Issue 48

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

2

2

s

s y

s

2

 M M

S

s x

s





M

S





xy

yz

xz

(15)

2

0

     N f





e

2

2

x y

  x

y

 

x

y





where

2

2

2

  

   

(  

)

(  

)

(  

)

b s w w

b s w w

b s w w

(16)

2

N Nx  

N

Ny





xy

2

2

x y

 

x

y





The stability equations in terms of displacement can be obtained by substituting equation (7) in equation (15). The equations obtained based on the present theory of refined shear deformation of the composite plates resting on two-parameter elastic foundation are four in number and are as follows:

2

2

2

3

3

u

u

v

w

w











s

s

66 2 ) s

0

0

0

s

s

(  

)

(  

0

A

A

A A

B

12 B B







11

66

12 66

11

2

2

3

2

x y

 

x

y

x

x y







 

2

2

2

3

3

u

v

v

w

w











s

s

66 2 ) s

0

0

0

s

s

(

)

(  

0

 A A

A

A

B

12 B B









12 66

66

22

22

2

2

3

2  

x y

 

x

y

y

x y







4

4

4

4

4

4

w

w

w

w

w

w













s

s

s

s

b

b

b

s

s

s

2(  

66 2 )

2(  

66 2 )

D

12 D D

D

D

12 D D

D









11

22

11

22

4

2

2

4

4

2 2

4

x

x y

y

x

x y

y



  





 



2

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

0

3

3

3

3

4

4

u

u

v

v

w

w













s

s

66 2 ) s

s

66 2 ) s

s

s

s

66 2 ) s

0

0

0

0

b

b

(  

(  

2(  

B

12 B B

12 B B

B

D

12 D D





11

22

11

3

2

2  

3

4

2 2

x

x y

x y

y

x

x y



 





 

4

4

4

4

2

2

w

w

w

w

w

w













s

s

s

s

s

s

s

b

s

s

s

s

s

(17)

2(

66 2 )

( K w w  W b s

)

D

H

12 H H 

H

A

A















22

11

22

55

44

4

4

2 2

4

2

2

y

x

x y

y

x

y





 







2 ( ) 0     b s w w N

K



g

T RIGONOMETRIC S OLUTION TO M ECHANICAL B UCKLING ectangular plates are generally classified in accordance with the type of support. We are here concerned with the exact solution of Eq. (17) for a simply supported rectangular composite plate (Figure 2a). The following boundary conditions are imposed for the present refined shear deformation theory at the side edges [21]:

R

       b s w v w w

s

(18a)

0, N M M at x a 

0

b

s

x

x

x

y



       b s w u w w

s y

(18b)

0, N M M at y b 

0

b

s

y

y

y



The following approximate solution is seen to satisfy both the differential equation and the boundary conditions

cos( ) sin( ) cos( ) sin( ) sin( ) sin( ) sin( ) sin( ) y x y y y        

0     0         b s w w u v

mn U x 

      

     

1 1  m n    

V

mn

(19)

  

bmn W x W x smn

where U mn , V mn , W bmn , and W smn are arbitrary parameters to be determined and λ = mπ/a and μ = nπ/b . Substituting Eq.

(19) into Eq. (17), one obtains

213