Issue 51

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

where  is the top surface, and e Pasternak foundation model:

f is the density of reaction force of foundation. For the

2



w x y

2 ( , )



( , ) k w x y k 

f

(10)

e

w

p



x

The equilibrium equations can be acquired using the Hamilton principle.

2

3

3



N u 







w

'2

x

0

0











0 : u

I

I

3 1 I k A

0

1

2

2

2 t x

2 t x



x



 

 

t

b x

2

2

2

3

4

4















M

w

w

u

w

0 N f

0

0

0





 











0 : w

q x t

I

I

I

I

( , )

0

(11)

e

0

1

2

4

5

2

2

2

2 t x

2 2

2 2







 

 

 

x

x

t

t x

t x



Q

s

2

2

4

4 t x  









M

u

xz

'

'2

'2

'

2 '4

x k A k A I k A  

0



 





5 1 I k A

6 1 I k A

:

0

1

1

3 1

2

2 t x

2

2

2 2



x



 





 

x

t

x

denote the resulting force in-plane,     , b s x x M M

denote the total moment resultants and   xz Q are

where   x N

transverse shear stress resultants and they are defined as

h

h

/2

/2





b x





, dz M zdz  

N

,

x

x

x





h

h

/2

/2

(12)

h

h

/2

/2





s x









M

( ) , f z dz Q

( ) g z dz

x

xz

xz





h

h

/2

/2

Following the Navier solution process, we assume the following solution form for 



0 0 , , u w  and that check the boundary

conditions,

cos( ) sin( ) e sin( )       

0 u            w  0 1       m  

U x W x x 

i t 

(13)



where , , U W and  are arbitrary parameters to be determined,  is the natural frequency, and m L   

. The transverse

load ( ) q x is also expanded in Fourier series as:



1    m

 

sin m Q x 

q x

(14)

( )

where

2 ( )sin( ) L

m Q q x  



x dx

(15)

L

0

In the case where a sinusoidally distributed load, we have

1 0 1 , m Q q  

(16)

In the case where uniform distributed the load, we have

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