Page 63 - Analysis and Design of Machine Elements

Notes: Strength of Machine Elements 41
1. When it is difficult to determine the possible variation of stresses, use r = Const. to
start the calculation.
2. An equivalent set of formulas also holds for cyclic shear stresses. For cases involving
fluctuating torsional shear stresses, substitute with .
3. When the number of cycles is less than the critical number of cycles, that is,
√
3
10 < N < N , the endurance strength at the number of cycle N is = m N 0 ,
rN
0
N r
instead of . Therefore, −1 should be substituted with −1N .
r
4. If the endurance limit is located around point F in the − diagram, both fatigue
a
m
1
strength and static strength need to be calculated.
5. The fatigue safety factor can be obtained by either a graphical or analytical approach.
In summary, fatigue strength analyses involve establishing a relationship between the
fatigue strength of the material or element and the working stress to determine safety
factors. The fatigue strength of the material or element is represented by a − dia-
m a
gram; while the working stress is the sum of mean stress and stress amplitude .
m a
2.3.5 Fatigue Strength for Uniaxial Stresses with Variable Amplitude
The discussion so far has dealt with fatigue behaviour of an element under uniform stress
amplitudes. In real engineering practice, nearly all machine elements are subject to a
spectrum of speeds or loads, giving rise to stresses varying in both amplitudes and mean

values. Automotive suspension and aircraft structural components are typical examples
of elements subject to a spectrum of variable stress amplitudes.
The variations in stress amplitudes make the direct use of standard S-N curves inap-
plicable because these curves are developed from constant stress amplitude operations.
Therefore, it becomes necessity to develop a verified theory to predict fatigue strength
for spectrum load operations using the constant amplitude S-N curves.
The linear cumulative damage rule, proposed by Palmgren of Sweden in 1924 and,
independently, by Miner of the United States in 1945, can be used to deal with fatigue
strength analysis for elements under variable uniaxial stresses. The rule assumes that
any stress amplitude greater than the endurance limit contributes certain fatigue dam-
age to the element. The amount of damage depends on the number of cycles at that
stress amplitude and the total number of cycles that would produce failure to a standard
specimen at the same stress amplitude. When the total accumulated damage generated
by different stress levels reaches a critical value, fatigue failure occurs [15].

2.3.5.1 Linear Cumulative Damage Rule (Miner’s Rule)
Assume that , , … arethe maximumstressofeachcycle; n , n , n … are the
2
1
3
2
1
3
number of cycles acting on the element at each stress level and N , N , N … are the
1
i
2
3
number of cycles to failureateachstresslevel acquired from the S-N curve.
i
For stresses greater than the endurance limit, each cycle will cause certain damage.
The damage contribution of n cycles at stress is assumed to be n /N ; while the
1
1
1
1
damage contribution of n cycles at stress is assumed to be n /N .... Similarly, the
2
2
2
2
damage caused by n cycles at stress is n /N . For stresses smaller than the endurance
i
i
i
i
limit, it is assumed that they will not cause any damage and can be neglected in the