A size-dependent shear deformation beam model based on the strain gradient elasticity theory

International Journal of Engineering Science 70 (2013) 1–14

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International Journal of Engineering Science

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A size-dependent shear deformation beam model basedon the strain gradient elasticity theory

0020-7225/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijengsci.2013.04.004

⇑ Corresponding author. Tel.: +90 242 3106319.E-mail address: [email protected] (Ö. Civalek).

1 Tel.: +90 242 3106319.

Bekir Akgöz 1, Ömer Civalek ⇑Akdeniz University, Civil Engineering Department, Division of Mechanics, Antalya, Turkey

a r t i c l e i n f o

Article history:Received in revised form 17 April 2013

This paper is dedicated to Professor J.N.Reddy on the occasion of his 68th birthday.

Keywords:Size effectModified strain gradient theorySinusoidal shear deformation theoryBendingVibrationMicrobeam

a b s t r a c t

A new size-dependent higher-order shear deformation beam model is developed based onmodified strain gradient theory. The model captures both the microstructural and sheardeformation effects without the need for any shear correction factors. The governing equa-tions and boundary conditions are derived by using Hamilton’s principle. The static bend-ing and free vibration behavior of simply supported microbeams are investigated.Analytical solutions including Poisson effect for deflections under point and uniform loadsand for first three natural frequencies are obtained by Navier solution. The results are com-pared with other beam theories and other classical and non-classical models. A detailedparametric study is carried out to show the influences of thickness-to-material length scaleparameter ratio, slenderness ratio and shear deformation on deflections and natural fre-quencies of microbeams. It is observed that effect of shear deformation becomes more sig-nificant for both smaller slenderness ratios and higher modes.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Motivated by developments in nanotechnology, microbeams, in which its thickness (or diameter) is generally on the or-der of microns and sub-microns, have been extensively used in many applications of micro- and nano-sized devices and sys-tems (e.g. atomic force microscopes (AFMs) (Kahrobaiyan, Rahaeifard, & Ahmadian, 2011), microsensors (Faris & Nayfeh,2007; Moser & Gijs, 2007), microactuators (Najar, Choura, El-Borgi, Abdel-Rahman, & Nayfeh, 2005), nano- and micro-electromechanical systems (NEMS and MEMS) (Li, Bhushan, Takashima, Baek, & Kim, 2003)). However, it has been experimentallyobserved for several materials that microstructural effects appear and have considerable effect on mechanical properties anddeformation behavior for smaller sizes (Lam, Yang, Chong, Wang, & Tong, 2003; McFarland & Colton, 2005; Poole, Ashby, &Fleck, 1996). The well-known classical continuum theories, that are independent of scale of the structure’s size, may becomeinadequate for estimation and explanation of size dependency in micro- and nano-scale structures. Therefore, several high-er-order continuum theories such as couple stress theory (Koiter, 1964; Mindlin & Tiersten, 1962; Toupin, 1964), micropolartheory (Eringen, 1967), nonlocal elasticity theory (Eringen, 1972, 1983) and strain gradient theories (Aifantis, 1999; Fleck &Hutchinson, 1993; Vardoulakis & Sulem, 1995), which contain at least one additional material length scale parameter, havebeen developed to predict the microstructural size dependency of these small-scale structures.

One of the higher-order continuum theories, named as strain gradient theory, developed by Fleck and Hutchinson (1993,2001), can be viewed as extended form of the Mindlin’s simplified theory (Mindlin, 1965). This theory requires five

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additional material length scale parameters related to second-order deformation gradients. Subsequently, Lam et al. (2003)proposed a more useful form of the strain gradient theory which is named as modified strain gradient theory (MSGT) andincludes three additional material length scale parameters for linear elastic isotropic materials. Recently, this modified the-ory has been employed by many researchers in order to analyze size-dependent structures. For instance, Kong, Zhou, Nie,and Wang (2009) and Wang, Zhao, and Zhou (2010) investigated static bending and free vibration behaviors of Bernoulli–Euler and Timoshenko homogeneous microbeams, respectively. Stability and bending responses of microbeams with variousboundary conditions was also investigated by present authors on the basis of Bernoulli–Euler beam model (Akgöz & Civalek,2011, 2012). A size-dependent Kirchhoff microplate model was presented by Wang, Zhou, Zhao, and Chen (2011) and Movas-sagh and Mahmoodi (2013). Static torsion and torsional vibration analysis of clamped–clamped and clamped-free microbarswere carried out by Kahrobaiyan, Tajalli, Movahhedy, Akbari, and Ahmadian (2011), and also longitudinal free vibrationproblem of microbars was analytically solved for clamped–clamped and clamped-free boundary conditions by presentauthors (Akgöz & Civalek, 2013a). Furthermore, Kahrobaiyan, Rahaeifard, Tajalli, and Ahmadian (2012) and Ansari, Gholami,and Sahmani (2011) introduced Bernoulli–Euler and Timoshenko beam models for functionally graded microbeams,respectively.

Moreover, size-dependent models for other type structures like microplates and microcylinders made of functionallygraded materials were presented by Sahmani and Ansari (2013), Sadeghi, Baghani, and Naghdabadi (2012), respectively.In addition to these, there are some papers on nonlinear analysis of microstructures based on MSGT in the scientific litera-ture (Asghari, Kahrobaiyan, Nikfar, & Ahmadian, 2012; Ghayesh, Amabili, & Farokhi, 2013; Kahrobaiyan, Asghari, Rahaeifard,& Ahmadian, 2011; Zhao, Zhou, Wang, & Wang, 2012). Also, other non-classical theories have been widely applied to inves-tigate the mechanical behavior of small-sized structures such as nonlocal elasticity theory (Amara, Tounsi, Mechab, & Adda-Bedia, 2010; Demir, Civalek, & Akgöz, 2010; Peddieson, Buchanan, & McNitt, 2003; Reddy, 2010; Reddy & Pang, 2008; Roque,Ferreira, & Reddy, 2011; S�ims�ek & Yurtcu, 2013) and modified couple stress theory (MCST) (Akgöz & Civalek, 2013b; Asghari,Kahrobaiyan, & Ahmadian, 2010; Farokhi, Ghayesh, & Amabili, 2013; Kahrobaiyan, Asghari, Rahaeifard, & Ahmadian, 2010;Ke & Wang, 2011; Kong, Zhou, Nie, & Wang, 2008; Lee & Chang, 2011; Ma, Gao, & Reddy, 2008; Park & Gao, 2006; Reddy,2011; Roque, Fidalgo, Ferreira, & Reddy, 2013; S�ims�ek, 2010).

Presently, various beam theories have been proposed and used to investigate the mechanical behaviors of beams. Influ-ences of shear deformation can be neglected for slender beams with a large aspect ratio. However, effects of shear deforma-tion and rotary inertia become more prominent and cannot be ignored for moderately thick beams and vibration responseson higher modes. In this manner, several shear deformation beam theories have been developed to account for the effects oftransverse shear. One of the earlier shear deformation beam theories is the first-order shear deformation beam theory (com-monly named as Timoshenko beam theory (TBT)) (Timoshenko, 1921). This theory assumes that shear stress and strain areconstant along the height of the beam. In fact, the distributions of these are not uniform, and also there are no transverseshear stress and strain at the top and bottom surfaces of the beam. For this reason, a shear correction factor is needed, asa disadvantage of the theory. After that, some higher-order shear deformation beam theories, which satisfy the conditionof no shear stress and strain without any shear correction factors, have been presented such as parabolic (third-order) beamtheory (Levinson, 1981; Reddy, 1984), trigonometric (sinusoidal) beam theory (Touratier, 1991), hyperbolic beam theory(Soldatos, 1992), exponential beam theory (Karama, Afaq, & Mistou, 2003) and a general exponential beam theory (Aydogdu,2009a). These theories have been used less than Euler–Bernoulli beam theory (EBT) and TBT on prediction of the mechanicalresponses of microstructures on the basis of the non-classical continuum theories (Aydogdu, 2009b; Nateghi, Salamat-talab,Rezapour, & Daneshian, 2012; Reddy, 2007; Salamat-talab, Nateghi, & Torabi, 2012; S�ims�ek & Reddy, 2013a, 2013b; Thai,2012; Thai & Kim, 2013; Thai & Vo, 2012, 2013; Tounsi, Semmah, & Bousahla, 2013).

In the present study, a new size-dependent trigonometric (sinusoidal) shear deformation beam model in conjunctionwith MSGT is developed. This model captures both the microstructural and shear deformation effects without the needfor any shear correction factors. The governing equations and boundary conditions are derived by using Hamilton’s principle.The static bending and free vibration behavior of simply supported microbeams are investigated. Analytical solutions fordeflections under point and uniform loads and for first three natural frequencies are presented. The results are comparedwith other beam theories as EBT and TBT and other classical and non-classical models as CT and MCST. A detailed parametricstudy is carried out to show the influences of thickness-to-material length scale parameter ratio, slenderness ratio and sheardeformation on deflections and natural frequencies of microbeams.

2. Theory and formulation

The modified strain gradient elasticity theory was proposed by Lam et al. (2003) in which contains a new additional equi-librium equation besides the classical equilibrium equations and also three additional material length scale parameters be-sides two classical ones for linear elastic materials. The strain energy U on the basis of the modified strain gradient elasticitytheory can be written by Lam et al. (2003)

U ¼ 12

Z L

0

ZA

rijeij þ pici þ sð1Þijk gð1Þijk þmsijv

sij

� �dA dx ð1Þ

B. Akgöz, Ö. Civalek / International Journal of Engineering Science 70 (2013) 1–14 3

eij ¼12ðui;j þ uj;iÞ ð2Þ

emm;i ¼ ci ð3Þ

gð1Þijk ¼13ðejk;i þ eki;j þ eij;kÞ �

115

dijðemm;k þ 2emk;mÞ þ djkðemm;i þ 2emi;mÞ þ dkiðemm;j þ 2emj;mÞ� �

ð4Þ

vsij ¼

12ðhi;j þ hj;iÞ ð5Þ

hi ¼12

eijkuk;j ð6Þ

where ui, hi, eij, ci, gð1Þijk and vs

ij denote the components of the displacement vector u, the rotation vector h, the strain tensor e,the dilatation gradient vector c, the deviatoric stretch gradient tensor g(1) and the symmetric rotation gradient tensor vs,respectively. Also, d is the symbol of Kronecker delta and eijk is the permutation symbol.

Furthermore, the components of the classical stress tensor r and the higher-order stress tensors p, s(1) and ms defined as(Lam et al., 2003)

rij ¼ kemmdij þ 2leij ð7Þ

pi ¼ 2ll20ci ð8Þ

sð1Þijk ¼ 2ll21gð1Þijk ð9Þ

msij ¼ 2ll2

2vsij ð10Þ

where l0, l1, l2 are additional material length scale parameters related to dilatation gradients, deviatoric stretch gradients androtation gradients, respectively. Furthermore, k and l are the Lamé constants defined as

k ¼ Evð1þ vÞð1� 2vÞ ; l ¼ E

2ð1þ vÞ ð11Þ

The displacement components of an initially straight beam on the basis of sinusoidal beam theory (see Fig. 1) can be writ-ten as (Touratier, 1991)

u1ðx; z; tÞ ¼ uðx; tÞ � z@wðx; tÞ@x

þ RðzÞ/ðx; tÞ

u2ðx; z; tÞ ¼ 0u3ðx; z; tÞ ¼ wðx; tÞ

ð12Þ

in which

/ðx; tÞ ¼ @wðx; tÞ@x

�uðx; tÞ ð13Þ

where u1, u2 and u3 are the x-, y- and z-components of the displacement vector, and also u and w are the axial and transversedisplacements, u is the angle of rotation of the cross-sections about y-axis of any point on the mid-plane of the beam, respec-tively. R(z) is a function which depends on z and plays a role in determination of the transverse shear strain and stress dis-tribution throughout the height of the beam. In order to satisfy no shear stress and strain condition at the upper (z = �h/2)and lower (z = h/2) surfaces of the beam, R(z) is selected as following without need for any shear correction factors

RðzÞ ¼ hp sin

pzh

� �ð14Þ

L

x

z

h

b

z

y

Fig. 1. Geometry of a simply supported beam and cross-section.


It can be noted that the displacement components for EBT and TBT will be obtained by setting R(z) in Eq. (12) equal to (0)and (z), respectively. Use of Eqs. (12)–(14) into Eq. (2), we obtain the non-zero strain components as

e11 ¼@u@x� z

@2w@x2 þ R

@/@x

; e13 ¼12

S/ ð15Þ

where

SðzÞ ¼ cospzh

� �ð16Þ

and from Eqs. (15) and (3), the components of dilatation gradient vector c are expressed as

c1 ¼@2u@x2 � z

@3w@x3 þ R

@2/@x2

c2 ¼ 0

c3 ¼ �@2w@x2 þ S

@/@x

ð17Þ

By inserting Eq. (15) in Eq. (4), the non-zero components of deviatoric stretch gradient tensor g(1) can be obtained as

gð1Þ111 ¼15

2@2u@x2 � z

@3w@x3 þ R

@2/@x2

!þ p2

h2 R/

" #

gð1Þ113 ¼ gð1Þ131 ¼ gð1Þ311 ¼ �4

15@2w@x2 � 2S

@/@x

!

gð1Þ122 ¼ gð1Þ212 ¼ gð1Þ221 ¼ �15

@2u@x2 � z

@3w@x3 þ R

@2/@x2 �

p2

3h2 R/

!

gð1Þ133 ¼ gð1Þ313 ¼ gð1Þ331 ¼ �15

@2u@x2 � z

@3w@x3 þ R

@2/@x2 þ

43

p2

h2 R/

!

gð1Þ223 ¼ gð1Þ232 ¼ gð1Þ322 ¼1

15@2w@x2 � 2S

@/@x

!; gð1Þ333 ¼

15

@2w@x2 � 2S

@/@x

!

ð18Þ

Also, using of Eq. (12) in Eq. (6) gives

h1 ¼ 0; h2 ¼ �@w@xþ 1

2S/; h3 ¼ 0 ð19Þ

and the non-zero components of the symmetric part of the rotation gradient tensor vs can be achieved by using of Eq. (19)into Eq. (5) as

vs12 ¼ vs

12 ¼ �12

@2w@x2 �

12

S@/@x

!; vs

23 ¼ vs32 ¼ �

p2

4h2 R/ ð20Þ

Use of Eq. (15) in Eq. (7), the non-zero components of classical stress tensor r can be written as

r11 ¼ E0@u@x� z

@2w@x2 þ R

@/@x

!; r13 ¼ lS/

r22 ¼ r33 ¼Ev

ð1þ vÞð1� 2vÞ@u@x� z

@2w@x2 þ R

@/@x

! ð21Þ

where

E0 ¼ Eð1� vÞð1þ vÞð1� 2vÞ ð22Þ

From Eqs. (8) and (15), the non-zero components of higher-order stress tensor p are obtained as

p1 ¼ 2ll20@2u@x2 � z

@3w@x3 þ R

@2/@x2

!; p3 ¼ �2ll20

@2w@x2 � S

@/@x

!ð23Þ

By inserting Eq. (18) in Eq. (9), we obtain the non-zero components of higher-order stress tensor s(1) as


sð1Þ111 ¼25ll2

1 2@2u@x2 � z

@3w@x3 þ R

@2/@x2

!þ p2

h2 R/

" #

sð1Þ113 ¼ sð1Þ131 ¼ sð1Þ311 ¼ �8

15ll2

1@2w@x2 � 2S

@/@x

!

sð1Þ122 ¼ sð1Þ212 ¼ sð1Þ221 ¼ �25ll2

1@2u@x2 � z

@3w@x3 þ R

@2/@x2 �

p2

3h2 R/

!

sð1Þ133 ¼ sð1Þ313 ¼ sð1Þ331 ¼ �25ll2

1@2u@x2 � z

@3w@x3 þ R

@2/@x2 þ

43

p2

h2 R/

!

sð1Þ223 ¼ sð1Þ232 ¼ sð1Þ322 ¼2

15ll2

1@2w@x2 � 2S

@/@x

!; sð1Þ333 ¼

25ll2

1@2w@x2 � 2S

@/@x

!

ð24Þ

Similarly, the non-zero components of higher-order stress tensor ms are determined by using of Eq. (20) into Eq. (10)

ms12 ¼ ms

12 ¼ �ll22@2w@x2 �

12

S@/@x

!; ms

23 ¼ ms32 ¼ �

ll22

2p2

h2 R/ ð25Þ

Substituting Eqs. (15)–(25) into Eq. (1), the first variation of strain energy of microbeam is expressed as

dU ¼Z L

0

ZA

rijdeij þ pidci þ sð1Þijk dgð1Þijk þmsijdvs

ij

� �dAdx

¼Z L

0

ZA

r11de11 þ 2r13de13 þ p1dc1 þ p3dc3þsð1Þ111dg

ð1Þ111 þ 3sð1Þ113dg



ð1Þ133

�þ3sð1Þ223dg

ð1Þ223 þ sð1Þ333dg

ð1Þ333 þ 2ms

12dvs12 þ 2ms

23dvs23

�dAdx

¼Z L

0A �E0

@2u@x2 þ 2lk1

@4u@x4

!duþ k2/� k3

@2/@x2 þ k4

@3w@x3 þ

12p2 lIk1

@4/@x4 �

4p@5w@x5

! !d/

"

þ k5@4w@x4 � k4

@3/@x3 � 2lIk1

@6w@x6 �

24p3

@5/@x5

! !dw

#dxþ A E0

@u@x� 2lk1

@3u@x3

!duþ 2lAk1

@2u@x2 d

@u@x

� �"

þ ðk3 þ k6Þ@/@x� ðk4 þ k7Þ

@2w@x2 �

12p2 lIk1

@3/@x3 �

4p@4w@x4

! !d/þ k6/þ

12p2 lIk1

@2/@x2 �

4p@3w@x3

! !d@/@x

� �

þ k4@2/@x2 � k5

@3w@x3 þ 2lIk1

@5w@x5 �

24p3

@4/@x4

! !dwþ k5

@2w@x2 � k4

@/@x� 2lIk1

@4w@x4 �

24p3

@3/@x3

! !d@w@x

� �

þ �k7/þ 2lIk1@3w@x3 �

24p3

@2/@x2

! !d@2w@x2

!#L

0

ð26Þ

where L is length of the beam, A is the area of cross-section, I is the second moment of area and

k1 ¼ l20 þ

25

l21; k2 ¼ lA12þ p2

h2

415

l21 þ

18

l22

� �� ; k3 ¼

6p2 E0I þ lA l20 þ

23

l21 þ

18

l22

� �

k4 ¼1p

24p2 E0I þ lA 4l2

0 þ43

l21 þ l22

� �� ; k5 ¼ E0I þ lA 2l2

0 þ8

15l21 þ l2

2

� �

k6 ¼15lAl2

1; k7 ¼4

5plAl21

ð27Þ

The kinetic energy of the beam is given by

K ¼Z L

0

ZA

12q

@u1

@t

� �2

þ @u2

@t

� �2

þ @u3

@t

� �2" #

dAdx ð28Þ

where q is the mass density. From Eqs. (12) and (28), the first variation of the kinetic energy can be expressed as

dK ¼Z L

0m0

@u@t

@du@tþ @w@t

@dw@t

� þm2

@2w@x@t

@2dw@x@t

� 24p3

@/@t

@2dw@x@t

þ @2w@x@t

@d/@t

!þ 6

p2

@/@t

@d/@t

" #( )dx ð29Þ

where (m0,m2) are the mass inertias as


ðm0;m2Þ ¼ qZ

Að1; z2ÞdA ð30Þ

The first variation of the work done by external forces can be written as

dW ¼Z L

0ðf duþ qdwÞdx ð31Þ

where f(x, t), q(x, t) and are the axial and transverse distributed, respectively. After that, with the aid of the Hamilton’s prin-ciple as (Reddy, 2002)

0 ¼Z T

0ðdK � dU þ dWÞdt ð32Þ

and substituting Eqs. (26), (29) and (31) into Eq. (32), integrating by parts and setting the coefficients du, d/ and dw equal tozero, the governing equations of motion of the microbeam based on SBT can be obtained as

du : �m0@2u@t2 þ A E0

@2u@x2 � 2lk1

@4u@x4

!þ f ¼ 0 ð33Þ

d/ :24p3 m2

@3w@x@t2 �

6p2 m2

@2/

@t2 � k2/þ k3@2/@x2 � k4

@3w@x3 �

12p2 lIk1

@4/@x4 �

4p@5w@x5

!¼ 0 ð34Þ

dw : �m0@2w@t2 þm2

@4w@x2@t2 �

24p3

@3/

@x@t2

!� k5

@4w@x4 þ k4

@3/@x3 þ 2lIk1

@6w@x6 �

24p3

@5/@x5

!þ q ¼ 0 ð35Þ

and boundary conditions at x = 0 and x = L

either A E0@u@x� 2lk1

@3u@x3

!¼ 0 or u ¼ 0 ð36Þ

either 2lAk1@2u@x2 ¼ 0 or

@u@x¼ 0 ð37Þ

either ðk3 þ k6Þ@/@x� ðk4 þ k7Þ

@2w@x2 �

12p2 lIk1

@3/@x3 �

4p@4w@x4

!¼ 0 or / ¼ 0 ð38Þ

either k6/þ12p2 lIk1

@2/@x2 �

4p@3w@x3

!¼ 0 or

@/@x¼ 0 ð39Þ

either k5@3w@x3 � k4

@2/@x2 � 2lIk1

@5w@x5 �

24p3

@4/@x4

!þm2

24p3

@2/

@t2 �@3w@x@t2

!¼ 0 or w ¼ 0 ð40Þ

either k5@2w@x2 � k4

@/@x� 2lIk1

@4w@x4 �

24p3

@3/@x3

!¼ 0 or

@w@x¼ 0 ð41Þ

either � k7/þ 2lIk1@3w@x3 �

24p3

@2/@x2

!¼ 0 or

@2w@x2 ¼ 0 ð42Þ

3. Analytical solutions for bending and vibration problems of simply supported microbeams

Here, in order to solve static bending and free vibration problems of simply supported microbeams, Navier’s method isused. The boundary conditions of a simply supported beam for two ends can be specified as

@/@x¼ ðk3 þ k6Þ

@/@x� ðk4 þ k7Þ

@2w@x2 �

12p2 lIk1

@3/@x3 �

4p@4w@x4

!¼ 0 ð43Þ


w ¼ @2w@x2 ¼ k5

@2w@x2 � k4

@/@x� 2lIk1

@4w@x4 �

24p3

@3/@x3

!¼ 0 ð44Þ

The following expansions of generalized displacements which include undetermined Fourier coefficients and certain trig-onometric functions can be successfully employed as

wðx; tÞ ¼X1n¼1

Wn sinnpx

Leixnt ð45Þ

/ðx; tÞ ¼X1n¼1

Hn cosnpx

Leixnt ð46Þ

where Wn and Hn are the undetermined Fourier coefficients and xn is natural frequency. This means that Eqs. (45) and (46)must satisfy the corresponding boundary conditions.

3.1. Static bending

The external applied force q can be expanded by Fourier series with Fourier coefficient Qn as following

qðxÞ ¼X1n¼1

Q n sinnpx

Lð47Þ

Q n ¼2L

Z L

0qðxÞ sin

npxL

dx ð48Þ

and Qn can be expressed as in the cases of uniform and point loads respectively

Q n ¼4q0

npfor n ¼ 1;3;5; . . . ð49Þ

Q n ¼2PL

sinnp2

for n ¼ 1;2;3; . . . ð50Þ

where q0 is the value of the load per unit length for uniformly distributed load and P is the magnitude of the point load actingon the midspan of the microbeam.

In the absence of any axial loads and time derivatives in Eqs. (34) and (35), the governing equations for static bending aregiven by

k2/� k3@2/@x2 þ k4

@3w@x3 þ

12p2 lIk1

@4/@x4 �

4p@5w@x5

!¼ 0 ð51Þ

k5@4w@x4 � k4

@3/@x3 � 2lIk1

@6w@x6 �

24p3

@5u@x5

!¼ q ð52Þ

Substituting Eqs. (44)–(46), into Eqs. (50) and (51), the following relation is achieved as

k2 þ a2k3 þ a4 12p2 lIk1 �a3 k4 þ a2 48

p3 lIk1 �

�a3 k4 þ a2 48p3 lIk1

�a4 k5 þ a22lIk1 �

" #Hn

Wn

� ¼

0Q n

� ð53Þ

where a = np/L. Then, Hn and Wn can be obtained by solving above algebraic equations system as follows

Hn ¼a3 k4 þ a2 48

p3 lIk1 �

k2 þ a2k3 þ a4 12p2 lIk1

�ða4ðk5 þ a22lIk1ÞÞ � a3ðk4 þ a2 48

p3 lIk1Þ �2 Q n ð54Þ

Wn ¼k2 þ a2k3 þ a4 12

p2 lIk1 �

k2 þ a2k3 þ a4 12p2 lIk1

�ða4ðk5 þ a22lIk1ÞÞ � a3 k4 þ a2 48

p3 lIk1 � �2 Q n ð55Þ

3.2. Free vibration

Substituting Eqs. (45) and (46) into Eqs. (35) and (36) as the governing equations for free vibration without all externalapplied forces, the following equation is obtained as

Table 1Dimens

h/l

1

5

10

Table 2Dimens

h/l

1

5

10


k2 þ a2k3 þ a4 12p2 lIk1 � 6

p2 m2x2n �a3 k4 þ a2 48

p3 lIk1 �

þ a 24p3 m2x2

n

�a3 k4 þ a2 48p3 lIk1

�þ a 24

p3 m2x2n a4 k5 þ a22lIk1

��x2

nðm0 þ a2m2Þ

" #Hn

Wn

� ¼

00

� ð56Þ

For a non-trivial solution, the determinant of coefficient matrix must be vanished and the characteristic equation can bereached by providing this condition. The eigenvalues are obtained by solving the characteristic equation. It can be noted thatthe smallest root of the characteristic equation gives the first natural (fundamental) frequency.

4. Numerical results and discussion

Bending and free vibration problems of a simply supported microbeam are analytically solved with Navier solution basedon SBT and MSGT. For illustration purpose, the microbeam is taken to be made of epoxy with the following material prop-erties: the Young’s modulus = 1.44 GPa, the Poisson’s ratio v = 0.38, the mass density q = 1220 kg/m3 and the material lengthscale parameter l = 17.6 lm (Ma et al., 2008; Park & Gao, 2006; Wang et al., 2010). Also, the shear correction factor is taken asks = 5/6 for TBT. The microbeam has a rectangular cross-section and the width-to-thickness ratio is taken to be constant as b/h = 2 while the length-to-thickness ratio is taken several values as L/h = 5–100. All material length scale parameters are con-sidered to be equal to each other as l0 = l1 = l2 = l. If the first two or all material length scale parameters equal to zero, themodified strain gradient model will be turned into modified couple stress model or classical model, respectively. The newresults are presented and compared with EBT and TBT corresponding to modified strain gradient model (Akgöz & Civalek,2012; Kong et al., 2009; Wang et al., 2010), modified couple stress model (Kong et al., 2008; Ma et al., 2008; Park & Gao,2006) and classical model.

Comparison of dimensionless maximum deflections under uniform and point loads with different thickness-to-materiallength scale parameters ratio and length-to-thickness ratio corresponding to different beam theories and models is given inTables 1 and 2, respectively. It is clearly shown that the dimensionless maximum deflections obtained by both CT and TBTare bigger than the other ones while those obtained by both MSGT and EBT are smaller than the other ones and an increase inh/l gives rise to a decrease in the difference between dimensionless maximum deflections related to classical and non-clas-sical models. On the other hand, difference between the results corresponding to EBT, TBT and SBT is more prominent forsmaller slenderness ratios (as L = 10h) but they are diminishing for larger slenderness ratios (asL = 100h). Moreover, it canbe concluded that shear deformation becomes more important for stubby beams.

Non-dimensional first three natural frequencies with various values of h/l and slenderness ratios corresponding to differ-ent beam theories and models are listed in Tables 3–5, respectively. It can be clearly observed from the tables that the non-

ionless maximum deflections ð�w ¼ 1000wEI=qL4Þ under uniform load (q0 = 10 lN/lm).

Theory L = 10h L = 30h L = 100h

CT MCST MSGT CT MCST MSGT CT MCST MSGT

EBT 6.9556 2.0934 0.7513 6.9556 2.0934 0.7550 6.9556 2.0934 0.7555TBT 7.3006 2.2384 1.0760 6.9940 2.1097 0.7937 6.9591 2.0949 0.7590SBT 7.2999 2.1277 0.7742 6.9939 2.0973 0.7576 6.9591 2.0938 0.7557

EBT 6.9556 6.3644 5.2285 6.9556 6.3644 5.2358 6.9556 6.3644 5.2366TBT 7.3006 6.6804 5.5740 6.9940 6.3995 5.2743 6.9591 6.3675 5.2401SBT 7.2999 6.6468 5.4576 6.9939 6.3958 5.2613 6.9591 6.3672 5.2389

EBT 6.9556 6.7978 6.4250 6.9556 6.7978 6.4278 6.9556 6.7978 6.4281TBT 7.3006 7.1349 6.7703 6.9940 6.8352 6.4662 6.9591 6.8011 6.4315SBT 7.2999 7.1247 6.7317 6.9939 6.8341 6.4619 6.9591 6.8010 6.4312

ionless maximum deflections ð�w ¼ 1000wEI=PL3Þ under point load (P = 100 lN).

Theory L = 10h L = 30h L = 100h


EBT 11.1290 3.3495 1.2007 11.1290 3.3495 1.2079 11.1290 3.3495 1.2087TBT 11.8176 3.6239 1.7662 11.2055 3.3813 1.2808 11.1359 3.3524 1.2156SBT 11.8060 3.4152 1.2438 11.2051 3.3570 1.2129 11.1359 3.3502 1.2092

EBT 11.1290 10.1830 8.3627 11.1290 10.1830 8.3769 11.1290 10.1830 8.3785TBT 11.8176 10.8023 9.0290 11.2055 10.2528 8.4529 11.1359 10.1893 8.3854SBT 11.8060 10.7319 8.8020 11.2051 10.2452 8.4272 11.1359 10.1886 8.3831

EBT 11.1290 10.8764 10.2789 11.1290 10.8764 10.2843 11.1290 10.8764 10.2849TBT 11.8176 11.5435 10.9565 11.2055 10.9511 10.3606 11.1359 10.8831 10.2918SBT 11.8060 11.5167 10.8749 11.2051 10.9486 10.3519 11.1359 10.8829 10.2911

Table 3Non-dimensional fundamental frequencies ð �x1 ¼ x1L2 ffiffiffiffiffiffiffiffiffiffiffiffiffi

m0=EIp

Þ.

h/l Theory L = 10h L = 30h L = 100h


1 EBT 13.4484 24.5137 40.9238 13.4975 24.6031 40.9674 13.5031 24.6133 40.9724TBT 13.1232 23.7053 34.1822 13.4595 24.5061 39.9336 13.4996 24.6045 40.8750SBT 13.1239 24.3157 40.3106 13.4595 24.5801 40.8962 13.4996 24.6112 40.9659

5 EBT 13.4484 14.0593 15.5118 13.4975 14.1105 15.5572 13.5031 14.1164 15.5624TBT 13.1232 13.7192 15.0180 13.4595 14.0707 15.4989 13.4996 14.1128 15.5571SBT 13.1239 13.7542 15.1796 13.4595 14.0749 15.5184 13.4996 14.1132 15.5589

10 EBT 13.4484 13.6037 13.9928 13.4975 13.6533 14.0408 13.5031 13.6590 14.0462TBT 13.1232 13.2748 13.6274 13.4595 13.6149 13.9979 13.4996 13.6555 14.0424SBT 13.1239 13.2844 13.6670 13.4595 13.6160 14.0027 13.4996 13.6556 14.0428

Table 4Non-dimensional natural frequencies ð �x2 ¼ x2L2 ffiffiffiffiffiffiffiffiffiffiffiffiffi

m0=EIp

Þ (second mode).



1 EBT 53.1473 96.8767 163.1234 53.9161 98.2779 163.8039 54.0056 98.4412 163.8836TBT 48.6751 86.4924 106.1615 53.3195 96.7708 149.3488 53.9507 98.3005 162.3472SBT 48.6922 94.1134 154.5577 53.3206 97.9163 162.6833 53.9507 98.4080 163.7807

5 EBT 53.1473 55.5613 61.4494 53.9161 56.3649 62.1605 54.0056 56.4586 62.2435TBT 48.6751 50.9056 54.8583 53.3195 55.7406 61.2486 53.9507 56.4010 62.1591SBT 48.6922 51.3623 56.8752 53.3206 55.8057 61.5514 53.9507 56.4071 62.1874

10 EBT 53.1473 53.7610 55.3397 53.9161 54.5386 56.0909 54.0056 54.6292 56.1785TBT 48.6751 49.2437 50.3506 53.3195 53.9352 55.4195 53.9507 54.5736 56.1165SBT 48.6922 49.3752 50.8635 53.3206 53.9529 55.4932 53.9507 54.5752 56.1234

Table 5Non-dimensional natural frequencies ð �x3 ¼ x3L2 ffiffiffiffiffiffiffiffiffiffiffiffiffi

m0=EIp

Þ (third mode).



1 EBT 117.2698 213.7589 365.0057 121.0359 220.6236 368.3141 121.4877 221.4471 368.7159TBT 98.8865 174.0724 202.6362 118.1086 213.3473 307.6397 121.2103 220.7383 361.1098SBT 98.9997 202.3317 329.1099 118.1148 218.8416 362.7956 121.2107 221.2794 368.1960

5 EBT 117.2698 122.5963 136.1297 121.0359 126.5334 139.6063 121.4877 127.0057 140.0248TBT 98.8865 103.5636 109.8844 118.1086 123.4725 135.1623 121.2103 126.7151 139.5987SBT 98.9997 105.3367 117.3126 118.1148 123.7882 136.6162 121.2107 126.7457 139.7416

10 EBT 117.2698 118.6239 122.2579 121.0359 122.4334 125.9355 121.4877 122.8904 126.3771TBT 98.8865 100.0818 101.9104 118.1086 119.4733 122.6462 121.2103 122.6098 126.0645SBT 98.9997 100.6267 103.8873 118.1148 119.5600 123.0029 121.2107 122.6180 126.0991


dimensional natural frequencies predicted by both CT and TBT are lower than the other ones while those obtained by bothMSGT and EBT are larger than the other ones. In other words, it can be said that the opposite situation in maximum deflec-tions is valid for natural frequencies. Also, an increase in h/l leads to a decrease in the difference between non-dimensionalnatural frequencies corresponding to classical and non-classical models and also this difference becomes more prominent forhigher modes. On the other hand, difference between the results corresponding to EBT, TBT and SBT is more significant forshort beams. This situation can be interpreted as the effect of shear deformation is minor for slender beams with a large slen-derness ratio.

In the Figs. 2 and 3, deflections of simply supported microbeam subjected to point and uniform loads are compared withclassical and non-classical models based on different beam theories, respectively. It is clearly shown from Figs. 2 and 3 thatdeflections evaluated by MSGT are always smaller than those obtained by CT and MCST and also deflections based on EBT arealways smaller than those based on the other beam theories. Moreover, it can be revealed that influence of shear deforma-tion is more prominent for short beams as = 5h.

Figs. 4 and 5 depict the effects of thickness-to-length scale parameter ratio h/l on variation of maximum dimensionlessdeflection of microbeam under point and uniform loads with classical and non-classical models based on different beam the-ories. It is evident that an increase in h/l leads to increase of maximum dimensionless deflections predicted by non-classical

(a) (b)

0 1 2 3 4 50

0.002

0.004

0.006

0.008

0.01

0.012

0.014

length / thickness

defle

ctio

n / t

hick

ness

EBT-CTEBT-MCSTEBT-MSGTTBT-CTTBT-MCSTTBT-MSGTSBT-CTSBT-MCSTSBT-MSGT

0 5 10 15 200

0.5

1

1.5

2

2.5

3

length / thickness

defle

ctio

n / t

hick

ness


Fig. 3. Deflection of simply supported microbeam subjected to uniform load with classical and non-classical models based on different beam theories (a)L = 5h (b) L = 20h.

(a) (b)

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

h/l

dim

ensi

onle

ss d

efle

ctio

n


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

h/l

dim

ensi

onle

ss d

efle

ctio

n


Fig. 4. Effect of thickness-to-length scale parameter ratio h/l on variation of maximum dimensionless deflection of microbeam under point load withclassical and non-classical models based on different beam theories (a) L = 10h (b) L = 60h.

(a) (b)

0 1 2 3 4 50

0.5

1

1.5

2

2.5x 10-3

length / thickness

defle

ctio

n / t

hick

ness


0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

length / thickness

defle

ctio

n / t

hick

ness


Fig. 2. Deflection of simply supported microbeam subjected to point load with classical and non-classical models based on different beam theories (a)L = 5h (b) L = 20h.


models while those obtained by classical model are stationary. From the figures, it can also be inferred that the values ofmaximum deflection predicted by classical and non-classical models become very close to each other for bigger values ofh/l as h/l > 10.

(a) (b)

1 2 3 4 5 6 7 8 9 1010

15

20

25

30

35

40

45

h/l

ϖ1


1 2 3 4 5 6 7 8 9 1010

15

20

25

30

35

40

45

h/l

ϖ1


Fig. 6. Effect of thickness-to-length scale parameter ratio h/l on variation of first dimensionless frequency (-1) of microbeam with classical and non-classical models based on different beam theories (a) L = 10h (b) L = 60h.

(a) (b)

1 2 3 4 5 6 7 8 9 1040

60

80

100

120

140

160

180

h/l

ϖ2


1 2 3 4 5 6 7 8 9 1040

60

80

100

120

140

160

180

h/l

ϖ2


Fig. 7. Effect of thickness-to-length scale parameter ratio h/l on variation of second dimensionless frequency (-2) of microbeam with classical and non-classical models based on different beam theories (a) L = 10h (b) L = 60h.

(a) (b)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

h/l

dim

ensi

onle

ss d

efle

ctio

n


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

h/l

dim

ensi

onle

ss d

efle

ctio

n


Fig. 5. Effect of thickness-to-length scale parameter ratio h/l on variation of maximum dimensionless deflection of microbeam under uniform load withclassical and non-classical models based on different beam theories (a) L = 10h (b) L = 60h.


Variation of first three dimensionless natural frequencies of microbeam with respect to thickness-to-length scale param-eter ratio h/l with classical and non-classical models based on different beam theories are illustrated in Figs. 6–8. These fig-

(a) (b)

1 2 3 4 5 6 7 8 9 10100

150

200

250

300

350

400

h/l

ϖ3


1 2 3 4 5 6 7 8 9 10100

150

200

250

300

350

400

h/l

ϖ3


Fig. 8. Effect of thickness-to-length scale parameter ratio h/l on variation of third dimensionless frequency (-3) of microbeam with classical and non-classical models based on different beam theories (a) L = 10h (b) L = 60h.


ures reveal that natural frequencies based on MSGT are always bigger than CT and MCST. Also, effects of shear deformationare more considerable for smaller values of L/h. In addition, similar trends are observed with Figs. 4 and 5 for the effect ofthickness-to-length scale parameter on natural frequencies.

5. Conclusions

In this study, a size-dependent sinusoidal shear deformation beam model in conjunction with modified strain gradientelasticity theory (MSGT) is developed. The model captures both the microstructural effects and shear deformation effectswithout any shear correction factors. The governing equations and boundary conditions are derived by using Hamilton’sprinciple. The static bending and free vibration behavior of simply supported microbeams are investigated. Analytical solu-tions for deflections under point and uniform loads and for first three natural frequencies are presented. The results are com-pared with other beam theories as EBT and TBT and other classical and non-classical models as CT and MCST. A detailedparametric study is carried out to show the influences of thickness-to-material length scale parameter ratio, slenderness ra-tio and shear deformation on deflections and natural frequencies of microbeams. The obtained results can be summarized as:

� Microbeams based on the non-classical theories, especially MSGT, are stiffer than based on the classical theory.� In static bending case, deflections predicted by both MSGT and EBT are always smaller than those predicted by the other

considered beam models and theories.� In free vibration case, natural frequencies obtained by both MSGT and EBT are always greater than those predicted by the

other considered beam models and theories.� The difference between the results about both deflections and natural frequencies decreases as the ratio of thickness to

material length scale parameter increases.� Effect of shear deformation becomes more considerable for both smaller slenderness ratios and higher modes.� If there is only one material length scale parameter (l0 = l1 = 0, l2 = l), the current non-classical model based on MSGT will

be another non-classical model based on MCST. Also, if all material length scale parameters equal to zero (l0 = l1 = l2 = 0),the present non-classical model will turn into the classical model.

Acknowledgments

This study has been supported by The Scientific and Technological Research Council of Turkey (TÜB_ITAK) with Project No:112M879. This support is gratefully acknowledged.

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A size-dependent shear deformation beam model based on the strain gradient elasticity theory