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where
c
any nonzero real number is an eigenvector for
λ
2
, and that any vector
d
=
3
d
e
x
+
5
d
e
y
−
d
e
z
(11)
λ
.
where
d
any nonzero real number is an eigenvector for
The numbers
c
and
d
are usually
3
taken as being equal to one.
Since the eigenvectors serve only to define principal directions, the full collection of
eigenvectors defined above, by the presence of scalars
b, c, d
is not needed. We choose the
first two eigenvectors by requiring that they have unit magnitude and define the third as
the cross product of the first two. This choice produces a right-handed orthogonal triad of
unit vectors
b
e
x
3
e
z
√
10
+
b
2
b
1
=
2
(
e
x
+
3
e
z
)
=
+
(
)
3
b
3
e
x
+
2
e
y
−
e
z
b
2
=
√
14
(12)
b
2
=
−
3
e
x
+
5
e
y
+
e
z
√
35
The matrix
T
in this case has three distinct eigenvalues. The vector subspace spanned by
the set of eigenvectors corresponding to any one eigenvalue is one dimensional. Therefore,
the normalization of the eigenvectors means that the only possible choices for the first
eigenvector are
b
1
or
b
3
=
b
1
×
−
b
1
, and the only possible choices for the second eigenvector are
b
2
or
b
2
. Once the first two eigenvectors have been chosen, the third is uniquely determined
as the right-handed cross product of the first two, because a right-handed orthogonal triad is
stipulated. In physical terms, if the principal planes are visualized as the faces of a cube, the
orientation of this cube in three-dimensional space is uniquely determined. This uniqueness
is not obtained when
T
has repeated eigenvalues.
−
1.7.3
Strength Theories
The stress-strain curve characteristics normally are determined by a tensile test of a bar.
At a certain level of stress, such as the yield stress
σ
σ
u
, the bar
undergoes a transition to inelastic behavior. Often the bar is considered to have “failed”,
hence the following theories are sometimes referred to as
failure theories
or
strength theories.
For complex states of stress, failure theories have been developed that provide relationships
between stresses in complicated (two or three dimensional) situations and the behavior of
a material in simple tension. Although a brief discussion of some failure theories are given
in this section, it may be helpful to consult an appropriate reference for a more detailed
discussion. The literature contains a variety of other theories, as well as specialized criteria
for phenomena such as anisotropic materials.
ys
or the ultimate stress
Maximum Stress Theory
The maximum stress, or
Rankine
21
, theory is based on the maximum stress being chosen as
the criterion of failure. Yield (or some other measure of “failure”) occurs when loading on
21
William John Macquorn Rankine (1820-1872) was a Scottish engineer at Glasgow University. Although he
is best known for his work in thermodynamics (Rankine cycle and Rankine absolute temperature scale), he
made numerous contributions to the theory of elasticity and to a variety of other fields. He developed the stress
transformation equations when he was 32. He was also interested in the theory of structural analysis and such
members as restraining walls and arches.
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