ISSN: 2180-1053        Vol. 7     No. 1    January - June 2015

Investigating The Surface Elasticity and Tension Effects on Critical Buckling Behaviour of Nanotubes 
Based on Differential Transformation Method

11

INVESTIGATING THE SURFACE ELASTICITY AND 
TENSION EFFECTS ON CRITICAL BUCKLING 

BEHAVIOUR OF NANOTUBES BASED ON DIFFERENTIAL 
TRANSFORMATION METHOD 

F. Ebrahimi1*, M. Boreiry1, G. Shaghaghi1 

1Department of Mechanical Engineering,
 Imam Khomeini International University

Qazvin, Iran

ABSTRACT

By considering the coupled effects of surface and nonlocal elasticity theory, 
the critical buckling load response of silicon/aluminium nanotubes is 
investigated in this paper. The nonlocal Eringen theory takes into account 
the effect of small scale size while the Gurtin-Murdoch model is used to 
incorporate the surface effects. Governing equations are derived through 
Hamilton’s principle. The differential transformation method (DTM) as 
an efficient and accurate numerical tool is employed to solve the governing 
equations of nanotubes subjected to different boundary conditions. The 
output results are compared favourably with available published works. 
The detailed mathematical derivations are presented and numerical 
investigations are performed while the emphasis is placed on investigating 
the effect of the nonlocal parameter, surface effect, aspect ratio, mode 
number and beam size on critical buckling loads of the nanotube in detail. 
The results show that increasing the nonlocal parameter increase the 
buckling ratio of the nanotubes.

KEYWORDS: Critical buckling behaviour, Nonlocal elasticity theory, 
Surface elasticity and tension effects, Differential transformation method 

1.0 INTRODUCTION

In order to study the mechanical behaviours of nanostructures, the 
surface effects and nonlocal elasticity theory are two important fields 
which are investigated by researchers separately, or simultaneously. 
The surface of a solid is a region with small thickness which has 
different properties from the bulk. If the surface energy-to-bulk energy 
ratio is large, for example in the case of nanostructures, the surface 
effects cannot be ignored (He et. al, 2004). On the other hand, the 
nonlocal elasticity theory which is initiated in the paper of Eringen 

* Corresponding author email: febrahimy@eng.ikiu.ac.ir



ISSN: 2180-1053        Vol. 7     No. 1    January - June 2015

Journal of Mechanical Engineering and Technology 

12

(1983) expresses that the stress at a point is a function of strains at all 
points in the continuum. To account for the effect of surfaces/interfaces 
on mechanical deformation, the surface elasticity theory is presented 
by modelling the surface as a two dimensional membrane adhering to 
the underlying bulk material without slipping (Gurtin and Murdoch 
1975),(Gurtin et al, 1998). There are many studies related to the wave 
propagation, static, buckling and free linear and nonlinear vibration 
analysis of nanobeams and carbon nanotubes based on different beam 
theories 

Gurtin et al (1998) established the theory of surface elasticity to 
explain various size-dependent phenomena at the nanoscale, and 
the predictions fit well with atomistic simulations and experimental 
measurements. Wang and Feng (2007) analysed the surface effects on the 
axial buckling of nanowires. By using the surface Cauchy–Born model 
(Park, 2009) analysed the size-dependent effect of the residual surface 
stress on the resonant frequencies of silicon nanowires under finite 
deformation. Hosseini et al. (2013) studied the surface and nonlocal 
effects on free vibration of nanobeam based on both Timoshenko and 
Euler-Bernoulli beam theory (EBT) for different boundary conditions. 
In similar work, Malekzadeh et al. (2013) studied surface and nonlocal 
effect on free nonlinear vibration of non-uniform nanobeams based on 
EBT and Timoshenko beam theory. They expressed that the influence 
of surface and nonlocal effects depends on the boundary conditions 
of the nanobeam. Also, Eltaher et al.  (2013) studied the coupling 
effects of nonlocal and surface energy on free vibration of nanobeam 
based on EBT for simply supported nanobeam. They showed that the 
surface effects depend on the size and the material of the nanobeam by 
calculating natural frequencies for two different materials. Recently, 
(Ansari and Sahmani, 2011) studied bending behaviour and buckling 
of nanobeams including surface stress corresponding to different beam 
theories without consideration of nonlocality effect.

Moreover, the governing motion equations are often solved by 
analytical method Hosseini et al. (2013) or finite element methods 
(Eltaher et al, 2013) or generalized differential quadrature (GDQ) 
method (Ansari and Sahmani, 2011) and other solutions which need 
high CPU time to solve. DTM is also used to find the exact solution 
of both linear and nonlinear equations and even partial differential 
equations with high precision and also is simpler in compare with other 
methods. Although this method comes from Taylor series expansion, 
but DTM is different from the traditional high order Taylor’s series 
method. Because the traditional Taylor expansion requires symbolic 
competition of the necessary derivatives of the data functions and 



ISSN: 2180-1053        Vol. 7     No. 1    January - June 2015

Investigating The Surface Elasticity and Tension Effects on Critical Buckling Behaviour of Nanotubes 
Based on Differential Transformation Method

13

thus is computationally taken long time for large orders, while DTM 
takes less time to solve polynomial series. In other words, by applying 
DTM, governing equations for different boundary conditions reduces 
to algebraic equations, and finally all the calculations turn into simple 
iterative process. Also as seen in the literature, DTM has been used for 
solving a vast range of problems in different fields of engineering. 

To the best knowledge of the authors, no research effort has been 
devoted so far to find the solution of critical buckling load of nanotubes 
considering both surface and small scale effects employing differential 
transformation method. Motivated by this fact, in this study, differential 
transformation method is applied in analysing the surface effects, 
including surface elasticity and stress, on critical buckling load of 
nanotubes, made of Aluminium and Silicon, using nonlocal elasticity 
theory. Hamilton’s principle is employed to derive the governing 
equation and corresponding boundary conditions. DTM is then used 
to obtain the critical buckling load of nanotubes with various boundary 
conditions. To this end, the output results are compared favourably 
with those published works and influences of the surface effect, 
nonlocal parameter and size of nanotube on the critical buckling load 
are investigated. 

2.0 THEORY AND FORMULATION

Nonlocal constitutive relation for Euler-Bernoulli beam is given as 
(Eringen, 1983):

 3 

2.0 THEORY AND FORMULATION 
 
Nonlocal constitutive relation for Euler-Bernoulli beam is given as (Eringen, 1983): 

2

2
xx

xx xxEx


  


 


 (1) 

where xx and xx  are the nonlocal stress and strain, respectively. The distributed 
transverse loading induced by the residual surface tension is (Farshi et al , 2010): 

2

0 2

w
q q H

x


 


 (2) 

and H is the surface effect constant is given by: 

2 oH D  (3) 

Effective flexural rigidity, *EI  for nanotube is given by: 
* 4 4 3 31 ( ) ( )

4
s

o i o iEI E R R E R R      (4) 

The general differential equation of Euler-Bernoulli beam based on nonlocal continuum 
model and surface effect are derived using the principle of Hamilton and expressed by 
(Reddy, (2007): 

*
2 2 2 2 2

2

2

2 2 2 2 2 4

0 02 2

2 4

2 22

[ ]
w w w w

EI N A I
x x x x x t t x

w w w
N A I

x x t t

w
q H q

x

w
H

xx

  

 








      
     

       

   
  
   


 

   
  
  

 
 
 

 (5) 

 
The coordinate system for nanotube is shown in Figure 1. 
 
 

 
Figure 1. Geometry of nanotube with Length L , inner and outer radii Ri and Ro 

 
 

3.0 DIFFERENTIAL TRANSFORMATION METHOD 
 
Differential transformation method is one of the useful techniques to solve the 
differential equations with small calculation errors and ability to solve nonlinear 
equations with boundary conditions value problems. Abdel-Halim Hassan (2002) 
applied the DTM on eigenvalues and normalized eigenfunctions.  

 
 
 
 

 

where 

 3 

2.0 THEORY AND FORMULATION 
 
Nonlocal constitutive relation for Euler-Bernoulli beam is given as (Eringen, 1983): 

2

2
xx

xx xxEx


  


 


 (1) 

where xx and xx  are the nonlocal stress and strain, respectively. The distributed 
transverse loading induced by the residual surface tension is (Farshi et al , 2010): 

2

0 2

w
q q H

x


 


 (2) 

and H is the surface effect constant is given by: 

2 oH D  (3) 

Effective flexural rigidity, *EI  for nanotube is given by: 
* 4 4 3 31 ( ) ( )

4
s

o i o iEI E R R E R R      (4) 

The general differential equation of Euler-Bernoulli beam based on nonlocal continuum 
model and surface effect are derived using the principle of Hamilton and expressed by 
(Reddy, (2007): 

*
2 2 2 2 2

2

2

2 2 2 2 2 4

0 02 2

2 4

2 22

[ ]
w w w w

EI N A I
x x x x x t t x

w w w
N A I

x x t t

w
q H q

x

w
H

xx

  

 








      
     

       

   
  
   


 

   
  
  

 
 
 

 (5) 

 
The coordinate system for nanotube is shown in Figure 1. 
 
 

 
Figure 1. Geometry of nanotube with Length L , inner and outer radii Ri and Ro 

 
 

3.0 DIFFERENTIAL TRANSFORMATION METHOD 
 
Differential transformation method is one of the useful techniques to solve the 
differential equations with small calculation errors and ability to solve nonlinear 
equations with boundary conditions value problems. Abdel-Halim Hassan (2002) 
applied the DTM on eigenvalues and normalized eigenfunctions.  

 
 
 
 

 

are the nonlocal stress and strain, respectively. The 
distributed transverse loading induced by the residual surface tension 
is (Farshi et al , 2010):

 3 

2.0 THEORY AND FORMULATION 
 
Nonlocal constitutive relation for Euler-Bernoulli beam is given as (Eringen, 1983): 

2

2
xx

xx xxEx


  


 


 (1) 

where xx and xx  are the nonlocal stress and strain, respectively. The distributed 
transverse loading induced by the residual surface tension is (Farshi et al , 2010): 

2

0 2

w
q q H

x


 


 (2) 

and H is the surface effect constant is given by: 

2 oH D  (3) 

Effective flexural rigidity, *EI  for nanotube is given by: 
* 4 4 3 31 ( ) ( )

4
s

o i o iEI E R R E R R      (4) 

The general differential equation of Euler-Bernoulli beam based on nonlocal continuum 
model and surface effect are derived using the principle of Hamilton and expressed by 
(Reddy, (2007): 

*
2 2 2 2 2

2

2

2 2 2 2 2 4

0 02 2

2 4

2 22

[ ]
w w w w

EI N A I
x x x x x t t x

w w w
N A I

x x t t

w
q H q

x

w
H

xx

  

 








      
     

       

   
  
   


 

   
  
  

 
 
 

 (5) 

 
The coordinate system for nanotube is shown in Figure 1. 
 
 

 
Figure 1. Geometry of nanotube with Length L , inner and outer radii Ri and Ro 

 
 

3.0 DIFFERENTIAL TRANSFORMATION METHOD 
 
Differential transformation method is one of the useful techniques to solve the 
differential equations with small calculation errors and ability to solve nonlinear 
equations with boundary conditions value problems. Abdel-Halim Hassan (2002) 
applied the DTM on eigenvalues and normalized eigenfunctions.  

 
 
 
 

 

Effective flexural rigidity, 

 3 

2.0 THEORY AND FORMULATION 
 
Nonlocal constitutive relation for Euler-Bernoulli beam is given as (Eringen, 1983): 

2

2
xx

xx xxEx


  


 


 (1) 

where xx and xx  are the nonlocal stress and strain, respectively. The distributed 
transverse loading induced by the residual surface tension is (Farshi et al , 2010): 

2

0 2

w
q q H

x


 


 (2) 

and H is the surface effect constant is given by: 

2 oH D  (3) 

Effective flexural rigidity, *EI  for nanotube is given by: 
* 4 4 3 31 ( ) ( )

4
s

o i o iEI E R R E R R      (4) 

The general differential equation of Euler-Bernoulli beam based on nonlocal continuum 
model and surface effect are derived using the principle of Hamilton and expressed by 
(Reddy, (2007): 

*
2 2 2 2 2

2

2

2 2 2 2 2 4

0 02 2

2 4

2 22

[ ]
w w w w

EI N A I
x x x x x t t x

w w w
N A I

x x t t

w
q H q

x

w
H

xx

  

 








      
     

       

   
  
   


 

   
  
  

 
 
 

 (5) 

 
The coordinate system for nanotube is shown in Figure 1. 
 
 

 
Figure 1. Geometry of nanotube with Length L , inner and outer radii Ri and Ro 

 
 

3.0 DIFFERENTIAL TRANSFORMATION METHOD 
 
Differential transformation method is one of the useful techniques to solve the 
differential equations with small calculation errors and ability to solve nonlinear 
equations with boundary conditions value problems. Abdel-Halim Hassan (2002) 
applied the DTM on eigenvalues and normalized eigenfunctions.  

 
 
 
 

 

for nanotube is given by:

 3 

2.0 THEORY AND FORMULATION 
 
Nonlocal constitutive relation for Euler-Bernoulli beam is given as (Eringen, 1983): 

2

2
xx

xx xxEx


  


 


 (1) 

where xx and xx  are the nonlocal stress and strain, respectively. The distributed 
transverse loading induced by the residual surface tension is (Farshi et al , 2010): 

2

0 2

w
q q H

x


 


 (2) 

and H is the surface effect constant is given by: 

2 oH D  (3) 

Effective flexural rigidity, *EI  for nanotube is given by: 
* 4 4 3 31 ( ) ( )

4
s

o i o iEI E R R E R R      (4) 

The general differential equation of Euler-Bernoulli beam based on nonlocal continuum 
model and surface effect are derived using the principle of Hamilton and expressed by 
(Reddy, (2007): 

*
2 2 2 2 2

2

2

2 2 2 2 2 4

0 02 2

2 4

2 22

[ ]
w w w w

EI N A I
x x x x x t t x

w w w
N A I

x x t t

w
q H q

x

w
H

xx

  

 








      
     

       

   
  
   


 

   
  
  

 
 
 

 (5) 

 
The coordinate system for nanotube is shown in Figure 1. 
 
 

 
Figure 1. Geometry of nanotube with Length L , inner and outer radii Ri and Ro 

 
 

3.0 DIFFERENTIAL TRANSFORMATION METHOD 
 
Differential transformation method is one of the useful techniques to solve the 
differential equations with small calculation errors and ability to solve nonlinear 
equations with boundary conditions value problems. Abdel-Halim Hassan (2002) 
applied the DTM on eigenvalues and normalized eigenfunctions.  

 
 
 
 

 



ISSN: 2180-1053        Vol. 7     No. 1    January - June 2015

Journal of Mechanical Engineering and Technology 

14

The general differential equation of Euler-Bernoulli beam based on 
nonlocal continuum model and surface effect are derived using the 
principle of Hamilton and expressed by (Reddy, (2007):

 3 

2.0 THEORY AND FORMULATION 
 
Nonlocal constitutive relation for Euler-Bernoulli beam is given as (Eringen, 1983): 

2

2
xx

xx xxEx


  


 


 (1) 

where xx and xx  are the nonlocal stress and strain, respectively. The distributed 
transverse loading induced by the residual surface tension is (Farshi et al , 2010): 

2

0 2

w
q q H

x


 


 (2) 

and H is the surface effect constant is given by: 

2 oH D  (3) 

Effective flexural rigidity, *EI  for nanotube is given by: 
* 4 4 3 31 ( ) ( )

4
s

o i o iEI E R R E R R      (4) 

The general differential equation of Euler-Bernoulli beam based on nonlocal continuum 
model and surface effect are derived using the principle of Hamilton and expressed by 
(Reddy, (2007): 

*
2 2 2 2 2

2

2

2 2 2 2 2 4

0 02 2

2 4

2 22

[ ]
w w w w

EI N A I
x x x x x t t x

w w w
N A I

x x t t

w
q H q

x

w
H

xx

  

 








      
     

       

   
  
   


 

   
  
  

 
 
 

 (5) 

 
The coordinate system for nanotube is shown in Figure 1. 
 
 

 
Figure 1. Geometry of nanotube with Length L , inner and outer radii Ri and Ro 

 
 

3.0 DIFFERENTIAL TRANSFORMATION METHOD 
 
Differential transformation method is one of the useful techniques to solve the 
differential equations with small calculation errors and ability to solve nonlinear 
equations with boundary conditions value problems. Abdel-Halim Hassan (2002) 
applied the DTM on eigenvalues and normalized eigenfunctions.  

 
 
 
 

 

The coordinate system for nanotube is shown in Figure 1.

 3 

2.0 THEORY AND FORMULATION 
 
Nonlocal constitutive relation for Euler-Bernoulli beam is given as (Eringen, 1983): 

2

2
xx

xx xxEx


  


 


 (1) 

where xx and xx  are the nonlocal stress and strain, respectively. The distributed 
transverse loading induced by the residual surface tension is (Farshi et al , 2010): 

2

0 2

w
q q H

x


 


 (2) 

and H is the surface effect constant is given by: 

2 oH D  (3) 

Effective flexural rigidity, *EI  for nanotube is given by: 
* 4 4 3 31 ( ) ( )

4
s

o i o iEI E R R E R R      (4) 

The general differential equation of Euler-Bernoulli beam based on nonlocal continuum 
model and surface effect are derived using the principle of Hamilton and expressed by 
(Reddy, (2007): 

*
2 2 2 2 2

2

2

2 2 2 2 2 4

0 02 2

2 4

2 22

[ ]
w w w w

EI N A I
x x x x x t t x

w w w
N A I

x x t t

w
q H q

x

w
H

xx

  

 








      
     

       

   
  
   


 

   
  
  

 
 
 

 (5) 

 
The coordinate system for nanotube is shown in Figure 1. 
 
 

 
Figure 1. Geometry of nanotube with Length L , inner and outer radii Ri and Ro 

 
 

3.0 DIFFERENTIAL TRANSFORMATION METHOD 
 
Differential transformation method is one of the useful techniques to solve the 
differential equations with small calculation errors and ability to solve nonlinear 
equations with boundary conditions value problems. Abdel-Halim Hassan (2002) 
applied the DTM on eigenvalues and normalized eigenfunctions.  

 
 
 
 

 

Figure 1. Geometry of nanotube with Length L , inner and outer radii 
Ri and Ro

3.0 DIFFERENTIAL TRANSFORMATION METHOD

Differential transformation method is one of the useful techniques to 
solve the differential equations with small calculation errors and ability 
to solve nonlinear equations with boundary conditions value problems. 
Abdel-Halim Hassan (2002) applied the DTM on eigenvalues and 
normalized eigenfunctions. 

Table 1. Some of the transformation rules of the one-dimensional DTM

 4 

 
Table 1. Some of the transformation rules of the one-dimensional DTM 

 
Original function  Transformed function 
    ( )f x g x h x        ( )F K G K H K   
   ( )f x g x     ( )F K G K  

    ( )f x g x h x      
0

( )
K

l

F K G K l H l


   

  ( )
n

n

d g x
f x

dx
    

 !
( )

!
k n

F K G K n
k


   

  nf x x      
1
0

k n
F K K n

k n



   


 

 
Table 2. Transformed boundary conditions (B.C.) based on DTM 

 
X=0    X=L   
Original 

B.C. 
 

Transformed 
B.C. 

 

Original 
B.C. 

 
Transformed B.C. 

f (0) 0  F[0] 0  f ( ) 0L   
0

[ ] 0
k

F k




  

df (0)
0

dx
   

dF[0]
0

dx
   

df ( )
0

dx
L

   
0

 [ ] 0
k

k F k




  
2

2

(0)
0

dx
d f

   
2

2

[0]
0

dx
d F

   
2

2

( )
0

dx
d f L

    
0

1  [ ] 0
k

k k F k




   

3

3

(0)
0

dx
d f

   
3

3

[0]
0

dx
d F

   
3

3

( )
0

dx
d f L

   
  

0

1 2 [ ] 0
k

k k k F k




  
 

 
In this method, certain transformation rules are applied to both the governing 
differential equations of motion and the boundary conditions of the system in order to 
transform them into a set of algebraic equations as presented in Table 1 and Table 2. 
The solution of these algebraic equations gives the desired results of the problem. The 
basic definitions and the application procedure of this method can be introduced as 
follows. The transformation equation of function ( )f x  can be defined as (Chen et al 
2004),  
 

 
0

1 ( )
( )

!

k

x xk
d f x

F k
k dx 

  (6) 

where ( )f x is the original function and [ ]F k is the transformed function. The inverse 
transformation is defined as: 

   0
0

( )  k
k

f x x x F k




   (7) 
Combining equations (6) and (7) one obtains: 

 
 

0

0

0

( )
( )

!

kk

x xk
k

d f xx x
f x

k dx







   (8) 

 
In actual application, the function f(x) is expressed by a finite series and Equation (8) 
can be written as follows: 



ISSN: 2180-1053        Vol. 7     No. 1    January - June 2015

Investigating The Surface Elasticity and Tension Effects on Critical Buckling Behaviour of Nanotubes 
Based on Differential Transformation Method

15

Table 2. Transformed boundary conditions (B.C.) based on DTM 

 4 

 
Table 1. Some of the transformation rules of the one-dimensional DTM 

 
Original function  Transformed function 
    ( )f x g x h x        ( )F K G K H K   
   ( )f x g x     ( )F K G K  

    ( )f x g x h x      
0

( )
K

l

F K G K l H l


   

  ( )
n

n

d g x
f x

dx
    

 !
( )

!
k n

F K G K n
k


   

  nf x x      
1
0

k n
F K K n

k n



   


 

 
Table 2. Transformed boundary conditions (B.C.) based on DTM 

 
X=0    X=L   
Original 

B.C. 
 

Transformed 
B.C. 

 

Original 
B.C. 

 
Transformed B.C. 

f (0) 0  F[0] 0  f ( ) 0L   
0

[ ] 0
k

F k




  

df (0)
0

dx
   

dF[0]
0

dx
   

df ( )
0

dx
L

   
0

 [ ] 0
k

k F k




  
2

2

(0)
0

dx
d f

   
2

2

[0]
0

dx
d F

   
2

2

( )
0

dx
d f L

    
0

1  [ ] 0
k

k k F k




   

3

3

(0)
0

dx
d f

   
3

3

[0]
0

dx
d F

   
3

3

( )
0

dx
d f L

   
  

0

1 2 [ ] 0
k

k k k F k




  
 

 
In this method, certain transformation rules are applied to both the governing 
differential equations of motion and the boundary conditions of the system in order to 
transform them into a set of algebraic equations as presented in Table 1 and Table 2. 
The solution of these algebraic equations gives the desired results of the problem. The 
basic definitions and the application procedure of this method can be introduced as 
follows. The transformation equation of function ( )f x  can be defined as (Chen et al 
2004),  
 

 
0

1 ( )
( )

!

k

x xk
d f x

F k
k dx 

  (6) 

where ( )f x is the original function and [ ]F k is the transformed function. The inverse 
transformation is defined as: 

   0
0

( )  k
k

f x x x F k




   (7) 
Combining equations (6) and (7) one obtains: 

 
 

0

0

0

( )
( )

!

kk

x xk
k

d f xx x
f x

k dx







   (8) 

 
In actual application, the function f(x) is expressed by a finite series and Equation (8) 
can be written as follows: 

In this method, certain transformation rules are applied to both 
the governing differential equations of motion and the boundary 
conditions of the system in order to transform them into a set of 
algebraic equations as presented in Table 1 and Table 2. The solution of 
these algebraic equations gives the desired results of the problem. The 
basic definitions and the application procedure of this method can be 
introduced as follows. The transformation equation of function   can be 
defined as (Chen et al 2004), 

 4 

 
Table 1. Some of the transformation rules of the one-dimensional DTM 

 
Original function  Transformed function 
    ( )f x g x h x        ( )F K G K H K   
   ( )f x g x     ( )F K G K  

    ( )f x g x h x      
0

( )
K

l

F K G K l H l


   

  ( )
n

n

d g x
f x

dx
    

 !
( )

!
k n

F K G K n
k


   

  nf x x      
1
0

k n
F K K n

k n



   


 

 
Table 2. Transformed boundary conditions (B.C.) based on DTM 

 
X=0    X=L   
Original 

B.C. 
 

Transformed 
B.C. 

 

Original 
B.C. 

 
Transformed B.C. 

f (0) 0  F[0] 0  f ( ) 0L   
0

[ ] 0
k

F k




  

df (0)
0

dx
   

dF[0]
0

dx
   

df ( )
0

dx
L

   
0

 [ ] 0
k

k F k




  
2

2

(0)
0

dx
d f

   
2

2

[0]
0

dx
d F

   
2

2

( )
0

dx
d f L

    
0

1  [ ] 0
k

k k F k




   

3

3

(0)
0

dx
d f

   
3

3

[0]
0

dx
d F

   
3

3

( )
0

dx
d f L

   
  

0

1 2 [ ] 0
k

k k k F k




  
 

 
In this method, certain transformation rules are applied to both the governing 
differential equations of motion and the boundary conditions of the system in order to 
transform them into a set of algebraic equations as presented in Table 1 and Table 2. 
The solution of these algebraic equations gives the desired results of the problem. The 
basic definitions and the application procedure of this method can be introduced as 
follows. The transformation equation of function ( )f x  can be defined as (Chen et al 
2004),  
 

 
0

1 ( )
( )

!

k

x xk
d f x

F k
k dx 

  (6) 

where ( )f x is the original function and [ ]F k is the transformed function. The inverse 
transformation is defined as: 

   0
0

( )  k
k

f x x x F k




   (7) 
Combining equations (6) and (7) one obtains: 

 
 

0

0

0

( )
( )

!

kk

x xk
k

d f xx x
f x

k dx







   (8) 

 
In actual application, the function f(x) is expressed by a finite series and Equation (8) 
can be written as follows: In actual application, the function f(x) is expressed by a finite series and 
Equation (8) can be written as follows:

 5 

 
 

0

0

0

( )
( )

!

kk

x x

N

k
k

d f xx x
f x

k dx 


   (9) 
which implies that the term in relation (9) is negligible: 

 
 

0

0

1

( )
( )

!

kk

x xk
k N

d f xx x
f x

k dx




 


 

 
(10) 

 
 

3.1 Implementation of Differential Transform Method 
 

While solving the Equation (5) authors preferred DTM approach which avoids solving 
complicated transcendental algebraic equations for general boundary conditions. In 
order to derive differential form of Equation (5) we refer Table 1 and the following 
expression is written as: 

 * ( 4)! ( 2)!( ) [ 4] [ 2] 0
! !

k k
EI H N W k N H W k

k k
 

 
          (11) 

 
And the various boundary condition for nanotube by using Table 2 can be expressed as: 
 Simply supported–Simply supported: 

   0 0 , W 2 0W    
(12) 

0 0

[ ] 0 , ( 1) [ ] 0
k k

W k k k W k
 

 

     

 Clamped–Clamped: 
   W 0 0 , W 1 0   

(13) 

0 0

[ ] 0 ,  [ ] 0
k k

W k k W k
 

 

    

 Clamped–Simply supported: 
   W 0 0 , W 1 0   

(14) 

0 0

[ ] 0 , ( 1) [ ] 0
k k

W k k k W k
 

 

     

 
By using Equation(11) and with the transformed boundary conditions one arrives at the 
following eigenvalue problem: 
 

 11 12
21 22

( ) ( )
0

( ) ( )
A N A N

C
A N A N


 
 
 

 (15) 

 
Where C correspond to the missing boundary conditions at x=0. For the non-trivial 
solutions of Equation(15), it is necessary that the determinant of the coefficient matrix is 
equal to zero: 
 

11 12

21 22

( ) ( )
0

( ) ( )
A N A N
A N A N

  (16) 



ISSN: 2180-1053        Vol. 7     No. 1    January - June 2015

Journal of Mechanical Engineering and Technology 

16

3.1	 Implementation	of	Differential	Transform	Method

While solving the Equation (5) authors preferred DTM approach which 
avoids solving complicated transcendental algebraic equations for 
general boundary conditions. In order to derive differential form of 
Equation (5) we refer Table 1 and the following expression is written as:

 5 

 
 

0

0

0

( )
( )

!

kk

x x

N

k
k

d f xx x
f x

k dx 


   (9) 
which implies that the term in relation (9) is negligible: 

 
 

0

0

1

( )
( )

!

kk

x xk
k N

d f xx x
f x

k dx




 


 

 
(10) 

 
 

3.1 Implementation of Differential Transform Method 
 

While solving the Equation (5) authors preferred DTM approach which avoids solving 
complicated transcendental algebraic equations for general boundary conditions. In 
order to derive differential form of Equation (5) we refer Table 1 and the following 
expression is written as: 

 * ( 4)! ( 2)!( ) [ 4] [ 2] 0
! !

k k
EI H N W k N H W k

k k
 

 
          (11) 

 
And the various boundary condition for nanotube by using Table 2 can be expressed as: 
 Simply supported–Simply supported: 

   0 0 , W 2 0W    
(12) 

0 0

[ ] 0 , ( 1) [ ] 0
k k

W k k k W k
 

 

     

 Clamped–Clamped: 
   W 0 0 , W 1 0   

(13) 

0 0

[ ] 0 ,  [ ] 0
k k

W k k W k
 

 

    

 Clamped–Simply supported: 
   W 0 0 , W 1 0   

(14) 

0 0

[ ] 0 , ( 1) [ ] 0
k k

W k k k W k
 

 

     

 
By using Equation(11) and with the transformed boundary conditions one arrives at the 
following eigenvalue problem: 
 

 11 12
21 22

( ) ( )
0

( ) ( )
A N A N

C
A N A N


 
 
 

 (15) 

 
Where C correspond to the missing boundary conditions at x=0. For the non-trivial 
solutions of Equation(15), it is necessary that the determinant of the coefficient matrix is 
equal to zero: 
 

11 12

21 22

( ) ( )
0

( ) ( )
A N A N
A N A N

  (16) 

And the various boundary condition for nanotube by using Table 2 can 
be expressed as:

• Simply supported–Simply supported:

 5 

 
 

0

0

0

( )
( )

!

kk

x x

N

k
k

d f xx x
f x

k dx 


   (9) 
which implies that the term in relation (9) is negligible: 

 
 

0

0

1

( )
( )

!

kk

x xk
k N

d f xx x
f x

k dx




 


 

 
(10) 

 
 

3.1 Implementation of Differential Transform Method 
 

While solving the Equation (5) authors preferred DTM approach which avoids solving 
complicated transcendental algebraic equations for general boundary conditions. In 
order to derive differential form of Equation (5) we refer Table 1 and the following 
expression is written as: 

 * ( 4)! ( 2)!( ) [ 4] [ 2] 0
! !

k k
EI H N W k N H W k

k k
 

 
          (11) 

 
And the various boundary condition for nanotube by using Table 2 can be expressed as: 
 Simply supported–Simply supported: 

   0 0 , W 2 0W    
(12) 

0 0

[ ] 0 , ( 1) [ ] 0
k k

W k k k W k
 

 

     

 Clamped–Clamped: 
   W 0 0 , W 1 0   

(13) 

0 0

[ ] 0 ,  [ ] 0
k k

W k k W k
 

 

    

 Clamped–Simply supported: 
   W 0 0 , W 1 0   

(14) 

0 0

[ ] 0 , ( 1) [ ] 0
k k

W k k k W k
 

 

     

 
By using Equation(11) and with the transformed boundary conditions one arrives at the 
following eigenvalue problem: 
 

 11 12
21 22

( ) ( )
0

( ) ( )
A N A N

C
A N A N


 
 
 

 (15) 

 
Where C correspond to the missing boundary conditions at x=0. For the non-trivial 
solutions of Equation(15), it is necessary that the determinant of the coefficient matrix is 
equal to zero: 
 

11 12

21 22

( ) ( )
0

( ) ( )
A N A N
A N A N

  (16) 

• Clamped–Clamped:

 5 

 
 

0

0

0

( )
( )

!

kk

x x

N

k
k

d f xx x
f x

k dx 


   (9) 
which implies that the term in relation (9) is negligible: 

 
 

0

0

1

( )
( )

!

kk

x xk
k N

d f xx x
f x

k dx




 


 

 
(10) 

 
 

3.1 Implementation of Differential Transform Method 
 

While solving the Equation (5) authors preferred DTM approach which avoids solving 
complicated transcendental algebraic equations for general boundary conditions. In 
order to derive differential form of Equation (5) we refer Table 1 and the following 
expression is written as: 

 * ( 4)! ( 2)!( ) [ 4] [ 2] 0
! !

k k
EI H N W k N H W k

k k
 

 
          (11) 

 
And the various boundary condition for nanotube by using Table 2 can be expressed as: 
 Simply supported–Simply supported: 

   0 0 , W 2 0W    
(12) 

0 0

[ ] 0 , ( 1) [ ] 0
k k

W k k k W k
 

 

     

 Clamped–Clamped: 
   W 0 0 , W 1 0   

(13) 

0 0

[ ] 0 ,  [ ] 0
k k

W k k W k
 

 

    

 Clamped–Simply supported: 
   W 0 0 , W 1 0   

(14) 

0 0

[ ] 0 , ( 1) [ ] 0
k k

W k k k W k
 

 

     

 
By using Equation(11) and with the transformed boundary conditions one arrives at the 
following eigenvalue problem: 
 

 11 12
21 22

( ) ( )
0

( ) ( )
A N A N

C
A N A N


 
 
 

 (15) 

 
Where C correspond to the missing boundary conditions at x=0. For the non-trivial 
solutions of Equation(15), it is necessary that the determinant of the coefficient matrix is 
equal to zero: 
 

11 12

21 22

( ) ( )
0

( ) ( )
A N A N
A N A N

  (16) 

• Clamped–Simply supported:

 5 

 
 

0

0

0

( )
( )

!

kk

x x

N

k
k

d f xx x
f x

k dx 


   (9) 
which implies that the term in relation (9) is negligible: 

 
 

0

0

1

( )
( )

!

kk

x xk
k N

d f xx x
f x

k dx




 


 

 
(10) 

 
 

3.1 Implementation of Differential Transform Method 
 

While solving the Equation (5) authors preferred DTM approach which avoids solving 
complicated transcendental algebraic equations for general boundary conditions. In 
order to derive differential form of Equation (5) we refer Table 1 and the following 
expression is written as: 

 * ( 4)! ( 2)!( ) [ 4] [ 2] 0
! !

k k
EI H N W k N H W k

k k
 

 
          (11) 

 
And the various boundary condition for nanotube by using Table 2 can be expressed as: 
 Simply supported–Simply supported: 

   0 0 , W 2 0W    
(12) 

0 0

[ ] 0 , ( 1) [ ] 0
k k

W k k k W k
 

 

     

 Clamped–Clamped: 
   W 0 0 , W 1 0   

(13) 

0 0

[ ] 0 ,  [ ] 0
k k

W k k W k
 

 

    

 Clamped–Simply supported: 
   W 0 0 , W 1 0   

(14) 

0 0

[ ] 0 , ( 1) [ ] 0
k k

W k k k W k
 

 

     

 
By using Equation(11) and with the transformed boundary conditions one arrives at the 
following eigenvalue problem: 
 

 11 12
21 22

( ) ( )
0

( ) ( )
A N A N

C
A N A N


 
 
 

 (15) 

 
Where C correspond to the missing boundary conditions at x=0. For the non-trivial 
solutions of Equation(15), it is necessary that the determinant of the coefficient matrix is 
equal to zero: 
 

11 12

21 22

( ) ( )
0

( ) ( )
A N A N
A N A N

  (16) 

By using Equation(11) and with the transformed boundary conditions 
one arrives at the following eigenvalue problem:

 5 

 
 

0

0

0

( )
( )

!

kk

x x

N

k
k

d f xx x
f x

k dx 


   (9) 
which implies that the term in relation (9) is negligible: 

 
 

0

0

1

( )
( )

!

kk

x xk
k N

d f xx x
f x

k dx




 


 

 
(10) 

 
 

3.1 Implementation of Differential Transform Method 
 

While solving the Equation (5) authors preferred DTM approach which avoids solving 
complicated transcendental algebraic equations for general boundary conditions. In 
order to derive differential form of Equation (5) we refer Table 1 and the following 
expression is written as: 

 * ( 4)! ( 2)!( ) [ 4] [ 2] 0
! !

k k
EI H N W k N H W k

k k
 

 
          (11) 

 
And the various boundary condition for nanotube by using Table 2 can be expressed as: 
 Simply supported–Simply supported: 

   0 0 , W 2 0W    
(12) 

0 0

[ ] 0 , ( 1) [ ] 0
k k

W k k k W k
 

 

     

 Clamped–Clamped: 
   W 0 0 , W 1 0   

(13) 

0 0

[ ] 0 ,  [ ] 0
k k

W k k W k
 

 

    

 Clamped–Simply supported: 
   W 0 0 , W 1 0   

(14) 

0 0

[ ] 0 , ( 1) [ ] 0
k k

W k k k W k
 

 

     

 
By using Equation(11) and with the transformed boundary conditions one arrives at the 
following eigenvalue problem: 
 

 11 12
21 22

( ) ( )
0

( ) ( )
A N A N

C
A N A N


 
 
 

 (15) 

 
Where C correspond to the missing boundary conditions at x=0. For the non-trivial 
solutions of Equation(15), it is necessary that the determinant of the coefficient matrix is 
equal to zero: 
 

11 12

21 22

( ) ( )
0

( ) ( )
A N A N
A N A N

  (16) 

Where correspond to the missing boundary conditions at x=0. For the 
non-trivial solutions of Equation(15), it is necessary that the determinant 
of the coefficient matrix is equal to zero:



ISSN: 2180-1053        Vol. 7     No. 1    January - June 2015

Investigating The Surface Elasticity and Tension Effects on Critical Buckling Behaviour of Nanotubes 
Based on Differential Transformation Method

17

 5 

 
 

0

0

0

( )
( )

!

kk

x x

N

k
k

d f xx x
f x

k dx 


   (9) 
which implies that the term in relation (9) is negligible: 

 
 

0

0

1

( )
( )

!

kk

x xk
k N

d f xx x
f x

k dx




 


 

 
(10) 

 
 

3.1 Implementation of Differential Transform Method 
 

While solving the Equation (5) authors preferred DTM approach which avoids solving 
complicated transcendental algebraic equations for general boundary conditions. In 
order to derive differential form of Equation (5) we refer Table 1 and the following 
expression is written as: 

 * ( 4)! ( 2)!( ) [ 4] [ 2] 0
! !

k k
EI H N W k N H W k

k k
 

 
          (11) 

 
And the various boundary condition for nanotube by using Table 2 can be expressed as: 
 Simply supported–Simply supported: 

   0 0 , W 2 0W    
(12) 

0 0

[ ] 0 , ( 1) [ ] 0
k k

W k k k W k
 

 

     

 Clamped–Clamped: 
   W 0 0 , W 1 0   

(13) 

0 0

[ ] 0 ,  [ ] 0
k k

W k k W k
 

 

    

 Clamped–Simply supported: 
   W 0 0 , W 1 0   

(14) 

0 0

[ ] 0 , ( 1) [ ] 0
k k

W k k k W k
 

 

     

 
By using Equation(11) and with the transformed boundary conditions one arrives at the 
following eigenvalue problem: 
 

 11 12
21 22

( ) ( )
0

( ) ( )
A N A N

C
A N A N


 
 
 

 (15) 

 
Where C correspond to the missing boundary conditions at x=0. For the non-trivial 
solutions of Equation(15), it is necessary that the determinant of the coefficient matrix is 
equal to zero: 
 

11 12

21 22

( ) ( )
0

( ) ( )
A N A N
A N A N

  (16) 

Solution of Equation (16) is simply a polynomial root finding problem. 
Many techniques such as Newton’s method, Laguerre’s method, etc. 
can be used to find the roots of this equation.

4.0 RESULTS AND DISCUSSIONS

A nanotube with circular cross section and two different materials, 
aluminium and silicon, are considered. The elastic bulk and surface 
properties of aluminium with crystallographic direction of [1 1 1] and 
silicon with crystallographic direction of [1 0 0] are tabulated in Table 
3. The nondimensional buckling load is defined as:

 6 

Solution of Equation (16) is simply a polynomial root finding problem. Many 
techniques such as Newton’s method, Laguerre’s method, etc. can be used to find the 
roots of this equation. 
 

 
4.0 RESULTS AND DISCUSSIONS 
 
A nanotube with circular cross section and two different materials, aluminium and 
silicon, are considered. The elastic bulk and surface properties of aluminium with 
crystallographic direction of [1 1 1] and silicon with crystallographic direction of [1 0 0] 
are tabulated in Table 3. The nondimensional buckling load is defined as: 
Table 4 presents the first critical buckling load where the length-to-thickness ratio is 10 

while varying the nonlocal parameter. As can be noted, the obtained results are in good 
agreement with those of Reddy (2007) and even more conservative than those presented 
by Thai (2012). The critical buckling load decreases as the nonlocal parameter increases. 
This emphasizes the significance of the nonlocal effect on the buckling response of 
beams. 

 
Table 3. Material properties of AL and Si  

 
Mater
ial 

E(Gpa) ρ(kg/m3)   Es(N/M) τo(N/M) 

AL 70 2700 0.3 5.1882 0.9108 
Si 210 2370 0.24 -10.6543 0.6048 

 
Table 4. The nondimensional buckling load for simply supported beam 

 

L/h µ Thai[19] Reddy[14] Present 

10 0 9.8696 9.8696 9.86960440 
 1 8.9830 8.9830 8.98301623 
 2 8.2426 8.2426 8.24258361 
 3 7.6149 7.6149 7.61491765 
 4 7.0761 7.0761 7.07607999 

 
 
The effects of nonlocal effect (NE), nonlocal parameter and nonlocal surface effect 
(NSE) on the first three nondimensional buckling loads with different boundary 
conditions are presented in Table 5. It should be noted that  0   corresponds to local 
beam theory. From obtained results, it can be deduced that, when the nonlocal 
parameter increases, the buckling load decrease. Figure 2,3 and 4 depict the variation of 
the normalized nondimensional buckling load versus the nanotube length for three 
boundary conditions, i.e. Simply–Simply (S–S), Clamped–Simply (C–S) and Clamped–
Clamped (C–C). It can be noted, buckling load ratio is decreased with increasing in the 
beam size. But, the increasing in nonlocality parameter leads to increase the buckling 
ratio for the same beam size. 

 2 *cr LN N EI  (17) 

Table 4 presents the first critical buckling load where the length-to-
thickness ratio is 10 while varying the nonlocal parameter. As can be 
noted, the obtained results are in good agreement with those of Reddy 
(2007) and even more conservative than those presented by Thai 
(2012). The critical buckling load decreases as the nonlocal parameter 
increases. This emphasizes the significance of the nonlocal effect on the 
buckling response of beams.

Table 3. Material properties of AL and Si

 6 

Solution of Equation (16) is simply a polynomial root finding problem. Many 
techniques such as Newton’s method, Laguerre’s method, etc. can be used to find the 
roots of this equation. 
 

 
4.0 RESULTS AND DISCUSSIONS 
 
A nanotube with circular cross section and two different materials, aluminium and 
silicon, are considered. The elastic bulk and surface properties of aluminium with 
crystallographic direction of [1 1 1] and silicon with crystallographic direction of [1 0 0] 
are tabulated in Table 3. The nondimensional buckling load is defined as: 
Table 4 presents the first critical buckling load where the length-to-thickness ratio is 10 

while varying the nonlocal parameter. As can be noted, the obtained results are in good 
agreement with those of Reddy (2007) and even more conservative than those presented 
by Thai (2012). The critical buckling load decreases as the nonlocal parameter increases. 
This emphasizes the significance of the nonlocal effect on the buckling response of 
beams. 

 
Table 3. Material properties of AL and Si  

 
Mater
ial 

E(Gpa) ρ(kg/m3)   Es(N/M) τo(N/M) 

AL 70 2700 0.3 5.1882 0.9108 
Si 210 2370 0.24 -10.6543 0.6048 

 
Table 4. The nondimensional buckling load for simply supported beam 

 

L/h µ Thai[19] Reddy[14] Present 

10 0 9.8696 9.8696 9.86960440 
 1 8.9830 8.9830 8.98301623 
 2 8.2426 8.2426 8.24258361 
 3 7.6149 7.6149 7.61491765 
 4 7.0761 7.0761 7.07607999 

 
 
The effects of nonlocal effect (NE), nonlocal parameter and nonlocal surface effect 
(NSE) on the first three nondimensional buckling loads with different boundary 
conditions are presented in Table 5. It should be noted that  0   corresponds to local 
beam theory. From obtained results, it can be deduced that, when the nonlocal 
parameter increases, the buckling load decrease. Figure 2,3 and 4 depict the variation of 
the normalized nondimensional buckling load versus the nanotube length for three 
boundary conditions, i.e. Simply–Simply (S–S), Clamped–Simply (C–S) and Clamped–
Clamped (C–C). It can be noted, buckling load ratio is decreased with increasing in the 
beam size. But, the increasing in nonlocality parameter leads to increase the buckling 
ratio for the same beam size. 

 2 *cr LN N EI  (17) 

Table 4. The nondimensional buckling load for simply supported beam

 6 

Solution of Equation (16) is simply a polynomial root finding problem. Many 
techniques such as Newton’s method, Laguerre’s method, etc. can be used to find the 
roots of this equation. 
 

 
4.0 RESULTS AND DISCUSSIONS 
 
A nanotube with circular cross section and two different materials, aluminium and 
silicon, are considered. The elastic bulk and surface properties of aluminium with 
crystallographic direction of [1 1 1] and silicon with crystallographic direction of [1 0 0] 
are tabulated in Table 3. The nondimensional buckling load is defined as: 
Table 4 presents the first critical buckling load where the length-to-thickness ratio is 10 

while varying the nonlocal parameter. As can be noted, the obtained results are in good 
agreement with those of Reddy (2007) and even more conservative than those presented 
by Thai (2012). The critical buckling load decreases as the nonlocal parameter increases. 
This emphasizes the significance of the nonlocal effect on the buckling response of 
beams. 

 
Table 3. Material properties of AL and Si  

 
Mater
ial 

E(Gpa) ρ(kg/m3)   Es(N/M) τo(N/M) 

AL 70 2700 0.3 5.1882 0.9108 
Si 210 2370 0.24 -10.6543 0.6048 

 
Table 4. The nondimensional buckling load for simply supported beam 

 

L/h µ Thai[19] Reddy[14] Present 

10 0 9.8696 9.8696 9.86960440 
 1 8.9830 8.9830 8.98301623 
 2 8.2426 8.2426 8.24258361 
 3 7.6149 7.6149 7.61491765 
 4 7.0761 7.0761 7.07607999 

 
 
The effects of nonlocal effect (NE), nonlocal parameter and nonlocal surface effect 
(NSE) on the first three nondimensional buckling loads with different boundary 
conditions are presented in Table 5. It should be noted that  0   corresponds to local 
beam theory. From obtained results, it can be deduced that, when the nonlocal 
parameter increases, the buckling load decrease. Figure 2,3 and 4 depict the variation of 
the normalized nondimensional buckling load versus the nanotube length for three 
boundary conditions, i.e. Simply–Simply (S–S), Clamped–Simply (C–S) and Clamped–
Clamped (C–C). It can be noted, buckling load ratio is decreased with increasing in the 
beam size. But, the increasing in nonlocality parameter leads to increase the buckling 
ratio for the same beam size. 

 2 *cr LN N EI  (17) 



ISSN: 2180-1053        Vol. 7     No. 1    January - June 2015

Journal of Mechanical Engineering and Technology 

18

The effects of nonlocal effect (NE), nonlocal parameter and nonlocal 
surface effect (NSE) on the first three nondimensional buckling loads 
with different boundary conditions are presented in Table 5. It should 
be noted that 

 6 

Solution of Equation (16) is simply a polynomial root finding problem. Many 
techniques such as Newton’s method, Laguerre’s method, etc. can be used to find the 
roots of this equation. 
 

 
4.0 RESULTS AND DISCUSSIONS 
 
A nanotube with circular cross section and two different materials, aluminium and 
silicon, are considered. The elastic bulk and surface properties of aluminium with 
crystallographic direction of [1 1 1] and silicon with crystallographic direction of [1 0 0] 
are tabulated in Table 3. The nondimensional buckling load is defined as: 
Table 4 presents the first critical buckling load where the length-to-thickness ratio is 10 

while varying the nonlocal parameter. As can be noted, the obtained results are in good 
agreement with those of Reddy (2007) and even more conservative than those presented 
by Thai (2012). The critical buckling load decreases as the nonlocal parameter increases. 
This emphasizes the significance of the nonlocal effect on the buckling response of 
beams. 

 
Table 3. Material properties of AL and Si  

 
Mater
ial 

E(Gpa) ρ(kg/m3)   Es(N/M) τo(N/M) 

AL 70 2700 0.3 5.1882 0.9108 
Si 210 2370 0.24 -10.6543 0.6048 

 
Table 4. The nondimensional buckling load for simply supported beam 

 

L/h µ Thai[19] Reddy[14] Present 

10 0 9.8696 9.8696 9.86960440 
 1 8.9830 8.9830 8.98301623 
 2 8.2426 8.2426 8.24258361 
 3 7.6149 7.6149 7.61491765 
 4 7.0761 7.0761 7.07607999 

 
 
The effects of nonlocal effect (NE), nonlocal parameter and nonlocal surface effect 
(NSE) on the first three nondimensional buckling loads with different boundary 
conditions are presented in Table 5. It should be noted that  0   corresponds to local 
beam theory. From obtained results, it can be deduced that, when the nonlocal 
parameter increases, the buckling load decrease. Figure 2,3 and 4 depict the variation of 
the normalized nondimensional buckling load versus the nanotube length for three 
boundary conditions, i.e. Simply–Simply (S–S), Clamped–Simply (C–S) and Clamped–
Clamped (C–C). It can be noted, buckling load ratio is decreased with increasing in the 
beam size. But, the increasing in nonlocality parameter leads to increase the buckling 
ratio for the same beam size. 

 2 *cr LN N EI  (17) 

 corresponds to local beam theory. From obtained 
results, it can be deduced that, when the nonlocal parameter increases, 
the buckling load decrease. Figure 2,3 and 4 depict the variation of the 
normalized nondimensional buckling load versus the nanotube length 
for three boundary conditions, i.e. Simply–Simply (S–S), Clamped–
Simply (C–S) and Clamped–Clamped (C–C). It can be noted, buckling 
load ratio is decreased with increasing in the beam size. But, the 
increasing in nonlocality parameter leads to increase the buckling ratio 
for the same beam size.

 7 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 

Figure 2.  Buckling load ratio of the nanotube with various length and nonlocal parameters (S-S) 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Figure 3.  Buckling load ratio of the nanotube with various length and nonlocal parameters(C-C) 
 

Figure 2.  Buckling load ratio of the nanotube with various length and 
nonlocal parameters (S-S)

 7 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 

Figure 2.  Buckling load ratio of the nanotube with various length and nonlocal parameters (S-S) 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Figure 3.  Buckling load ratio of the nanotube with various length and nonlocal parameters(C-C) 
 

Figure 3.  Buckling load ratio of the nanotube with various length and 
nonlocal parameters(C-C)



ISSN: 2180-1053        Vol. 7     No. 1    January - June 2015

Investigating The Surface Elasticity and Tension Effects on Critical Buckling Behaviour of Nanotubes 
Based on Differential Transformation Method

19

 8 

 

 
 
 

Figure 4. Buckling load ratio of the nanotube with various length and nonlocal parameters(C-S) 
 
Table 5. The first three critical buckling loads versus nonlocal parameters for nanotubes with 

various boundary conditions 
 

µ 
iN  

S-S 
 NE NSE(AL) NSE(Si) 
0 i=1 9.8696 42.9089 34.2707 
 i=2 39.4784 72.5177 63.7895 
 i=3 88.8264 121.8660 113.2280 
     
2 i=1 8.2425 41.2819 32.6437 
 i=2 22.4976 55.0996 46.4614 
 i=3 31.9919 65.0312 56.3930 
     
4 i=1 7.0760 40.1154 31.4772 
 i=2 15.3068 48.3461 39.7079 
 i=3 19.5092 52.5485 43.9103 

µ 
iN  

S-S 
 NE NSE(AL) NSE(Si) 
0 i=1 20.1907 53.2300 44.5918 
 i=2 59.6759 92.7188 84.0806 
 i=3 118.9000 151.9390 143.3010 
     
2 i=1 14.3828 47.3828 38.7839 
 i=2 27.2063 60.2456 51.6074 
 i=3 35.1838 68.2376 59.5994 
     
4 i=1 11.1697 44.2090 35.5708 
 i=2 17.6192 50.6585 42.0203 

Figure 4. Buckling load ratio of the nanotube with various length and 
nonlocal parameters(C-S)

Table 5. The first three critical buckling loads versus nonlocal 
parameters for nanotubes with various boundary conditions

 8 

 

 
 
 

Figure 4. Buckling load ratio of the nanotube with various length and nonlocal parameters(C-S) 
 
Table 5. The first three critical buckling loads versus nonlocal parameters for nanotubes with 

various boundary conditions 
 

µ 
iN  

S-S 
 NE NSE(AL) NSE(Si) 
0 i=1 9.8696 42.9089 34.2707 
 i=2 39.4784 72.5177 63.7895 
 i=3 88.8264 121.8660 113.2280 
     

2 i=1 8.2425 41.2819 32.6437 
 i=2 22.4976 55.0996 46.4614 
 i=3 31.9919 65.0312 56.3930 
     

4 i=1 7.0760 40.1154 31.4772 
 i=2 15.3068 48.3461 39.7079 
 i=3 19.5092 52.5485 43.9103 

µ 
iN  

S-S 
 NE NSE(AL) NSE(Si) 
0 i=1 20.1907 53.2300 44.5918 
 i=2 59.6759 92.7188 84.0806 
 i=3 118.9000 151.9390 143.3010 
     

2 i=1 14.3828 47.3828 38.7839 
 i=2 27.2063 60.2456 51.6074 
 i=3 35.1838 68.2376 59.5994 
     

4 i=1 11.1697 44.2090 35.5708 
 i=2 17.6192 50.6585 42.0203 



ISSN: 2180-1053        Vol. 7     No. 1    January - June 2015

Journal of Mechanical Engineering and Technology 

20

 9 

 i=3 20.6567 53.6960 45.0578 
 

µ 
iN  

C-S 
 NE NSE(AL) NSE(Si) 
0 i=1 39.4784 72.5177 63.8795 
 i=2 80.7629 113.8020 105.1640 
 i=3 157.914 190.9530 182.3150 
     
2 i=1 22.0603 55.0996 46.4614 
 i=2 30.8814 63.9207 55.2825 
 i=3 37.9758 71.0151 62.3769 
     
4 i=1 15.3068 48.3461 39.7079 
 i=2 19.0906 52.1298 43.4917 
 i=3 21.5831 54.6224 45.9842 

 
 
 
5.0 CONCLUSIONS 
 
In the present study, the buckling behaviour of nanotubes including the effect of surface 
stress was predicted via linear partial differential equations of motion and related 
boundary conditions were derived. The nanotubes are considered to be made of Al with 
positive surface elasticity and Si with negative surface elasticity. Afterward, the 
differential transformation method as an efficient and accurate numerical tool was 
applied to solve the linear equations of nanotubes subjected to different boundary 
conditions. The good agreement between the results of this article and those available in 
literature validated the presented approach. Numerical results demonstrate that the small 
scale effects play an important role on the buckling behaviour of the nanotube. Also, it 
is observed that increasing the nonlocal parameter increased the buckling ratio of the 
nanotubes. 

 
 

NOMENCLATURES 

 

*EI   Effective flexural rigidity 
q       Distributed transverse loading 

N     Critical buckling load 
A      Cross sectional area 
I       Mass moment of inertia  

sE    Surface elasticity modulus 
E      Elasticity Modulus 

Greek symbols 
    Nonlocal parameter   

    Mass density  
0    Surface tension 

     Poisson’s ratio 
 
 

 
 
 

5.0 CONCLUSIONS

In the present study, the buckling behaviour of nanotubes including 
the effect of surface stress was predicted via linear partial differential 
equations of motion and related boundary conditions were derived. 
The nanotubes are considered to be made of Al with positive surface 
elasticity and Si with negative surface elasticity. Afterward, the 
differential transformation method as an efficient and accurate 
numerical tool was applied to solve the linear equations of nanotubes 
subjected to different boundary conditions. The good agreement 
between the results of this article and those available in literature 
validated the presented approach. Numerical results demonstrate that 
the small scale effects play an important role on the buckling behaviour 
of the nanotube. Also, it is observed that increasing the nonlocal 
parameter increased the buckling ratio of the nanotubes.

NOMENCLATURES

  

 9 

 i=3 20.6567 53.6960 45.0578 
 

µ 
iN  

C-S 
 NE NSE(AL) NSE(Si) 
0 i=1 39.4784 72.5177 63.8795 
 i=2 80.7629 113.8020 105.1640 
 i=3 157.914 190.9530 182.3150 
     
2 i=1 22.0603 55.0996 46.4614 
 i=2 30.8814 63.9207 55.2825 
 i=3 37.9758 71.0151 62.3769 
     
4 i=1 15.3068 48.3461 39.7079 
 i=2 19.0906 52.1298 43.4917 
 i=3 21.5831 54.6224 45.9842 

 
 
 
5.0 CONCLUSIONS 
 
In the present study, the buckling behaviour of nanotubes including the effect of surface 
stress was predicted via linear partial differential equations of motion and related 
boundary conditions were derived. The nanotubes are considered to be made of Al with 
positive surface elasticity and Si with negative surface elasticity. Afterward, the 
differential transformation method as an efficient and accurate numerical tool was 
applied to solve the linear equations of nanotubes subjected to different boundary 
conditions. The good agreement between the results of this article and those available in 
literature validated the presented approach. Numerical results demonstrate that the small 
scale effects play an important role on the buckling behaviour of the nanotube. Also, it 
is observed that increasing the nonlocal parameter increased the buckling ratio of the 
nanotubes. 

 
 

NOMENCLATURES 

 

*EI   Effective flexural rigidity 
q       Distributed transverse loading 

N     Critical buckling load 
A      Cross sectional area 
I       Mass moment of inertia  

sE    Surface elasticity modulus 
E      Elasticity Modulus 

Greek symbols 
    Nonlocal parameter   

    Mass density  
0    Surface tension 

     Poisson’s ratio 
 
 

 
 
 

  



ISSN: 2180-1053        Vol. 7     No. 1    January - June 2015

Investigating The Surface Elasticity and Tension Effects on Critical Buckling Behaviour of Nanotubes 
Based on Differential Transformation Method

21

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Journal of Mechanical Engineering and Technology 

22

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