Page 461 - Mechanical Behavior of Materials

Section 9.7 Mean Stresses 461

Similarly, calculating all of the values and plotting versus N f , while also including the
Table E9.1 data and the Eq. 9.22 line, gives Fig. E9.5(b).
The correlation of the data with the line for zero mean stress is now much improved, and no

clear trends are seen except for some scatter. Hence, the equation of Morrow with σ provides a
f
reasonably accurate representation of these data. (Ans.)
9.7.5 Safety Factors with Mean Stress

The discussion in Section 9.2.4 on safety factors can be generalized to include cases with
nonzero mean stress. One option is to apply the logic of Fig. 9.6 with the stress amplitude
σ a simply replaced by an equivalent completely reversed stress amplitude, σ ar . The stress–life
curve is then σ ar = f (N f ), and safety factors in stress and life are calculated by generalizing
Eqs. 9.9 and 9.10:

σ ar1 N f 2
X S = , X N = (a, b) (9.25)
ˆ
ˆ σ ar N f = ˆ N N σ ar =ˆσ ar
The value of ˆσ ar is calculated from the stress amplitude ˆσ a and mean stress ˆσ m expected to occur
in actual service, with the use of Eq. 9.18 or 9.21 or other similar mean stress relationship. Also,
−b
for stress–life curves of the form of Eq. 9.7, the two safety factors are related by X S = X from
N
Eq. 9.12.
A second option is to multiply ˆσ a and ˆσ m by load factors Y a and Y m , respectively, to calculate
a value of equivalent completely reversed stress σ that cannot exceed the stress–life curve at the
ar1
ˆ
ˆ
desired service life N, so that σ ≤ f (N) is required. For example, the Morrow σ ar expression of
ar1
Eq. 9.21 is used with the stress–life curve of Eq. 9.22 as follows:
Y a ˆσ a b

ˆ
σ ar1 = , σ ar1 ≤ σ (2N) (a, b) (9.26)
f
Y m ˆσ m
1 −
σ
f
The SWT expression of Eq. 9.18, with σ max = σ m + σ a substituted, is employed as
b

σ ar1 = (Y m ˆσ m + Y a ˆσ a )Y a ˆσ a , σ ar1 ≤ σ (2N) (a, b) (9.27)
ˆ
f
The load factor approach has the advantage that different values can be assigned to Y a and Y m ,
which may be desirable if the value of one of ˆσ a or ˆσ m is more uncertain than the other.
Assume that the same load factor is applied for both the stress amplitude and mean, Y = Y a =

Y m . This common load factor can be factored out of Eq. 9.27, so that σ = Y σ max ˆσ a = Y ˆσ ar .
ˆ
ar1
Comparison with Eq. 9.25(a) gives Y = X S , so that the load factor and the safety factor in stress are
equivalent if SWT is employed. Such Y = X S equivalence also applies for the Walker mean stress
relationship, Eq. 9.19, but not for the Morrow or Goodman equations, as the latter mathematical
forms do not allow similar factoring.