Issue 51

A. Chikh, Frattura ed Integrità Strutturale, 51 (2020) 115-126; DOI: 10.3221/IGF-ESIS.51.09

q

4





0

1 , m Q 





m

(17)

,

1, 3, 5...

m

m 

In the case where static problems, we have the following equation:       K F  

(18)

and   K is the symmetric matrix given by

where     

 , , t U W 

  S S S K S S S S S S      11 12 12 22

    

13

(19)

23



13

23

33

In the case of free vibration problem problems, the analytical solutions can be obtained by:



  

  K

 

2 M 



 

0

(20)

where   M is the symmetric matrix given by

  m m m M m m m m m m      11 12 12 22

    

13

(21)

23



13

23

33

For buckling problems, can be expressed as       0 K N   

(22)

in which:

2 

3 

11 S A S B S k A D S E k k N k S k A F S k A A k A G                    11 12 11 13 1 11 4 2 2 22 11 1 0 2 2 3 2 2 2 , , ' , ' , ' ' p w

,

(23)

s

23

1

11 33

1

55

1

11

11 1 12 m I m I m k A I m I I m I m k A I              2 13 1 3 22 1 4 2 2 4 2 , , ' , , ' ,



23

5

33

1

6

where

h

2











2

2

11 11 11 11 11 11 A B D E F G C z f z z zf z f z dz  11 , , , , , 1, , ( ), , ( ), ( )

,



h

2

(24)

h

2



s

2

( ) A C g z dz 

55

55



h

2

h

2









h   

2

2

( ) 1, , ( ), , ( ), ( ) z z f z z zf z f z dz 

1 2 3 4 5 6 , , , , , I I I I I I

(25)

2

119