PSI - Issue 81

Denys Rudavskyi et al. / Procedia Structural Integrity 81 (2026) 151–155

154

a

b

Fig. 5. Numerically calculated shear stress distribution near the crack front in the reinforced arch region after injection ( E 2 / E 1 =0.01): ( a ) – crack plane orientation  =45  , ( b ) –  =20  . The analytical approach was considered, and the following formulations (1) – (3) for SIFs K II approximation at the deepest point A on the semi-elliptical crack front (Fig. 6) were proposed. In case of empty crack and  =90 

  2 2 2 2 k b a b   , k a b  ' .

2

2

  k k E k k K k a k '2 '2 2 max       

a t

t a

   2

  



 

1 0,12

tg

, where

(1)

K

 





II



In case of a filled crack and  =90 





2

 a k E E 1 2 1

2

a t

t a

  

   2

  





   max

tg

1 0,12

, where

a c  .

(2)

K

 





 





II

2

'2

'2

2



   1

 k k E k k K k k 

E E

2 1

In case of a filled crack and  =45 





2

 a k E E 1 2 1

2

a t

t a

  

   2

  





   max

tg

1 0,237

.

(3)

K

 





 





II

2

'2

'2

2



   1

 k k E k k K k k 

E E

2 1

The numerically obtained SIFs K II values at the point A on the crack front (Fig. 6) were compared against the analytical formulations (1) – (3). This comparison utilised identical boundary conditions, geometry, and material characteristics.

Fig. 6. Representation of the most stressed volume of the bridge arch, showing a semi-elliptical surface crack inclined at an angle  relative to the local tangential direction.