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assumptions usually have been confirmed by experiment, and generally the limitations of
these theories are reasonably well known and documented.
Before considering approximate theories for various structures, the general equations of
the theory of elasticity which are the basis of all elastic engineering models will be discussed.
1.1
Fundamental Definitions
To describe structural problems it will be useful to employ one of the two right-handed
Cartesian
1
coordinate systems of Fig. 1.5. The coordinate system of Fig. 1.5b will normally
be used in connection with structural members, e.g., for beams, plates, and shells.
Scalar, vector, and matrix notations are to be used. Einstein's
2
summation convention
will often be employed to simplify the notation. This involves omitting summation signs
and agreeing to add over
repeated indices
. These indices are “dummy indices” in that they
can be arbitrarily renamed.
Thus,
3
a
i
b
i
=
a
i
b
i
=
a
1
b
1
+
a
2
b
2
+
a
3
b
3
=
a
j
b
j
=
a
k
b
k
i
=
1
3
3
σ
=
1
σ
=
σ
+
σ
+
σ
+
σ
+
σ
+
σ
(1.1)
ik
ik
ik
ik
11
11
12
12
13
13
21
21
22
22
23
23
i
=
1
k
=
+
σ
31
31
+
σ
32
32
+
σ
33
33
The indices assume the values 1, 2, and 3 for a three-dimensional continuum, whereas they
are 1 and 2 for a structural member with two significant directions.
Abbreviations for derivatives will be
Ordinary derivatives:
da
k
dx
=
a
k
d
x
a
k
=
a
k,x
=
(1.2a)
Partial derivatives:
∂
a
k
∂
a
k
∂
a
k
x
i
=
∂
i
a
k
=
a
k,i
or
x
=
∂
x
a
k
=
a
k,x
=
(1.2b)
∂
The notation used to define the state of stress in a three-dimensional continuum is indi-
cated in Fig. 1.6. The stress components normal to a coordinate plane are denoted by
σ
x
,
σ
y
,
and
σ
z
and the shear stress components by
τ
xy
,
τ
xz
,
τ
yz
,
τ
yx
,
τ
zx
,
and
τ
zy
. The single subscript
1
Rene Descartes (1596-1650) was a French philosopher, physicist, and mathematician. He was a contemporary of
Galileo and is probably best known for his philosophical works. His mathematical prowess was first recognized
when, as a soldier, he solved a geometrical problem which was posted as a challenge on a placard on a street in
Holland. His chief contributions in mathematics were his theories of analytical geometry and vortices.
2
Albert Einstein (1879-1955), the father of the general theories of relativity and gravitation, was one of the first
scientists to make extensive use of the index notation. His reputation as a slow child, pacifist, nuclear bomb
politician, Zionist, physicist, and Nobel prize winner is well-known.
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