Issue 52

N. Hebbar et alii, Frattura ed Integrità Strutturale, 52 (2020) 230-246; DOI: 10.3221/IGF-ESIS.52.18

x      xz      z 

x         z   xz   

11 Q z Q z Q z Q z 13 ( ) ( )

( ) 0 ( ) 0

    

    

  

(11)

13

33

55 Q z

0 0

( )

with ( x , z , xz ) and ( x  , z  , xz  ) are respectively the stresses and the deformations. The expressions ij Q depends on the normal deformation z  : In the case of two-dimensional shear deformation (2D) the normal deformation z  = 0, therefore:

  2(1 ) E z  

  Q z E z    11

 



Q z

55 and

(12)

In the case of quasi-three-dimensional shear deformation (quasi-3D) the normal deformation z   0, therefore:

 

 

1 E z 

E z

  Q z Q z   

  Q z Q z    

 





55 Q z

2 ,

and

(13)

11

33

13

11

2(1 )  



E QUATION OF MOTION

T

he Hamilton principle is used in this study to derive the equations of motion; it can be given in the following analytic form [65]:

 T U V K dt      

(14)

0

0

where: δU is the variation of the strain energy , δV is the variation of the kinetic energy and δK is the variation of the potential energy. The variation of the deformation energy of the beam can be defined as follows:

/2 /2 0 /2 /2 b h      L b h



 

 





U

dzdydx

(

)

(15)

x x xz xz

0  L

2 d w 

d u 



d

0 ( U N M M     

0



) M Q dx  

(16)

x

z

b

s

2

dx

dx

dx

with: N x , M b , M s and Q are the resultants of the stress in terms of axial force, bending moment, higher-order moment and shear force, respectively:

h

/2









 

h   

x dz 

, N M M ,

z f z

(17a)

1, ,

x

b

s

/2

235