Efficient Plant October 2017 - 30

feature | reliability strategies

WEIBULL
PROBABILITY
DENSITY FUNCTION:

f(T) = [β/η][(T- γ)/η]
(β-1)
[e-(T- γ)β]

Bathtub Curve
Infant
Mortality
Failures

End of Life,
Wear and Tear
Failures

Zone I
Failure Rate

Fig. 1. As shown here, a typical
bathtub curve is composed of
three distinct parts or zones
that play an important role in
tracking and predicting equipment failure.

β = 0.5

Zone III
Normal Service Life

β = 3.0

Random Failures
Zone II

where :

f(T) = probability

β = 1.0

density function

β = shape function or
Weibull slope

T = time in service

or number of
operation cycles

η = scaling factor that

defines the scale
for this equipment

γ = factor based on

location, i.e.,
environmental
conditions, often
set equal to zero.

30

Service Time, Cycles

has passed through Zone I and is beginning to
operate in Zone II.
 Zone II has a Weibull β shape factor equal to
about 1.0. This portion of the bathtub curve
is typically associated with normal service
life and the failure rate in this zone is relatively low. It is basically a random failure rate
function that is more or less linear. (Referring
to the accompanying Sidebar, note how the
equation simplifies when β = 1 and γ = 0.) The
failure-rate-versus-time curve in Zone II is
either flat, with little increase over time, or the
failure rate increase over time is gradual. This
is the longest portion of the bathtub curve and
represents the period of time when the equipment provides reliable service with normal
maintenance and care.
 Zone III has a Weibull function β shape factor
equal to about 3.0. A significant change in the
β shape factor signals a significant change in
the failure rate. This portion of the bathtub
curve is associated with end-of-life failures due
to age and wear. As the equipment is operated further into this region of the curve, the
failure rate increases exponentially.

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Some contend that the optimum time to replace
or refurbish equipment to maximize its service time
and minimize risk, based upon the bathtub curve, is
when the failure rate in Zone III becomes equal to
or slightly greater than the maximum failure rate in
Zone I. That means replacing equipment when the
chance of failure due to age effects becomes equal
to or slightly greater than the chance of failure due
to infant mortality. Otherwise, according to this
argument, there's a greater risk of failure in replacing
the equipment than just letting it remain in place.
While inspection of the bathtub curve in Fig. 1 indicates the basic merit of this idea, at least as a starting
point, there are two factors to consider. They could
lead to a modification of this strategy.
During the Zone 1 infant mortality period,
the equipment often isn't actually in service. In
recognition of the potential for installation errors,
assembly errors, or defective parts, many organizations require that equipment be run or tested
without being in a production mode to check for
such latent deficiencies.
This is the reasoning behind static-pressure
testing, load testing, post-maintenance testing, and
circuit-board burn-in periods. The idea is to stress

OCTOBER 2017
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Table of Contents for the Digital Edition of Efficient Plant October 2017
