Issue 61

K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25

VIRTUAL WORK PRINCIPLE

T

he static equations of the theory can be derived from the virtual work principle [9] by expressing the strain energy variation as follows:   0                    xx xx yy yy xy xy xz xz yz yz V V dV q WdV (5)

According to the substitution of equations (2) in the static equation (5) we obtain:

   









   

   

   

   

  

  

  

  

2

2

























u

w

v

w

4 h

4 h

y

y

 A

x

x

 N M

  P xx





 N M

  P





xx

xx

yy

yy

yy

2

2

2

2













x

x

x

y

y

y





x

y

3

3









   

   

  

  

2

  

  

  

  









    u y





  v



 x

w

w

4 h

  

  

y

y

x

x

 



  P xy







 



N

M

Q

2

(6)

xy

xy

xx

x

2









 



x

y

x

y

x

x y

3

   

   

  

  

  

  



 x









w

w

w

4

4

  

  

  

  









 q W dA







   R







R

Q

0

xx

x

yy

y

yy

y

2

2







y

y

h

h

where the resultants forces are defined as follows:

    

    

             xx yy xy

xx N M P N M P N M P xx xx yy yy yy

        xz yz

xx Q R Q R

  

  

n h

n h





  2 1,





xx

3 z z dz

k

k





z dz

1, ,

;

(7)

h

h



k



1

k

1

yy

yy





k

1

k

1

xy

xy

xy

Therefore, from the eq (7) and eq (3), we obtain the generalized relations resultants forces as follows [9]:

0                 ij ij ij A B E

          N M sym D F P sym sym H                                      s S S ij ij S s ij A D Q R sym F                         

  

  

0            ij ij

2  

      ij

(8)

where:

T

T

 

    

 

0

0

0

0

0

0

2

2

2

0

0

2

        

  

1

2

6

1

2

6

1

2

6

T

T

  s s

 

    

 

0

0

2

2

   

4

5

4

5

1     k k n h k h









2 3 4 6 C z z z z z dz , , 1, , , , ,

1, 2,6  i j

, , , ij A B D E F H , , ij ij ij ij

ij

ij



1

1   k k n h k h 









, S S S ij ij ij

2 4 C z z dz i j 1, , , ,





, A D F

4, 5

ij



1

377