Chapter 1

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1

Chapter 1. Introduction and Mathematical Preliminaries


1.

Scope


Behavior of structures deals with


(a) Micro
state of stresses, strains, and displacements at a point


(b) Macro
global behavior, collapse

mechanism, etc

Theory of elasticity is concerned with

(a) Equilibrium of forces (although equations may be expressed in terms of stress, a state




of equ
i
librium

must be established by forces)

(b) Kinematics and compatibility
examine strain
-
displacement relationship

(c) Constitutive equations
stress
-
strain relationship

(d) Boundary conditions
domain

(e) Uniqueness
applicability of solutions

2.

Vector

Algebra



where

are scalar components of the vector and

are unit vectors. Unit vectors
are mutually orthogonal only in the right
-
hand Cartesian coordinate system.

Unless
specifically
mentioned the coordinate system adopted, the coordinate system is assumed to be the right
-
hand
Cartesian coordinate system in all subsequent discussions herein.


length of a vector


scalar
quantity


termed to be dot product or scalar product


2


vector projection on to the x
-
axis

Particularly, if

is a unit vector,
, then the dot product of two unit vectors is equal

to the
direction cosine of the angle between these two unit vectors.



The
s
e are
procedure
s

frequently used in elementary mechanics introduced in statics and
mechanics of materials.


vector
; referred to as vector product or cross product


area of a parallelogram

3.

Scalar and Vector Fields


temperature, potential, etc.


vector fields, velocity, e
tc
.


vector




scalar


vector



3

$

Integral Theorem


Two integral theorems relating vector fields are particularly useful in structural
mechanics for
transforming between contour, area, and volume integral.

Green’s theorem
: Consider two functions

and

which are continuous

and have continuous first partial derivatives (

co
ntinuity) in a domain
D
, Green’s

theorem states that


where
A

is a closed region of
D

bounded by
C
. It should be noted that
A

should not have any
holes in it. This Green’s theorem is the basis of an old instrument, planimeter,
measuring the
area enclosed by a closed contour.


Divergence theorem
: Consider a continuously differentiable vector point function G in D.
The divergence theorem states that



where
v

is the volume bounded by the oriented surfa
ce
A

and n is the positive (outward) normal
to
A
. It should be noted that
v

should not have any void in it.

Extra handout!

4.

Indicial Notation


A mathematical agreement to simplify write
-
ups.

$

free index
: unrepeated indices are known as free indices.
Tensorial rank of a given term
is equal to the number of free indices appearing in that term.

$

dummy index
: when an index appears twice in a term, that index is understood to take
on all the values of its range, and the resulting terms summed unless other
wise noted. In

4

this so
-
called summation convention, repeated indices are referred to as dummy indices,
since their replacement by any other letter not appearing as a free index does not change
the meaning of the term in which they appear.

Example


The com
ponent of a first order tensor (vector) in three
-
space (range) may be shown as



The 2nd order tensor may be shown as



In general, for a range on N, an nth order tensor will have

components. Hence, for a range

of 3 on both i and j, the indicial equation



represents in expanded form, the three equations



For a range of 2 on i and j, the indicial equation



represents in expanded form, the 4 equations (
)


5



5.

Coordinate Transformation


Fig. 1.1 Cartesian coordinate systems


In the study of deformable body mechanics
, there will be many occasions to consider the


coordinate transformation from the original to the deformed configuration
s, or from the local to

the
global coordinate

system.

Consider the point
P

with coordinates

in the
unprime
d system and

in the primed system. The linear transformation
between the two coordinates of
P

is given by























6



or in indicial notation



where
j

is the repeated index and hence

a dummy index that i
nvokes a summation convention.
E
ach of the nine (
)

is the cosine of the angle between the
ith primed and the jth unprimed
axis.



Direction cosines ar
e arranged in tabular form for computation:




















It is emphasized that, in general,

for
.
Although the direction cosine between the
primed and unprimed axes is denoted by
, it is misleading in that a better representation might
be
.

Hen
ce, it can be readily understood that

for
.
According
ly

the angle
between two axes

may have the relationship
:
.


From a computational standpoint, it is often convenient to ca
rry out the transformations
in matrix form as



where


7




is called a rotation matrix. In Cartesian coordinates, the rotation matrix is also an
orthogonal matrix having a prop
erty;
. From the arrangement of the direction
cosines shown above, it is obvious that




















7.

Cartesian Tensors


A tensor of order
n

is a set of
quantities which transforms from one coordinate to
another by a specified transformation law, as follows:

n

order



transformation law

0

zero (scalar)


1

one (vector)



2

two (dyadic)



3

three



4

four





Second
-
order tensors

(dyadics) are particularly prevalent in elasticity and they are

transformed by



as in the case of following matrix stiffness relationships:


8



Hence,





8.

Operational Tensor

$

Kronecker
-
Delta



$

Permutation Symbol (or Alternating Tensor)



Thus



$

Application






contraction
; reducing the order by two




th component




9

if

i=j





normality property of direction cosines

if i
j


orthogonality


equivalent statement

9.

Computational Examples

(1) Show th
at

Proof:



(2) Show that

Proof:


10



(3) Prove that the product of two first order tensors is a second order tensor.

Proof:


Let

and

be two first order tensors and
be their product.

Then



note
are dummy indices

Hence



QED

Exercises

1.2 P
rove that if
, then
is a second order tensor.

Proof:


QED

1.4 If two first order tensors are related by
, prove that
is a
second order tensor.

Proof:



Hence,
is a second order tensor.


11

HW#1 1.1,1.3, and 1.5.

1.1 Show that (a)

and (b)

(a)


(
b)


1.3 Prove that the product of a first order and second order tensor is a third order tensor.

Proof:


Let
be a first order tensor,
is a second order tensor and
is a third order tensor

so
that
. Then


QED

1.5 Show that if
is a first order tensor,
is a second order tensor.

Proof:





Hence,
is a second order tensor.

(ref. Text pp. 6, Eq 1
-
22)



H.W.
2


The transformation matrix of a plane frame member is given by



12



Using Maple
®
, symbolic computation program, prove that this rotation matrix is an orthogonal
matrix, i.e., prove
.