On size-dependent Timoshenko beam element based on modified couple stress theory

International Journal of Engineering Science 107 (2016) 134–148

Contents lists available at ScienceDirect

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

On size-dependent Timoshenko beam element based on modified couple stress theory Amir Mehdi Dehrouyeh-Semnani a,∗, Arian Bahrami b a b

School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran Department of Mechanical Engineering, Eastern Mediterranean University, G. Magosa, TRNC, Mersin 10, Turkey

a r t i c l e

i n f o

Article history: Received 2 June 2016 Accepted 13 July 2016

Keywords: Couple stress Timoshenko beam Finite element method Small scale Material length scale parameter

a b s t r a c t Due to complexity of geometry of small-scale structures as well as size effect phenomenon in such scale, size-dependent finite element method can be employed as a powerful numerical technique to investigate the mechanical behavior of such structures. Recently, two distinct size-dependent Timoshenko beam elements have been proposed based on modified couple stress theory. The first beam element is a two-node element which has 3-DOF (degrees of freedom) at each node. The second beam element is also a two-node element, but it has 2-DOF at each node. Since enough verification and convergence studies have not been performed on the proposed beam elements, the present study aims to examine the accuracy, reliability and stability of aforementioned beam elements in the static bending. To that end, the cantilevered, simply supported and doubly clamped Timoshenko beams are chosen as the case studies. It is observed that the results obtained via the 6-DOF beam element are in excellent agreement with those obtained via the other solutions for the three case studies. In addition, it is found that the rate of convergence increases by ascending the influence of size-dependency. Although the 4-DOF beam element presents stable solutions, the acquired results reveal that the 4-DOF beam element is incapable of verifying the results obtained based on the other solutions for the three case studies. Moreover, it is found that the 4-DOF beam element error has an ascending trend with respect to the size-dependent shear deformation. Finally the reason for inaccuracy of the 4-DOF beam element is discussed. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Yang, Chong, Lam, and Tong (2002) developed modified couple stress theory as a linear elastic constitutive law for isotropic couple stress materials based on strain gradient theory. The new higher-order elastic theory has been employed by many researchers to study the size effect phenomenon in the microstructures where the attempts of the classical models have been failed. Microbeams, as the most widely utilized continuous elements in micron and sub-micron scales, can be found in various micro-electro-mechanical systems. Hence, many researchers have developed governing equations of beam-type microstructures and have investigated mechanical behavior of such structures based on modified couple stress theory. Some of these works can be outlined as: an Euler–Bernoulli beam model for static bending analysis and comparison with experimental data by Park and Gao (2006), a microstructure-dependent Timoshenko beam model for static ∗

Corresponding author. Fax: +98 11420701. E-mail address: [email protected] (A.M. Dehrouyeh-Semnani).

http://dx.doi.org/10.1016/j.ijengsci.2016.07.006 0020-7225/© 2016 Elsevier Ltd. All rights reserved.

A.M. Dehrouyeh-Semnani, A. Bahrami / International Journal of Engineering Science 107 (2016) 134–148

135

bending and free vibration analysis by Ma, Gao, and Reddy (2008), a nonlinear Euler–Bernoulli beam model for static bending, free oscillation and post buckling analysis by Xia, Wang, and Yin (2010), functionally graded Timoshenko beam models for static and free vibration analysis by Asghari, Rahaeifard, Kahrobaiyan, and Ahmadian (2011), a new model for bending analysis of composite laminated microbeams with first order shear deformation by Chen, Li, and Xu (2011), buckling analysis of axially loaded micro-scaled beams with different boundary conditions based on Euler–Bernoulli beam theory by Akgöz and Civalek (2011), nonlinear free vibration analysis of extensible functionally graded microbeams by Ke, Wang, Yang, and Kitipornchai (2012), static bending and dynamic analysis of third-order shear deformation functionally graded microbeams by Salamat-talab, Nateghi, and Torabi (2012), three-dimensional nonlinear size-dependent behaviour of Timoshenko microbeams by Ghayesh, Amabili, and Farokhi (2013), static and dynamic stability analysis of a functionally graded microbeam under electrostatic force by Abbasnejad, Rezazadeh, and Shabani (2013), static bending and free vibration analysis of functionally graded microbeams using a new higher order beam theory by S¸ ims¸ ek and Reddy (2013), study of thermal postbuckling behavior of size-dependent functionally graded Timoshenko microbeams by Ansari, Faghih Shojaei, Gholami, Mohammadi, and Darabi (2013), thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium based on sinusoidal shear deformation beam theory by Akgöz and Civalek (2014), micro-inertia effects on the dynamic characteristics of micro-beams by Fathalilou, Sadeghi, and Rezazadeh (2014), investigation of dynamic pull-in instability of geometrically nonlinear actuated microbeams by Sedighi, Chan-Gizian, and Noghreha-Badi (2014), nonlinear thermal stability and vibration analysis of pre/post-buckled temperature- and microstructure-dependent functionally graded beams resting on elastic foundation by Komijani, Esfahani, Reddy, Liu, and Eslami (2014), nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using He’s variational method by S¸ ims¸ ek (2014), nonlinear-electrostatic analysis of micro-actuated beams incorporating surface elasticity by Shaat and Mohamed (2014), three-dimensional vibration analysis of curved microbeams under fluid force induced by external flow by Tang, Ni, Wang, Luo, and Wang (2014), a functionally graded sandwich microbeam model for static bending, free vibration and buckling by Thai, Vo, Nguyen, and Lee (2015), developing a nonlinear model for cantilevered microbeams and also exploring the nonlinear dynamics i.e., frequency–response curves, phase portraits and time histories by Dai, Wang, and Wang (2015), a parametric study on nonlinear dynamics of microbeams by considering different parameters due to time-dependent longitudinal excitation load by Ghayesh, Farokhi, and Alici (2015), new models for viscoelastically damped sandwich microbeams as well as study of resonant frequency and loss factor by Dehrouyeh-Semnani, Dehrouyeh, Torabi-Kafshgari, and Nikkhah-Bahrami (2015a, 2015b), study of thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams by Farokhi and Ghayesh (2015), thermal buckling analysis of microcomposite laminated based on Euler-Bernoulli, Timoshenko and Reddy beam theories by Mohammadabadi, Daneshmehr, and Homayounfard (2015), complex sub and supercritical global dynamics of a parametrically excited microbeam subject to a time-dependent axial load with special consideration to chaotic motion by Ghayesh and Farokhi, (2015), an exact solution for vibrations of postbuckled microscale beams by Ansari, Ashrafi, and Arjangpay (2015), frequency and stability analysis of axially moving microbeams with constant velocity using different beam theories by Dehrouyeh-Semnani, Dehrouyeh, Zafari-Kolukhi, and Ghamami (2015), a new nonlinear model for dynamical performance analysis of microgyroscope (Ghayesh, Farokhi, & Alici, 2016), free vibrations and stability of spinning microbeam based on Euler–Bernoulli and Timoshenko beam theories by Ilkhani and Hosseini-Hashemi (2016), postbuckling analysis of functionally graded small scale beams under general beam theory by Akbarzadeh Khorshidi, Shariati, and Emam (2016), out-of-plane free vibration analysis of rotary tapered microbeams made of axially functionally graded materials using Euler-Bernoulli beam theory by Shafiei, Kazemi, and Ghadiri (2016b), nonlinear free vibration analysis of axially functionally graded tapered microbeams with different boundary conditions based on Euler-Bernoulli beam theory and von-Kármán’s geometric nonlinearity by Shafiei, Kazemi, and Ghadiri (2016a), a new nonlinear model for coupled three dimensional microbeam as well as static bending and free oscillation analysis of a microbridge by Mojahedi and Rahaeifard (2016), nonlinear static and forced vibration analysis of CNT-based resonators under AC and DC actuations based on a fully nonlinear Euler-Bernoulli beam model by Farokhi, Païdoussis, and Misra, 2016, investigation of free flexural vibration of geometrically imperfect functionally graded microbeams with different boundary conditions by Dehrouyeh-Semnani, Mostafaei, and Nikkhah-Bahrami (2016), study of pull-in behavior of functionally graded sandwich bridges subjected to electrostatic actuation effect and intermolecular Casimir forces using Euler–Bernoulli, Timoshenko, and Reddy beam theories by Shojaeian and Zeighampour (2016), nonlinear vibration analysis of porous and imperfect functionally graded tapered microbeams by Shafiei, Mousavi, and Ghadiri (2016), investigation of large-amplitude dynamical behaviour of microcantilevers considering different sources of nonlinearity by Farokhi, Ghayesh, and Hussain (2016). The above mentioned works deal with the size-dependent mechanical behavior of simple beam-type microstructures. In many real applications, the geometry of beam-type microstructures is not simple, therefore, employing the size-dependent beam elements to investigate the mechanical behavior of such structures sounds to be essential. Recently, two distinct sizedependent Timoshenko beam elements have been developed by Arbind and Reddy (2013) and Kahrobaiyan, Asghari, and Ahmadian (2014) for both the static and dynamic analysis. Considering only the bending deformation, both the elements are a two-node beam element, and the beam element developed by Arbind and Reddy (2013) has 3-DOF (degrees of freedom) at each node, whereas, the beam element developed by Kahrobaiyan et al. (2014) has 2-DOF at each node. In other words, the beam element proposed by Kahrobaiyan et al. (2014) with fewer DOF is capable of predicting the size-dependent mechanical behavior of Timoshenko microbeam. Since enough verification and convergence studies have not been performed on the proposed beam elements, it seems necessary to examine the accuracy, reliability and stability of both the beam elements. In this study, using exact,

136

A.M. Dehrouyeh-Semnani, A. Bahrami / International Journal of Engineering Science 107 (2016) 134–148

analytical and semi-analytical solutions for the static bending behavior of Timoshenko microbeams with different boundary conditions, the size-dependent beam elements are scrutinized by incorporating different effective parameters. The outline of the article is as follows: In Section 2, the governing equations and related boundary conditions of Timoshenko microbeam based on modified couple stress theory are reviewed. In Section 3, 6-DOF and 4-DOF Timoshenko beam elements are reviewed. Afterwards, the explicit form of stiffness matrices and force vectors are developed. In Section 4, the accuracy and stability of the 6-DOF and the 4-DOF beam elements are explored. To that end, three different case studies are considered. In Section 5, the reason for inaccuracy of the 4-DOF beam element is discussed. The article concludes with Section 6, where the work is summarized and the final remarks are provided. 2. Size-dependent Timoshenko beam model Consider a straight microbeam of length L, cross-sectional area A, second moment of cross-sectional area I, Young’s modulus E, shear modulus G, shear correction factor ks , material length scale parameter , subjected to an external load q. The non-zero components of displacement vector based on Timoshenko beam theory can be expressed as (Reddy, 2011):

u1 = z φ , u3 = w

(1)

in which w and ψ stand for the transverse deformation and the rotation angle of microbeam, respectively. The non-zero components of strain ɛ, curvature χ, stress σ , and couple stress m tensors can be written as (Reddy, 2011)







εxx

dψ 1 =z , εxz = εzx = dx 2

σxx

dψ dw G2 = Ez , τxz = τzx = ks G + ψ , mxy = myx = dx dx 2

dw 1 + ψ , χxy = χyx = dx 4



d2 w dψ − dx dx2







d2 w dψ − dx dx2

 (2)

The potential energy of the microbeam due to bending deformation U is given by (Reddy, 2011):



1 U= 2

V

(σ

1 : ε + m : χ )dV = 2



L 0

 

dψ EI dx

2



dw + ks GA +ψ dx

2

GA2 + 4



d2 w dψ − dx dx2

2  dx

(3)

The work done by the forces applied on the microbeam can be written as:



W =

L 0



qw dx +

L 0

(V w + Mψ )|xx==0L dx

(4)

in which V, M are, respectively, the applied transverse force and bending moment at the two ends of the microbeam. Using Hamilton’s principle, the static equilibrium equations and corresponding boundary conditions (at x = 0 and L) of a Timoshenko microbeam based on modified couple stress theory can be obtained by (Reddy, 2011):



dψ d2 w ks GA + dx dx2



1 + GA2 4



d4 w d3 ψ − dx3 dx4



=q

   2  d3 w d2 ψ dw 1 2 d ψ EI 2 − ks GA ψ + + GA − =0 dx 4 dx dx2 dx3    2  d3 w dw 1 d ψ ks GA ψ + + GA2 − = V or w = ws dx 4 dx2 dx3   d2 w dψ 1 dψ EI + GA2 − = M or ψ = ψs dx 4 dx dx2     1 ∂ 2w dw dw 2 dψ 4

GA

dx

−

dx2

= 0 or

dx

=

dx

(5)

(6)

s

The governing equations and boundary conditions of the classical Timoshenko beam model can be recovered from Eqs. (5) and (6) by setting  = 0. In comparison with the classical Timoshenko beam model (i.e.,  = 0), the order of transverse deformation w and rotation angle ψ increase by 2 and 1, respectively. In addition, the number of boundary conditions increases by 2. 3. Size-dependent Timoshenko beam elements 3.1. 6-DOF beam element Arbind and Reddy (2013) developed a two-node Timoshenko beam element based on modified couple stress theory. The beam element has 3-DOF at each node i.e., the transverse deformation (w), the slope (-dw/dx), and the rotation angle

A.M. Dehrouyeh-Semnani, A. Bahrami / International Journal of Engineering Science 107 (2016) 134–148

137

(ψ ). Arbind and Reddy (2013) derived the weak form formulation of the equations and then by considering Hermit cubic approximation for the transverse deformation w and Lagrange linear approximation for the rotation angle ψ developed the finite element model. The proposed element has 6-DOF, whereas, the classical two-node Timoshenko beam element has 4-DOF. The weak form formulation of Eq. (5) is given by Arbind and Reddy (2013); Dehrouyeh-Semnani (2015); DehrouyehSemnani, BehboodiJouybari, and Dehrouyeh (2016):

   1 dw dNiw 2 dψ ks GA ψ + − GA dx dx 4 dx 0     L ψ ∂ψ ∂ Ni ∂w ψ EI − ks GA ψ + N − ∂x ∂x ∂x i 0 

L



d2 w − dx2 1 GA2 4





d2 Niw + qw + V ∗ + Y ∗ dx = 0, i = 1, .., 4 dx2

   ψ ∂ w ∂ Ni ψ− + M∗ dx = 0, i = 1, 2 ∂x ∂x

(7)

where

 V =

 Y∗ =



dw −ks GA φ + dx

∗

1 − GA2 4





1 + GA2 4

d2 w dψ − dx dx2



dNiw dx



d3 w d2 ψ − dx2 dx3



 + qw

Niw

 x=L x=0 

 x=L    x=L d2 w dψ 1 ψ 2 dψ , M∗ = EI + GA  − Ni 2 dx 4 dx dx x=0 x=0

(8)

The finite element formulation of 6-DOF beam element for static solution is given by (Arbind & Reddy, 2013):



  w

K11

K12

K21

K22

 F1

=

ψ

(9)

0

where



L

ij K11

=

ij K21

ji = K21

ij K22

0

 =

L



w 2 w dN w dN j d2 Niw d N j 1 ks GA i + GA2 dx 4 dx2 dx2 dx dx

 L 2 w dN ψ dNiw ψ 1 d N j i = ks GA N − GA2 dx dx j 4 dx2 dx 0





0

 F1 =



L

0

 dNψ dN 1 j ψ ψ i EI + GA2 + ks GANi N j dx 4 dx dx



ψ

q Niw

w = w1

T

−



dx + V1

dw1 dx

M1

w2

−

V2

dw2 dx

T

M2

(10)

T

,ψ =



ψ1

ψ2

T

in which Nw and Nψ denote Lagrange linear and Hermite cubic interpolations, respectively. In addition, V1 and V2 are applied transverse force at the element’s nodes, and also M1 and M2 are applied bending moment at the element’s nodes.

N1w

=2

 x 3

N3w = −2

L

−3

 x 3 L

 x 2

+3

L

+

 x 2 L

1, N2w

=L

, N4w = L

 3 x L

 3 x L

−

−2

 x 2 L

 x 2 

x L+ L



L

x ψ x ψ N1 = 1− , N2 = L L

(11)

Eq. (9) can be rewritten as follows:



[K ]{δ} = {F }, {δ} = w1

−dw1 /dx

ψ1

w2

−dw2 /dx

ψ2

T

(12)

138

A.M. Dehrouyeh-Semnani, A. Bahrami / International Journal of Engineering Science 107 (2016) 134–148

The explicit form of stiffness matrix based on Eq. (12) can be obtained by:

⎡

6λ1 3λ2 + 3 ⎢ 5L L

λ1 10

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ K=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

+

3λ2 2L2

−

2Lλ1 λ2 + 15 L

λ1

−

2

Lλ1 λ2 − 12 4L

6λ1 3λ2 − 3 5L L

−

Lλ1 λ2 λ3 + + 3 4L L

λ1 3λ2 10

−

2L2

λ1 2 6λ1 3λ2 + 3 5L L

λ1 10

+

3λ2 2L2

−

Lλ1 λ2 + 30 2L

−

Lλ1 λ2 + 12 4L

−

λ1 3λ2 10

−

2L2

2Lλ1 λ2 + 15 L Sym

−

⎤

λ1

⎥ ⎥ Lλ1 λ2 ⎥ ⎥ − + 12 4L ⎥ ⎥ ⎥ Lλ1 λ2 λ3 ⎥ − − ⎥ 6 4L L ⎥ ⎥ ⎥ λ1 ⎥ ⎥ 2 ⎥ ⎥ Lλ1 λ2 ⎥ ⎥ − 12 4L ⎥ ⎥ ⎦ Lλ1 λ2 λ3 3

+

2

4L

+

(13)

L

where

λ1 = ks GA, λ2 = GA2 , λ3 = EI

(14)

The force vector for a uniform distribution load q based on Eq. (12) is given by:



F=q

L2 12

L 2

L 2

0

−

L2 12

T

(15)

0

In addition, the force vector for a linearly varying distributed load (q = q0 (1 − x/L )) based on Eq. (12) is found to be equal to:

F=

q0 L  21 60

3L

0

9

−2L

T

0

(16)

3.2. 4-DOF beam element Kahrobaiyan et al., 2014 developed a two-node Timoshenko beam element which has 2-DOF at each node. The shape functions of the new beam element derived by directly solving the static equilibrium equations of modified couple stress Timoshenko beam equations. It should be pointed out that the governing equations used by Kahrobaiyan et al., 2014 can be obtained from Eq. (5) by letting –ψ instead of ψ . (Kahrobaiyan et al., 2014) neglected the effect of shear correction factor i.e., ks = 1. However, the influence of shear correction factor for the 4-DOF beam element is considered in this study. The governing equation can be rewritten as follows (Kahrobaiyan et al., 2014):

1 d2 θ GA2 2 = c1 4 dx   1  2 d4 θ d2 θ 2 c1 L2 − (1 + α ) 2 = . 4 4 L dx EI dx

ks GAγ +

(17a-b)

in which c1 is a constant and also γ , θ and α can be obtained by (Kahrobaiyan et al., 2014):

θ=

dw dw GA2 + ψ, γ = − ψ, α = dx dx EI

(18)

Kahrobaiyan et al., 2014 concluded that the coefficient of d4 θ /dx4 is negligible comparing with the coefficient of d2 θ /dx2 , hence Eq. (18) can be reduced to:

d2 θ 2 c1 L2 = 2 EI(1 + α ) dx

(19)

Solving the new system of equations (Eqs. (17a) and (19)) leads to determination of the transverse deformation w and the rotation angle ψ as follows (Kahrobaiyan et al., 2014):

w=

ψ



c1 c1 1 + α /2 1 − x2 + c2 x + c3 − 2 EI (1 + α ) ks GA 1 + α

1 = 2







c1 c1 1 + α /2 c2 x2 x− x3 + + c3 x ks GA 1 + α 3EI (1 + α ) 2

+ c4

(20)

By applying the following boundary conditions (Kahrobaiyan et al., 2014),

w|x=0 = w1 , w|x=L = w2 ,

ψ |x=0 = ψ1 , ψ |x=L = ψ2

(21)

A.M. Dehrouyeh-Semnani, A. Bahrami / International Journal of Engineering Science 107 (2016) 134–148

Eq. (20) can be rewritten as:



w = N1w

N2w

 ψ = N1ψ

N3w

ψ

N4w

ψ

N2

ψ

N3

N4

 

w1

ψ1

w2

ψ2

w1

ψ1

w2

ψ2

139

T T

(22)

The shape functions of 4-DOF beam element can be derived as (Kahrobaiyan et al., 2014):

  3

1 x 2 (1 + φ ) L

N1w = 1 + N3w =

  2 x

1 3 (1 + φ ) L

−2

 

−3



 x 2

 x 3 L

−φ

L



+φ



  3

x L x , N2w = 2 L 2 (1 + φ ) L

  3 x

x L , N4w = 2 L 2 (1 + φ ) L





− (4 + φ )

+ (φ − 2 )

 

 x 2 L

 x 2 L

−φ

+ (2 + φ ) x L



x L



6 x x 3 x x ψ ψ N1 = − 1− , N2 = 1 − 1− L (1 + φ ) L L L (1 + φ ) L  x    x   x  x 3 6 ψ ψ N3 = 1− , N4 = 1− L (1 + φ ) L L L (1 + φ ) L where

φ=



12EI α 1+ 2 ks GAL2

(23)

 (24)

By letting  = 0, the shape functions of classical Timoshenko beam element can be recovered (Friedman & Kosmatka, 1993). The finite element formulation of 4-DOF beam element for static solution can be achieved by (Kahrobaiyan et al., 2014):

[K ]{δ} = {F }, {δ} =



w1

ψ2



ψ2

w2

(25)

Substituting Eq. (22) into Eqs. (3) and (4), the stiffness matrix and force vector of 4-DOF beam element can be obtained by:

 Kij =

L

EI 0

 F =



L 0





ψ

ψ dNi dN j

dNiw ψ + ks GA − Ni dx dx

dx

q Niw

T



dx + V1

M1

V2

M2



dN w j dx

ψ

− Nj

 +

ψ μA2 dNi 4

d2 Niw + dx dx2

T



ψ

dN j dx

+

d2 Nw j dx2

 dx

(26)

in which V1 and V2 are applied transverse force at the element’s nodes, and also M1 and M2 are applied bending moment at the element’s nodes. Based on the work of Kahrobaiyan et al., 2014, the explicit form of 4-DOF stiffness matrix can be written as:

⎡ ⎢

12

EI (1 + α ) ⎢ ⎢ K= 3 L (1 + φ ) ⎢

6L

−12

( 4 + φ )L2

−6L

⎣

12

Sym

6L

⎤

⎥ ( 2 − φ )L2 ⎥ ⎥ ⎥ −6L ⎦ ( 4 + φ )L2

(27)

The solution procedure proposed by Kahrobaiyan et al. (2014) to derive the shape functions of 4-DOF beam element was checked and it is revealed that the shape functions of 4-DOF beam element were correctly derived. However, it is found that the explicit form of stiffness matrix Eq. (27) had been incorrectly reported by Kahrobaiyan et al. (2014). The corrected form of stiffness matrix can be obtained by:

⎡

12λ1

⎢ ⎢ ⎢ K= 2⎢ 3 L (1 + φ ) ⎣ EI

−12λ1

6Lλ1

L2 λ2

−6Lλ1

L2 λ3

12λ1

Sym where

6Lλ1

⎤

⎥ ⎥ ⎥ ⎥ −6Lλ1 ⎦ L2 λ2

     α α  2 λ1 = 1 + α + φ 1 + , λ2 = 3 1 + φ 1 + + ( 1 + φ )2 + 3 + ( 1 + φ ) α 2 2     α  2 2 λ3 = 3 1 + φ 1 + − (1 + φ ) + 3 − (1 + φ ) α 2

(28)

(29)

The classical Timoshenko beam element (Friedman & Kosmatka, 1993) can be recovered from Eq. (28) by neglecting the influence of couple stress i.e., α = 0. The derivation procedure for K11 and K22 can be found in detail in Appendix A.

140

A.M. Dehrouyeh-Semnani, A. Bahrami / International Journal of Engineering Science 107 (2016) 134–148

Table 1 Comparison of maximum dimensionless deflection 102 w/L for a cantilevered microbeam.

Solution method

∗

L/h

12

 /h

1/3

2/3

4.3780 4.3780 4.3889 (0.2490)∗

2.2175 2.2175 2.2372 (0.8884)

Exact 6-DOF beam element 4-DOF beam element

10 1.0 1.2196 1.2196 1.2427 (1.8941)

8

1/3

2/3

1.0

3.0477 3.0477 3.0587 (0.3609)

1.5446 1.5446 1.5644 (1.2819)

0.8506 0.8506 0.8738 (2.7275)

1/3

2/3

1.9592 1.9592 1.9703 (0.5666)

0.9941 0.9941 1.0140 (2.0018)

1.0 0.5487 0.5487 0.5720 (4.2464)

Error (%) Table 2 Convergence study of maximum dimensionless deflection 102 w/L for a cantilever microbeam based on the 6-DOF beam element.

Number of elements

L/h

12

/h

1/3

2/3

1.0

1/3

10 2/3

1.0

1/3

8 2/3

1.0

2 10 20 40 80 120 140 250 500

4.1107 4.3693 4.3761 4.3775 4.3779 4.3780 4.3780 4.3780 4.3780

2.0949 2.2152 2.2170 2.2174 2.2175 2.2175 2.2175 2.2175 2.2175

1.1627 1.2189 1.2194 1.2195 1.2196 1.2196 1.2196 1.2196 1.2196

2.8635 3.0419 3.0464 3.0474 3.0476 3.0477 3.0477 3.0477 3.0477

1.4633 1.5432 1.5443 1.5446 1.5446 1.5446 1.5446 1.5446 1.5446

0.8149 0.8501 0.8505 0.8506 0.8506 0.8506 0.8506 0.8506 0.8506

1.8429 1.9557 1.9584 1.9591 1.9592 1.9592 1.9592 1.9592 1.9592

0.9459 0.9932 0.9939 0.9941 0.9941 0.9941 0.9941 0.9941 0.9941

0.5292 0.5484 0.5486 0.5487 0.5487 0.5487 0.5487 0.5487 0.5487

The force vector for a uniform distribution load q can be expressed as:



F=q

L 2

L2 12

L 2

−

L2 12

T (30)

Moreover, the force vector for a linearly varying distributed load (q = q0 (1 − x/L )) can be expressed as:

F=

 q0 L 42 + 40φ 120(1 + φ )

( 6 + 5 φ )L

18 + 20φ

− ( 4 + 5 φ )L

T

(31)

4. Comparison and convergence studies Modified couple stress Timoshenko beam model presents an appropriate model when the roles of size dependency and shear deformation become significant in the mechanical behavior of microbeams. (Dehrouyeh-Semnani & Nikkhah-Bahrami, 2015b) presented dimensionless form of Eq. (5) and investigated the influence of size-dependent shear deformation on the mechanical behavior of microbeams. It was indicated that the slenderness ratio L/ I/A and the ratio of material length scale parameter to thickness /h increase the role of size dependent shear deformation in the mechanical behavior of microbeams and also boundary conditions play a prominent role. Hence, these parameters are used to compare the numerical results obtained based on the beam elements with those obtained based on the exact, analytical, and semi-analytical solutions of cantilevered, simply supported and doubly clamped microbeams, respectively. For numerical solutions, the epoxy microbeam is considered with the following material properties and geometrical parameters: Young’s modulus E = 1.44GPa, Poisson ratio v = 0.38 and width b = 2 h (h is the thickness of microbeam). The shear coefficient ks is taken to be (5 + 5v)/(6 + 5v), which was shown to be the best expression for a rectangular crosssection beam (Kaneko, 1975). In addition, the point √ load for all case studies is considered 100 μN. The slenderness ratio for rectangular cross section can be obtained by 12L/h, therefore, the dimensionless parameter L/h is used instead of the slenderness ratio. 4.1. Cantilevered microbeam The exact solution for static bending of a cantilevered microbeam subjected to a point load at its free end proposed by Dehrouyeh-Semnani and Nikkhah-Bahram, (2015a). The exact solution is employed to examine the accuracy and reliability of the 4-DOF and 6-DOF beam elements. The results are tabulated in Tables 1–3 for different dimensionless parameters. The listed results in Table 1 illustrate that the outcomes of 6-DOF beam element are in excellent agreement with those obtained based on the exact solution. Moreover, it can be observed from Table 2, when the number of elements increases the solution is stable. The convergence study shows that by increasing the ratio of material length scale parameter to thickness /h or decreasing the ratio of length to thickness L/h, the convergence rate increases. In other words, by increasing the role of size-dependent shear deformation, the final solution can be obtained by use of fewer elements.

A.M. Dehrouyeh-Semnani, A. Bahrami / International Journal of Engineering Science 107 (2016) 134–148

141

Table 3 Convergence study of maximum dimensionless deflection 102 w/L for a cantilever microbeam based on the 4-DOF beam element.

Number of elements

L/h

12

10

8

/h

1/3

2/3

1.0

1/3

2/3

1.0

1/3

2/3

1.0

2 10 20 40 50 250 500

4.3880 4.3886 4.3888 4.3889 4.3889 4.3889 4.3889

2.2338 2.2363 2.2370 2.2372 2.2372 2.2372 2.2372

1.2377 1.2416 1.2424 1.2427 1.2427 1.2427 1.2427

3.0578 3.0585 3.0586 3.0587 3.0587 3.0587 3.0587

1.5612 1.5637 1.5643 1.5644 1.5644 1.5644 1.5644

0.8691 0.8730 0.8736 0.8738 0.8738 0.8738 0.8738

1.9695 1.9701 1.9703 1.9703 1.9703 1.9703 1.9703

1.0110 1.0135 1.0139 1.0140 1.0140 1.0140 1.0140

0.5677 0.5714 0.5719 0.5720 0.5720 0.5720 0.5720

Table 4 Comparison study of maximum dimensionless deflection 102 w/L for a simply supported microbeam.

Solution method

∗

L/h

12

 /h

1/3

2/3

0.2781 0.2781 0.2809 (1.0068)∗

0.1415 0.1415 0.1465 (3.5336)

Analytical 6-DOF beam element 4-DOF beam element

10 1.0 0.0785 0.0785 0.0843 (7.3885)

8

1/3

2/3

1.0

0.1950 0.1950 0.1978 (1.4359)

0.0994 0.0994 0.1044 (5.0302)

0.0554 0.0554 0.0612 (10.4693)

1/3

2/3

0.1269 0.1269 0.1298 (2.2853)

0.0649 0.0649 0.0700 (7.8582)

1.0 0.0365 0.0365 0.0424 (16.1644)

Error (%) Table 5 Convergence study of maximum dimensionless deflection 102 w/L for a simply supported microbeam based on 6-DOF beam element.

Number of elements

L/h

12

10

8

/h

1/3

2/3

1.0

1/3

2/3

1.0

1/3

2/3

1.0

2 10 20 40 50 60 250 500

0.2114 0.2760 0.2777 0.2780 0.2781 0.2781 0.2781 0.2781

0.1106 0.1409 0.1413 0.1415 0.1415 0.1415 0.1415 0.1415

0.0639 0.0783 0.0784 0.0785 0.0785 0.0785 0.0785 0.0785

0.1490 0.1936 0.1947 0.1950 0.1950 0.1950 0.1950 0.1950

0.0789 0.0990 0.0993 0.0994 0.0994 0.0994 0.0994 0.0994

0.0463 0.0553 0.0553 0.0554 0.0554 0.0554 0.0554 0.0554

0.0979 0.1261 0.1267 0.1269 0.1269 0.1269 0.1269 0.1269

0.0529 0.0647 0.0649 0.0649 0.0649 0.0649 0.0649 0.0649

0.0316 0.0364 0.0365 0.0365 0.0365 0.0365 0.0365 0.0365

Now, the 4-DOF beam element developed by Kahrobaiyan et al. (2014) is studied. It can be inferred from Table 1, by increasing /h and decreasing L/h i.e., the role of size-dependent shear deformation increases, the 4-DOF beam element error increases. For example, when L/A = 8 and /h = 1 i.e. the role of size-dependent shear deformation is significant, the beam element error is about 4.25%. Moreover, the tabulated results illustrate that the values of static deflection predicted based on the 4-DOF beam element is higher than those predicted based on the exact solution i.e., the beam element considers less bending rigidity for cantilevered microbeams. Table 3 demonstrates that the solution is stable as the number of elements increases and therefore, the beam element error does not lessen by increasing the number of elements.

4.2. Simply supported microbeam In this subsection, static bending of a simply supported microbeam subjected to a point load at its center is studied. To that end, the analytical solution proposed by Ma et al. (2008) is employed. The numerical and analytical results are listed in Table 4, and moreover Tables 5 and 6 illustrate the convergence study for the 6-DOF and 4-DOF beam elements, respectively. Table 4 shows that the predicted results based on the 6-DOF beam element are in excellent agreement with the analytical results achieved based on Navier solution producer (Ma et al., 2008) and also the convergence study in Table 5 shows that the beam element presents a stable solution. Moreover, the results tabulated in Table 5 indicate that the convergence rate has an ascending trend with respect to the size-dependent shear deformation. Table 6 illustrates that the numerical results obtained based on the 4-DOF beam element are stable, however, Table 4 illustrates the beam 4-DOF element predicts values of static deflection greater than those predicted by using analytical solution. It can be deduced from Table 4, the beam element error has an ascending trend with respect to size-dependent shear deformation. For instance, when L/A = 8 and /h = 1, i.e., the effect of size-dependent shear deformation is significant, the beam element error is about 16.16% and also the error does not diminish by increasing the number of elements.

142

A.M. Dehrouyeh-Semnani, A. Bahrami / International Journal of Engineering Science 107 (2016) 134–148 Table 6 Convergence study of maximum dimensionless deflection 102 w/L for a simply supported microbeam based on the 4-DOF beam element.

Number of elements

L/h

12

10

8

/h

1/3

2/3

1.0

1/3

2/3

1.0

1/3

2/3

1.0

2 10 20 40 50 250 500

0.2807 0.2809 0.2809 0.2809 0.2809 0.2809 0.2809

0.1456 0.1462 0.1464 0.1465 0.1465 0.1465 0.1465

0.0831 0.0840 0.0842 0.0843 0.0843 0.0843 0.0843

0.1976 0.1978 0.1978 0.1978 0.1978 0.1978 0.1978

0.1036 0.1042 0.1044 0.1044 0.1044 0.1044 0.1044

0.0601 0.0610 0.0612 0.0612 0.0612 0.0612 0.0612

0.1296 0.1297 0.1298 0.1298 0.1298 0.1298 0.1298

0.0693 0.0699 0.0700 0.0700 0.0700 0.0700 0.0700

0.0413 0.0423 0.0424 0.0424 0.0424 0.0424 0.0424

Table 7 Comparison study of maximum dimensionless deflection 102 w/L of a doubly clamped microbeam.

Solution method

∗

L/h

12

10

 /h

1/3

2/3

Galerkin method 6-DOF beam element 4-DOF beam element

0.0739 0.0739 0.0769 (4.0595)∗

0.0381 0.0381 0.0432 (13.3858)

1.0 0.0217 0.0217 0.0277 (27.6498)

8

1/3

2/3

1.0

0.0531 0.0531 0.0561 (5.6497)

0.0275 0.0275 0.0327 (18.9091)

0.0159 0.0159 0.0219 (37.7358)

1/3

2/3

0.0360 0.0360 0.0391 (8.6111)

0.0189 0.0189 0.0241 (27.5132)

1.0 0.0111 0.0111 0.0172 (54.9550)

Error (%)

Table 8 Convergence study of maximum dimensionless deflection 102 w/L for a doubly clamped microbeam based on the 6-DOF beam element.

Number of elements

L/h

12

/h

1/3

2/3

1.0

1/3

10 2/3

1.0

1/3

8 2/3

1.0

2 10 20 40 50 60 250 500

0.0073 0.0719 0.0735 0.0738 0.0739 0.0739 0.0739 0.0739

0.0071 0.0376 0.0380 0.0381 0.0381 0.0381 0.0381 0.0381

0.0068 0.0216 0.0217 0.0217 0.0217 0.0217 0.0217 0.0217

0.0073 0.0517 0.0528 0.0531 0.0531 0.0531 0.0531 0.0531

0.0070 0.0272 0.0275 0.0275 0.0275 0.0275 0.0275 0.0275

0.0066 0.0158 0.0159 0.0159 0.0159 0.0159 0.0159 0.0159

0.0072 0.0352 0.0358 0.0359 0.0360 0.0360 0.0360 0.0360

0.0068 0.0187 0.0188 0.0188 0.0189 0.0189 0.0189 0.0189

0.0063 0.0111 0.0111 0.0111 0.0111 0.0111 0.0111 0.0111

Table 9 Convergence study of maximum dimensionless deflection 102 w/L for a doubly clamped microbeam based on the 4-DOF beam element.

Number of elements

L/h

12

/h

1/3

2/3

1.0

1/3

10 2/3

1.0

1/3

8 2/3

1.0

2 10 20 40 50 250 500

0.0767 0.0768 0.0769 0.0769 0.0769 0.0769 0.0769

0.0424 0.0430 0.0432 0.0432 0.0432 0.0432 0.0432

0.0265 0.0274 0.0276 0.0277 0.0277 0.0277 0.0277

0.0559 0.0560 0.0561 0.0561 0.0561 0.0561 0.0561

0.0319 0.0326 0.0327 0.0327 0.0327 0.0327 0.0327

0.0208 0.0217 0.0219 0.0219 0.0219 0.0219 0.0219

0.0389 0.0390 0.0391 0.0391 0.0391 0.0391 0.0391

0.0234 0.0240 0.0241 0.0241 0.0241 0.0241 0.0241

0.0162 0.0171 0.0172 0.0172 0.0172 0.0172 0.0172

4.3. Doubly clamped microbeam The accuracy and stability of the 6-DOF and 4-DOF beam elements for static bending of a doubly clamped Timoshenko microbeam subjected to a point load at its center is investigated. Galerkin method as a powerful semi-analytical method is used to solve the problem. Afterwards, the obtained results are employed to explore the accuracy and stability of the beam elements. The solution producer based on Galerkin method is given in Appendix B. Table 7 shows comparison studies between numerical and semi-analytical results, and the convergence studies for the 6-DOF and 4-DOF beam elements are tabulated in Table 8 and 9, respectively.

A.M. Dehrouyeh-Semnani, A. Bahrami / International Journal of Engineering Science 107 (2016) 134–148

143

Fig. 1. Error of 4-DOF beam element as a function of the material length to thickness ratio l/h for different values of the length to thickness ratio L/h and different boundary conditions.

There is an excellent agreement between the results achieved by using the 6-DOF beam element and those obtained by using Galerkin method and, moreover, the tabulated results in Table 8 demonstrate that the solution is stable. It can be observed from Table 8 that the convergence rate ascends by increasing the size-dependent shear deformation. As seen from Table 7, the values of error for the 4-DOF beam element increase meaningfully in comparison with the previous case studies. It can be seen from Table 7, when L/A = 8 and /h = 1, the beam element error is about 55% whereas the beam element error for the simply supported and cantilevered microbeams with the identical geometry parameters and material properties are about 4.25% and 16.16%, respectively. Moreover, the results listed in Table 9 show that the solution is stable. Dehrouyeh-Semnani and Nikkhah-Bahrami (2015a) indicated that when the slenderness ratio is decreased or the ratio of material length scale parameter to thickness is increased, the influence of size-dependent shear deformation on the mechanical behavior of microbeam is risen. Moreover, they concluded that the effect of size-dependent shear deformation on the mechanical behavior of microbeam is strongly dependent on the boundary conditions of microbeam. It was reported that the size-dependent shear deformation has the highest influence on the mechanical behavior of the microbeam with doubly clamped boundary conditions followed by simply supported and cantilevered boundary conditions, respectively. The errors of 4-DOF beam element as a function of /h for different values of L/h and different boundary conditions are depicted in Fig. 1. The plots show that when the role of size-dependent shear deformation is increased the error of 4-DOF beam element ascends i.e., by increasing L/h or decreasing /h the error increases and also it is observed the doubly clamped boundary conditions has the maximum error followed by simply supported and cantilevered boundary conditions, respectively.

144

A.M. Dehrouyeh-Semnani, A. Bahrami / International Journal of Engineering Science 107 (2016) 134–148

Kahrobaiyan et al. (2014) employed the 4-DOF beam element to validate the experimental data reported by Lam, Yang, Chong, Wang, and Tong, (2003). The obtained results indicate that the beam element results are in good agreement with the experimental data. Hence, they deduced that the beam element is valid, reliable and can be successfully employed to deal with the mechanical problems in micron and sub-microns scales. The size-dependent shear deformation has the lowest influence on the mechanical behavior of microbeams with cantilevered boundary conditions; therefore, the Timoshenko beam element cannot be adequately verified by employing the cantilevered microbeam results. Hence, unlike the claim made by Kahrobaiyan et al. (2014) which the 4-DOF beam element is reliable and valid to deal with the mechanical problems in micron and sub-microns scales, the 4-DOF beam element is incapable of dealing with the simple case studies. Finally, it should be pointed out that employing the stiffness matrix reported by Kahrobaiyan et al. (2014) yields the error values reduce. However, the error values are significant and the trend of error with respect to the size-dependent shear deformation does not change.

5. On inaccuracy of 4-DOF beam element To solve the static equilibrium equations of modified couple stress Timoshenko beam equations, (Kahrobaiyan et al., 2014) simplified one of the equations (Eq. (17a-b)) by changing the order of the equation to two whereas the order of the equation is of order four (Eq. 19). This simplification yields the shape functions of the new beam element to be polynomial. But, this simplification forced them to consider the transverse deformation and the rotation angle as degrees of freedom at each node. Since the size-dependent Timoshenko beam model has three boundary conditions at each end (see Eq. (6)); it is obvious that the 4-DOF beam element is incapable of considering all the boundary conditions. For instance, the boundary conditions of a clamped end are as follows:

w = 0, ψ = 0,

dw =0 dx

(32)

It is evident that the 4-DOF beam element is incapable of considering the third condition (i.e., dw/dx = 0). The 4-DOF beam element considers only the boundary conditions related to the transverse deformation and the rotation angle or the transverse force and bending moment, and it neglects the boundary conditions associated with the first derivation of transverse deformation or the moment due to the couple stress. In other words, the 4-DOF beam element neglects the nonclassical boundary conditions. The exact solution of Eq. (17a–b) can be obtained by:

θ = c2 sinh (ζ x ) + c3 cosh (ζ x ) + c4 + c5 x + ξ x , γ 2

1 d2 θ c1 = − 2 2 , GA 4 dx



ζ =2

(1 + α ) , (/L )2

ξ =−

c1 L2 EI(1 + α )

(33)

Using Eqs. (18) and (33) the transverse deformation w and the rotation angle ψ can be obtained by:

w=

1 2



(θ + γ )dx + c6 , ψ =

1 (θ − γ ) 2

(34)

By selecting w, −dw/dx and ψ as DOF of the beam element and also by applying the appropriate boundary conditions as follows:



w|x=0 = w1 ,

dw dx



=

x=0

dw1 dw , ψ|x=0 = ψ1 , w|x=L = w2 , dx dx

=

x=L

dw2 , ψ|x=L = ψ2 dx

(35)

Using Eq. (35), Eq. (34) can be written as:



w = N1w

 ψ = N1ψ

N2w ψ

N2

N3w ψ

N3

N4w ψ

N4

N5w ψ

N5

N6w



ψ

N6

w1



w1

−dw1 /dx −dw1 /dx

ψ1 ψ1

w2 w2

−dw2 /dx −dw2 /dx

ψ2

T

ψ2

T

(36)

in which Nw and Nψ are the transverse and rotation shape functions of the new beam element. Using the achieved shape functions, one can obtain the stiffness matrix and the force vector of the beam element by substitution of Nw and Nψ into Eq. (26). It is so obvious that the final form of the beam element can be expressed as:



[K ]{δ} = {F }, {δ} = w1

−dw1 /dx

ψ1

w2

−dw2 /dx

ψ2

T

(37)

Since the new beam element has 3 DOF at each node, it is capable of considering all the boundary conditions of the Timoshenko beam model, unlike the beam element proposed by Kahrobaiyan et al. (2014). It should be pointed out that Eq. (33) indicates the shape functions of new beam element are not polynomial, unlike the classical Timoshenko beam element and also the non-classical Timoshenko beam elements proposed by Kahrobaiyan et al. (2014) and Arbind and Reddy (2013).

A.M. Dehrouyeh-Semnani, A. Bahrami / International Journal of Engineering Science 107 (2016) 134–148

145

6. Conclusion The accuracy and stability of two different size-dependent Timoshenko beam elements developed by Kahrobaiyan et al. (2014) and Arbind and Reddy (2013) are studied. Both the elements proposed by Kahrobaiyan et al. (2014) and Arbind and Reddy (2013) are two-node beam elements, but have 2- and 3-DOF at each node respectively. The exact, analytical and semianalytical solutions for static behavior of a microbeam with three different boundary conditions are employed to investigate the accuracy of the beam elements. In addition, convergence study is performed to study the stability of beam elements. The three case studies illustrate that the 6-DOF beam element is capable of predicting size-dependent static behavior of Timoshenko microbeam and also presents a stable solution. Although the 4-DOF beam element presents stable solutions, the obtained results indicate that it is incapable of verifying the results achieved based on the other solutions for the three case studies. The results show that by increasing the dimensionless material length scale parameter or decreasing the ratio of length to thickness, the beam element error increase. Moreover, the results obtained for doubly clamped boundary conditions have the most error followed by simply supported and cantilevered boundary conditions, respectively. In other words, by increasing the influence of size-dependent shear deformation, the inaccuracy of 4-DOF beam element increases. Appendix A On the basis of Eq. (26), Kij can be calculated by the following equation:

Kij = Kij1 + Kij2 + Kij3  Kij1 = EI Kij2

0

= GA

ψ

ψ dNi dN j

L

L

dx 

ψ



GA2 Kij3 = 4

L





− Ni

dx

0

dx

dx

dNiw

dN w j

0

− Nj

dx

ψ

d2 Niw + dx dx2

dNi



  2

ψ

dN2w dx

=

  2 x

1 6 2 (1 + φ ) L

−6

L

d2 Nw j

+

dx

x

(A1)

dx

dN j

Based on Eq. (23), we have

dN1w 1 x = 6 dx L (1 + φ ) L



ψ

dx

dx2



  

ψ

dN1 d2 N1w 6 x = = 2 2 dx L d2 x L (1 + φ )

−φ ,

x

− 2 (4 + φ )

L



+ (2 + φ ) ,

−1



ψ

  

dN2 d2 N2w 1 x = = 6 dx L (1 + φ ) L d2 x

ψ



(A2)

− (4 + φ )

ψ

Substituting N1 from Eq. (23) and dN1w /dx, d2 N1w /dx2 , and dN1 /dx from Eq. (A2) into Eq. (A1) results in: 1 K11 =

12EI L3 ( 1 + φ )

2

GAφ 2

2 ,K11 =

L (1 + φ )

2

12GA2

3 , K11 =

(A3)

L3 ( 1 + φ )

2

1 + K 2 can be expressed as Considering the definition of variable φ (Eq. (24)), K11 11 1 2 K11 + K11 =

12EI + GAL2 φ 2 L3 ( 1 + φ )

2

=



12EI L3 ( 1 + φ )

2



1+φ 1+

α 

(A4)

2

3 can be rewritten as Considering the definition of variable α (Eq. (18)), K11 3 K11 =

12GA2 L3 ( 1 + φ )

2

=

12EI L3 ( 1 + φ )

2

α

(A5)

Therefore, K11 can be obtained by:

K11 =



12EI L3 ( 1 + φ )

2



1+φ 1+

α  2

+



L3



12EI 12EI α α= 1+α+φ 1+ 2 2 (1 + φ ) L3 ( 1 + φ )

 (A6)

The expression for K11 based on the published work by Kahrobaiyan et al. (2014) is as follows (see Eq. (27)):

K11 =

12EI (1 + α ) L3 ( 1 + φ )

(A7)

Comparing Eqs. (A6) and (A7) shows that K11 was incorrectly reported by Kahrobaiyan et al. (2014). ψ ψ Substitution of N1 from Eq. (23) and dN2w /dx, d2 N2w /dx2 , and dN2 /dx from Eq. (A2) into Eq. (A1) yields



1 K22 = EI



3 L (1 + φ )

2

+

1 GALφ 2 2 3 ,K22 = , K22 = GA2 2 L 4 (1 + φ )



3 L (1 + φ )

2

+

1 L



(A8)

146

A.M. Dehrouyeh-Semnani, A. Bahrami / International Journal of Engineering Science 107 (2016) 134–148

Considering the definition of variables φ and α (Eqs. (24) and (18)), we have



GALφ 2

1 2 K22 + K22 =

4 (1 + φ )

2

 3 K22 = GA2

3 L (1 + φ )

2

+

L (1 + φ )

2



1 L

=

Hence, K22 can be obtained by:

 

EI

K22 =

L3

(1 + φ ) EI

=



3 1+φ 1+

2

 



3 1+φ 1+

L3 ( 1 + φ )

2



3

+ EI



EI L3 ( 1 + φ )

2

α 

 

3 + (1 + φ )

2

+ (1 + φ )

2

2

α 



1 EI α = 3 1+φ 1+ 2 L 2 L3 ( 1 + φ )

+



L2 +





2

2



L2

(A9)

(A10)



(1 + φ )

2

+ (1 + φ ) + 3 + (1 + φ ) 2

+ (1 + φ )

α L2 EI

L3



2

3 + (1 + φ )

2



α L2

 2 α L

(A11)

The expression for K22 based on the published work by Kahrobaiyan et al. (2014) is as follows (see Eq. (27)):

12EI (4 + φ )(1 + α )L2 L3 ( 1 + φ )

K22 =

(A12)

Comparing Eqs. (A11) and (A12) indicates that K22 was incorrectly reported by Kahrobaiyan et al. (2014). Appendix B Consider a doubly clamped microbeam subjected to a point load (F) at its center. The governing equations and associated boundary conditions based on the size-dependent Timoshenko beam model can be obtained by (see Eqs. (5) and (6)):



ks GA EI

dψ d2 w + dx dx2



+

1 GA2 4



d4 w d3 ψ − 3 dx dx4



= F δ (x − L/2 )

   2  d3 w dw 1 d ψ + GA2 − =0 ψ+ 2 3

d2 ψ − ks GA dx2

w|x=0 =

dx

4

dx

dx

dw dw |x=0 = ψ|x=0 = |x=L = w|x=L = ψ|x=L = 0 dx dx

(B1)

where δ is Kronecker delta. To solve Eq. (B1), the following expansions for the deflection w and the rotation angle ψ are selected as follows: n

αi wi (x ), wi = xi+1 (x − L )2

w= i=1 m

ψ=

βi ψi (x ), ψi = xi (x − L )

(B2)

i=1

where wi satisfies all the boundary conditions related to w and dw/dx and also ψ i satisfies all the boundary conditions associated with ψ . In addition, α i and β i are unknown coefficients. It is supposed that m is equal to n. By applying Galerkin method, we have





L

0





wi ks GA L

dψ d2 w + dx dx2

+

1 GA2 4



d4 w d3 ψ − 3 dx dx4





− F δ (x − L/2 ) dx = 0

 2    2  d3 w d ψ dw 1 d ψ ψi EI 2 − ks GA ψ + + GA2 − dx = 0 2 3 dx

dx

0



4

dx

dx

(B3)

Eq. (B3) can be written in the matrix form as follows:

! !

K 11

K 21

where

"

!

"

!

P1i

K 22

" "

  1  P {α} =   {β} P2

   L  4 3 d2 w j dψ j 1 1 2 d wj 12 2 d ψj − GA  dx , K = w k GA + GA  dx s i ij 4 dx 4 dx2 dx4 dx3 0 0      L    L dw d3 w d2 ψ j 1 1 =− ψi ks GA j + GA2 3 j dx, Kij22 = ψi EI + GA2 − k GA ψ dx s j dx 4 4 dx dx2 0 0  L = wi F δ (x − L/2 )dx = F wi (L/2 ), P2i = 0, i, j = 1, 2, . . . , n 

L

Kij11 = Kij21

K 12

(B4)



wi ks GA

0

Solving Eq. (B4) leads to determination of the unknown coefficients α i and β i .

(B5)

A.M. Dehrouyeh-Semnani, A. Bahrami / International Journal of Engineering Science 107 (2016) 134–148

147

References Abbasnejad, B., Rezazadeh, G., & Shabani, R. (2013). Stability analysis of a capacitive fgm micro-beam using modified couple stress theory. Acta Mechanica Solida Sinica, 26(4), 427–440. Akbarzadeh Khorshidi, M., Shariati, M., & Emam, S. A. (2016). Postbuckling of functionally graded nanobeams based on modified couple stress theory under general beam theory. International Journal of Mechanical Sciences, 110, 160–169. Akgöz, B., & Civalek, Ö. (2011). Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. International Journal of Engineering Science, 49(11), 1268–1280. Akgöz, B., & Civalek, Ö. (2014). Thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium. International Journal of Engineering Science, 85(0), 90–104. Ansari, R., Ashrafi, M. A., & Arjangpay, A. (2015). An exact solution for vibrations of postbuckled microscale beams based on the modified couple stress theory. Applied Mathematical Modelling, 39(10–11), 3050–3062. Ansari, R., Faghih Shojaei, M., Gholami, R., Mohammadi, V., & Darabi, M. A. (2013). Thermal postbuckling behavior of size-dependent functionally graded Timoshenko microbeams. International Journal of Non-Linear Mechanics, 50(0), 127–135. Arbind, A., & Reddy, J. N. (2013). Nonlinear analysis of functionally graded microstructure-dependent beams. Composite Structures, 98, 272–281. Asghari, M., Rahaeifard, M., Kahrobaiyan, M. H., & Ahmadian, M. T. (2011). The modified couple stress functionally graded Timoshenko beam formulation. Materials & Design, 32(3), 1435–1443. Chen, W., Li, L., & Xu, M. (2011). A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation. Composite Structures, 93(11), 2723–2732. Dai, H. L., Wang, Y. K., & Wang, L. (2015). Nonlinear dynamics of cantilevered microbeams based on modified couple stress theory. International Journal of Engineering Science, 94(0), 103–112. Dehrouyeh-Semnani, A. M. (2015). The influence of size effect on flapwise vibration of rotating microbeams. International Journal of Engineering Science, 94(0), 150–163. Dehrouyeh-Semnani, A. M., BehboodiJouybari, M., & Dehrouyeh, M. (2016). On size-dependent lead-lag vibration of rotating microcantilevers. International Journal of Engineering Science, 101, 50–63. Dehrouyeh-Semnani, A. M., Dehrouyeh, M., Torabi-Kafshgari, M., & Nikkhah-Bahrami, M. (2015a). A damped sandwich beam model based on symmetric-deviatoric couple stress theory. International Journal of Engineering Science, 92, 83–94. Dehrouyeh-Semnani, A. M., Dehrouyeh, M., Torabi-Kafshgari, M., & Nikkhah-Bahrami, M. (2015b). An investigation into size-dependent vibration damping characteristics of functionally graded viscoelastically damped sandwich microbeams. International Journal of Engineering Science, 96, 68–85. Dehrouyeh-Semnani, A. M., Dehrouyeh, M., Zafari-Kolukhi, H., & Ghamami, M. (2015). Size-dependent frequency and stability characteristics of axially moving microbeams based on modified couple stress theory. International Journal of Engineering Science, 97, 98–112. Dehrouyeh-Semnani, A. M., Mostafaei, H., & Nikkhah-Bahrami, M. (2016). Free flexural vibration of geometrically imperfect functionally graded microbeams. International Journal of Engineering Science, 105, 56–79. Dehrouyeh-Semnani, A. M., & Nikkhah-Bahrami, M. (2015a). A discussion on incorporating the Poisson effect in microbeam models based on modified couple stress theory. International Journal of Engineering Science, 86(0), 20–25. Dehrouyeh-Semnani, A. M., & Nikkhah-Bahrami, M. (2015b). The influence of size-dependent shear deformation on mechanical behavior of microstructures-dependent beam based on modified couple stress theory. Composite Structures, 123(0), 325–336. Farokhi, H., & Ghayesh, M. H. (2015). Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams. International Journal of Engineering Science, 91(0), 12–33. Farokhi, H., Païdoussis, M. P., & Misra, A. K. (2016). A new nonlinear model for analyzing the behaviour of carbon nanotube-based resonators. Journal of Sound and Vibration, 378, 56–75. Farokhi, H., Ghayesh, M. H., & Hussain, S. (2016). Large-amplitude dynamical behaviour of microcantilevers. International Journal of Engineering Science, 106, 29–41. Fathalilou, M., Sadeghi, M., & Rezazadeh, G. (2014). Micro-inertia effects on the dynamic characteristics of micro-beams considering the couple stress theory. Mechanics Research Communications, 60(0), 74–80. Friedman, Z., & Kosmatka, J. B. (1993). An improved two-node timoshenko beam finite element. Computers & Structures, 47(3), 473–481. Ghayesh, M. H., Amabili, M., & Farokhi, H. (2013). Three-dimensional nonlinear size-dependent behaviour of Timoshenko microbeams. International Journal of Engineering Science, 71(0), 1–14. Ghayesh, M. H., & Farokhi, H. (2015). Chaotic motion of a parametrically excited microbeam. International Journal of Engineering Science, 96, 34–45. Ghayesh, M. H., Farokhi, H., & Alici, G. (2015). Subcritical parametric dynamics of microbeams. International Journal of Engineering Science, 95(0), 36–48. Ghayesh, M. H., Farokhi, H., & Alici, G. (2016). Size-dependent performance of microgyroscopes. International Journal of Engineering Science, 100, 99–111. Ilkhani, M., & Hosseini-Hashemi, S. (2016). Size dependent vibro-buckling of rotating beam based on modified couple stress theory. Composite Structures, 143, 75–83. Kahrobaiyan, M., Asghari, M., & Ahmadian, M. (2014). A Timoshenko beam element based on the modified couple stress theory. International Journal of Mechanical Sciences, 79, 75–83. Kaneko, T. (1975). On Timoshenko’s correction for shear in vibrating beams. Journal of Physics D: Applied Physics, 8(16), 1927. Ke, L.-L., Wang, Y.-S., Yang, J., & Kitipornchai, S. (2012). Nonlinear free vibration of size-dependent functionally graded microbeams. International Journal of Engineering Science, 50(1), 256–267. Komijani, M., Esfahani, S. E., Reddy, J. N., Liu, Y. P., & Eslami, M. R. (2014). Nonlinear thermal stability and vibration of pre/post-buckled temperature- and microstructure-dependent functionally graded beams resting on elastic foundation. Composite Structures, 112, 292–307. Lam, D., Yang, F., Chong, A., Wang, J., & Tong, P. (2003). Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids, 51(8), 1477–1508. Ma, H. M., Gao, X. L., & Reddy, J. N. (2008). A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. Journal of the Mechanics and Physics of Solids, 56(12), 3379–3391. Mohammadabadi, M., Daneshmehr, A. R., & Homayounfard, M. (2015). Size-dependent thermal buckling analysis of micro composite laminated beams using modified couple stress theory. International Journal of Engineering Science, 92(0), 47–62. Mojahedi, M., & Rahaeifard, M. (2016). A size-dependent model for coupled 3D deformations of nonlinear microbridges. International Journal of Engineering Science, 100, 171–182. Park, S. K., & Gao, X. L. (2006). Bernoulli–Euler beam model based on a modified couple stress theory. Journal of Micromechanics and Microengineering, 16(11), 2355. Reddy, J. N. (2011). Microstructure-dependent couple stress theories of functionally graded beams. Journal of the Mechanics and Physics of Solids, 59(11), 2382–2399. Salamat-talab, M., Nateghi, A., & Torabi, J. (2012). Static and dynamic analysis of third-order shear deformation FG micro beam based on modified couple stress theory. International Journal of Mechanical Sciences, 57(1), 63–73. Sedighi, H. M., Chan-Gizian, M., & Noghreha-Badi, A. (2014). Dynamic pull-in instability of geometrically nonlinear actuated micro-beams based on the modified couple stress theory. Latin American Journal of Solids and Structures, 11(5), 810–825. Shaat, M., & Mohamed, S. A. (2014). Nonlinear-electrostatic analysis of micro-actuated beams based on couple stress and surface elasticity theories. International Journal of Mechanical Sciences, 84(0), 208–217.

148

A.M. Dehrouyeh-Semnani, A. Bahrami / International Journal of Engineering Science 107 (2016) 134–148

Shafiei, N., Kazemi, M., & Ghadiri, M. (2016a). Nonlinear vibration of axially functionally graded tapered microbeams. International Journal of Engineering Science, 102, 12–26. Shafiei, N., Kazemi, M., & Ghadiri, M. (2016b). On size-dependent vibration of rotary axially functionally graded microbeam. International Journal of Engineering Science, 101, 29–44. Shafiei, N., Mousavi, A., & Ghadiri, M. (2016). On size-dependent nonlinear vibration of porous and imperfect functionally graded tapered microbeams. International Journal of Engineering Science, 106, 42–56. Shojaeian, M., & Zeighampour, H. (2016). Size dependent pull-in behavior of functionally graded sandwich nanobridges using higher order shear deformation theory. Composite Structures, 143, 117–129. S¸ ims¸ ek, M. (2014). Nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using modified couple stress theory and He’s variational method. Composite Structures, 112(0), 264–272. S¸ ims¸ ek, M., & Reddy, J. N. (2013). Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory. International Journal of Engineering Science, 64(0), 37–53. Tang, M., Ni, Q., Wang, L., Luo, Y., & Wang, Y. (2014). Size-dependent vibration analysis of a microbeam in flow based on modified couple stress theory. International Journal of Engineering Science, 85(0), 20–30. Thai, H.-T., Vo, T. P., Nguyen, T.-K., & Lee, J. (2015). Size-dependent behavior of functionally graded sandwich microbeams based on the modified couple stress theory. Composite Structures, 123(0), 337–349. Xia, W., Wang, L., & Yin, L. (2010). Nonlinear non-classical microscale beams: Static bending, postbuckling and free vibration. International Journal of Engineering Science, 48(12), 2044–2053. Yang, F., Chong, A., Lam, D., & Tong, P. (2002). Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39(10), 2731–2743.