PSI - Issue 22

Elise Zgheib et al. / Procedia Structural Integrity 22 (2019) 25–32 Elise Zgheib – Wassim Raphael / Structural Integrity Procedia 00 (2019) 000 – 000

29

5

3. Bayesian Linear Regression (BLR) method To overcome this inaccurate estimation, correction coefficients are implemented in Eurocode 2 formula. Different approaches were elaborated including multiple proposition for the correction coefficient. However, the correction coefficient expression incorporating the percentage of admixtures has led to the most accurate results as shown in the below equation.

         

         

0.3

        

        

    

    

     0.7

1.4





(7)

   

   

   

 t t 0

 10x 3 h 0 100 RH x35 0.7

 0.2 ) (0.1 t 0 16.8x35 0.2

f cm 1

f cm 1

100 SF

 t 0 ) ( φ(t,

)x

x

x

x

E x

0.5

   

     





  

   

1.5 1 0.012RH 18 h 0

f cm 250 35

t t 0

where SF refers to the percentage of silica fume. To quantify the correction coefficient E, Bayesian inference, which is a way to get sharper predictions from data, is applied in this study. The Bayesian inference consists of multiplying the expert knowledge already known and named as the prior distribution, by the likelihood function coming from the database. Therefore, a posterior distribution is defined which is an update of the knowledge already known using the latest database, (Box and Tiao (1992), Riddle and Muehleisen (2014) and Heo et al. (2012)). More specifically, the Bayesian Linear Regression (BLR) method is applied in this study, (Bolstad and Curran (2016)), which is an approach to linear regression in which the statistical analysis is undertaken within the context of Bayesian inference. The detailed procedure and calculations are presented in the following. First, a linearization of Eq. (7) shall be established, to apply the BLR method. Therefore, and since the linearization does not affect the results, the logarithmic transformation is applied to Eq. (7). Also, by t aking the error ε i into consideration in calculations, the equation of the i th observation is:

i i 1       ai

(8)

Where:   t 0 ) φ(t, i ln   ;

    

    

   

   

  

  

100 SF

 0.2 ) t 0 (0.1 x16.8 35 0.2

xCst

 

E x

Cst



1 ln

;

          

          

         

         

0.3

        

        

    

    

1.4

     0.7





   

   

   

 t t 0

 10x 3 h 0 100 RH x35 0.7

f cm 1

f cm 1

x

x

 

ai ln

0.5

   

     





    f cm 250 35 1.5 1 0.012RH 18 h 0   

t t 0

ai    1 and a

By considering that the likelihood is normally distributed (Raphael et al. (2006)), with a mean of

2 , then the likelihood function for β

a is:

variance of σ ε

   

   

 n i 1  

 2 2 1

  2 2 1 











(9)

p a / 1

exp

  

    

i

1

ai 2

Moreover, the correction coefficient shall be positive since creep compliances are always positive, then the prior of θ 1 is normally distributed with a mean of µ θ1 and a variance of σ θ1 2 , and the a priori function for θ 1 can be written as:

     

     

 2 2 1

   2 2 1

 





(10)

p 1

exp

 

   

1 1 2

1

1