Page 64 - Analysis and Design of Machine Elements

Analysis and Design of Machine Elements
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calculation. When the cumulative damage approaches 1 or 100%, fatigue failure ensues.
The linear cumulative fatigue damage rule or Miner’s rule can be expressed as
n 1 n 2 n i n z
+ +···+ +···+ = 1 (2.32)
N 1 N 2 N i N z
For a more general case,
z
∑ n
i
= 1 (2.33)
N
i=1 i
2.3.5.2 Prediction of Cumulative Fatigue Damage
We have introduced fatigue prediction for constant stresses amplitude. For variable
amplitude stresses, we need first to convert them to constant amplitude stresses and
then use the previously introduced method to predict fatigue.
From the S-N curve, we have
m
m
N = N = Const. (2.34)
i i −1 0
Therefore
( ) m
−1
N = N (2.35)
i 0

i
Substituting Eq. (2.35) into the linear cumulative-damage rule in Eq. (2.33), we have
z
∑ m m
n = N (2.36)
0 −1
i i
i=1
When Eq. (2.36) is satisfied, it indicates that the element reaches the endurance limit.
Assuming the fatigue effect of variable amplitude stresses is equivalent to that of a con-
stant amplitude stress (usually select = ) that operates the equivalent number of
v
1
v
cycles N ,wethenhave
v
z
∑ m m m
n = N • = N • 1 (2.37)
v
v
v
i i
i=1
The equivalent number of cycles N is then derived as
v
z ( ) m
∑ i
N = n i (2.38)
v
i=1 1
Corresponding to the equivalent number of cycles N , the endurance strength for
v
completely reversed stress is
−1Nv
m
m N = N (2.39)
−1Nv v −1 0
Therefore
√ √
N 0 √ N 0
√
= m = √ (2.40)
−1Nv −1 −1 m √ z
N ( ) m
v √ ∑ i
n i
i=1 1