Issue 51

S. Merdaci et alii, Frattura ed Integrità Strutturale, 51 (2020) 199-214; DOI: 10.3221/IGF-ESIS.51.16

The external force according to Navier’s solution can be expressed as

1 1     m n  

sin( )sin( ) mn x y  

q x y

q

( , )

(17)

/ n b   

/ m a   

and

, « m » and « n »are mode numbers. For the case of a sinusoidally distributed load, we

where

have

1 m n   and 11 0 q q 

(18)

where q 0 represents the intensity of the load at the plate center. Following the Navier solution procedure, we assume the following form of solution for ( u,v,w b ,w s

) that satisfies the

boundary conditions

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

b          s u          v    

mn U x V x mn

      

,

(19)

bmn w W x w W x

smn

where U mn

, V mn

,W bmn

, and W smn

are arbitrary parameters. Eqn.(15) in combination with Eqn. (16) can be combined into a

system of first order equations as:       , K F  

(20)

where    and   F denotes the columns

   T mn mn bmn smn U V W W   , , ,



and    T F



0, 0, q q    ,

(21)

,

mn

mn

and

11 12 a a a a a a a a a a a a a a a a 13 12 22 23 13 23 33

     

     

14

  K

24



(22)

34

14

24

34

44

= a ji of the coefficient matrix [K]. The elements of the symmetric matrix [K] presented in Eqn. (23) are

The elements a ij

given by





2 A A   11

2 

 

a

11

66





A A   

 

a

12

12

66

2 

2  66 2 ) ]





(  

a

B

12 B B

[

13

11

s

s

2  66 2 ) ] s

2 





(  

a

B

12 B B

[

(23)

14

11

205