Vehicle and Traffic
Consideration
Highways
Airports
03/29/17
CEE 320
1
Steve Muench
Load Quantification For
Highways
Equivalent Single Axle Load (ESAL)
Converts wheel loads of various magnitudes and
Atoms, Elements and Minerals
A. Changing scales to looking at the
elements of the earth and its crust (8 most common)
B. Introduction to minerals that comprise rocks
(11 most common)
C. The silicate
Engineering Geology
Rock
structures are subjected to different
stresses (uniform or directional), arising from
tectonic forces acting on them.
The type and nature of stresses results in
different ge
MODULE 2: FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS
Lecture 5
28
Compatible Systems and Charpits Method
In this lecture, we shall study compatible systems of rst-order PDEs and the Charpits
method fo
Rocks
A rock is a naturally occurring, solid aggregate
of minerals.
Fig 4.1
Three Classes of Rocks
Igneous (made by fire) - Solidified from
molten rock (i.e., magma).
Sedimentary - Deposited and burie
What is geomorphology?
geo = earth
morph = form
-ology = study of
1
Study of landforms and landscapes
Study of surface processes responsible
for landforms / landscapes
Relationship between landform sc
Elementary properties of Complex numbers
Lecture 1
Complex Analysis
Introduction
Let us consider the quadratic equation x 2 + 1 = 0.
It has no real root.
Let i(iota) be the solution of the above equat
Chapter 1
MINERALS and ROCKS
Minerals: Building blocks of rocks
Definition of a mineral:
Naturally occurring
Inorganic solid
Ordered internal molecular structure
Definite chemical composition
De
ASSESSMENT OF
WEATHERING
Dr. C Mahanta
Department of Civil Engineering
1) SLAKE DURABILITY TEST:
Rock material, in the form of pieces ( size of around
5 cm or more) chosen
0.5 kg of the oven dried sam
Introduction to Earthquakes
C. Mahanta
Department of Civil Engineering
IIT Guwahati
What are Earthquakes?
The shaking or trembling caused by the sudden
release of energy
Usually associated with faul
Hydrogeology
STUDY OF OCCURANCE,DISTRIBUTION, MOVEMENT OF GROUND
WATER
USES
Possible directions of groundwater flow can
be drawn on absolute elevation water table
contour maps.
Areas can be differen
Analytic functions
Lecture 5
Analytic functions
Analytic functions
Definition: A function f is called analytic at a point z0 C if there exist
r > 0 such that f is differentiable at every point z B(z0
Elementary functions
Lecture 6
Elementary functions
The Exponential Function
Recall:
Eulers Formula: For y R, e iy = cos y + i sin y
and for any x, y R, e x+y = e x e y .
Definition: If z = x + iy , t
Zeros of analytic functions
Lecture 14
Zeros of analytic functions
Zeros of analytic functions
Suppose that f : D C is analytic on an open set D C.
A point z0 D is called zero of f if f (z0 ) = 0.
The
Complex Integration
Lecture 7
Complex Integration
Complex Integration
Integral of a complex valued function of real variable:
Definition: Let f : [a, b] C be a function. Then f (t) = u(t) + iv (t)
whe
Open Set and Closed set
Lecture 2
Open Set & Closed set
Some Basic Definitions
Open disc: Let z0 C and r > 0 then, B(z0 , r ) = cfw_z C : |z z0 | < r
is an open disc centered at z0 with radius r .
De
DIFFICULTIES FACED DURING
BRIDGE CONSTRUCTION IN
BRAHAMAPUTRA RIVER
Group Project Members:
Gagandeep Singh Randhawa 120104023
Goparaju VNS Manohar Uttej 120104024
Gupta Himanshu 120104025
Himanshu Gup
Maximum Modulus Theorem and Laurent Series
Lecture 15
Maximum Modulus Theorem and Laurent Series
Maximum Modulus Theorem
Maximum Modulus Theorem: Let D C be a domain and f : D C is
analytic. If there
Singularities
Lecture 16
Singularities
Singularities
Behavior of following functions f at 0:
1
f (z) = 9
z
sin z
f (z) =
z
ez 1
f (z) =
z
1
f (z) =
sin( z1 )
f (z) = Log z
1
f (z) = e z
In the above w
Rocks
Important types of rocks.
The diagram in the next slide represents the ROCK
CYCLEa scheme that represents the processes of
continuous changes that connect the three major groups
of rocks:
SEDIME
Engineering properties and
classification of Rock Masses
Rock masses
Rock
masses different minerals of
varying strength and susceptible to
weathering.
deformation structural features
develop in Rocks
Evaluation of integrals
Lecture 18
Evaluation of integrals
Evaluation of certain contour integrals: Type I
Type I: Integrals of the form
Z
2
F (cos , sin ) d
0
dz
.
iz
Substituting for sin , cos and d
Differentiability
Lecture 4
Differentiability
Differentiability
Recall: Let A be a nonempty open subset of R. x0 A. Then we say f is
differentiable at x0 if the limit
lim
h0
f (x0 + h) f (x0 )
h
exist
Cauchys Theorem
Lecture 1
Cauchys Theorem
Complex integration
Question: Under what conditions on f we can guarantee the existence of F
such that F 0 = f ?
Definition:(Simply connected domain) A domain
Residues theorem and its Applications
Lecture 17
Residues theorem and its Applications
Residues
Question: Let is a simple closed contour in a simply connected domain D
and let z0 doesnt
lie on . If f
Complex Integration
Lecture 8
Complex Integration
Recall
Definition: Let (t); t [a, b], be a contour and f be complex valued
continuous function defined on a set containing then the line integral or t
Power Series
Lecture 12
Power Series
Series of complex numbers
If zn C for all n 0 then the series
X
zn converges
n=0
to
> 0 there exists an positive integer N such that
mz if for every
X
zn z < for
MA 201
COMPLEX ANALYSIS
ASSIGNMENT5
Z
(1) Evaluate
|z 2 |= 2
z
dz.
cos z
Z
(z 2 + 3z + 2)
dz.
(z 3 z 2 )
(2) Using the Cauchys residue theorem, evaluate
Z |z|=2
1
(3) Using the argument principle, eva
Taylors Theorem
Lecture 13
Taylors Theorem
Taylor Series
Question: Let f : B(z0 , R) C analytic. Can we represent f as a power series
around z0 ?
(Taylors Theorem:) Let f be analytic on D = B(z0 , R).
Applications of Cauchys Integral Formula
Lecture 11
Applications of Cauchys Integral Formula
Cauchys estimate
Cauchys estimate: Suppose that f is analytic on a simply connected domain
D and B(z0 , R)