PSI - Issue 72

Victor Rizov / Procedia Structural Integrity 72 (2025) 128–134

130

where D 1 , D 2 and D 3 are parts of the path, D , in Fig. 1. J stD 1 , J stD 2 and J stD 3 are

i n 

1

  

  

x u

x v

 

 

  

  

 

(4)

cos

,

J

u

p

p

ds









1

0 1

1 D xD i 1

1 yD i

1

stD

D i

D

1

i



1

D

i n 

2

  

  

x u

x v

 

 

  

  

 

,

cos

J

u

p

p

ds









2

0 2

2

2 xD i

2 yD i

2

stD

D i

D

D

1

i



2

D

(5)

i n 

3

  

  

x u

x v

 

 

  

  

 

,

(6)

cos

J

u

p

p

ds









3

0 3

3 D xD i 3

3 yD i

3

stD

D i

D

1

i



3

D

where n 1 , n 2 and n 3 are the numbers of layers in parts, D 1 , D 2 and D 3 , of the path, u 0 D 1i , u 0 D 2i and u 0 D 3i are the specific energies, α D 1 , α D 2 and α D 3 are the inclinations, p xD 1 i , p xD 2 i and p xD 3 i are the longitudinal stresses, p yD 1 i , p yD 2 i and p yD 1 3 are the transversal stresses, u and v are the horizontal and vertical displacements. Equation (7) defines u 0 D 1i .

2 1

(7)

u

,



  D i 1

0 1

D i

where

(8)

i D i E  1





  n z z 1 1 1    

(9)

The curvature, κ 1 , and the coordinate of neutral axis, z 1 n , are determined from the longitudinal force, N , and the bending moment, M , by Eqs. (10) and (11).

1 i       i       1 z i 1 1 i n 1 1 1 z i i n z i

N b 

1 1 dz

(10)

D i

M b 

1 1 1 z dz

(11)

,

D i

1 z i

where (refer to Fig. 1) 0  N ,

(12)

P

(13)

M

a

.



2

Equation (14) is used to define the variation of the modulus of elasticity, Ei, transversally to the layer.