This document provides an overview of soil dynamics and vibratory motion. It discusses periodic and non-periodic motion, describes simple harmonic motion using trigonometric and complex notation, and defines displacement, velocity and acceleration for vibratory systems. Fourier series are also summarized, including their trigonometric, exponential and discrete transform forms. Examples are provided to illustrate Fourier analysis and the power spectrum.
In this presentation, following topics are covered:
1- Introduction to soil liquifaction.
2- Causes and effects of soil liquifaction
3- Methods to remove soil liquifaction.
4- Mechanism of soil liquifaction.
5- Conclusion.
In this presentation, following topics are covered:
1- Introduction to soil liquifaction.
2- Causes and effects of soil liquifaction
3- Methods to remove soil liquifaction.
4- Mechanism of soil liquifaction.
5- Conclusion.
Class notes of Geotechnical Engineering course I used to teach at UET Lahore. Feel free to download the slide show.
Anyone looking to modify these files and use them for their own teaching purposes can contact me directly to get hold of editable version.
Lecture 11 Shear Strength of Soil CE240Wajahat Ullah
Shear Strength of Soil
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Introduction
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Mohr-Coulomb criterion
Introduction
Lab tests for getting the shear strength
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Shear Strength of soil and behaviour of soil under shear actionsatish dulla
it contains details of property and theory of soil under shear action.Even the experiments to test the soil strength has given with illstrations
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The process in which the response of the soil influences the motion of the structure and the motion of the structure influences the response of the soil is termed as soil-structure interaction (SSI)
Class notes of Geotechnical Engineering course I used to teach at UET Lahore. Feel free to download the slide show.
Anyone looking to modify these files and use them for their own teaching purposes can contact me directly to get hold of editable version.
Lecture 11 Shear Strength of Soil CE240Wajahat Ullah
Shear Strength of Soil
Shear strength in soils
Introduction
Definitions
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Introduction
Lab tests for getting the shear strength
Direct shear test
Introduction
Procedure & calculation
Critical void ratio
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it contains details of property and theory of soil under shear action.Even the experiments to test the soil strength has given with illstrations
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http://movie-rulz.xyz/category/hollywood-movies/2016-english-movies/
http://movie-rulz.xyz/
http://movie-rulz.xyz/category/telugu-movies/2016-telugu-movies/
The process in which the response of the soil influences the motion of the structure and the motion of the structure influences the response of the soil is termed as soil-structure interaction (SSI)
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Soil Dynamics
1. Soil Dynamics
1. Vibratory Motion
Cristian Soriano Camelo1
1Federal University of Rio de Janeiro
Geotechnical Engineering
June 30th, 2017
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 1 / 33
2. Outline
1 Introduction
Vibratory Motion
2 Complex Notation for Simple Harmonic Motion
Vibratory motion
3 Other measures of motion
Displacement, Velocity, Acceleration
4 Fourier Series
Trigonometric, Exponential, Discrete transform, Power spectrum
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 2 / 33
3. Outline
1 Introduction
Vibratory Motion
2 Complex Notation for Simple Harmonic Motion
Vibratory motion
3 Other measures of motion
Displacement, Velocity, Acceleration
4 Fourier Series
Trigonometric, Exponential, Discrete transform, Power spectrum
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 3 / 33
4. Introduction
Different types of
dynamic loading
Vibratory motion of
soils and structures
Problems of dynamic response
To solve these problems it is necessary to understand and describe the
dynamic events.
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 4 / 33
5. Types of vibratory motion
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 5 / 33
6. Periodic motion
u(t) is periodic if there exists some period Tf , then u(t + Tf ) = u(t)
For example, simple harmonic motion.
Simple harmonic motion can be characterized by three quantities:
amplitude, frequency and phase.
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 6 / 33
7. Nonperiodic motion
Examples of non periodic motion are impulsive loads (explosions,
falling weights) or longer-duration transient loadings (earthquakes or
traffic).
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 7 / 33
8. Simple harmonic motion - Trigonometric notation
Simple harmonic motion can be expressed in terms of a displacement
as:
u(t) = Asin(ωt + φ)
Where, A= amplitude, ω=circular frequency (angular speed), φ=phase
angle.
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 8 / 33
9. Simple harmonic motion - Trigonometric notation
The phase angle describes the amount of time by which the peaks (and
zero points) are shifted from those of a pure sine function.
Another common measure is the number of cycles per unit of time,
known as frequency, expressed as the reciprocal of the period.
f = 1
T = ω
2π
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 9 / 33
10. Simple harmonic motion - Trigonometric notation
Simple harmonic motion can also be described as a sum of sine and
cosine function, u(t) = acos(ωt) + bsin(ωt)
Using the rotating vector representation, the sum of the sine and cosine
function can be expressed as:
Therefore, u(t) = Asin(ωt + φ)
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 10 / 33
11. Outline
1 Introduction
Vibratory Motion
2 Complex Notation for Simple Harmonic Motion
Vibratory motion
3 Other measures of motion
Displacement, Velocity, Acceleration
4 Fourier Series
Trigonometric, Exponential, Discrete transform, Power spectrum
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 11 / 33
12. Simple harmonic motion - Complex notation
For many dynamic analyses trigonometric notation leads to long
equations. Complex notation can be derived from trigonometric
notation using Euler’s law.
u(t) = a−ib
2 eiωt + a+ib
2 e
−iωt
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 12 / 33
13. Outline
1 Introduction
Vibratory Motion
2 Complex Notation for Simple Harmonic Motion
Vibratory motion
3 Other measures of motion
Displacement, Velocity, Acceleration
4 Fourier Series
Trigonometric, Exponential, Discrete transform, Power spectrum
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 13 / 33
14. Other measures of motion
If the displacement with time is known, other parameters of interest
can be determined:
Trigonometric Complex -
u(t) = Asin(ωt) u(t) = Ae
iωt
Displacement
u′(t) = ωAcos(ωt) = ωAsin(ωt + π/2) u′(t) = iωAe
iωt
Velocity
u′′(t) = −ω2Asin(ωt) = ω2Asin(ωt + π) i2ω2Ae
iωt
Acceleration
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 14 / 33
15. Outline
1 Introduction
Vibratory Motion
2 Complex Notation for Simple Harmonic Motion
Vibratory motion
3 Other measures of motion
Displacement, Velocity, Acceleration
4 Fourier Series
Trigonometric, Exponential, Discrete transform, Power spectrum
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 15 / 33
16. Fourier series
In geotechnical earthquake engineering, it is possible to break down
complicated loading functions (earthquakes) into the sum of simple
harmonic functions.
Trigonometric Form: Is expressed as a sum of simple harmonic
functions.
x(t) = a0 + ∞
n=1(ancos(ωnt) + bnsin(ωnt))
Where,
a0 = 1
Tf
Tf
0 x(t)dt,
an = 2
Tf
Tf
0 x(t)cos(ωnt)dt,
bn = 2
Tf
Tf
0 x(t)sin(ωnt)dt.
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 16 / 33
17. Fourier series
Example 1 (Mathematica Software): Given the following piecewise
function, obtain the Fourier expansion for 5 and 10 coefficients and
compare the results with the current function.
x[t ]:=Piecewise[{{Sin[t], 0 < t < Pi}, {0, −Pi < t < 0}}]
-3 -2 -1 1 2 3
Time
0.2
0.4
0.6
0.8
1.0
x
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 17 / 33
21. Fourier series
Exponential form: using Eulers law is possible to obtain the Fourier
series in exponential form.
xt = a0 + (an+ibn
2 eiwnt + an+ibn
2 e−iwnt)
The function in Example 1 expressed in complex notation and using 5
coefficients is:
x(t)=1
4 ie−it − 1
4ieit + 1
π − e−2it
3π − e2it
3π − e−4it
15π − e4it
15π
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 21 / 33
22. Fourier series
Discrete Fourier Transform (DFT): in many applications (e.g
earthquake engineering), loading motion parameters are be described
by a finite number of data points.
DFT is an algorithm that takes a signal and determines the frequency
content of the signal. To determine that frequency content, DFT
decomposes the signal into simpler parts.
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 22 / 33
23. Fourier series
The DFT is given by:
X(ω) = N−1
n=0 x(n)e
−iπωn/N
Where x(n) is the n-value of the different points of the signal and ω
varies from 0 to N-1.
Breaking the previous equation into sines and cosines, it is possible to
identify two correlation calculations composed by a real and an
imaginary components of X(ω) :
X(ω) = N−1
n=0 x(n)cos 2πωn
N − i N−1
n=0 x(n)sin 2πωn
N
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 23 / 33
24. Fourier series
Example 2 - DFT: Given the following signal samples, determine the
DFT.
20 40 60 80 100
w
-2
-1
1
2
Amplitude
Signal x(n)
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 24 / 33
27. Fourier series
The final DFT presents four peaks that represent the real and
imaginary components of X(ω). For example, at ω=10 there is a peak
in the real component (signal similar to a cosine), and at ω=3 there is
a peak in the imaginary part (signal similar to a sine).
20 40 60 80 100
w
1
2
3
4
5
Magnitude
x(w)
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 27 / 33
28. Fourier series
Correlation: Correlation is a measure of how two signals are. In
general correlation can be expressed as:
N
n=0 x(n)y(n)
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 28 / 33
29. Fourier series
Example 3 - DFT: The following is a list of 200 elements containing
a periodic signal with random noise added.
data=Table N RandomReal[] − 1
2 + sin 30 2πn
200 , {n, 200} ;
50 100 150 200
w
-1.5
-1.0
-0.5
0.5
1.0
1.5
Amplitude
Signal x(t)
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 29 / 33
30. Fourier series
The DFT in this case shows a peak at 30+1 and a symmetric peak at
201-30.
50 100 150 200
w
1
2
Q
4
5
6
U
Magnitude
x(w)
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 30 / 33
31. Fourier series
Power Spectrum: The power spectrum answers the question ”How
much of the signal is at a frequency ω”?. In the frequency domain, this
is the square of the DFT’s magnitude:
P(ω) = Re(X(ω))2 + Im(X(ω))2
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 31 / 33
32. Fourier series
Power spectra are often used to describe earthquake-induced ground
motions:
The figure indicates that most of the energy in the accelerogram is in
the range of 0.1 to 20 Hz and the largest amplotude at approximately 6
Hz.
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 32 / 33
33. For Further Reading I
Steven L. Kramer.
Geotechnical Earthquake Engineering.
Prentice Hall, 1996.
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 33 / 33