Issue 50

H. Saidi et alii, Frattura ed Integrità Strutturale, 50 (2019) 286-299; DOI: 10.3221/IGF-ESIS.50.24

where K are the shear moduli of the subgrade (shear layer foundation stiffness). If foundation is homogeneous and isotropic, we will get 1 2 S S S K K K   . If the shear layer foundation stiffness is neglected, Pasternak foundation becomes a Winkler foundation. The variation of kinetic energy of the plate can be expressed as: W K is the modulus of subgrade reaction (elastic coefficient of the foundation) and 1 S K and 2 S









( ) K u u v v w w z dV             

V







A 



0 0 0 0 0 I u u v v w w            0 0

 

  



0 w w x x       



0 w w    y   y  

0

0





u v   



I u  

v



1 0

0 0

0









 

  

     

y      







u v   





J u  

v



1 0

0 0

0

x x

y

  



  



  

  

  

  

w w w w        

  

  

0

0

0

0









I

K

 

2

2

(13)  

y  

x x y y    

x x

y

  



  



  

  

 





0 w w   



















w 

w 

0

0

0









J

dA

2

x x x x    

    

y

y

y

y



where dot-superscript convention indicates the differentiation with respect to the time variable t ; ( ) z  is the mass density given by Eq. (1b); and ( i I , i J , i K ) are mass inertias expressed by

h

/2









h   

2

( ) z z z dz 

I I I

0 1 2 , ,

1, ,

(14a)

/2

h

/2









h   

2

( ) z dz 

J J K

, , f z f f

1 2 2 , ,

(14b)

/2

Substituting Eqs. (9), (11), and (13) into Eq. (8), integrating by parts, and collecting the coefficients of 0 u  , 0 v  , 0 w  and   ; the following equations of motion are obtained:



0 0 1 xy N N    I u I   



w

 





x

0





0 : u

J

1







x

y

x

x 



N N 



w







xy

y

0

0 0 1 I v I    





0 : v

J

1









x

y

y

y

b

b

2

2



2 M M y M M      2 s s  x y xy

b x

2

  

  



   

M

0 u v x y  



y

2

2

0      I w J 



  





0 : w

f

I w I 

2

(15)

e

0 0 1

2

0

2

2



x

s

s

2

M S S  



s x

2

 

0 u v x y        0   



xy

y

xz

yz



:  



  

J

2

1

2

2

 



x y 

x y





y

x

2      J w K 



2



2

0

2

2

2

2

2

2

where x y        is the Laplacian operator in two-dimensional Cartesian coordinate system. / /

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