Page 53 - Analysis and Design of Machine Elements

Strength of Machine Elements
For a general stress state, three principal stresses can be determined and ordered such 31
that ≥ ≥ .Themaximum shearstressisthen max = ( − )/2. Thus, for a gen-
2
1
1
3
3
eral state of stress, the maximum shear stress theory predicts yield strength as
− 3 s
1
max = ≤ or − ≤ s (2.6)
3
1
2 2
This implies that the shear yield strength is given by
s
= (2.7)
s
2
2.2.2.2 Maximum Distortion Energy Theory
The maximum distortion energy theory predicts that yield occurs when the distortion
energy per unit volume reaches or exceeds the distortion energy per unit volume for
yield in uniaxial tension of the same material [6, 7]. It is also called von Mises or von
Mises–Hencky theory, giving credit to R. von Mises and H. Hencky for the development
of it.
When applying this theory, it is convenient to use an equivalent stress ,orvon Mises
e
stress, to transform multiaxial stresses into an equivalent uniaxial stress. The equivalent
stress is the value of a uniaxial tensile stress that would produce the same level of dis-
tortion energy as the actual stresses involved [6, 7]. By the distortion energy theory, the
equivalent stress can be derived and expressed by principal stresses as
1 2 2 2 1∕2
= √ [( − ) +( − ) +( − ) ] (2.8)
1
e
2
2
3
3
1
2
Thus, the yield strength criterion is expressed as
1 2 2 2 1∕2
√ [( − ) +( − ) +( − ) ] ≤ s (2.9)
1
3
3
2
1
2
2
When using xyz components of the stress, the von Mises stress and yield strength
criterion can be rewritten as
1 2 2 2 2 2 2 1∕2
√ [( − ) +( − ) +( − ) + 6( + + )] ≤ s (2.10)
z
zx
z
yz
y
x
x
xy
y
2
For pure shear in plane problem where = = 0, the yield strength can be obtained
x y
from Eq. (2.10) as
s
xy
≤ √ (2.11)
3
Thus, the shear yield strength predicted by the maximum distortion energy theory is
s
= √ = 0.577 s (2.12)
s
3
Both the maximum shear stress theory and the maximum distortion energy theory
can be applied in the analysis and design of a machine element. The maximum shear
stress theory gives a simple and moderately conservative approach; while the maximum
distortion energy theory provides a more accurate prediction [7].