Civil Engineering Reference
In-Depth Information
σ
II
σ
II
σ
Y
σ
I
0,
σ
II
0
σ
I
0,
σ
II
0
σ
Y
σ
II
σ
I
σ
Y
σ
I
σ
Y
σ
I
σ
Y
σ
Y
σ
I
σ
Y
σ
I
0,
σ
II
0
σ
I
σ
II
σ
Y
σ
Y
σ
I
0,
σ
II
0
F
IGURE
14.20
Yield locus for the
Tresca theory of elastic failure
σ
II
σ
Y
σ
II
σ
Y
Tresca yield locus
45
°
σ
I
σ
Y
σ
Y
σ
Y
F
IGURE
14.21
Yield locus for the
von Mises theory
45
◦
and an intercept on the
σ
II
axis of
σ
Y
. Clearly all combinations of
σ
I
and
σ
II
that
lie inside the locus will not cause failure, while all combinations of
σ
I
and
σ
II
on or
outside the locus will. Thus the inside of the locus represents elastic conditions while
the outside represents plastic conditions. Note that for the purposes of a yield locus,
σ
I
and
σ
II
are interchangeable.
The shear strain energy (von Mises) theory for a two-dimensional stress system is
represented by Eq. (14.55). This equation may be shown to be that of an ellipse whose
major andminor axes are inclined at 45
◦
to the axes of
σ
I
and
σ
II
as shown in Fig. 14.21.
It may also be shown that the ellipse passes through the six corners of the Tresca yield
locus so that at these points the two theories give identical results. However, for other
combinations of
σ
I
and
σ
II
the Tresca theory predicts failure where the von Mises
theory does not so that the Tresca theory is the more conservative of the two.
The value of the yield loci lies in their use in experimental work on the validation of
the different theories. Structural members fabricated from different materials may be
subjected to a complete range of combinations of
σ
I
and
σ
II
each producing failure. The
results are then plotted on the yield loci and the accuracy of each theory is determined
for different materials.
