) e.sin Dt m If, x(0) 0, x(0) 0 t d x( t) h( t ). f ( ) d Duhamel's Integral t sn t) h( t ). f ( ) d D t 0 Prof. Deepankar Choudhury, Department of Civil Engineering, IIT Bombay, Mumbai, India 1
Example Problem 1
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 k 1 = k 2 = 90 N/m and m 1 = m 2 = 10 kg. And c 1 = c 2 = 6 N-s/m. Estimate the damped frequency, damped period, damping ratio of the system. 2
Module 3 Engineering Seismology IIT Bombay, DC 22
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 IIT Bombay, DC 23
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.
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 IIT Bombay, DC 25
Focus and Epicenter of Earthquake
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 IIT Bombay, DC 27
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.
Where do Earthquakes occur? Faults Plate boundaries IIT Bombay, DC 29
Release of Accumulated energy IIT Bombay, DC 30
The Focus and Epicenter of an Earthquake The point within Earth where faulting begins is the focus, or hypocenter The point directly above the focus on the surface is the epicenter IIT Bombay, DC 31