Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

(b) What is the value of the Mises equivalent stress for the given state of plane stress?
(c) What is the factor of safety predicted by the failure criterion of the maximum-distortionenergy
theory of failure? According to this theory, does the component fail?
Plan the Solution
The principal stresses will be determined for the given state of stress. With these stresses,
the maximum-shear-stress theory and the maximum-distortion-energy theory will be
used to investigate the potential for failure.
SolutioN
The principal stresses can be calculated from the stress transformation equations [Equation
(12.12)] or from Mohr’s circle, as discussed in Section 12.10. Equation 12.12 will be
used here. From the stress element, the values to be used in the stress transformation
equations are s x = +75 MPa, s y = 0 MPa, and τ xy = +90 MPa. The in-plane principal
stresses are calculated as
s
p1, p2
s + s ⎛s − s
= ± ⎜
2 ⎝ 2
x y x y
2
⎞
⎟ + τ
⎠
75 MPa + 0 MPa ⎛ 75 MPa − 0 MPa ⎞
=
± ⎜
⎟ + (90 MPa)
2
⎝ 2 ⎠
= 135.0 MPa, −60.0 MPa
2
xy
2
2
(a) Maximum-Shear-Stress theory
Since s p1 is positive and s p2 is negative, failure will occur if s p1 − s p2 ≥ s Y . For the principal
stresses existing in the component,
s
− s = 135.0 MPa − ( − 60.0 MPa) = 195.0 MPa < 270 MPa
p1 p2
Therefore, according to the maximum-shear-stress theory, the component does not fail.
The factor of safety associated with this state of stress is
270 MPa
FS = = 1.385
Ans.
195.0 MPa
(b) Mises Equivalent Stress
The Mises equivalent stress s M associated with the maximum-distortion-energy theory
can be calculated from Equation (15.8) for the plane stress state considered here:
2
s = [ s − s s + s ]
2 1/2
M p1
p1 p2 p2
= [(135.0 MPa) − (135.0 MPa)( − 60.0 MPa) + ( −60.0 MPa) ]
2 2 1/2
= 173.0 MPa
Ans.
(c) Maximum-Distortion-Energy theory Factor of Safety
The factor of safety for the maximum-distortion-energy theory can be calculated from the
Mises equivalent stress:
270 MPa
FS = = 1.561
Ans.
173.0 MPa
According to the maximum-distortion-energy theory, the component does not fail.
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