StressFailureCriteriaofPlaneStress

StressFailureCriteriaofPlaneStress - Stress Failure...

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Unformatted text preview: Stress Failure Criteria of Plane Stress 1. Standard Coupon Test We can state that the element or component will be safe as long as |0'l|<0'y and |02|<0y (maximum normal stress criterion), where oy is the yield strength of the tested specimen. However. . Since this above state is different from the state of uniaxial stress found in a specimen subjected to a tensile test, it is clearly not possible to predict directly from such a test whether or not the structural element or machine component under investigation will fail. Stress Failure Criteria -1- 10/09/2008 2. Maximum-Shearing—Stress Criterion (Tresca Yield Criterion) This criterion is based on the observation that yield in ductile materials is caused by slippage of the material along oblique surfaces and is due primarily to shearing stresses. According to this criterion, a given structural component is safe as long as the maximum value of the shearing stress in that component remains smaller than the corresponding value of the maximum shearing stress in a tensile-test specimen of the same material as the specimen starts to yield. For a uniaxial tension coupon test (think about the corresponding Mohr is circle): 0' R =—y (1) coupon 2 For a component (also think about the corresponding Mohr is circle): If the principal stresses 6] and 62 have the same sign, the maximum-shearing-stress criterion gives: R=|01|<R =_0y:>lo-|<o' (2) 2 coupon 2 l y o- 0' R = l—22‘l— < Rcoupon : 7y Z) '02! < 6y If the principal stresses 61 and (52 have opposite signs, the maximum—shearing-stress criterion gives: IO‘l—O'zl 0' <R 2—i:>laJ—02|<0'y (4) R = . 2 coupon 2 Graphically: Given a combination of principal stresses and an allowable yield stress, and using the Tresca yield criterion, one can check if the material has exceeded the allowable yield stress. This surface (originally defined by Tresca) does not require that 61>02. However, using the definition that the first principal stress is greater than the second (i.e. 61>62), the upper left side of the dashed line in the above figure can not happen. Hence, only the bottom right needs to be considered. Stress Failure Criteria -2- 10/09/2008 3. Maximum-Distortion-Energy Criterion (von Mises Yield Criterion) This criterion is based on the determination of the distortion energy in a given material. According to this criterion, a given structural component is safe as long as the maximum value of the distortion energy per unit volume in that material remains smaller than the distortion energy per unit volume required to cause yield in a tensile—test specimen of the same material. The distortion of energy per unit volume (derivation of which is beyond the scope of this course) is : l ud=6—5(012—0,02+022) (5) In the particular case ofa tensile—test specimen which is starting to yield, we have 1 2 (ad), — 35% (6) Thus, the von Mises criterion indicates that the structural component is safe as long as ud < (ud )y , which gives: 0'12 —0'10'2 + 022 < 0'; (7) Graphically, The shaded area is bounded by the ellipse of equation 012— 0'10'2 + 0'22 = of (8) Again, given a combination of principal stresses and an allowable yield stress, and using the von Mises yield criterion, one can check if the material has exceeded the allowable yield stress. This surface (originally defined by von Mises) does not require that 01>02. However, using the definition that the first principal stress is greater than the second (i.e. 61>62), the upper left side of the dashed line in the above figure can not happen. Hence, only the bottom right needs to be considered. Stress Failure Criteria -3- 10/09/2008 4. Tresca Criterion vs. von Mises Criterion The Tresca criterion and the von Mises criterion are compared in the following figure. We note that the ellipse passes through the vertices of the hexagon. Thus, for the state of stress represented by these six points, the two criteria give the same results. For any other state of stress, the Tresca criterion is more conservative than the von Mises criterion, since the hexagon is located within the ellipse. A state of stress of particular interest is that associated with yield in a pure shear stress condition (recall mohr’s circle). In that case (0'] =—0'2 ), the corresponding points are located on the bisector of the second and forth quadrants. It follows that yield occurs in pure shear stress condition when 012—02 20.50'y according to the Tresca criterion; and when 0', 2 ~02 = 0.5770} according to the von Mises criterion. Note that based on the experimental results of various ductile materials, the von Mises criterion appears more somewhat more accurate. 5. 3D Cases Tresca criterion: < “y” (9) Von Mises Criterion: (a,—a2)2+(al—a3)2+(az—a3)2 =2ay2 (10) Stress Failure Criteria -4— 10/09/2008 at ‘40 M Pt] < (7", +< 7 MM 7 7» (TH L’- (r y T 250 MPa 4* a, kl (n 250 M PH 1' (r SAMPLE PROBLEM 7.4 The state of plane stress shown occurs at a critical point of a steel machine component. As a result of several tensile tests, it has been found that the ten, sile yield strength is (rY : 250 MPa for the grade of steel used Determine the (actor ot safety with respect to yield. using (a) the maxinnnn—shearirig—stress criterion. and (b) the maximurn—distortion-energy criterion. SOLUTION Mohr’s Circle. We construct Mohr’s circle for the given state of stress and find (so 7 40) : 20 Mm Ni.— (rw 1‘ ()C : H0". + cry) : 7m : R : \V/(ic??: (FX) N : \w/(eo‘)? Vi (25ftj : (,5 mm Principal Stresses 01: 0C + CA : 20 + ()5 : +85 MP3 2 0C # BC : 20 — 65 S "’45 MP2) (1'2 (1. hrliiximumShearing-Stress Criterion. Since for the grade ol‘ steel used the tensile strength is (Ty : 250 Ml’a, the corresponding shearing stress at yield is (250 MPa) : l25 MPzi NH l SOY: TY : 7Y7 _ i25 Mi): * : 1.92 4 T MPa m For TW :7 (55 lVlPll‘. PS. I ['13. b. Maximom—Distortion-Energy Criterion. Introducing a factor of safety into Eq. (7.26), we write t >2 PS : +85 MPa, (r; : '45 MPa, and (Ty : 250 MPa, we have 2 n g a!“ e 0,072+ 0'3 For (I; (85)2 e- <85>e45> + W 3 1 14.3 f , ES. Comment. For a ductile material with (Ty :' 250 MPa, we have drawn the hexagon associated with the maximum—sheenrig—stress criterion and the 61- lipse associated with the maximum—(iistortion-energy criterion, The given state of plane stress is represented by point H of coordinates a: : 85 MPa and (5;, : #45 MPa We note that the straight line drawn through points 0 anti H intersects the hexagon at point T and the ellipse at point M. For each amen?“ the value obtained for F5. can be verified by measuring the line segments 10' dicated and computing their ratios: OT F.S. : i! = 1.92 0H (d) (b) («15. : in : 2.19 ...
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StressFailureCriteriaofPlaneStress - Stress Failure...

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