PSI - Issue 48

Mohamed El Amine Ben Seghier et al. / Procedia Structural Integrity 48 (2023) 356–362 Ben Seghier et al / Structural Integrity Procedia 00 (2023) 000 – 000

359

4

( ) ( ) k k U U

g g

 

α



(9)

k

The HL-RF method without controlling the search direction may fail to converge with the oscillating points when the limit state surface is highly concave. Thus, in this current paper, the FORM formula is improved based on the conjugate search direction by the following relation (Keshtegar 2016) :

( )  U U U U α ( ) C C k k T C C k k g g 

( ) C

T

g



(10)

C

C

U

α



k

1

k

k



C k α can be defined as the conjugate direction or conjugate unit vector which can be expressed as follows: C C k k k C k k      U d α U d

where,

(11)

In which, k d is the conjugate gradient vector, which is defined as: 2 2

( ) C U

( ) C U

g g

g





k

k

( ) min{ C U

,2}

d

,

d

g  

(12)

1

k

k

k



2

T

|

(

) |

C

d

U

g



(

)

C

U



1

1

k

k





1

k



The conjugate gradient vector ( k d ) at the design point ( C

k U ) is computed based on a hybrid scalar factor using

2

2

1 ( ) ( C k C k  U U

g g

 

( ) C U

g   T

k

and

to consider the effect of the previous conjugate gradient vector. The

2

|

(

) |

C

d

U

g

)

1

1

k

k





conjugate unit normal ( C

k α ) in not parallel to the unit normal vector ( k α ) thus, the parallel risk between C k α and

other previous unite normal vectors are reduced in the proposed method. In Eq. (11),  is the finite-step size, which should with a larger value as 1   . If  is well-defined, then the convergence of conjugate FORM can be obtained, thus it is defined in the current work based on the adaptive following relation:

100 ( ) g 

0 



(13)

2



In which, ( )  g  is the gradient vector at the mean point. To control the instability of the proposed conjugate method, the sufficient descent condition is used to adjust the finite step size as below:

k       

1 U -U U -U U -U U -U C C C C k k k k C C C C   

if if

1



(14)

k

k

 

1

1

k

k

k

k





2

4. Comparative results In order to implement the above methodology an example of non-normal highly performance function was used (Zhao, Asce, and Lu 2007). The applied performance function is expressed as follows: 50 ( ) 2 2 4 1    g X x x (15) where, x 1 ~ Lognormal (   5 ,   l) and x 2 ~ Gumbel (   10 ,   10). The reliability index extracted from