Issue 48

Y. Khalfi et alii, Frattura ed Integrità Strutturale, 48 (2019) 208-221; DOI: 10.3221/IGF-ESIS.48.22

where A ij

,D ij , etc., are the plate stiffness, defined by

2

2

      

     

    

     

s

s

s

11 2 (1, , ( ), ( ), ( )) (1, , ( ), ( ), ( )) (1, , ( ), ( ), ( )) Q z f z zf z f z Q z f z zf z f z dz Q z f z zf z f z 2 12 2 2

11 11 11 A D B D H A D B D H A D B D H 11 12 12 12 12 66 66 66 s s s s 66

11

/2    h



s

12

/2

h



s

66

66

22 22 22 22 22 A D B D H A D 11 11 11 11 , , , ) s s s B D H 11 ) ( ,  ( , , , , s s s

 2

/2 44 55 / /2    h s s h A A



(10)

44 ( ) Q g z dz

E QUILIBRIUM AND S TABILITY E QUATIONS

T

he equilibrium equations of the rectangular composite plate resting on the Pasternak elastic foundation under mechanical loadings may be derived on the basis of the stationary potential energy. The total potential energy of the plate, V, may be written in the form V = U + U F (11) Here, U is the total strain energy of the plate, and is calculated as

1 2

/2 0 0 /2 a b h h 

  

 

 

(12)

U

dzdydx

x x  

   

 

 

 







y y xy xy yz yz

xz xz

and U f is the strain energy due to the Pasternak elastic foundation, which is given by [20]

1 2

a b

 

( f w w dydx  ) e b s

U



(13)

f

0 0

where fe is the density of reaction force of foundation. For the Pasternak foundation model:

2

(14)

( f K w w K w w      ) ( W b s g b s e

)

where K W

is the Winkler foundation stiffness and K g

is a constant showing the effect of the shear interactions of

the vertical elements. Using Eqs. (2), (3), and (7) and employing the virtual work principle to minimize the functional of total potential energy function result in the expressions for the equilibrium equations of plate resting on two parameters elastic foundation as

xy N N

 

x

0

 

  x

y

  xy N N x y



   y

0

2

2

b

b y

2

b

 M M



M



xy

x

2

0

   N f





e

2

2

x y

 

x

y





212