Civil Engineering Reference
In-Depth Information
stress analysis, the coordinate system for each element is the same as the global
coordinate system. It is a fact of human nature, especially of engineers, that we
select the simplest frame in which to describe a particular occurrence or event.
This is a way of saying that we tend to choose a coordinate system for conve-
nience and that convenience is most often related to the geometry of the problem
at hand. The selected coordinate system seldom, if ever, corresponds to maxi-
mum loading conditions. Specifically, if we consider the element stress calcula-
tion represented by Equation 9.120, the stress components are referred, and
calculated with reference, to a specified Cartesian coordinate system. To deter-
mine the critical loading on any model, we must apply one of the so-called fail-
ure theories. As we limit the discussion to linearly elastic behavior, the “failure”
in our context is yielding of the material. There are several commonly accepted
failure theories for yielding in a general state of stress. The two most commonly
applied are the
maximum shear stress theory
and the
distortion energy theory.
We
discuss each of these briefly. In a general, three-dimensional state of stress, the
principal stresses
1
,
2
,
and
3
are given by the roots of the cubic equation rep-
resented by the determinant [2]
x
−
xy
xz
=
0
(9.121)
xy
y
−
yz
xz
yz
z
−
Customarily, the principal stresses are ordered so that
1
>
2
>
3
. Via the
usual convention, a positive normal stress corresponds to tension, while a nega-
tive normal stress is compressive. So, while
3
is algebraically the smallest of the
three principal stresses, it may represent a compressive stress having signifi-
cantly large magnitude. Also recall that the principal stresses occur on mutually
orthogonal planes (the
principal planes
) and the shear stress components on
those planes are zero.
Having computed the principal stress components, the maximum shear
stress is
largest of
|
1
−
2
|
2
,
|
1
−
3
|
2
,
|
2
−
3
|
2
max
=
(9.122)
The three shear stress components in Equation 9.122 are known to occur on
planes oriented 45
◦
from the principal planes.
The maximum shear stress theory (MSST) holds that failure (yielding) in a
general state of stress occurs when the maximum shear stress as given by Equa-
tion 9.122 equals or exceeds the maximum shear stress occurring in a uniaxial
tension test at yielding. It is quite easy to show that the maximum shear stress
in a tensile test at yielding has value equal to one-half the tensile yield strength
of the material. Hence, the failure value in the MSST is
max
=
S
y
/
2
=
S
ys
.
In
this notation,
S
y
is tensile yield strength and
S
ys
represents yield strength in shear.
The distortion energy theory (DET) is based on the strain energy stored in a
material under a given state of stress. The theory holds that a uniform tensile or
