PSI - Issue 59

6

Ivan Shatskyi et al. / Procedia Structural Integrity 59 (2024) 246–252 Shatskyi et al. / Structural Integrity Procedia 00 (0000) 000 – 000

251

2 2 /( ) 2 0 l m h E 

  is a normalizing factor. It is also taken into account that with a change of  from 0

Here



to 1/2  changes from 1 to 9/7. 3.3. Other analytical results

Approximate analytical estimates of the ultimate bending loads can also be obtained in the presence of a crack assembly, plate boundary, or elastic base. To do this, in formulas (11) – (13), the following changes should be made: to calculate | | * m we should take / ( ) 0  m F instead of 0 m , and to calculate | |  m we should use / ( ) 0  m F instead of 0 m respectively. The expressions for changes ( )  F and ( )  F are given below. For bending plates with two collinear cracks (Shatskii (1990))

      1

   ,

1

( ) ( )  

K E



( ) F F

  ( )

 

 

1



where l d 2 /   , d is the distance between the centers of the cracks; ( )  K , ( )  E are the complete elliptic integrals of the first and second kind. In the case of plate bending with a periodic system of parallel contact cracks (Shatskii (1991), Dalyak (2004))

2(3 2 ) (1 2 )   

 (1 )

th

A

th

 

 

A

( )  

A

 A

( )  

F



,

;

,

,

F

3





A

A

where l d 2 /   , d is the distance between the centers of the cracks For an internal closable crack perpendicular to the free edge of a semi- infinite plate (Shats’kii and Perepichka (1991))

2

5   2

(3 2 )(3 ) 5 5 2 2    

1

1

128 11 4 2  

7 2









4   

 

 

2 

3 

( ) 6

( ) 1 

F

O

  ,  

;

 









8

16

128

 

2

2

5   2

3 1

1

1

128 11 4 2  

7 2

 











4    

 

  

  

2 

3 

( ) 6

( ) 1 

F

O

  ,  

,

 







8

16

128



where l d 2 /   , d is the distance from the center of the crack to the edge of the plate. For bending of the cracked plate on elastic Winkler’s foundation (Shats’ kyi and Makoviichuk (2003))

(3 )(1 ) 32 3 2 3 2 2         

2

2

(3 2 )(1 ) 64 3 2 3         

4

4

  ; F

  ,

( ) 1 

( ln )

( ) 1 

( ln )

O

F

O

 



 



where 1/4 ( / ) l k D   , k is a foundation stiffness coefficient, D is a bending stiffness of the plate.

4. Conclusions Taking into account the contact of the crack edges in the plate bending increases the design strength by tens of percent. The lower estimates of the ultimate bending load according to criteria (8) or (9) differ from the upper estimate according to criterion (10) by more than a factor of two. To predict the destructive load corresponding to