PSI - Issue 19

Edrissa Gassama et al. / Procedia Structural Integrity 19 (2019) 711–718 Gassama et al./ Structural Integrity Procedia 00 (2019) 000 – 000

714

4

for some function ( ) g  . Fundamentally, the idea behind fitting models that contain censored observations is to replace the censored points with their expected values. There have been many different forms of ( ) g  discussed in the literature, however, one of the more prominent forms, and that selected in this study, follows from the use of the lower truncated normal expectation model (Schmee and Hahn (1979), Chatterjee and McLeish (1981)). In the lower truncated normal expectation model:

( ) z



( )

T 1 n −

n

1

− + y

−

=   X

g



(7)

( ) z

2

2

1

−

1 n − indicates parameters from the previous iteration (see Section 2.2) and:

where the superscript

T 2 − = X z y   2

(8)

The probability density function (pdf) and cumulative distribution function (cdf) for the standard normal distribution are as follows:

1 exp

(

) 2 2 2

( ) z

− X



=

(9)

2



( ) −  =  y z

( ) 2 z

dz 

(10)

2  is the variance: $ ( ) 2 2 m n − = −   y y

and

(11)

2.2. Solution procedure

Hahn and Schmee proposed an iterative method to estimate the parameters of a linear model with censored data (Schmee and Hahn (1979)). In their method, the iterative solution procedure is described as follows: • STEP 1: Generate a least-squares fit of the data, including the run-out data, by treating the run-out data as if they failed at their censored times. • STEP 2: Substitute the estimated parameters and variance into Equation (6) to get an estimated failure time for the censored data. • STEP 3: Use the estimated lives from STEP 2, in place of the run-out points, and obtain a new least-squares fit of the data using Equation (5). • STEP 4: Repeat STEP 2 and STEP 3 until convergence of the fit is achieved. Hahn and Schmee recommend that convergence be based on the slope and intercept estimates agreeing to 3 decimal places on two consecutive iterations.

2.3. Master S-N fatigue curve

The form of the Master S-N curve presented in the 2016 edition of API 579 is as follows: