Postbuckling analysis of Timoshenko nanobeams including surface stress effect

International Journal of Engineering Science 75 (2014) 1–10

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International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci

Postbuckling analysis of Timoshenko nanobeams including surface stress effect R. Ansari a, V. Mohammadi a, M. Faghih Shojaei a, R. Gholami b,⇑, S. Sahmani a a b

Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran Department of Mechanical Engineering, Lahijan Branch, Islamic Azad University, P.O. Box 1616, Lahijan, Iran

a r t i c l e

i n f o

Article history: Received 17 May 2013 Received in revised form 12 October 2013 Accepted 14 October 2013 Available online 15 November 2013 Keywords: Nanobeams Surface stress Postbuckling Size effect Generalized differential quadrature method

a b s t r a c t A modified continuum model is developed to predict the postbuckling deflection of nanobeams incorporating the effect of surface stress. To have this problem in view, the classical Timoshenko beam theory in conjunction with the Gurtin–Murdoch elasticity theory is utilized to propose non-classical beam model taking surface stress effect into account. The geometrical nonlinearity is considered in the analysis using the von Karman assumption. By employing the principle of virtual work, the size-dependent governing differential equations and related boundary conditions are derived. On the basis of the shifted Chebyshev–Gauss–Lobatto grid points, the generalized differential quadrature (GDQ) method is adopted as an accurate, simple and computational efficient numerical solution to discretize the non-classical governing differential equations along with various end supports. Selected numerical results are worked out to demonstrate the nonlinear equilibrium paths of the postbuckling behavior of nanobeams corresponding to different values of beam thickness, buckling mode number, surface elastic constants, and various types of boundary conditions. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction During the last three centuries, buckling characteristics of different elastic structures have been studied. In a straight beam, which is subjected to an axial compressive load, the buckling phenomenon may occur when the applied compressive load exceeds a critical value that can be led to turn the beam into an unstable condition and then it will buckle into one of several stable curves (Magnucka-Blandzi & Magnucki, 2011). However, the beam is still able to carry considerable load beyond buckling, namely as postbuckling response. Various investigations have been carried out to analyze the postbuckling behavior of beam structures. The thermal buckling and postbuckling behaviors of a composite beam with embedded shape memory were investigated analytically by Lee and Choi (1999). Zhang and Taheri (2003) simulated the dynamic pulsebuckling and postbuckling responses of fiber-reinforced plastic laminated beams, having initial geometric imperfections, subjected to axial impact. Abu-Salih and Elata (2005) analyzed the postbuckling characteristics of an infinite beam that is bonded to a linear elastic foundation and they assumed that the beam is subjected to an internal compressive stress. The nonlinear large deflection-small strain analysis and postbuckling behavior of Timoshenko beam-columns of symmetrical cross-section with semi-rigid connectors subjected to conservative and non-conservative loads were presented by Aristizabal-Ochoa (2007). Machado (2008) studied nonlinear buckling and postbuckling of thin-walled composite beams, considering shear

⇑ Corresponding author. Tel./fax: +98 141 2222906. E-mail address: [email protected] (R. Gholami). 0020-7225/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijengsci.2013.10.002

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deformation and geometrical nonlinear coupling, subjected to transverse external force. Emam (2009) presented static and dynamic responses of postbuckling of geometrically imperfect composite beams. Later, Dourakopoulos and Sapountzakis (2010) explained postbuckling analysis of beams of arbitrary cross-section taking into account moderate large displacements, large angles of twist and adopting second order approximations for the deflection-curvature relations. The nonlinear response of composite beams modeled according to higher-order shear deformation theories in postbuckling was studied by Emam (2011). More recently, Vosoughi, Malekzadeh, Banan, and Banan (2012) investigated thermal buckling and postbuckling behaviors of laminated composite beams with temperature-dependent material properties. However, all of the previous investigations are based on the classical continuum theory which has not the capability to consider small scale effects. To incorporate the size effect that exists at nanostructures, it is needed to refine the classical continuum model. Modified continuum models are one of the most applied theoretical approaches to investigate the mechanical behaviors of structures at nanoscale because of their computational efficiency and the capability to produce accurate results which are comparable to those of atomistic models. Surface stress effect is one of such considerable size effects. Various atomistic simulation results have demonstrated that mechanical behaviors of nanostructures are different from ones at conventional size due to the effect of surface elasticity (Streitz, Cammarata, & Sieradzki, 1994; Dingreville, Qu, & Cherkaoui, 2005). Gurtin and Murdoch (1975, 1978) formulated a generic model of surface elasticity based on the continuum mechanics including surface stress effect to represent the significant influence of surface stress on elastic behaviors of nanostructures. Their proposed model has an excellent capability to incorporate the surface stress effects into the mechanical response of nanostructures and has been applied in many studies during the past several years. Lu, Lim, and Chen (2009) examined the surface stress effects on the bending and free vibration responses of functionally graded ultra-thin films. The classical generalized shear deformable theory is adopted to model the film bulk. By adopting the Gurtin–Murdoch continuum model, the elastic field of an isotropic elastic layer bonded to a rigid base was analyzed by Zhao and Rajapakse (2009). Intarit, Senjuntichai, and Rajapakse (2010) presented analytical solutions for shear and opening dislocations in an elastic half-plane with surface stresses by using the Gurtin–Murdoch continuum theory of elastic material surfaces. Fu, Zhang, and Jiang (2010) demonstrated the effect of surface energies on the nonlinear static and dynamic responses of nanobeams. A non-classical solution was developed by Ansari and Sahmani (2011a) to analyze the bending and buckling behaviors of nanobeams including surface stress effects corresponding to different types of the classical beam theory. Ansari and Sahmani (2011b) presented a first attempt to predict free vibration characteristics of nanoplates including surface stress effects based on the continuum modeling approach. For this purpose, they implemented the Gurtin–Murdoch elasticity theory into the various classical plate theories. Recently, Ansari, Gholami, Faghih Shojaei, Mohammadi, and Sahmani (2013) developed a non-classical circular plate model to study the surface stress effect on the free vibration behavior of circular nanoplates under various edge supports based on the Gurtin–Murdoch elasticity theory and first-order shear deformation plate theory. Also, Ansari, Hosseini, Darvizeh, and Daneshian (2013) developed a non-classical model for free vibrations of nanobeams accounting for surface stress effects based on Gurtin–Murdoch elasticity theory and the classical Euler–Bernoulli beam theory. In the present investigation, the postbuckling characteristics of nanobeams including surface stress effect are studied. To this end, the Gurtin–Murdoch elasticity theory in the framework of Timoshenko beam theory is utilized to develop a nonclassical beam model incorporating surface stress effect. The geometric imperfections are taken into account based on the nonlinear von Karman relations. On the basis of the principle of virtual work, the size-dependent governing differential equations and associated boundary conditions are derived. Afterwards, generalized differential quadrature (GDQ) method is employed to discretize the governing differential equations along with various end supports by using the shifted Chebyshev–Gauss–Lobatto grid points. 2. Formulation of governing equations and corresponding boundary conditions A schematic diagram of a nanobeam with outer surface layer, length L, width b, and height h is depicted in Fig. 1. The nanobeam generally consists of a bulk part and an additional outer thin surface layer. The mechanical properties of the bulk part are k, and l which denote Lame constants. Also, the surface Lame constants are introduced by ks and ls. Furthermore, the parameter ss represents the surface residual stress.

Fig. 1. A schematic nanobeam with surface layer.

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Based on the Timoshenko beam theory, the total displacement of an arbitrary point in the x- and z- directions which are denoted by ux(x, z) and uz(x, z), respectively can be written as

ux ¼ uðxÞ  zwðxÞ;

uy ¼ 0; uz ¼ wðxÞ;

ð1Þ

in which u(x), w(x) and w(x) respectively represent the axial displacement of the center of sections, the lateral deflection of the beam, and the rotation angle of the cross section with respect to the vertical direction. Therefore, the nonzero nonlinear components of strain tensor for a Timoshenko nanobeam subjected to large amplitude vibrations will be obtained by von-Karman relation as

exx ¼

 2 du dw 1 dw z þ ; dx dx 2 dx

exz ¼

  1 dw w 2 dx

ð2Þ

Moreover, according to the linear elasticity, the stress components can be expressed as

rxx

"  2 # du dw 1 dw ; ¼ ðk þ 2lÞ z þ dx dx 2 dx

rxz ¼ lks

  dw w dx

ð3Þ

The resulting in-plane loads lead to surface stresses which can be calculated using surface constitutive equations as

rsab ¼ ss dab þ ðss þ ks Þecc dab þ 2ðls  ss Þeab þ ss usa;b ða; b ¼ x; yÞ rsaz ¼ ss usz;a

ð4Þ

Based on the following relation, the surface stress components can be derived as

rsxx ¼ ðks þ 2ls Þ

" #  2   du 1 dw dw ss dw 2  þ z þ ss ; dx 2 dx dx 2 dx

rsxz ¼ ss

dw dx

ð5Þ

It is clear that the value of stress component rzz of beams is small compared to the values of rxx and rxz. Therefore, in the classical beam theories, the stress component of rzz is neglected. However, the surface conditions described by Gurtin– Murdoch model cannot be satisfied with this assumption. To solve this problem, it is assumed that the stress component rzz varies linearly through the beam thickness and satisfies the balance conditions on the surfaces (Lu, He, Lee, & Lu, 2006). According to this assumption, rzz can be expressed as

rzz ¼

drsþ xz dx

þ 2

drs xz dx

þ

drsþ xz dx

 h

drs xz dx

z

ð6Þ

Using Eq.(5), rzz can be derived as follows 2

rzz ¼

2zss d w h dx2

ð7Þ

Therefore, by including rzz based on Eq. (7) in the components of stress given by Eq. (3), the stress components for the bulk of the nanobeam can be modified as

rxx ¼ ðk þ 2lÞ

"  2 # 2 du dw 1 dw 2mzss d w þ z þ ; dx dx 2 dx ð1  mÞh dx2

rxz ¼ lks

  dw w : dx

ð8Þ

The strain energy of a nanobeam incorporating the surface effects based on the continuum surface elasticity theory can be expressed as

Ps ¼

1 2

Z Z x

A

rij eij dAdx þ

1 2

Z S

rsij eij dSþ þ þ

Z S

rsij eij dS 



)  2 !   Z ( 1 e dw dx e xx Þ du þ 1 dw e xx Þ dw þ Q dw  w þ Q  ðMxx þ M ðNxx þ N ¼ 2 x dx 2 dx dx dx dx

ð9Þ

in which

"  2 #  2 dw e xx ¼ 2ðb þ hÞðks þ 2l Þ du þ 1 dw N s þ 2ðb þ hÞss :  ðb þ hÞ s s dx 2 dx dx ! 2 3 e ¼ 2ðb þ hÞss dw : e xx ¼ ðks þ 2l Þ bh þ h dw ; Q M s dx 2 6 dx

ð10aÞ

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R. Ansari et al. / International Journal of Engineering Science 75 (2014) 1–10

Nxx ¼ bhðk þ 2lÞ

"  2 # du 1 dw ; þ dx 2 dx

3

Q ¼ bhlks

2

  dw w dx

2

ð10bÞ

bh ðk þ 2lÞ dw mbh ss d w þ 12 dx 6ð1  mÞ dx2

M xx ¼ 

The work PP made by the axial force N0x can be calculated as

Pp ¼

Z

1 2

L

0

N0x

 2 dw dx dx

ð11Þ

By employing the principle of virtual work as

dðPp  Ps Þ ¼ 0

ð12Þ

one can obtain the governing stability equations and associated boundary conditions of a nanobeam with surface layer as follow

e xx Þ dðN xx þ N ¼0 dx  2 eÞ d  dðQ þ Q e xx Þ dw  N 0x d w ¼ 0 þ ðNxx þ N 2 dx dx dx dx Q

ð13Þ

e xx Þ dðM xx þ M ¼0 dx

Also, the associated boundary conditions can be written as

e xx Þ ¼ 0 du ¼ 0 or dðN xx þ N

ð14aÞ

eÞ ¼ 0 e xx Þ dw þ Q þ Q dw ¼ 0 or dððNxx þ N dx e xx Þ ¼ 0 dw ¼ 0 or dðM xx þ M

ð14bÞ ð14cÞ

Substitution of Eq. (10) into Eq. (13) leads to the following governing differential equations as 2

d u

A11

2

dx

þ

! 2 2 dw d w dw d w  A33 ¼0 2 dx dx dx dx2

ð15aÞ

! " #  2 2 2 dw d w du 1 dw d w A13  þ ðA11  A33 Þ þ A33 þ A33 2 þ A11 2 2 dx dx 2 dx dx dx dx " # 2 2 2 d u dw d w dw d w þ A11 2 þ ðA11  A33 Þ  N ¼0 0x 2 dx dx2 dx dx dx 2

d w

A13

  2 3 dw d w d w  w þ D11 2  E11 3 ¼ 0 dx dx dx

ð15bÞ

ð15cÞ

where

A11 ¼ bhðk þ 2lÞ þ 2ðb þ hÞðks þ 2ls Þ;

A33 ¼ 2ðb þ hÞss ; A13 ¼ bhlks ; ! bh bh h mbh2 ss ¼ ðk þ 2lÞ þ ðks þ 2ls Þ þ ; E11 ¼ 12 2 6 6ð1  mÞ 3

D11

2

3

ð16Þ

In this study, nanobeams with possible combination of the following boundary conditions are considered  Simply supported boundary condition (SS)

e xx ¼ 0 at ends u ¼ w ¼ Mxx þ M

ð17Þ

 Clamped boundary condition (C)

u ¼ w ¼ w ¼ 0 at ends

ð18Þ

R. Ansari et al. / International Journal of Engineering Science 75 (2014) 1–10

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3. Solution procedure By introducing the following dimensionless parameters

u ¼ LU;

w ¼ hW;

D11

d11 ¼

A110 h

X ¼ x=L;

e11 ¼

; 2

E11 A110 h

; 2

n ¼ h=L;

fa11 ; a13 ; a33 g ¼

fA11 ; A13 ; A33 g ; A110

e x ¼ N0x N A110

ð19Þ

in which A110 = bh(k + 2l). The non-dimensional part of governing equations for postbuckling of a nanobeam including surface stress effect can be defined as follows 2

a11

d U dX

2

2

þ ða11  a33 Þn2

dW d W ¼0 dX dX 2

2

ða13 þ 2a33 Þn

d W dX

2

ð20aÞ 2

 a13

2

dw dU d W d U dW þ a11 n þ 2 dX dX dX 2 dX dX

!

 2 2 2 3 d W dW e xn d W ¼ 0 þ ða11  a33 Þn3 N 2 2 2 dX dX dX

  2 3 dW d w d W a13 n  w þ d11 n2 2  e11 n3 ¼0 3 dX dX dX

ð20bÞ

ð20cÞ

Boundary conditions also can be expressed in non-dimensional forms in a similar way as  For simply supported boundary condition (SS) 2

u ¼ w ¼ e11 n

d w 2

dx

 d11

dw ¼0 dx

ð21Þ

 For clamped boundary condition (C)

u¼w¼w¼0

ð22Þ

3.1. Generalized differential quadrature method On the basis of the generalized differential quadrature (GDQ) method (Bellman, Kashef, & Casti, 1972), the r-th derivative of a function f(x) can be obtained as a linear sum of the function values as follows

 @ r fðxÞ @xr 

¼ x¼xi

N X Crij fðxj Þ

ð23Þ

j¼1

in which N is the number of total discrete grid points used in the process of approximation in the x direction. A column vector F can be defined as

F ¼ ½F j  ¼ ½fðxj Þ ¼ ½fðx1 Þ; fðx2 Þ; . . . ; fðxN ÞT

ð24Þ

in which f(xj) denotes the nodal value of f(x) at xj. A differential matrix operator based on Eq. (24) can be expressed as follows

  @r ðFÞ ¼ Drx F ¼ Drx i;j fF j g @xr

ð25Þ

  Drx ¼ Drx i;j ¼ Crij ;

ð26Þ

where

i; j ¼ 1 : N

in which r is the order of differentiation and Crij are weighting coefficients obtained by

8 Ix > > > pðxi Þ > > > ðxi xj Þpðxj Þ > >   > < Cr1 1 r1 ij Crij ¼ r Cij Cii  xi xj > > > > N > X > > > Crij > :

r¼0 i – j and i; j ¼ 1; . . . ; N and r ¼ 1 i – j and i; j ¼ 1; . . . ; N and r ¼ 2; 3; . . . N  1 i ¼ j and i; j ¼ 1; . . . ; N and r ¼ 1; 2; 3; . . . N  1

j¼1;j–i

in which pðxi Þ ¼

QN

j¼1;j–i ðxi

 xj Þ, and Ix is a N  N identity matrix.

ð27Þ

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R. Ansari et al. / International Journal of Engineering Science 75 (2014) 1–10

3.2. Discretization It is considered that the one dimensional functions U(X), W(X) and w(X) are defined on 0 6 X 6 1. Therefore, the mesh can be generated by a shifted Chebyshev–Gauss–Lobatto grid points as

Xi ¼

  1 i1 1  cos p ; 2 N1

i¼1:N

ð28Þ

The column vectors U,W and W can be defined as follows

UT ¼ ½U 1 ; . . . ; U N ;

WT ¼ ½W 1 ; . . . ; W N ;

WT ¼ ½W1 ; . . . ; WN :

ð29Þ

in which Ui = U(Xi), Wi = W(Xi), Wi = w(Xi). Now, using the GDQ method, the equilibrium differential equations (Eq. (20)) can be discretized as follows





a11 D2X U þ ða11  a33 Þn2 D2X W  D1X W ¼ 0 ða13 þ 2a33 ÞnD2X W  a13 D1X W þ a11 n e x nD2 W ¼ 0 N X

ð30aÞ









3





D2X U  D1X W þ D2X W  D1X U þ ða11  a33 Þn3 D2X W  D1X W  D1X W 2 ð30bÞ



a13 nD1X W  W þ d11 n2 D2X W  e11 n3 D3X W ¼ 0

ð30cÞ

in which  denotes the Hadamard product. Similarly, the corresponding boundary conditions can be discretized in the same way. The set of nonlinear equations of the domain can be expressed as

H : R3Nþ1 ! R3N ;

e x ; XÞ ¼ 0; Hð N

ZT ¼ ½UT ; WT ; WT 

ð31Þ

e x , is treated as a parameter. where the axial buckling load, N Now, the preceding parameterized equation can be solved on the basis of the pseudo arc-length continuation method. However, this point should be considered that in the procedure of nonlinear solution, the residual of equations related to e x ; ZÞ so as to take the boundary conditions boundaries must be inserted in the residual vector corresponding to domain Hð N into account. In order to consider this concept, the components of the residual vector relevant to the grid points at the boundaries are replaced with those attained from the discretized forms of boundary conditions. 4. Results and discussion To analyze the postbuckling characteristics of nanobeams including surface stress effect, the following geometrical and material properties are assumed which are obtained via the atomic examinations accomplished by experimental or numerical methods (Ogata, Li, & Yip, 2002; Zhu, Pan, & Chung, 2006; Miller & Shenoy, 2000). For Silicon (Si) material:

m ¼ 0:24; n ¼ h=L ¼ 1=8; b=h ¼ 1; N N Es ¼ 10:036 ; ss ¼ 0:605 ; ms ¼ m: m m

E ¼ 210GPa;

For Aluminum (Al) material:

E ¼ 68:5GPa; N Es ¼ 6:09 ; m

m ¼ 0:35; n ¼ h=L ¼ 1=8; b=h ¼ 1; N m

ss ¼ 0:91 ; ms ¼ m:

Tables 1 and 2 present the first four critical buckling loads of nanobeams subjected to various end supports, with and without consideration of surface stress effect corresponding to Si and Al material properties, respectively. It can be found Table 1 The first four critical buckling load obtained via the classical and non-classical beam models for Si-nanobeam (h = 1 nm). Mode no.

1 2 3 4

C-C

SS-C

SS-SS

Non-Classical

Classical

Non-Classical

Classical

Non-Classical

Classical

0.0406 0.0655 0.1044 0.1326

0.0442 0.0771 0.1247 0.1543

0.0267 0.0537 0.0863 0.1197

0.0241 0.0620 0.1034 0.1414

0.0188 0.0406 0.0710 0.1044

0.0123 0.0442 0.0847 0.1247

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R. Ansari et al. / International Journal of Engineering Science 75 (2014) 1–10 Table 2 The first four critical buckling load obtained via the classical and non-classical beam models for Al-nanobeam (h = 1 nm). Mode no.

1 2 3 4

C-C

SS-C

SS-SS

Non-Classical

Classical

Non-Classical

Classical

Non-Classical

Classical

0.1240 0.1679 0.2249 0.2514

0.0432 0.0737 0.1169 0.1422

0.0910 0.1494 0.2014 0.2409

0.0238 0.0599 0.0979 0.1314

0.0703 0.1240 0.1798 0.2249

0.0123 0.0432 0.0810 0.1169

Fig. 2. Postbuckling equilibrium paths of Si- and Al-nanobeams obtained by the nonclassical (solid line) and classical (dashed line) beam models (h = 2 nm).

Fig. 3. Surface stress effect on the postbuckling equilibrium path of Si-nanobeams with various beam thicknesses.

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R. Ansari et al. / International Journal of Engineering Science 75 (2014) 1–10

Fig. 4. The effect of Es on the postbuckling equilibrium path of Si-nanobeams.

τ τ τ

Fig. 5. The effect of ss on the postbuckling equilibrium path of Si-nanobeams.

out that in the case of Aluminum material properties, the critical buckling loads predicted by non-classical beam model are higher than those of the classical ones for all buckling mode numbers and boundary conditions. However, for the nanobeams with Silicon material properties, the effect of surface stress may lead to increase or decrease of critical buckling load which depends on the value of mode number and type of boundary conditions. Plotted in Fig. 2 are the postbuckling equilibrium paths of nanobeams obtained via the both classical and non-classical beam models corresponding to different material properties, mode numbers and boundary conditions. It is observed that surface stress has completely contrariwise influence on the postbuckling response of nanobeams with the two different

R. Ansari et al. / International Journal of Engineering Science 75 (2014) 1–10

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material properties. This pattern is due to different signs of Es for Si and Al materials. Moreover, it is shown that in the cases of Si nanobeams subjected to simply supported-clamped and clamped–clamped end supports, there is an intersection point for the postbuckling paths predicted by the classical and non-classical beam models, after which the effect of surface stress is fully inversed. This phenomenon may be because of the nonlinearity behavior of the developed model. Before this point, the linear terms have the major effects on the postbuckling equilibrium path, so ss is the dominant property. Nevertheless, after that point, the deflection of beam increases which causes to intensify the influence of nonlinear terms on the postbuckling response and Es is the dominant property. Fig. 3 depicts postbuckling characteristics of nanobeams achieved by the classical and non-classical beam models corresponding to various beam thicknesses. It is assumed that the nanobeams are made of Silicon material. It can be seen that by increasing the beam thickness, surface stress effect plays less important role in the postbuckling response of nanobeams. This anticipation is the same for all values of axial load, buckling mode number and all possible types of boundary conditions. Figs. 4 and 5 illustrate postbuckling equilibrium paths of nanobeams with various values surface elastic constants. It is revealed that the effect of surface stress can lead to higher or lower stiffness against postbuckling phenomenon which depends on the sign of surface elastic constants. This pattern is the same for all buckling mode numbers and boundary conditions. 5. Concluding remarks The postbuckling characteristics of nanobeams incorporating the effect of surface stress were predicted in the present study. To this end, the Gurtin–Murdoch elasticity theory in the framework of Timoshenko beam theory was utilized to develop non-classical beam model including surface stress effect. Afterwards, based on the principle of virtual work, the sizedependent governing differential equations and related boundary conditions were derived. In order to solve the problem accurately as well as computational efficiently, GDQ method was employed to discretize the differential equations along with various end supports on the basis of the shifted Chebyshev–Gauss–Lobatto grid points. It was found that the critical buckling load predicted by the proposed non-classical beam model may be higher or lower than that of the classical one which depends on the material properties, buckling mode number and boundary conditions. Furthermore, it was demonstrated that by increasing the beam thickness, surface stress effect plays less important role in the postbuckling response of nanobeams. 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