PSI - Issue 45

Moaz Sibtain et al. / Procedia Structural Integrity 45 (2023) 132–139

135

4

Sibtain et al. / Structural Integrity Procedia 00 (2019) 000 – 000 Based on the Euler-Bernoulli beam theory, the non-zero, linear strain and stress components are 2

2 ( , )

v x t



( , )

v x t



(5)

y

,

z







x

xx

x

x





  

  

2

2 ( , )

v x t



( , )

( )

v x t

E z



y

,

z







(6)

x

r

xx

2

1

x

x









where x v and y v denotes axial and transverse displacements, respectively, and z is in the thickness direction, as shown in Fig. 1, and r E is the Young’s modulus of the FGCNT reinforced beam. The kinetic energy   E K of the system (possess terms related to the axial speed) is 2 2 (7) The potential energy   t  of system is the sum of the strain energy   E  and the spring stiffness energy   s  in which k is the elastic spring stiffness. The following dimensionless parameters were introduced: 0 ( , ) ( , ) ( , ) ( , ) 1 2 ( ) z 1    , L y y x x E r  v x t v x t v x t v x t K U U dAdx t x t x                                

v

A

v

x

x

z

L h

y

*

*

*

*

3,

,

,

,

,

,

,

m

y 



v

v

x

x

z

t

t

















0

x

0

x

4

h

h

L

L

h

, xx m I L

(8)

2

, xx m I L

2

3

. A h

A

I

Ah

kL

*

,

,

,

,

,

,









A

A

A

I

k

c U 











3

1

2

xx

1

2

3

xx

A

A

A

I

A

A

3,

3,

3,

, xx m

3,

3,

m

m

m

m

m

where h denotes the beam’s thickness, and the subscript m represents the matrix’s material properties. By using the generalised Hamilton principle, the following dimensionless equations of motion are determined:       2 2 4 1 2 2 , , y x v x t v x t A A                   





x

x

x

x









 



(9)

  ,

  ,

  ,

   

2     c   

2 c       

   

2

2

2

v x t

v x t

v x t







  z

0,

I





x

x

x

xx

2

2

t

x t

x



 







  ,

  ,

3       A

   

2

v x t



2

2

v x t

  

  







y

2 

A



x

2

2

2

2

x

x

x

x









(10)

   

2     c   

2 c       

   

2

2

2

( , )

( , )

( , )

v x t

v x t

v x t











y

y

y

0,

I

kv x x 





 

0

x

x

y

2

2

t

x t

x



 



where star notation is dropped for simplicity. where xx I is a coefficient that denotes the mass moment of inertia       . SW m S x A W m x I V z dA       

(11)

3 A are stiffness parameters and are defined as

and 1 A ,

2 A , and

   2 1, ,

   



  



1 2 , A A A ,

, z z A d

cE E V 

z E 

(12)

3

1

SW m SW

m

A

3. Solution method The modal decomposition technique was used in order to solve the coupled dimensionless equation of motion; more details for general form of this method for a system of equations of motion is depicted in Zhai et al. (2021).