Issue 57

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

In the space of physical variables X , the limit state function is denoted   G X and in the standard space, we denote it   H u . The second-order Taylor series expansion of the limit state function   H u around the point   k P is written as follows:



         + - k O u u

    = + k

2

  k

  H u H u

 

      - k u u



H u

(32)

u

* U is a solution of the following optimization problem:

The search for the design point

    T u u





(33)

    i j g x u 0 min 

HL

Under constraint    0 H u The design point (or most probable point of failure) is the point on the limit state surface where the probability density of U is maximum, it is also defined as the point on the limit state surface closest to the origin. In this study the constrained minimization problem will be solved, using the Hasofer-lind-Rackwitz-Fiessler algorithm which is an adaptation of a first order optimization algorithm to the design point search problem.

Figure 8: Safe and Unsafe Regions approach for computing Hasofer Lind Reliability Index.

The equation of the tangent hyper-plane to   H u at the point   k+1 P is defined as follows:





     1 k

    k

     k u

      k u

 



k

1



 



H u

H u

H u

0

(34)

u

  k P we

     k u H u and by introducing the direction cosines of H at the point

By dividing the equation by the norm

obtain:

    k

H u

                 1 k k k u u

0

(35)

    k



H u

u

205