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

30 6

By multiplying the likelihood and the prior , the posterior will be equal to:

     

     

 n i 1  

1

  2 2 1 

  2 2 1





 

  









(11)

       p 1 / a p a / 1 xp 1

exp

    



   

i

1

ai 2

1 1 2

2

  

1

1

This posterior can be re-expressed as a normal distribution, by using some algebra. Since terms outside the exponential are normalizing constants with respect to θ 1 , they can be dropped and the terms inside the exponential can be written as follows:

       

       

 n i 1  

 n i 1





     

     

       i 2 1

n 1 2

  

 

(12)

2 2

i

ai 2

ai

          2 1

 n i 1  

1 1 1 2

 2 1 1 2 1      

  2 2 1 

 2 2 1 









    





i

1

ai 2

  2

1

1

Any term that does not include θ 1 can be viewed as a proportionality constant, therefore it can be factored out of the exponent, and dropped. By using algebra and dropping constants with respect to θ 1 , equation (12) can be written as:

                

                

2

      

      

 n i 1  



2 σ θ

2 σ ε

i Δ

ai β

1 μ θ





1

  1

2 1 θ n σ

2 σ ε

       

       

 n i 1





       i 2 1

n 1 2

 

(13)

2

ai

       2 2 1 1 1

1

1









2 σ ε

2 1 σ θ

  2

2

2

1

2 1 θ n σ

2 σ ε



Therefore, the posterior of θ 1 is normally distributed with mean µ θ 1p and variance σ θ 1p

2 that can be calculated as

follows:

 n i 1  

 2 σ ε

2 σ ε

2 σ θ

ai β i Δ 

1 μ θ



1

1p μ θ



(14)

2 1 θ n σ



2 σ ε

2 1 σ θ

2 1p σ θ



(15)

2 1 θ n σ

2 σ ε



Knowing μ θ1p and since θ 1p = μ θ1p then the correction coefficient E is equal to:

  100 Cstx SF 1p θ exp

E 

(16)

4. Results To quantify the correction coefficient E using the Bayesian Linear Regression (BLR) method, some tests having the same following properties: age of concrete at loading (t 0 ), notional size (h 0 ), relative humidity (RH) and duration of loading (t-t 0 ) have been selected from the database. The results of the application of the BLR method are shown in the following tables.