Civil Engineering Reference
In-Depth Information
Thus the theory proposes that failure in a complex system will occur when
σ
max
−
σ
min
σ
Y
2
=
2
or
σ
max
−
σ
min
=
σ
Y
(14.39)
Let us now examine stress systems having different relative values of
σ
I
,
σ
II
and
σ
III
.
First suppose that
σ
I
>σ
II
>σ
III
>
0. From Eq. (14.39) failure occurs when
σ
I
−
σ
III
=
σ
Y
(14.40)
Second, suppose that
σ
I
>σ
II
>
0 but
σ
III
=
0. In this case the three-dimensional stress
system of Fig. 14.18 reduces to a two-dimensional stress system but
is still acting on a
three-dimensional element
. Thus Eq. (14.39) becomes
σ
I
−
0
=
σ
Y
or
σ
I
=
σ
Y
(14.41)
Here we see an apparent contradiction of Eq. (14.12) where the maximum shear stress
in a two-dimensional stress system is equal to half the difference of
σ
I
and
σ
II
. However,
the maximum shear stress in that case occurs in the plane of the two-dimensional
element, i.e. in the plane of
σ
I
and
σ
II
. In this case we have a three-dimensional
element so that the maximum shear stress will lie in the plane of
σ
I
and
σ
III
.
Finally,
0. Again we have a two-
dimensional stress system acting on a three-dimensional element but now
σ
II
is a
compressive stress and algebraically less than
σ
III
. Thus Eq. (14.39) becomes
let us suppose that
σ
I
>
0,
σ
II
<
0 and
σ
III
=
σ
I
−
σ
II
=
σ
Y
(14.42)
Shear strain energy theory
This particular theory of elastic failure was established independently by von Mises,
Maxwell and Hencky but is now generally referred to as the von Mises criterion. The
theory proposes that:
Failure will occur when the shear or distortion strain energy in the material reaches the
equivalent value at yielding in simple tension.
In 1904 Huber proposed that the total strain energy,
U
t
, of an element of material
could be regarded as comprising two separate parts: that due to change in volume and
that due to change in shape. The former is termed the volumetric strain energy,
U
v
,
the latter the distortion or shear strain energy,
U
s
. Thus
U
t
=
U
v
+
U
s
(14.43)
