PSI - Issue 75

Carl-Fredrik Lind et al. / Procedia Structural Integrity 75 (2025) 519–529 Carl-Fredrik Lind et al./ Structural Integrity Procedia (2025)

523

5

3 6 l l l l 6 3

    

     

, y i          , y i , y k     , y k F F f f L

, y i       , y k f f

 

(2)

And the same for the moment:

, x i L M m l l m                                 , x i , x i , x k , x k , x k 3 6 l l 6 3 M m m

(3)

Using these equations along with Eq.(1) gives the following results:

F F

  

, y i            , x i 6

M M

i       k  

1

1   L 

(4)

 

t

t

2

, x k  

, y k   

The weld representation elements, e 1 and e 2 are each subdivided into n elements e 1 , 1 , e 1 , 2, ... e 1 ,n , and e 2 , 1 ,e 2 , 2 ... e 2 ,n with nodes j 1 , j 2 , ... j n − 1 , as shown in Fig. 4c. Considering Eq.(4) and by using equations Eq.(2) and Eq.(3) along with Eq.(1) gives the following results:

1           s σ F M 2 1 6 L t t

(5)













Where: M And L is obtained from (Liu et al., 2024). For any node w j where  w  1 2 , y i , 1 , 2 , y jn 1 , j k ...      , ... F F F F F , T T i j j jn jk y j y j     s σ F

... M M M M M

T

, x i

, 1

,2

, x jn

1

, x k

x j

j



 1  the

jw  is defined as follows:

1,2,...,

n

2 1

6

4 1

6

2 1

6





(

)

(

)

(

)

F

M

, y jw F M 

F

M













1 w w l l t  

t

1 w w l l t  

t

1 w w l l t  

t

, y jw

1

, x jw

1

, x jw

, y jw

1

, x jw

1

jw









2

2

2

(6) This is the average of Eq.(4) over two elements, equivalent to computing element-wise and averaging the nodes for an accurate nodal solution. It is noted that this does not apply to edge nodes. Due to the complexity of identifying the third node, a trade-off is made in favour of efficiency. Therefore, for edge nodes , i k , Eq.(4) is used without averaging. To compensate for this potential underestimation of stress, a safety factor is applied later during nodal life evaluation. In conclusion, , , s i i W    from which an associated lifespan can be calculated.

3.2. Master S-N curve

0  a a in the weld that is subject to Paris law:

Assume an initial crack with length

da dN

( )( n kn C M K 

)



(7)

m

n

n K  is the stress strength factor

Where N is the number of cycles, C is the crack extension parameter of the material,