Page 347 - Shigley's Mechanical Engineering Design

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322 Mechanical Engineering Design
hidden cycle is applied but once. If the all-positive stress cycle is applied alternately
with the all-negative stress cycle, the hidden cycle is applied 100 times.
To ensure that the hidden cycle is not lost, begin on the snapshot with the largest
(or smallest) stress and add previous history to the right side, as was done in Fig. 6–33b.
Characterization of a cycle takes on a max–min–same max (or min–max–same min)
form. We identify the hidden cycle first by moving along the dashed-line trace in
Fig. 6–33b identifying a cycle with an 80-kpsi max, a 60-kpsi min, and returning to
80 kpsi. Mentally deleting the used part of the trace (the dashed line) leaves a 40, 60,
40 cycle and a −40, −20, −40 cycle. Since failure loci are expressed in terms of stress
amplitude component σ a and steady component σ m , we use Eq. (6–36) to construct the
table below:

Cycle Number max min a m

1 80 60 70 10
2 60 40 10 50
3 20 40 10 30

The most damaging cycle is number 1. It could have been lost.
Methods for counting cycles include:
• Number of tensile peaks to failure.
• All maxima above the waveform mean, all minima below.
• The global maxima between crossings above the mean and the global minima
between crossings below the mean.
• All positive slope crossings of levels above the mean, and all negative slope cross-
ings of levels below the mean.
• A modification of the preceding method with only one count made between succes-
sive crossings of a level associated with each counting level.
• Each local max–min excursion is counted as a half-cycle, and the associated ampli-
tude is half-range.
• The preceding method plus consideration of the local mean.
• Rain-flow counting technique.

The method used here amounts to a variation of the rain-flow counting technique.
The Palmgren-Miner 24 cycle-ratio summation rule, also called Miner’s rule, is
written

n i

= c (6–57)
N i
where n i is the number of cycles at stress level σ i and N i is the number of cycles to fail-
ure at stress level σ i . The parameter c has been determined by experiment; it is usually
found in the range 0.7 < c < 2.2 with an average value near unity.

24 A. Palmgren, “Die Lebensdauer von Kugellagern,” ZVDI, vol. 68, pp. 339–341, 1924; M. A. Miner,
“Cumulative Damage in Fatigue,” J. Appl. Mech., vol. 12, Trans. ASME, vol. 67, pp. A159–A164, 1945.