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

ExAmpLE 15.9
The stresses on the free surface of a machine component are illustrated on the stress element
shown. The component is made of a brittle material with an ultimate tensile strength
of 200 MPa and an ultimate compressive strength of 500 MPa. Use the Mohr failure
criterion to determine whether this component is safe for the state of stress shown.
Plan the Solution
The principal stresses will be determined for the given state of stress. With these stresses,
the Mohr failure criterion will be used to investigate the potential for failure.
SolutioN
The principal stresses can be calculated from Equation 12.12:
y
260 MPa
120 MPa
x
60 MPa
s
p1, p2
s + s ⎛s − s
= ± ⎜
2 ⎝ 2
x y x y
60 MPa + ( −260 MPa)
=
±
2
= 100 MPa, −300 MPa
2
⎞
⎟ + τ
⎠
2
xy
⎛60 MPa − ( −260 MPa) ⎞
⎜
⎟ + ( −120 MPa)
⎝ 2 ⎠
2
2
Mohr Failure Criterion
Since σ p1 is positive and σ p2 is negative, failure will occur if
σ
σ
p1 p2
UT
σ
−
σ
UC
≥ 1
For the principal stresses existing in the component,
σ
σ
σ p 100 MPa ( 300 MPa)
− = − − = 0.5 − ( − 0.6) = 1.1 > 1 Ans.
σ 200 MPa 500 MPa
p1 2
UT
UC
Therefore, according to the Mohr failure criterion, the component fails.
pRoBLEmS
p15.52 The stresses on the surface of a structural steel beam are
σ x = 38 ksi, σ y = 0, and τ xy = 16.5 ksi. The steel has a yield strength
of σ Y = 50 ksi.
(a) What is the factor of safety predicted by the maximum-shearstress
theory of failure for the stress state presented? Does the
beam fail according to this theory?
(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-distortion-energy theory of failure?
According to this theory, does the beam fail?
p15.53 The stresses on the surface of a cast iron machine component
are σ x = 85 MPa, σ y = −230 MPa, and τ xy = 125 MPa. The
cast iron has a yield strength of σ Y = 350 MPa.
(a) What is the factor of safety predicted by the maximum-shearstress
theory of failure for the stress state presented? According
to this theory, does the component fail?
(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-distortion-energy theory of failure? According
to this theory, does the component fail?
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