Civil Engineering Reference
In-Depth Information
According to the von Mises criterion, failure occurs when
U
s
, given by Eq. (14.52),
reaches the value of
U
s
, given by Eq. (14.53), i.e. when
σ
II
)
2
σ
III
)
2
σ
I
)
2
2
σ
Y
(
σ
I
−
+
(
σ
II
−
+
(
σ
III
−
=
(14.54)
For a two-dimensional stress system in which
σ
III
=
0, Eq. (14.54) becomes
σ
I
σ
II
−
σ
Y
+
σ
I
σ
II
=
(14.55)
Design application
Codes of Practice for the use of structural steel in building use the von Mises crite-
rion for a two-dimensional stress system (Eq. (14.55)) in determining an equivalent
allowable stress for members subjected to bending and shear. Thus if
σ
x
and
τ
xy
are
the direct and shear stresses, respectively, at a point in a member subjected to bending
and shear, then the principal stresses at the point are, from Eqs (14.8) and (14.9)
σ
x
σ
x
2
1
2
4
τ
xy
σ
I
=
+
+
σ
x
σ
x
2
1
2
4
τ
xy
σ
II
=
−
+
Substituting these expressions in Eq. (14.55) and simplifying we obtain
σ
x
3
τ
xy
σ
Y
=
+
(14.56)
In Codes of Practice
σ
Y
is termed an equivalent stress and allowable values are given
for a series of different structural members.
Yield loci
Equations (14.39) and (14.54) may be plotted graphically for a two-dimensional stress
system in which
σ
III
=
0 and in which it is assumed that the yield stress,
σ
Y
, is the same
in tension and compression.
Figure 14.20 shows the yield locus for the maximum shear stress or Tresca theory of
elastic failure. In the first and third quadrants, when
σ
I
and
σ
II
have the same sign,
failure occurs when either
σ
I
=
σ
Y
(see Eq. (14.41)) depending on which
principal stress attains the value
σ
Y
first. For example, a structural member may be
subjected to loads that produce a given value of
σ
II
(
<σ
Y
) and varying values of
σ
I
.If
the loads were increased, failure would occur when
σ
I
reached the value
σ
Y
. Similarly
for a fixed value of
σ
I
and varying
σ
II
. In the second and third quadrants where
σ
I
and
σ
II
have opposite signs, failure occurs when
σ
I
−
σ
Y
or
σ
II
=
σ
Y
(see
Eq. (14.42)). Both these equations represent straight lines, each having a gradient of
σ
II
=
σ
Y
or
σ
II
−
σ
I
=
