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.
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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
.
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