Page 419 - Mechanical Behavior of Materials

420 Chapter 9 Fatigue of Materials: Introduction and Stress-Based Approach

Cyclic stressing with zero mean can be specified by giving the amplitude σ a ,orbygivingthe
numerically equal maximum stress, σ max . If the mean stress is not zero, two independent values are
needed to specify the loading. Some combinations that may be used are σ a and σ m , σ max and R,
σ and R, σ max and σ min , and σ a and A.The term completely reversed cycling is used to describe
a situation of σ m = 0, or R =−1, as in Fig. 9.2(a). Also, zero-to-tension cycling refers to cases
of σ min = 0, or R = 0, as in Fig. 9.2(c).
The same system of subscripts and the prefix are used in an analogous manner for other
variables, such as strain ε, force P, bending moment M, and nominal stress S. For example, P max
and P min are maximum and minimum force, P is force range, P m is mean force, and P a is force
amplitude. If there is any possibility of confusion as to what variable is used with the ratios R or A,
a subscript should be employed, such as R ε for strain ratio.

9.2.2 Point Stresses Versus Nominal Stresses
It is important to distinguish between the stress at a point, σ, and the nominal or average stress, S,
and for this reason we use two different symbols. Nominal stress is calculated from force or moment
or their combination as a matter of convenience and is only equal to σ in certain situations. Consider
the three cases of Fig. 9.3. For simple axial loading (a), the stress σ is the same everywhere and so
is equal to the average value S = P/A, where A is the cross-sectional area.
For bending, it is conventional to calculate S from the elastic bending equation, S = Mc/I,
where c is the distance from neutral axis to edge and I is the area moment of inertia about the
bending axis. Hence, σ = S at the edge of the bending member, with σ, of course, being less
elsewhere, as illustrated in Fig. 9.3(b). However, if yielding occurs, the actual stress distribution
becomes nonlinear, and σ at the edge of the member is no longer equal to S. This is also illustrated
in Fig. 9.3(b). Despite the limitation to elastic behavior, such values of S are often calculated beyond
yielding, and this can lead to confusion. Stresses σ for bending beyond yielding can be obtained by
replacing the elastic bending formula with more general analysis, as described in Chapter 13.
For notched members, nominal stress S is conventionally calculated from the net area remaining
after removal of the notch. (The term notch is used in a generic sense to indicate any stress raiser,
including holes, grooves, fillets, etc.) If the loading is axial, S = P/A is used, and for bending,
S = Mc/I is calculated on the basis of bending across the net area. Due to the stress raiser effect,
such an S needs to be multiplied by an elastic stress concentration factor, k t , to obtain the peak stress
at the notch, σ = k t S, as illustrated in Fig. 9.3(c). (Values of k t and corresponding definitions of S
are given for some representative cases in Appendix A, Figs. A.11 and A.12.) Note that k t is based
on linear-elastic materials behavior, and the value does not apply if there is yielding. Where yielding
occurs even locally at the notch, the actual stress σ is lower than k t S, as also illustrated in Fig. 9.3(c).
Stresses in notched members and other complex geometries may be determined from finite
element analysis or other numerical methods. Such analysis most commonly considers only linear-
elastic materials behavior, so calculated stresses are, again, not correct if yielding occurs—that is, if
the calculated stress exceeds the yield strength σ o .
To avoid confusion, we will strictly observe the distinction between the stress σ at a point of
interest and nominal stress S. For axial loading of unnotched members, where σ = S, we will use σ.
However, for bending and notched members, S or k t S are employed, except where it is truly
appropriate to use σ.