Budynas-Nisbett:Shigley's ll.Failure Prevention 5.Failures Resulting from ©The McGraw-Hil 219 Mechanical Engineering Static Loading Companies,2008 Design,Eighth Edition Failures Resulting from Static Loading 215 Figure 5-9 The distortion-energy (DE) theory for plane stress states. This is a plot of points obtained from Eq.(5-13) with a'=Sy. -S. Pure shear load line (a=-0s=T) DE ---MSS Equation(5-13)is a rotated ellipse in the oA.og plane,as shown in Fig.5-9 with '=Sy.The dotted lines in the figure represent the MSS theory,which can be seen to be more restrictive,hence,more conservative. Using xyz components of three-dimensional stress,the von Mises stress can be written as g'= 万[a,-,P+a,-P+a:-aP+66+最+】p (5-14④ and for plane stress, a'=(a2-aa,+a+3r)2 (5-15) 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 equal to the hydrostatic stress.There are eight surfaces symmetric to the principal directions that contain this stress.This forms an octahedron as shown in Fig.5-10.The shear stresses on these surfaces are equal and are called the octahedral shear stresses (Fig.5-10 has only one of the octahedral surfaces labeled).Through coordinate trans- formations the octahedral shear stress is given by ta=3[a1-022+(o2-0)2+(a3-o1)2]2 (5-16) The three-dimensional equations for DE and MSS can be plotted relative to three-dimensional. 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. SFor a derivation,see Arthur P.Boresi,op.cit.,pp.36-37.Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition II. Failure Prevention 5. Failures Resulting from Static Loading © The McGraw−Hill 219 Companies, 2008 Failures Resulting from Static Loading 215 Equation (5–13) is a rotated ellipse in the σA, σB plane, as shown in Fig. 5–9 with σ = Sy . 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 σ = 1 √2 (σx − σy ) 2 + (σy − σz) 2 + (σz − σx ) 2 + 6  τ 2 xy + τ 2 yz + τ 2 zx 1/2 (5–14) and for plane stress, σ =  σ2 x − σxσy + σ2 y + 3τ 2 xy 1/2 (5–15) 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 equal to the hydrostatic stress σav. There are eight surfaces symmetric to the principal directions that contain this stress. This forms an octahedron as shown in Fig. 5–10. The shear stresses on these surfaces are equal and are called the octahedral shear stresses (Fig. 5–10 has only one of the octahedral surfaces labeled). Through coordinate trans￾formations the octahedral shear stress is given by5 τoct = 1 3 (σ1 − σ2) 2 + (σ2 − σ3) 2 + (σ3 − σ1) 21/2 (5–16) 4 The three-dimensional equations for DE and MSS can be plotted relative to three-dimensional σ1, σ2, σ3, 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. 5 For a derivation, see Arthur P. Boresi, op. cit., pp. 36–37. Figure 5–9 The distortion-energy (DE) theory for plane stress states. This is a plot of points obtained from Eq. (5–13) with σ = Sy . –Sy –Sy Sy Sy B A DE MSS Pure shear load line (A B )