PSI - Issue 63

Kamila Kotrasova et al. / Procedia Structural Integrity 63 (2024) 27–34

31

The transverse resultants are      N n h xz n xz n f z dz V 1 

   

   

   

   

   

   

   2

   2

(16)

  

  

4 5

z

z

N

N

  E xz n 55 

  E yz n 56 

1

1

V

dz

dz









xz

2

2

h

h

1

1

n

n





n

n

h

h

N

  f z dz

  n h n 1

n

V





yz

yz

   

   

   

   

   

   

   2

   2

.

(17)

  

  

4 5

z

z

N

N

  E xz n 65 

  E yz n 66 

1

1

V

dz

dz









yz

2

2

h

h

1

1

n

n





n

n

h

h

The constitutive equations for shear forces are

yz xz yz V A A   66 56   .

(18)

xz xz V A A   56 55  

yz

For the components of A gilt       N n n n E z z A 1

4 5

h 4





  

  

i , j = 5, 6.

(19)

  n 1

 3 1 3 n

n

z

z



ij

ij

2

3

This approach yields for the case of single layer with E 55 = E 66 = G , E 56 = 0 and shear correction factor k * = 5/6 for the shear stiffness G ꞏ h . A second method to determine shear correction factors consists of considering the strain energy per unit area of the composite.

   

   

2 1

2 1

2 1

(20)





1

1





  V F E F V V A V   T h t T T dz 1

E   t T



W

dz





  h

where



 1 

~

~

 

 

 

1   F A A B B D z z z   h z dz m m n n z 1 ~    E E A E   



 z z m 1 

 

1

n



2

h



z

2 1 2 n 

n

2 1

z

z

~



.

(21)

1 m  



 1 2 z

 

  B E

2

m

m

n

E

E



z zdz

z







2

1

n



2

h



The first-order shear deformation theory yields mostly sufficient accurate results for the displacement and for the in-plane stresses. However, it may be recalled, that transverse shear and transverse normal stresses are main factors that cause delamination failure of laminates and therefore an accurate determination of the transverse stresses is needed. In this section it was demonstrated that one way to calculate the transverse stresses is an equilibrium approach in the frame of an extended 2D-modelling. Another simple method is to expand the first-order shear deformation theory from five to six unknown function or degrees of freedom, respectively, by including a z -dependent term into the polynomial representation of the out-of-plane displacement w ( x , y , z ). 3. Results and discussion The simply supported square laminate plate L x L : 20 cm x 20 cm with the thickness h = 1 cm is considered, see Fig. 2. The uniform transverse pressure load q = 1 MPa is subjected on the laminate plate. The lay-up is made from three layers, all layers are with the same thickness. Each layer has the same orthotropic material properties: