GC 2009267: TEACHING VON MISES STRESS: FROM PRINCIPAL AXES TO
NONPRINCIPAL AXES
IngChang Jong, University of Arkansas
IngChang Jong serves as Professor of Mechanical Engineering at the University of Arkansas. He
received a BSCE in 1961 from the National Taiwan University, an MSCE in 1963 from South
Dakota School of Mines and Technology, and a Ph.D. in Theoretical and Applied Mechanics in
1965 from Northwestern University. He and Dr. Bruce G. Rogers coauthored the textbook
Engineering Mechanics: Statics and Dynamics, Oxford University Press (1991). Dr. Jong was
Chair of the Mechanics Division, ASEE, in 199697. His research interests are in mechanics and
engineering education.
William Springer, University of Arkansas
William T. Springer serves as Associate Professor of Mechanical Engineering at the University of
Arkansas. He received a BSME in 1974, MSME in 1979, and Ph.D. in Mechanical Engineering in
1982 from the University of Texas at Arlington. Dr. Springer has been an active participant of the
NDE Engineering Division ASME for 27 years and severed as Chair of the division from 2001 to
2003. He has also been an active member of the Society for Experimental Mechanics where he
served as the Chair of the Modal Analysis and Dynamic Systems Technical Division from 1982
to 1994. He received the ASME Dedicated Service Award in 2006, was elected to Fellow
Member status in 2008, and was recognized as an Outstanding Mentor by Honors College of the
University of Arkansas in 2006. His research interests are in machine component design,
experimental modal analysis, structural dynamics, structural integrity monitoring, nondestructive
evaluation, and vibration testing.
© American Society for Engineering Education, 2009
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View Full DocumentTeaching von Mises Stress: From Principal Axes
To NonPrincipal Axes
Abstract
The von Mises stress is an equivalent or effective stress at which yielding is predicted to occur in
ductile materials. In most textbooks for machine design,
17
such a stress is derived using principal
axes in terms of the principal stresses
1
σ
,
2
, and
3
as
1/2
2 22
12
23
31
1
(
)(
)
2
σσ
′
=
−
+−
In their latest editions, some of these textbooks for machine design began to show that the von
Mises stress with respect to nonprincipal axes can
also
be expressed as
2
2
2
222
1
(
)6
(
)
2
x
y
y
z
z
x
xy
yz
zx
τττ
′
=
−
+
−
+
−
+
++
However, these textbooks do
not
provide an explanation regarding how the former formula is
evolved into the latter formula. Lacking a good explanation for the latter formula in the text
books or by the instructors in classrooms, students are often made to simply take it
on faith
that
these two formulas are somehow equivalent to each other. This paper is written to share with
educators of machine design and other readers two alternative paths that will arrive at the latter
general form of the von Mises stress: (
a
) by way of eigenvalues of the stress matrix, (
b
) by way
of stress invariants of the stress matrix. When used with the existing material presented in the
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 Spring '08
 Ingchang,J
 mechanics, Stress, Yield surface

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