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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. MaximumShearing—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 tensiletest 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 maximumshearingstress
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—shearingstress
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. MaximumDistortionEnergy 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—distortionenergy 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. hrliiximumShearingStress 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—DistortionEnergy 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—(iistortionenergy 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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 Spring '06
 Elghandour

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