GC : TEACHING VON MISES STRESS: FROM PRINCIPAL AXES TO NONPRINCIPAL AXES

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1 GC : TEACHING VON MISES STRESS: FROM PRINCIPAL AXES TO NONPRINCIPAL AXES Ing-Chang Jong, Universit of Arkansas Ing-Chang Jong serves as Professor of Mechanical Engineering at the Universit of Arkansas. He received a BSCE in 96 from the National Taiwan Universit, an MSCE in 963 from South Dakota School of Mines and Technolog, and a Ph.D. in Theoretical and Applied Mechanics in 965 from Northwestern Universit. He and Dr. Bruce G. Rogers coauthored the textbook Engineering Mechanics: Statics and Dnamics, Oxford Universit Press (99). Dr. Jong was Chair of the Mechanics Division, ASEE, in His research interests are in mechanics and engineering education. William Springer, Universit of Arkansas William T. Springer serves as Associate Professor of Mechanical Engineering at the Universit of Arkansas. He received a BSME in 974, MSME in 979, and Ph.D. in Mechanical Engineering in 98 from the Universit of Texas at Arlington. Dr. Springer has been an active participant of the NDE Engineering Division ASME for 7 ears and severed as Chair of the division from 00 to 003. He has also been an active member of the Societ for Experimental Mechanics where he served as the Chair of the Modal Analsis and Dnamic Sstems Technical Division from 98 to 994. He received the ASME Dedicated Service Award in 006, was elected to Fellow Member status in 008, and was recognized as an Outstanding Mentor b Honors College of the Universit of Arkansas in 006. His research interests are in machine component design, experimental modal analsis, structural dnamics, structural integrit monitoring, nondestructive evaluation, and vibration testing. American Societ for Engineering Education, 009

2 Teaching von Mises Stress: From Principal Axes To Non-Principal Axes Abstract The von Mises stress is an equivalent or effective stress at which ielding is predicted to occur in ductile materials. In most textbooks for machine design, -7 such a stress is derived using principal axes in terms of the principal stresses σ, σ, and σ 3 as / σ = ( σ σ) + ( σ σ3) + ( σ3 σ) 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 / σ = ( σx σ ) + ( σ σz) + ( σz σx) + 6( x + z + zx) 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 textbooks or b the instructors in classrooms, students are often made to simpl 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) b wa of eigenvalues of the stress matrix, (b) b wa of stress invariants of the stress matrix. When used with the existing material presented in the textbooks, either of these two paths will provide students with a much better understanding of the general form of the von Mises stress. The contributed work is aimed at enhancing the teaching and learning of one of the important failure theories usuall covered in a senior design course in most engineering curricula. I. Introduction Poncelet and Saint-Venant 8 proposed using the strain-energ in a ductile material to determine when failure of the material would occur. Maxwell 8 expanded the idea b showing that the total strain energ could be divided into distortional and volumetric terms, but he never extended the idea further. A distortion-energ theor was prompted from the observation that ductile materials stressed hdrostaticall exhibited ield strengths greatl in excess of the values given b simple tension tests. It was then postulated that ielding was related somehow to the angular distortion of the stressed element, rather than that ielding was a simple tensile or compressive phenomenon. Nowadas, the distortion-energ theor for ductile materials states that ielding occurs when the distortion strain energ per unit volume reaches or exceeds the distortion strain energ per unit volume for ield in simple tension or compression of the same material. Using principal axes, we can first decompose the state of stress at a point that is given in terms of the principal stresses σ, σ, and σ 3 into the sum of two states: (a) a state of hdrostatic stress

3 due to the stresses σ av acting in each of the principal directions and causing onl volume change, (b) a state of deviatoric stress causing angular distortion without volume change, where σ σ + σ + σ 3 3 av = () Such a decomposition of state of stress is illustrated in Fig.. The decomposition of a state of stress into a hdrostatic state and a deviatoric state is ver useful in applications in plasticit, thermoelasticit, fluid dnamics, and other fields of engineering science. (a) (b) (c) Fig. Decomposition of state of stress into a hdrostatic state and a deviatoric state In simple tension, we have, b Hooke s law, σ = Eε () where σ is the stress, ε is the strain, and E is the modulus of elasticit. Therefore, the strain energ per unit volume for simple tension is ε ε (3) u = σdε = Eεdε = Eε = εσ 0 0 For the state of stress in Fig. (a), we readil write the total strain energ per unit volume as where u = ( εσ + εσ + εσ 3 3) (4) ε = [ σ νσ ( + σ3) ] (5) E ε = [ σ νσ ( 3 + σ) ] (6) E ε3 = [ σ3 νσ ( + σ) ] (7) E in which ν is the Poison s ratio. Substituting Eqs. (5) through (7) into Eq. (4), we get the total strain energ as u = ( ) E σ + σ + σ ν σ σ + σ σ + σ σ (8) B letting σ= σ= σ3= σav in Eq. (8), we obtain the strain energ associated with hdrostatic loading, or onl volume change, as

4 u = ν σ + σ + σ 3 + ( σ σ + σ σ σ σ ) v 6E (9) Clearl, the distortion energ per unit volume is We obtain that u d ud = u u v + ν ( σ σ) + ( σ σ3) + ( σ3 σ) = 3E Note that u d = 0 if σ= σ = σ3; i.e., no distortion exists in hdrostatic state of stress. For simple tensile test of a ductile material, we have σ = S, σ= σ3= 0, and Eq. (0) reduces to +ν ud = S () 3E where S is the ield stress in tension for the material. B the distortion-energ theor, as stated earlier, ield is predicted to occur if the value of u d in Eq. (0) equals or exceeds the value of u d in Eq. (). In other words, ield occurs whenever ( σ σ ) + ( σ σ ) + ( σ σ ) 3 3 Therefore, the left side of Eq. () represents an equivalent or effective stress at which ielding of an ductile material is predicted to occur. This stress is usuall denoted as σ and is known as the von Mises stress: σ = σ σ + σ σ + σ σ / S (0) () / ( ) ( 3) ( 3 ) (3) The derivation of this form of the von Mises stress is based on the principal axes and arrives at the final result that is, of course, expressed in terms of the principal stresses σ, σ, and σ 3. However, normal and shear stresses ma be present at man locations of the machine parts that are subjected to either static or dnamic loads. Thus, the state of stress is often expressed in terms of the normal stresses σ x, σ, and σ z, and the shear stresses x, z, and zx, where the coordinate axes are non-principal axes. Since it is desirable in design to expeditiousl determine the factor of safet against either static or fatigue failure, an effective wa is needed to properl combine normal and shear stresses, at a point, into a single value (equivalent to σ ), which can be used to compare with the ield strength of the material. Recentl published machine design textbooks began to show, without explanation, that the von Mises stress with respect to non-principal axes can also be expressed as

5 / σ = ( σx σ ) + ( σ σz) + ( σz σx) + 6( x + z + zx) (4) Most students have little idea how Eq. (3) is evolved into Eq. (4) for the general form of σ. In this paper, we present two alternative paths that will lead from Eq. (3) to Eq. (4). II. Arriving at the General Form of σ via Eigenvalues of the Stress Matrix Neglecting couple stresses (or bod moments), we know that cross-shears are equal; i.e., x = x z = z xz zx = (5) Thus, the state of stress at point O of the coordinate sstem Oxz in the material is given b the smmetric stress matrix σ x x zx = x σ z (6) zx z σ z where the xz axes are generall not principal axes. The principal stresses σ, σ, and σ 3 at O of the material are given b the eigenvalues of the stress matrix, which are simpl the roots of the characteristic equation σ λ x x zx σ λ = x z σ λ zx z z Upon expansion, Eq. (7) becomes the cubic equation 0 (7) where λ + λ λ + = (8) 3 I I I3 0 I = σx + σ + σz (9) I σ z σx σ zx x x = + + σ σ σ z z zx z x = σ σ + σ σ + σ σ x z z x x z zx (0) I 3 σ x x zx = σ () x z σ zx z z The roots of Eq. (8) are the principal stresses σ, σ, and σ 3 at O of the material. The values of the principal stresses depend onl on the applied loads and cannot be influenced b the choice of orientation of the xz coordinate axes at point O. This means that the values of I, I, and I 3

6 given b Eqs. (9) through () must remain unchanged for all choices of orientations of the xz coordinate axes at point O. If the xz coordinate axes at point O coincide with the principal axes at point O, the crossshears vanish and the values of I, I, and I 3 can be expressed in terms of just the principal stresses as Based on Eqs. (9) and (), we write Based on Eqs. (0) and (3), we write I = σ+ σ + σ3 () I = σσ + σσ 3 + σσ 3 (3) I = σσσ (4) 3 3 σ+ σ + σ3 = σx + σ + σz (5) σσ + σσ + σσ = σ σ + σ σ + σ σ (6) 3 3 x z z x x z zx Squaring both sides of Eq. (5) and using Eq. (6), we write ( ) σ + σ + σ + σ σ + σ σ + σ σ σx σ σz σxσ σσz σzσx σx σ σz σ σ σ σ σ σ x z zx = ( ) ( 3 3 ) ( ) = (7) From Eq. (7), we get the relation ( ) σ + σ + σ = σ + σ + σ (8) 3 x z x z zx Based on the von Mises stress in Eq. (3) and Eqs. (8) and (6), we write ( σ ) = ( σ σ ) + ( σ σ ) + ( σ σ ) 3 3 = ( σ + σ + σ3) ( σσ + σσ 3 + σσ 3 ) = ( σx + σ + σz) + 4( + x + z zx) σxσ + σσz + σzσx + + x + z zx ( ) ( ) = σσ + σσ) + 6( + + ) ( σx σ σz) ( σxσ z z x x z zx = σ σ + σ σ + σ σ ( x ) ( z) ( z x) 6( x z zx) Therefore, we conclude that the following general form of the von Mises stress has been proved and is true: σ = σ σ + σ σ + σ σ (9) / ( x ) ( z) ( z x) 6( x z zx) (30) Q.E.D.

7 III. Arriving at the General Form of σ via Invariants of the Stress Matrix The fact that the values of I, I, and I 3, as shown in Eqs. (9) through (4), must and do remain unchanged under an choice of orientation of the xz coordinate axes (i.e., under an coordinate transformation) at point O means that the are invariants of the stress matrix at point O. This is well-known to those who are familiar with the field of continuum mechanics. 9- Using principal axes and principal stresses σ, σ, and σ 3 at O, we have earlier derived and established the von Mises stress as σ = σ σ + σ σ + σ σ / ( ) ( 3) ( 3 ) (3) (Repeated) Using the first two stress invariants in Eqs. () and (3), we write From Eqs. (3) and (3), we see that ( ) ( ) I 6I = σ + σ + σ 6 σσ + σσ + σσ ( σ σ σ3 ) ( σσ σσ3 σ3σ) = = ( σ σ ) + ( σ σ ) + ( σ σ ) 3 3 σ = I 6I Thus, the von Mises stress can be expressed in terms of stress invariants I and I as follows: / (3) σ = I 3 I (3) Note that Eq. (3) is independent of the choice of the coordinate sstem and is the most general form of the von Mises stress. Using the stress invariants in Eqs. (9) and (0), we write I 3I ( σx σ σ ) 3 ( σxσ + σσz + σzσx x z z zx) σx σ σz ( σxσ σσz σzσx) 3( x + z + zx) = + + = ( ) ( ) ( ) 6( σx σ σ σz σz σx x z zx ) = (33) Substituting Eq. (33) into Eq. (3), we conclude that the following general form of the von Mises stress has been proved and is true: σ = σ σ + σ σ + σ σ / ( x ) ( z) ( z x) 6( x z zx) (34) Q.E.D.

8 IV. Machine Element Design: Use of von Mises Stress and Assessment At the institution where the authors teach in the Department of Mechanical Engineering, the course MEEG 403 Machine Element Design is a course for mainl seniors in the curriculum. The specified prerequisite to this course is MEEG 303 Mechanics of Materials. In Machine Element Design, we explain and emphasize the theories of failure resulting from either static loading or fatigue (i.e., dnamic loading) as well as the design of components commonl used in modern machines. The tension test is uniaxial (i.e., simple ) and elongations are largest in the axial direction; therefore, strains and stresses can be measured and inferred up to failure. However, the state of stress at a point of a machine component is usuall not simple. Toda the generall accepted failure theories (or ield criteria) used to evaluate machine components manufactured from ductile materials are maximum shear stress theor, distortion energ theor, Coulomb-Mohr theor. In the distortion energ theor, ielding occurs when the von Mises stress σ is reached, or exceeded, b a state of stress in the machine component. We endeavor to present the derivations explaining how the von Mises stress in Eq. (3) is evolved into the general form in Eq. (4) in the course MEEG 403. Such an endeavor enables us to achieve the following two educational benefits: (a) students see the importance and are convinced via derivations that the two forms of the von Mises stress σ in Eqs. (3) and (4) are trul equivalent, (b) an uneasiness or compelled faith in the minds of the students, who use the general form of the von Mises stress in Eq. (4), is dispelled. The students appreciate having been exposed to the alternative stories behind the general form of the von Mises stress in Eq. (4). Compared with students who took the same course where the explanations and derivations for Eq. (4) were omitted, the students exposed to these derivations have now an increased sense of confidence in this class, and the did better in the quizzes and tests involving the use of von Mises stress, such as in computing the factor of safet for the machine components subjected to static or dnamic loads. V. Concluding Remarks When the xz coordinate axes at point O coincide with the principal axes, the cross-shears vanish and the normal stresses σx, σ, and σ z become the principal stresses σ, σ, and σ 3 on the principal planes. Therefore, the general form of the von Mises stress obtained in Eqs. (30) and (34) nicel degenerates into the form in Eq. (3). In other words, the path from Eqs. (30) and (34) to Eq. (3) is an eas and direct one-wa street. Nevertheless, there is no obvious clue as to how one can go from Eq. (3) to Eqs. (30) and (34).

9 For man ears, machine design textbooks presented onl the von Mises stress in the form of Eq. (3). Onl in recent ears, machine design textbooks began to show, without explanation, that the von Mises stress with respect to non-principal axes can also be expressed in the form of Eq. (4). This paper is written to share with educators of machine design and other readers a couple of was that ma be used to explain how Eq. (3) is evolved to arrive at Eq. (4). It is aimed at enhancing the teaching and learning of one of the important failure theories usuall covered in a senior design course in most engineering curricula. References. R. G. Budnas and J. K. Nisbett, Shigle s Mechanical Engineering Design (8 th Edition), McGraw-Hill, New York, NY, R. L. Norton, Machine Design An Integrated Approach ( nd Edition), Prentice-Hall Inc., Upper Saddle River, NJ, A. C. Ugural, Mechanical Design: An Integrated Approach, McGraw-Hill, New York, NY, A. D. Dimarogonas, Machine Design: A CAD Approach, John Wile & Sons, Inc., New York, NY, R. C. Juvinall and K. M. Marshek, Fundamentals of Machine Component Design (4 th Edition), John Wile & Sons, Inc., New York, NY, B. J. Hamrock, S. R. Schmid, and B. Jacobson, Fundamentals of Machine Elements ( nd Edition), McGraw-Hill, New York, NY, H. O. Fuchs and R. I. Stephens, Metal Fatigue in Engineering, John Wile & Sons, Inc., New York, NY, S. P. Timoshenko, Histor of Strength of Materials: With a Brief Account of the Histor of Theor of Elasticit and Theor of Structures, McGraw-Hill, New York, NY, A. P. Boresi and R. J. Schmidt, Advanced Mechanics of Materials (6 th Edition), John Wile & Sons, Inc., Hoboken, NJ, T. J. Chung, Continuum Mechanics, Prentice-Hall, Inc., Englewood Cliffs, NJ, Y. C. Fung, A First Course In Continuum Mechanics, Prentice-Hall, Inc., Englewood Cliffs, NJ, D. Frederick and T. S. Chang, Continuum Mechanics, Alln and Bacon, Inc., Boston, MA, 965.