Page 277 - Handbook of Materials Failure Analysis

6 Parametric Accelerated Life Testing 273

6.2 SAMPLE SIZE EQUATION
The Cumulative Distribution Function (CDF) in Weibull can be expressed as:
β
ðÞ
t
η (11.6)
FtðÞ ¼ 1 e
Where t is time and η is characteristic life.
The Weibull reliability function, R(t), is expressed as:
t β
ðÞ (11.7)
η
RtðÞ ¼ e
The characteristic life η MLE from the Maximum Likelihood Estimate (MLE) can be
derived as:
X β
β n
η ¼ t i (11.8)
MLE
i¼1 r
Where n is sample size, t i is testing duration for each sample, and r is the failure
numbers.
If confidence level is 100(1 α) and the number of failure is r 1, characteristic
life, η α , would be estimated from Equation 11.8,

β 2r β 2 X β
n
η ¼ 2 η ¼ 2 t for r 1 (11.9)
α
ð
χ 2r +2Þ MLE χ 2r +2Þ i¼1 i
ð
α
α
Presuming there is no failures, p-value is α and In(1/α) is mathematically equivalent
2
χ 2ðÞ
α
to Chi-Squared value, . Characteristic life, η α , would be represented as:
2
β 2 X β 1 X β
n
n
η ¼¼ t ¼ t for r ¼ 0 (11.10)
α
2
χ 2ðÞ i¼1 i ln 1 i¼1 i
α
α
We know that Equation 11.9 will be established to all cases r 0 and redefined as
follows:
β 2 X β
n
η ¼ t for r 0 (11.11)
α
2
χ 2r +2ð Þ i
α i¼1
To evaluate the Weibull reliability function in Equation 11.7, the characteristic life
can be converted into B X life as follows:
β
ðÞ
L BX
η
RtðÞ ¼ e ¼ 1 x (11.12)
After logarithmic transformation, Equation 11.12 can be expressed as:

β 1 β
¼ ln η (11.13)
L
BX
1 x
where L BX is B X life and x is cumulative failure rate till lifetime (x¼X/100).