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Nomenclature/Preliminaries Expectation Operator EX 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