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Nomenclature/Preliminaries
Expectation Operator
EX  

 x f x  dx

f (x) - PDF
x (fracture stress)
Nomenclature/Preliminaries
Various Types of Probability Density Functions Available
1. Gauss (Normal)
2. Lognormal
3. Beta
4. Poisson
5. Log Pearson
6. Extreme Value Distributions
PDF Must Satisfy 2 Conditions
f (x)  0

1 
 f x  dx

7. Type I
• Maximum
• Minimum
8. Type II
• Maximum - Frechet
• Minimum
9. Type III
• Maximum
• Minimum - Weibull
Nomenclature/Preliminaries
The Normal Distribution
The normal probability density function (PDF) is given by the expression
 x   2 
 1 
f (x)= 
 exp 
2
2

  2 


for   x    . Here s (a scatter parameter) is the standard deviation and m (a
central location parameter) is the mean.
Nomenclature/Preliminaries
TWO PARAMETER WEIBULL DISTRIBUTION
The Weibull probability density function (PDF) is given by the expression
m
f (  ) = 
 
 
 
  
(m-1)



 
exp - 
   



m



for  > 0 , and
f ( ) = 0
for   0 . Here m (a scatter parameter) and sq (a central location parameter) are
distribution parameters that define the Weibull distribution in much the same way
as the mean (a central location parameter) and standard deviation (a scatter
parameter) are parameters that define the Gaussian (normal) distribution.
Nomenclature/Preliminaries
The cumulative distribution function for the two-parameter Weibull distribution is
given by the expression
F(  )
=
 
1 - exp- 
   
m






for  > 0, and
F(  ) = 0
for   0. Note that a three-parameter formulation exists for the Weibull
distribution. However, the two-parameter formulation yields conservative results.
In addition, the three-parameter formulation is not used unless there is
overwhelming evidence of threshold behavior.
Nomenclature/Preliminaries
The Lognormal Distribution
The normal probability density function (PDF) is given by the expression
2






x
 1  1 
  1 
f (x)= 
 ~  exp -  ~ 2  ln  ~  
 x 2  x 
  2 x    x0  
~
for   x  0 . Here ~
x0 found from the following two expressions:
xand are
 ~
x2 
~
 X = x0  exp   
 2 
2
 X = ~
x0  exp  ~
x 2   exp  ~
x 2  1 
Where sX (a scatter parameter) is the standard deviation and mX (a central
location parameter) is the mean. These parameters are estimated from data,
x0 are found from the expressions
x and ~
and the distribution parameter ~
above.
Nomenclature/Preliminaries
The Reliability Index b
fM
M 0
Failure
M 0
Safe
0
-75
  
Pf     M 
 M 
    
-25
25
M
 M
75
M
125
M
Nomenclature/Preliminaries