Issue 57

K. Benyahi et alii, Frattura ed Integrità Strutturale, 57 (2021) 195-222; DOI: 10.3221/IGF-ESIS.57.16

It results:

    k

H u



      k

       k k



k

1





u

u

(36)

    k



H u

u

        k k

 

H u H u

the vector of direction cosines (or the vector of the normalized gradient) of H at the point   k P .       k d u and         u if the algorithm is convergent at iteration   k , let us set:

with    k



u

Ultimately when   k ,

     1 k u

         . k k



      . k

  k



k

1

(37)

 

  

u

Solving the problem gives a value of  which corresponds to the reliability index known as of Hasofer and Lind. Which leads to the iterative relation giving the reliability index:

    k

H u

       k k

  k



 



u

(38)

    k



H u

u

The reliability index search algorithm stops when the standard            1 k k u u t .

where t : the fixed tolerance. and      1 k u , is deduced by replacing Eqn. 38 in Eqn. 37 by:

    k

H u

              k k k u

    1 k u

     k

 



(39)

    k



H u

u

In standardized space, the approximate FORM/SORM (First/Second Order Reliability Method) consists of substituting for the failure surface a hyperplane tangent to this surface at the design point. An estimate of the probability of failure is then obtained by:

  

  

2

u

1 2











C





  



(40)

P

exp

du

f

C

2





In this present study the Hasofer-Lind index will be used because it allows to solve the problem caused by the non invariance. To overcome this problem, Hasofer and Lind proposed not to place themselves in the space of physical variables, but to perform a change of variable, towards a new space of statistically independent Gaussian variables with zero means and unit standard deviations:  i i X U Gaussian vector   0,1  ,

        0, 1, 0, , i i U U ij i j

(41)

In the case of independent Gaussian variables, the transformation of the physical space into the standard normal space is iso-probabilistic:

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