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Failures Resulting from Static Loading 223
Figure 5–9 B

The distortion-energy (DE) S y
theory yield envelope for plane
stress states. This is a plot
of points obtained from

Eq. (5–13) with σ = S y . Nonyield region
–S S A
y y

Pure shear load line ( )
B
A
–S y DE
MSS

called the von Mises stress, σ , named after Dr. R. von Mises, who contributed to the
theory. Thus Eq. (5–10), for yield, can be written as
(5–11)
σ ≥ S y
where the von Mises stress is
2 2 2 1/2
(σ 1 − σ 2 ) + (σ 2 − σ 3 ) + (σ 3 − σ 1 )

σ = (5–12)
2
For plane stress, the von Mises stress can be represented by the principal stresses
σ A , σ B , and zero. Then from Eq. (5–12), we get
2 2 1/2

σ = σ − σ A σ B + σ (5–13)

A B
Equation (5–13) is a rotated ellipse in the σ A , σ B plane, as shown in Fig. 5–9 with

σ = S y . The dotted lines in the figure represent the MSS theory, which can be seen to
be more restrictive, hence, more conservative. 4
Using xyz components of three-dimensional stress, the von Mises stress can be
written as
1/2
1 2 2 2 2 2 2

σ = √ (σ x − σ y ) + (σ y − σ z ) + (σ z − σ x ) + 6 τ xy + τ + τ zx (5–14)
yz
2
and for plane stress,
1/2
2 2 2
σ = σ − σ x σ y + σ + 3τ (5–15)

x y xy
The distortion-energy theory is also called:
• The von Mises or von Mises–Hencky theory
• The shear-energy theory
• The octahedral-shear-stress theory
Understanding octahedral shear stress will shed some light on why the MSS is conser-
vative. Consider an isolated element in which the normal stresses on each surface are

4 The three-dimensional equations for DE and MSS can be plotted relative to three-dimensional σ 1 , σ 2 , σ 3 ,
coordinate axes. The failure surface for DE is a circular cylinder with an axis inclined at 45° from each
principal stress axis, whereas the surface for MSS is a hexagon inscribed within the cylinder. See Arthur P.
Boresi and Richard J. Schmidt, Advanced Mechanics of Materials, 6th ed., John Wiley & Sons, New York,
2003, Sec. 4.4.