218 Budynas-Nisbett:Shigley's ll.Failure Prevention 5.Failures Resulting from T©The McGraw-Hil Mechanical Engineering Static Loading Companies,2008 Design,Eighth Edition 214 Mechanical Engineering Design subtracted from them,resulting in the stress state shown in Fig.5-8c.This element is subjected to pure angular distortion,that is,no volume change. The strain energy per unit volume for simple tension is u=eo.For the element of Fig.5-8a the strain energy per unit volume is u=e101+e202+e303]. Substituting Eq.(3-19)for the principal strains gives =2+听+G-2a+oa+oo】 (6 The strain energy for producing only volume change u can be obtained by substitut- ing oav for 01,o2,and o3 in Eq.(b).The result is 3蓝1-2w) v=2 (d) If we now substitute the square of Eq.(a)in Eq.(c)and simplify the expression,we get ,=6E(o+号+听+2012+20203+2030)） 1-2v (5-7刀 Then the distortion energy is obtained by subtracting Eq.(5-7)from Eq.(b).This gives 1+v「(o1-02)2+(o2-03)2+(a3-0)2 ld u-lv (5-8) 3E 2 Note that the distortion energy is zero if o1 =o2 =03. For the simple tensile test,at yield,o1=Sy and o2 =03 =0,and from Eq.(5-8) the distortion energy is 1+v =3E号 (5-9 So for the general state of stress given by Eq.(5-8),yield is predicted if Eq.(5-8) equals or exceeds Eg.(5-9).This gives T1-22+@2-o2+o-n2]p ≥S (5-10) 2 If we had a simple case of tension o,then yield would occur when S.Thus,the left of Eq.(5-10)can be thought of as a single,equivalent,or effective stress for the entire general state of stress given by o,o2,and o3.This effective stress is usually called the von Mises stress,o',named after Dr.R.von Mises,who contributed to the theory.Thus Eq.(5-10),for yield,can be written as a'≥S, (5-11) where the von Mises stress is 0= 「o1-22+(@2-32+(a3-m)27p 2 (5-12) For plane stress,let oA and og be the two nonzero principal stresses.Then from Eq.(5-12),we get '=(oi-aaag+ag)i (5-13)Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition II. Failure Prevention 5. Failures Resulting from Static Loading 218 © The McGraw−Hill Companies, 2008 214 Mechanical Engineering Design subtracted from them, resulting in the stress state shown in Fig. 5–8c. This element is subjected to pure angular distortion, that is, no volume change. The strain energy per unit volume for simple tension is u = 1 2 σ . For the element of Fig. 5–8a the strain energy per unit volume is u = 1 2 [1σ1 + 2σ2 + 3σ3]. Substituting Eq. (3–19) for the principal strains gives u = 1 2E σ2 1 + σ2 2 + σ2 3 − 2ν(σ1σ2 + σ2σ3 + σ3σ1) (b) The strain energy for producing only volume change uv can be obtained by substitut￾ing σav for σ1, σ2, and σ3 in Eq. (b). The result is uv = 3σ2 av 2E (1 − 2ν) (c) If we now substitute the square of Eq. (a) in Eq. (c) and simplify the expression, we get uv = 1 − 2ν 6E  σ2 1 + σ2 2 + σ2 3 + 2σ1σ2 + 2σ2σ3 + 2σ3σ1  (5–7) Then the distortion energy is obtained by subtracting Eq. (5–7) from Eq. (b). This gives ud = u − uv = 1 + ν 3E  (σ1 − σ2) 2 + (σ2 − σ3) 2 + (σ3 − σ1) 2 2  (5–8) Note that the distortion energy is zero if σ1 = σ2 = σ3. For the simple tensile test, at yield, σ1 = Sy and σ2 = σ3 = 0, and from Eq. (5–8) the distortion energy is ud = 1 + ν 3E S2 y (5–9) So for the general state of stress given by Eq. (5–8), yield is predicted if Eq. (5–8) equals or exceeds Eq. (5–9). This gives  (σ1 − σ2) 2 + (σ2 − σ3) 2 + (σ3 − σ1) 2 2 1/2 ≥ Sy (5–10) If we had a simple case of tension σ , then yield would occur when σ ≥ Sy . Thus, the left of Eq. (5–10) can be thought of as a single, equivalent, or effective stress for the entire general state of stress given by σ1, σ2, and σ3. This effective stress is usually 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 σ ≥ Sy (5–11) where the von Mises stress is σ =  (σ1 − σ2) 2 + (σ2 − σ3) 2 + (σ3 − σ1) 2 2 1/2 (5–12) For plane stress, let σA and σB be the two nonzero principal stresses. Then from Eq. (5–12), we get σ =  σ2 A − σAσB + σ2 B 1/2 (5–13)