Page 264 - Marks Calculation for Machine Design

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January 4, 2005
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Brown.cls
Brown˙C06
STRENGTH OF MACHINES
246
where again the factors-of-safety are smaller than that obtained with the distortion-energy
theory. This is what is meant by being more conservative, or more restrictive.
The fact that three of the four combinations in Example 1 resulted in unsafe designs,
and the fourth was literally borderline, indicates that a stronger material should be used for
these machine elements. For example, the yield stress (S y ) could be doubled by choosing
cast iron, or tripled by choosing a structural steel like ASTM-A36.
Comparison of Maximum-Shear-Stress Theory and Distortion-Energy Theory. In
Eq. (6.4), repeated here, the maximum shear stress (τ max ) associated with the maximum-
shear-stress theory was found to be related to the yield stress (S y ) as
σ 1 − σ 2 S y − 0 S y
τ max = = = = S sy (6.4)
2 2 2
or in decimal form
τ max = 0.5 S y (6.10)
For special cases of torsion, which is a pure shear condition where the maximum principal
stress (σ 1 ) is the shear stress (τ) and the minimum principal stress (σ 2 ) is the negative of
the shear stress (−τ), the distortion-energy theory from Eq. (6.8), repeated here, gives
2 2 2
σ + σ − σ 1 σ 2 = S (6.8)
1 2 y
where the inequality sign has been replaced by an equal to sign. Substituting the shear stress
(τ), which would actually be the maximum shear stress (τ max ), and (−τ) gives
2 2 2 2 2 2 2
(τ) + (−τ) − (τ)(−τ) = τ + τ + τ = 3 τ = S y
S 2
2 y
τ = (6.11)
3
S y
τ = √ = 0.577 S y
3
Summarizing Eqs. (6.10) and (6.11) gives

0.5 S y maximum-shear-stress theory
τ max = (6.12)
0.577 S y distortion-energy theory
It is common to see the distortion-energy theory rounded to (0.60 S y ) instead of the three
decimal place result given in Eq. (6.12).

6.1.2 Static Design for Brittle Materials
In contrast to ductile materials, brittle materials exhibit a true strain at fracture of less than
5 percent. Failure of a machine element made of a brittle material is usually associated
with the element suddenly fracturing. Therefore, the important strength for determining if
the design of the machine element under static conditions is safe, is the ultimate strength
(S u ). As mentioned earlier, brittle materials have an ultimate strength in compression,
designated (S uc ), significantly greater than its ultimate strength in tension, designated (S ut ).
(In Figs. 6.10 through 6.17 that follow, S uc = 3S ut .)