PSI - Issue 63

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

30

2

2

2

2

h

h

h

h









 

 

 

 









2 z z dz

 N E

ε

E

κ

 M E

ε

E

κ

z dz

z zdz

z zdz





2

2

2

2

h

h

h

h









   h t z dz γ k E .   2 2 h

(9)

V    

N y

M yx

x

V yz

N yx

M y

y z

V xz

N x

N x

M xy

M xy

M x

N xy

N xy

M x

V xz

M y

N yx

M yx

V yz

N y

Fig. 1. Internal force resultants. With stretching, coupling, bending and transverse shear stiffness matrix               N n n N n z z n h h h dz z dz n n 1 1 2 2 1 n E E A E

(10)

n

2

h



z

 2 1 2 n

n

z

z



(11)

 N

N

 

 



n

n

 B E

E

E

z zdz

zdz





2

1

1

n

n





1

n



2

h



z

n

2

h



z

 3 1 3 n

n

z

z



(12)

 N

N

 

 



z z dz 2

z dz 2

n

n

 D E

E

E





3

1

1

n

n





1

n



2

h



z

        N n h t z dz 1 2

(13)

n t n

 A E

E

h

2

h

where n E is the elastic matrix in the in-plane direction and n E t is the elastic matrix in the transversal direction. The constitutive equation can by written in the condensed hypermatrix form

0 0

γ κ ε

B D A B 0 0

V M N

    

    

    

    

    

    

.

(14)



A

The shear stiffness values can be improved with help of shear correction factors. In this case the part of the constitutive equation relating to the resultants N , M is not modified. The other part relating to transverse shear resultants V is modified by replacing the stiffness � by k * � . The parameters � ∗ � are the shear correction factors. A very simply approach is to introduce a weighting function f ( z ) for the distribution of the transverse shear stresses through the thickness h . Assume a parabolic function     2 1 z f z . (15)

   

   

  

  

4 5

2

h