Hydrocarbon Processing - March 2021 - 13

Engineering
Case Histories

A. SOFRONAS, CONSULTING ENGINEER
http://mechanicalengineeringhelp.com

Case 111: Predict the cause of cracks
in gear teeth and time to failure
In the practice of engineering, we
sometimes wish for better hindsight or
foresight. This may occur when management questions why something has
occurred, and what the risk is to operate
until the next planned turnaround. In
such a scenario, management is essentially asking for a prediction of the past
and the future.
Consider the following case history:
A crack is noticed on a gear tooth after
a sudden power outage. The risk is that
if the tooth breaks off and falls into the
gear mesh, a major wreck could occur,
resulting in a long outage. Knowing
when and why the crack started is also
important to ensure that a repeat event
will not occur. Starting up after a gear set
replacement without knowing the cause
of the crack is always risky.
Crack growth-type equations can be
used on ductile steels. Knowing or assuming an initial crack size and knowing
the cyclic stress trying to open the crack,
the cycles to grow the crack to a certain
length can be estimated.
For this case, it is assumed that some
event started a crack in the tooth root
across the tooth face, as shown on FIG. 1.
It takes a very large shock load of many
times the allowable stress to cause a
crack across the face of a case-hardened
gear tooth.1 The case depth may be ai
= 0.05 in. deep in the tooth face width
(b). When the crack was first noticed, it
appeared to have grown to af = 0.13 in.
length. When the crack progresses halfway through the tooth as shown, it will
break off in a bending-type failure at af =
0.5 in. This happens because as the crack
grows, it reduces the support material.
The cycles (N ) to grow the crack can
be approximated by integrating the Paris
Differential Equation, shown in Eq. 1:
da ÷ dN = C × (ΔK)m

(1)

where:
dN = Cycle length
da = Crack growth during the cycle
ΔK = Stress intensity at the crack tip
C, m = Material constants.2
For a large plate where the cross-section does not significantly change as the
crack grows, the constant surface bending stress (Δσ) from zero to peak stress
can be used, as shown in Eq. 2:
N = [8.3 × 108 ÷ Δσ 3.25 ] ×
[1 ÷ ai0.625 - 1 ÷ af 0.625]

(2)

The calculation is different for a
rectangular beam-shaped gear tooth, as
shown in FIG. 2. The stress becomes an
iterative procedure at each growth step,
since the bending resistance of the supporting material decreases as the crack
grows. Only two growth steps are used
in this example, with the average stress in
each interval labeled a. TABLE 1 shows the
results from the calculations.
From this data, it appears that the
power outage could have been the cause,
since sudden power outages can result
in a large gear tooth shock load.3 Once
started, the crack grows very quickly.
Caution suggests not starting up the unit
without a gear set replacement. This is
an approximate method of looking into
the past and future, but it does help lower the risk in making a decision.

NOTE
Case 110 was published in HP November
2020. For past cases, please visit
www.HydrocarbonProcessing.com.
LITERATURE CITED
Handschuh, R., B. A. Lerch and C. S. Burke,
" Investigation of low-cycle bending fatigue of AISI
9310 steel spur gears, " U.S. NASA/U.S. Army
Research Laboratory, TM-2007-214914, July 2007.
2
Sofronas, A., Analytical Troubleshooting of Processing
Machinery and Pressure Vessels Including Real-World
Case Studies, Wiley, Hoboken, New Jersey, 2006.
3
Sofronas, A., Unique Engineering Methods for
Analyzing Failures and Catastrophic Events: A
Practical Guide for Engineers, Ch. 7.2: " Sudden Power
Interruption To A System, " Wiley, Hoboken, New
Jersey, To be published, 2021.
1

TONY SOFRONAS, D. Eng, was the
worldwide lead mechanical engineer
for ExxonMobil Chemicals before
retiring. He now owns Engineered
Products, which provides consulting
and engineering seminars on
machinery and pressure vessels.
Dr. Sofronas has authored several engineering books
and numerous technical articles on analytical methods.

FIG. 1. Crack in a normal gear tooth.

TABLE 1. Crack growth with stress
Cyclic stress
on tooth

ai

af

N

Life at
1,200
rpm

Δσ = 11.6 ksi
to start

0.05
in.

0.13
in.

8.7 × 105
cycles

12 hr
ago

0.13
in.

0.50 1.8 × 105
cycles
in.

2.5 hr
to fail

a = 0.09 in.
average
Δσ = 16.9 ksi
to fail
a = 0.31 in.
average

FIG. 2. Crack in a rectangular beam-shaped
gear tooth.
Hydrocarbon Processing | MARCH 2021

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