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88 • using ansys for finite eLement anaLysis
Where:
∧ ∑ N � = i 1 lnX i
m =
N
If the location parameter is known, it can be subtracted from the origi-
nal data points before computing the maximum likelihood estimates of the
shape and scale parameters.
3.2.4.3 Comments
The lognormal distribution is a basic and commonly used distribution.
It is typically used to describe the scatter of the measurement data
of physical phenomena, where the logarithm of the data would fol-
low a normal distribution. The lognormal distribution is very suitable
for phenomena that arise from the multiplication of a large number of
error effects. It is also correct to use the lognormal distribution for a
random variable that is the result of multiplying two or more random
effects (if the effects that get multiplied are also lognormally distrib-
uted). If is often used for lifetime distributions; for example, the scatter
of the strain amplitude of a cyclic loading that a material can endure
until low-cycle-fatigue occurs is very often described by a lognormal
distribution.
The lognormal distribution is used extensively in reliability appli-
cations to model failure times. The lognormal and Weibull distribu-
tions are probably the most commonly used distributions in reliability
applications.
3.2.5 WeibULL DiSTRibUTion
3.2.5.1 Probability Density Function
The formula for the pdf of the general Weibull distribution is:
(
g x − m (g − ) 1 x − ) ) )
g
fx () = exp − ( ( ma / x ≥ mg a , > 0
;
a a 
