09gc.Budapest - GC 2009-267 TEACHING VON MISES STRESS FROM...

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GC 2009-267: TEACHING VON MISES STRESS: FROM PRINCIPAL AXES TO NONPRINCIPAL AXES Ing-Chang Jong, University of Arkansas Ing-Chang 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 1996-97. 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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Teaching von Mises Stress: From Principal Axes To Non-Principal 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, 1-7 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 non-principal 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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09gc.Budapest - GC 2009-267 TEACHING VON MISES STRESS FROM...

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