Introduction

• Most of geotechnical and foundation design is based on the soil’s behavior under staticloads.

• Important structures, however, require that highly competent engineers know how to analyze structures under complex dynamic loads.

• Examples of these dynamic loads: • Earthquakes.• The effects from bomb blasts.• Operation of very heavy or unbalanced

machinery, mining, construction (suchas pile driving, deep dynamic compaction, etc), heavy traffic, wind and wave actions.Ground motions result in increased settlements, and tilting of the foundations.

• The damage that ensues from the liquefaction of the soil during a seismic event. The search for oil in deeper and deeper oceans has meant designing offshore platforms that are subject to extreme wave and wind loads.

• In extreme cases earthquake loads are added to very high wave loadings

Why study soil dynamics?• The most common problems that engineers

encounter in the field of soil dynamics include:• Seismic induced ground movements and wave

propagation;• Foundations for heavy or vibrating machinery;• The changes of the bearing capacity of

foundations under dynamic loads;• The change in load capacity of deep foundations

under dynamic loads;• The changes of settlement due to dynamic loads;

• Increased lateral earth pressures due to dynamic loads;

• The potential for a soil to “liquefy” when subjected to dynamic loads;

• The potential for collapse of earth embankments under dynamic loads.

• What is the new failure criteria?• How should failure be defined?• What is an acceptable dynamic factor of safety?• How is it related to the static factor of safety?• How do soil parameters (φ, c, γ, etc) change

under dynamic loads?

Theory of Vibrations

Definitions :

Period : The time elapsed in repeating a periodic motion once. Cycle : Motion completed during a period is referred to as a cycle.

Frequency : The number of cycles of motion in a unit of time. (cf. Hz : cycle/sec)

• Natural frequency : The frequency with which an elastic system vibrates under the action of forces inherent in the system.

• Forced vibrations : Vibrations that occur under the excitation of external forces.

Forced vibrations occur at the frequency of the exciting force.

• Degrees of freedom : The number of independent

coordinates necessary to describe the motion of a system.

• Rosonance : If the frequency of excitation coincides with any one of the natural frequencies of the system, resonance is said to occur.

• Principal modes of vibration : In a principal mode, each point in the system vibrates with the same frequency. The vibration of a multi degree freedom system can always be represented by the superposition of principal modes.

Fundamental frequency : the lowest frequency among the principal (=the frequency of the first mode) modes of the vibrating system

• Normal mode of vibrations : when the amplitude of some point of the system vibrating in one of the principal modes is made equal to unity(i.e. normalized), the motion is called the normal mode of vibration.

Harmonic motion • The simplest form of periodic motion, occurring under the

influence of elastic restoring force in the absence of all friction, which is represented by sine or cosine functions

• (Phase plane representation of a harmonic motion) • The harmonic motion

• The displacement, sinxXw= … ① • w : circular (angular velocity) frequency in rads / unit time • One cycle is completed when

• 2wtπ= • t cy = T = 2wπ T : time period of motion • then, 12wfTπ== f

Free vibrations of a spring-mass system