Vibration analysis of pre-twisted functionally graded carbon nanotube reinforced composite beams in thermal environment

Accepted Manuscript Vibration analysis of pre-twisted functionally graded carbon nanotube reinforced composite beams in thermal environment Amin Ghorbani Shenas, Parviz Malekzadeh, Sima Ziaee PII: DOI: Reference:

S0263-8223(16)32490-4 http://dx.doi.org/10.1016/j.compstruct.2016.12.009 COST 8064

To appear in:

Composite Structures

Received Date: Accepted Date:

10 November 2016 5 December 2016

Please cite this article as: Ghorbani Shenas, A., Malekzadeh, P., Ziaee, S., Vibration analysis of pre-twisted functionally graded carbon nanotube reinforced composite beams in thermal environment, Composite Structures (2016), doi: http://dx.doi.org/10.1016/j.compstruct.2016.12.009

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Vibration analysis of pre-twisted functionally graded carbon nanotube reinforced composite beams in thermal environment Amin Ghorbani Shenas a, Parviz Malekzadeh b, 1, Sima Ziaee a a

b

Department of Mechanical Engineering, Yasouj University, Yasouj 75914-353, Iran Department of Mechanical Engineering, Persian Gulf University, Bushehr 7516913798, Iran

Abstract As a first endeavor, the free vibration behavior of the pre-twisted functionally graded carbon nanotube reinforced composite (FG-CNTRC) beams in thermal environment is studied. The governing equations are derived based on the higher-order shear deformation theory of beams by considering the temperature dependence of material properties and the initial thermal stresses. The free vibration eigenvalue equations are extracted by using the Chebyshev–Ritz method. In this regard, Chebyshev polynomials together with appropriate boundary functions are utilized as admissible functions of the Ritz method, which enables one to handle the problem with different sets of boundary conditions. The fast rate of convergence of the method is demonstrated numerically and its accuracy is verified by comparing the results in the limit cases with existing solutions in the literature. The effects of pre-twist angle together with carbon nanotubes (CNTs) distribution in thickness direction, the temperature dependence of material properties, the temperature rise, the geometrical shape parameters and boundary conditions on the frequency parameters are investigated. It is shown that the effects of the pre-twist angle on the natural frequencies depend on the beam boundary conditions and also the mode number.

Keywords: Vibration; pre-twisted beams; functionally graded; carbon nanotube reinforced composite; thermal environment; Chebyshev–Ritz method 1

Corresponding author. Tel.: +98 77 31222150; fax: +98 77 33440376. E-mail addresses: [email protected], [email protected] (Parviz Malekzadeh). 1

1. Introduction The pre-twisted beams as one of important structural elements have been extensively used in different branches of modern industrials such as aerospace, automobile, marine and nuclear technologies. The stiffness and strength to weight ratios are the main features that restrict the use of conventional materials in producing these types of structural elements. In the past years, usually the conventional composite materials composed of a matrix reinforced with micro-sized fibers such as glass, Kevlar and carbon fibers have been used to generate the pre-twisted beams for use in high performance devices. However, the research to improve the behavior of composite materials has been continued. On the other hand, it has been shown that carbon nanotubes (CNTs) have superior mechanical, electrical and thermal properties over the aforementioned conventional micro-sized fibers [1,2]. This motivates the scientist to use the CNTs as a new reinforcement material to produce high performance polymer composites in recent years [1,3,4]. Due to limitation of CNTs volume fraction amount for production of nanocomposites [5], it has been suggested to use their engineered gradients in the preferred direction to generate advanced composite materials, namely, the functionally graded CNTs reinforced composites (FG-CNTRCs) [6]. It is well known that if the structural elements made of FG-CNTRCs operate in thermal environments, their mechanical properties will degenerate [7-14]. In addition, their vibration characteristics will be affected by the induced thermal stresses [9,12,14]. Hence, in studying their vibrational behavior, the influences of the thermal environment can not be ignored. Since accurate design and manufacture of structural elements require knowledge of their vibrational behavior, the free and forced vibrations of FG-CNTRC beams, plates and shells of different shapes have been studied by some researchers in the last years; for example see Refs.

2

[4,9,10,14-33]. However, to the best of authors' knowledge, there is no research work on the vibration analysis of the pre-twisted FG-CNTRC beams in the open literature yet. In the following, some of the recently published works, the subjects of which are relevant to that of the present work, are reviewed. Jam and Kiani [20] analyzed the response of the FG-CNTRC beams under the action of an impacting mass based on the FSDT of beams. The governing equations of motion were solved by employing the conventional polynomial Ritz method and Runge–Kutta method. Chaudhari and Lal [21] investigated the nonlinear free vibration behavior of elastically supported FG-CNTRC beams subjected to thermal loading by using the finite element method based on a higher order shear deformation theory. They modeled the elastic foundation as a three-parameter nonlinear one composed of two parameters Pasternak foundation and Winkler cubic nonlinearity. Rafiee et al. [22] developed a computational model to study the nonlinear steady state response and free vibration of thin-walled CNTRC beams and blades. They established a set of nonlinear intrinsic equations which describe the response of rotating cantilever CNTRC beams undergoing large deformations. The CNTs were assumed to be uniformly distributed and randomly oriented through the epoxy resin matrix. The natural frequencies and nonlinear steady state response of the CNTRC beams and blades were calculated by discretizing the governing equations using the Galerkin method. Wu et al. [23] presented the nonlinear vibration of FG-CNTRC beams with initial imperfection based on the first-order shear deformation beam theory in conjunction with von Kármán geometric nonlinearity assumptions. A one-dimensional imperfection model in the form of the product of trigonometric and hyperbolic functions were used to describe the various possible geometric imperfections. The nonlinear eigenfrequency equations were derived by employing the Ritz method and then solved by an iteration procedure. Nejati et al. [24] studied

3

the buckling and vibration behavior of the FG-CNTRC cantilever beams. The stability and motion equations were derived based on the two-dimensional elasticity theory using Hamilton’s principle and were solved by employing the GDQM. Wattanasakulpong and Mao [25] investigated the stability and vibration characteristics of CNTRC beams with classical and nonclassical boundary conditions. They formulated the problem by using the FSDT and solved the obtained equations via Chebyshev collocation method. Mohammadimehr and Shahedi [26] studied the buckling and free vibration behaviors of two types sandwich beams including AL or PVC-foam flexible core and CNTRC face sheets. The governing equations were derived by employing the FSDT for face sheets and a third-order shear deformation theory for the core. They utilized the generalized differential quadrature method (GDQM) to solve the resulting equations of motion. In the present work, for the first time, the free vibration behavior of the pre-twisted FGCNTRC beams in thermal environment is studied. In addition to the temperature dependence of the material properties, which are assumed to be graded in the beams thickness direction, the initial thermal stresses are considered. In order to include the transverse shear deformations and rotary inertia effects accurately and efficiently, and in addition to avoid the use of shear correction factor, the governing equations are derived based on Reddy's third-order shear deformation theory. The system of algebraic eigenfrequency equations is derived by using the Hamilton’s principle in conjunction with the Chebyshev–Ritz method as a computationally efficient and accurate method for solving complicated structural problems [34-37]. After showing the fast rate of convergence and accuracy of the method, the influences of the linear and nonlinear variation of the twist angle along the beam axis, temperature dependence of material properties, temperature changes, material gradient index, and beam thickness-to-length ratio on

4

the non-dimensional frequencies of the pre-twisted FG-CNTRC beams under different boundary conditions are investigated.

2. Mathematical modeling The geometry and global coordinate system X-Y-Z of the pre-twisted FG-CNTRC beams under consideration are shown in Fig. 1. Also, the local coordinate system x-y-z (in the lower cases) which its x-axis is coincident with the global X-axis is shown in Fig. 2. As obvious from Fig.2, both the global X-axis and local x-axis pass through the centroid of beam cross-section and perpendicular to it. Hence, these axes represent the axis of twist of the beam cross section (see Fig. 2). As it can be seen from Fig. 1, the pre-twisted FG-CNTRC beam has a length L , constant thickness h, width b, and the angle of twist along the x-axis φ (see Fig. 2). The beam is made of a matrix reinforced with straight single walled carbon nanotubes (SWCNTs) which can be distributed uniformly or non-uniformly in the beam thickness direction (i.e., along the z-axis, see Fig.3). The matrix is assumed to be homogeneous and isotropic, and has linear elastic behavior. In the following subsections, the basic relations together with equations and the solution method are presented.

2.1. The effective material properties of FG-CNTRCs beams In this work, without loss of generality of the formulation and method of solution, the extended rule of mixture as a simple and convenient micromechanics model is adopted to estimate the longitudinal and transverse Young’s modulus, and also, shear modulus of the pretwisted FG-CNTRC beams as [4,6-14] E11 = η1 VCNT ( z ) E11CNT + Vm ( z ) E m ,

η2 E22

=

VCNT ( z ) Vm ( z ) η3 VCNT ( z ) Vm ( z ) + m , = CNT + m CNT E22 E G12 G12 G

5

(1a-c)

CNT where E11CNT and E22 are the longitudinal and transverse Young’s modulus of CNTs, respectively,

and G12CNT is the shear modulus of CNTs; E m and G m are the corresponding properties of matrix;

ηi (i = 1, 2, 3) are the efficiency parameter of the CNTs, which are evaluated by comparing the effective properties of CNTRC obtained by using the molecular dynamic simulation [38] with the counterparts computed by the rule of mixture. Also, VCNT and Vm are the volume fraction of the CNTs and matrix, respectively. The thermal expansion coefficients of fibrous composite materials can be determined using the mechanics of material approach with acceptable accuracy [39]. This approach has been employed by some researchers [9, 12-14] to evaluate the thermal expansion coefficients of CNTRCs. In this study, the same approach is employed to evaluate it [9, 12-14]

α11 =

VCNT ( z ) E11CNT α11CNT + Vm ( z ) E m α m VCNT ( z ) E11CNT + Vm ( z ) E m

(2)

where α11CNT is the thermal expansion coefficients of CNTs in their principal material direction and α m is the matrix thermal expansion coefficient. In order to study the effects of the CNTs distribution on the vibrational behavior of the pretwisted FG-CNTRCs beams, five different types of the CNTs distribution in the thickness direction are considered, which include the uniform distribution (UD), FG-V shape, FG-Λ shape, FG-O shape and FG-X shape distributions (see Fig. 3). The corresponding CNTs volume fractions are, respectively [13, 14] * UD: VCNT ( z ) = VCNT

(3)

 2z  * FG-V: VCNT ( z ) = 1 + VCNT h  

(4)

6

 2z  * FG- Λ : VCNT ( z ) = 1 − VCNT h  

(5)

 2z FG-O: VCNT ( z ) = 21 − h 

(6)

 * VCNT  

z * FG-X: VCNT ( z ) = 4 VCNT h

(7)

In which, * VCNT =

wCNT + (ρ

CNT

wCNT ρ m ) − (ρ CNT ρ m )wCNT

(8)

where wCNT is the mass fraction of the CNTs, ρ CNT and ρ m are the mass density of CNTs and matrix, respectively. The mass density and Poisson’s ratio of the FG-CNTRC beams are evaluated by using the rule of mixture, the results of which are, respectively,

ρ = VCNT ( z ) ρ CNT + Vm ( z ) ρ m , ν 12 = VCNT ( z )ν 12CNT + Vm ( z )ν m

(9a, b)

where ν 12CNT and ν m are the CNTs and matrix Poisson’s ratios, respectively.

2.2. Free vibration analysis It is assumed that the FG-CNTRC pre-twisted beam start to vibrate around its equilibrium state in thermal environment. The displacement components of an arbitrary material point of the pre-twisted FG-CNTRC beam measured with respect to its position in this equilibrium state are denoted by ui ( x, y, z, t ) with i=x, y, z. In the following the strains, stresses and consequently, the equations of motion due to this vibratory motion are formulated. In this regards, in order to consider the transverse shear deformation without using the shear correction factor, which is a

7

challenging parameter for the FG structural elements, the Reddy's third-order shear deformation theory of beams is employed to approximate the variation of the displacement components

(ui ) along

the thickness direction. Based on this theory, the displacement components of an

arbitrary material point of the pre-twisted FG-CNTRC beam are approximated as u x (x, y , z , t ) = u (x, t ) − z θ (x, t ) +

4C1 3  4C 3  ∂w( x, t )  ∂v( x, t )  z θ (x, t ) −  − y ϕ ( x, t ) + 21 y  ϕ ( x, t ) − , 2 ∂x  ∂x  3h 3b  

u y ( x, y, z , t ) = v ( x, t ) , u z ( x, y , z , t ) = w ( x, t )

(10a-c)

where u ( x, t ) , v(x, t ) and w( x, t ) are the displacement components of the material point (x,y) on the mid-plane of the beam cross section along the x-, y- and z-axes, respectively; θ (x, t ) and

ϕ (x, t ) are the bending rotations of the beam cross-section about the y- and z-axes (see Fig. 2), respectively. One can observe that by setting C1 = 0 , the formulation degenerates to those of the FSDT (or Timoshenko beam theory); on the other hand, by letting C1 = 1 , the displacement relations according to the Reddy's TSDT of beam are obtained. Based on this theory, the shear stresses vanish on the lateral surfaces of the pre-twisted FG-CNTRC beams. Using the three-dimensional linear elasticity strain-displacement relations together with the displacement relations (10), the nonzero components of the strain tensor (i.e., ε ij ; i, j = x, y, z ) become,

ε xx =

∂u  ∂ϕ   ∂θ  4C − y + Qθ  − z  − Qϕ  + 21 ∂x  ∂x   ∂x  h

 2  ∂w  z 3  ∂θ ∂ 2 w   Q z y θ − −  +   ∂x  3  ∂x ∂x 2   

∂v  β 2 z 3  ∂ϕ ∂ 2v    1 2C   ∂v   − β 2Q y 2 z  ϕ −  + − 2  , ε xy = ε yx =  − 21 y 2   − ϕ  , ∂x  3  ∂x ∂x   2 b   ∂x   1 2

ε xz = ε zx =  −

2C1 2   ∂w  z  −θ  2 h   ∂x 

(11a-c)

8

where Q =

h dφ and β = . b dx

The three-dimensional constitutive relations for the FG-CNTRC beams based on the TSDT can be reduced to

σ xx = Q11 ( z ) ε xx , σ xy = Q55 ( z) γ xy , σ xz = Q44 ( z ) γ xz

(12a-c)

where Q11 , Q55 and Q44 are the reduced stiffnesses of the FG-CNTRC beam and are given by Q11 ( z ) =

E11 ( z ) , Q55 ( z ) = G12 ( z ) , Q44 ( z ) = G13 ( z ) 1 −ν 12 ( z )ν 21 ( z )

(13a-c)

In this study, the Ritz method together with the Hamilton’s principle are used to derive the eigenfrequency free vibration equations. For the free vibration analysis, the Hamilton’s principle takes the following form [14]

∫ (δUˆ − δKˆ )dt = 0 t2

(14)

t1

where t1 and t2 are two arbitrary times and δ is the variational operator; also, Uˆ and Kˆ are the

potential and kinetic energies of the FG-CNTRC beam, which are defined as, respectively 2 2 1 1 L Th  ∂w   ∂v   ˆ U = ∫ (σ xxε xx + 2σ xyε xy + 2σ xzε xz ) dV + ∫ N xx   +    dx 2 V 2 0  ∂x   ∂x  

(15)

 ∂u 2  ∂u y  2  ∂u  2  1 ˆ  +  z   K = ∫ ρ  x  +  V 2  ∂t   ∂t   ∂t  

(16)

In Eq. (15), the thermal resultant force N xxTh is defined to be

N xxTh = −∫

b/ 2

∫

h/ 2

−b / 2 − h / 2

Q11 ( z ) α ∆T dz dy

(17)

where ∆T = T − T0 is the temperature change, T is the temperature at an arbitrary material point of the FG-CNTRC beam, and T0 is the free stress temperature of the FG-CNTRC beam which in this study is assumed to be the room temperature (T0 = 300 K ) .

9

( )

By using Eqs. (11) and (12), the total potential energy Uˆ of the FG-CNTRC beam in thermal environment based on the TSDT can be stated as,  ∂θ ∂ 2 w  1 L ∂u ∂ϕ  ∂w    ∂θ  z  Uˆ = ∫  N xx − M xy  Qθ + − M xz  − Qϕ  + Pxzy θ −  + Pxzz  − 2    2 0 ∂x ∂x  ∂x    ∂x    ∂x ∂x   ∂ϕ ∂ 2v  ∂v   ∂v   ∂w  y  − Pxyz − 2  + N xy − M xyy  − ϕ  + N xz − M xzz  − θ  dx  ϕ −  + Pxyy  ∂x    ∂x   ∂x   ∂x ∂x 

(

+

)

(

)

2 2 1 L Th  ∂w   ∂v   N +     dx xx  2 ∫0  ∂x   ∂x  

(18)

The various stress resultants introduced in Eq. (18) are defined in Appendix A. Also, using the displacement relations given in (10a-c), the kinetic energy Kˆ of the beam becomes 16 C1  ∂θ ∂ 2 w  16 C1  ∂ϕ ∂ 2 v  1 L   ∂u   ∂θ   ∂ϕ      Kˆ = ∫  I 00   + I 20   + I 60 − + I + I −   02 06 2 0   ∂t  9 h 4  ∂t ∂t∂x  9 b 4  ∂t ∂t∂x   ∂t   ∂t  2

2

2

2

2

8C  ∂u  ∂θ ∂ 2 w  8C  ∂u  ∂ϕ ∂ 2v   ∂u  ∂θ   ∂u  ∂ϕ   − 2 I 01    − 2 I10    + I 30 12   − −  + I 03 21   3 h  ∂t  ∂t ∂t∂x  3 b  ∂t  ∂t ∂t∂x   ∂t  ∂t   ∂t  ∂t 

− I 40

I 31

8 C1  ∂θ  3 h 2  ∂t

2 8C1  ∂θ  ∂θ ∂ w   ∂θ  ∂ϕ   + 2 I 11   −   − I 13 2  3 b  ∂t  ∂t ∂t∂x   ∂t  ∂t 

8 C1  ∂ϕ  ∂θ ∂ 2 w  32C1  + I 33 −  2  3 h  ∂t  ∂t ∂t∂x  9 h2 b2 2

 ∂v   ∂w  + I 00   + I 00    ∂t   ∂t 

2

2  ∂ϕ ∂ v   − −   ∂t ∂t∂x 

 ∂θ ∂ 2 w   ∂ϕ ∂ 2 v  8C  ∂ϕ  ∂ϕ ∂ 2 v      − I 04 21   − − −  3b  ∂t  ∂t ∂t∂x   ∂t ∂t∂x   ∂t ∂t∂x 

  dx 

(19)

where the inertia terms in Eq. (19) are defined as I ij = ∫

b/2

∫

h/2

−b / 2 − h / 2

ρ z i y j dz dy

for i,j=0, 1, 2, 3, 4, 5, 6

(20)

In order to simplify the governing equations and also to easily follow and conduct the parameter studies, the following dimensionless quantities are introduced

10

ξ=

Bij Dij I ij u w v 2x N xxTh Th − 1 , U = , W = , V = , bij = , d = , , , I = N = ij ij h h h Bm hi + j Bm hi + j I m hi + j Bm L

Λ =ωL

Im Bm

(21a-i)

where Bm and I m is the value of B00 (see Appendix A) and I 00 of a homogeneous beam made of matrix; also ω is the circular natural frequency of the FG-CNTRCs beam. It is well recognized that the computational efficiency and accuracy of the Ritz method depend on its basis (or admissible) functions. On the other hand, it has been shown that the Ritz method becomes numerically stable when the Chebyshev polynomials, which have the orthogonal properties, are chosen as its basis functions [34-37]. Therefore, in the present work a set of Chebyshev polynomials multiplied by suitable boundary functions, which assure the satisfaction of the essential geometric boundary conditions, is used as the basis functions of the displacement and rotation components. Accordingly, the displacement and rotation components can be approximated in the whole physical domain as, Nu Nv     U (ξ , t ) =  Fu (ξ )∑ Ai Pi (ξ ) sin (ωt + ϕ0 ) , V (ξ , t ) =  Fv (ξ )∑ Di Pi (ξ ) sin (ωt + ϕ0 ) , i =1 i =1     Nw Nθ     W (ξ , t ) =  Fw (ξ )∑ Ei Pi (ξ ) sin (ωt + ϕ0 ) , Θ(ξ , t ) =  Fθ (ξ )∑ Bi Pi (ξ ) sin (ωt + ϕ 0 ) , i =1 i =1     Nϕ   Φ (ξ , t ) =  Fϕ (ξ )∑ C i Pi (ξ ) sin (ωt + ϕ 0 ) i =1  

(22a-e)

where ϕ0 is the initial phase angle and Pα (ξ ) ( α = i, j, k , l , m ) is the α -th order one-dimensional Chebyshev polynomial of the first kind; N i (i = u , v, w,θ , ϕ ) is the number of Chebyshev

polynomial for the field variable i; also, the functions Fα (ξ ) (α = u, v, w,θ , ϕ ) are the boundary characteristic functions which can be expressed as,

11

Fα (ξ ) = (1 − ξ ) α (1 + ξ ) α R

S

for α = u , v, w, θ and ϕ

(23)

where the parameters Rα and S α are used to specify the state of the field variable α

(α = u, θ , ϕ , v, w)

at the beam edges ξ = 1 and ξ = −1 , respectively. The values of these

parameters for beams with simply supported (S), clamped (C), free (F) and combination of these boundary conditions are presented in Table 1. Evaluating Uˆ and Kˆ using the displacement relations given in Eq. (22) and substituting the results into Hamilton's principle (i.e., Eq. (14)), and also by considering the linear independence of the coefficients of the basis functions, one can represent the eigenfrequency equations in the matrix form as

([K ] − Λ [M ]){D} = 0 2

(24)

where [K ] and [M ] are the stiffness and mass matrices, respectively, and {D} is the vector of unknown coefficients of the basis functions,

[ ] [ ] [ ] [ ] [ ]  [M ]  [M ] [M ] = [M ]  [M ]  [M ] 

 K aa  ab T K T [K ] =  K ac  ad T K  K ae T 

aa

ab T ac T

ad T ae T

[K ] [K ] [K ] [K ] [K ] [K ] [K ] [K ] [K ] [K ] [K ] [K ] , [K ] [K ] [K ] [K ] [K ] [K ] [K ] [K ] [M ] [M ] [M ] [M ]  {A }   {B } [M ] [M ] [M ] [M ]   [M ] [M ] [M ] [M ] , {D} = {C }  {D } [M ] [M ] [M ] [M ]   {E } [M ] [M ] [M ] [M ] ab

ac

ad

ae

bb

bc

bd

be

bc T

cc

cd

ce

bd T

cd T

dd

de

be T

ce T

de T

ee

ab

ac

ad

ae

bb

bc

bd

be

bc T

cc

cd

ce

bd T

cd T

dd

de

be T

ce T

de T

ee

i

j

k l

m

The elements of the stiffness and mass matrices are given in Appendix B.

3. Numerical results

12

(25a-c)

In this section, at first, the fast rate of convergence and accuracy of the present method are shown. Then, parametric studies are conducted to display the effects of geometrical parameters, different types of CNTs distributions and temperature rise on the vibrational behavior of the pretwisted FG-CNTRC beams subjected to different boundary conditions. If not mentioned otherwise, the matrix phase is assumed to be Poly (methyl methacrylate) (PMMA) with temperature-dependent material properties as follows [7,14] E m = (3.52 - 0.0034T ) GPa , α m = 45(1 + 0.0005∆T ) × 10 −6 / K, ρ m = 1150 kg/m3 ,

ν m = 0.34

(26a-d)

where T = T0 + ∆T and T0 is the room temperature which is assumed to be T0 = 300 K . For the SWCNTs with the chiral index (10,10), the CNTs efficiency parameters are given in Table 2, and also, their temperature-dependent material properties are presented in Table 3. By fitting a thirdorder polynomial to these data, the material proprieties of the SWCNTs at any temperature other than those specified in Table 3 are estimated as

(

P (T ) = P0 1 + P1∆T + P2∆T 2 + P3∆T 3

)

(27)

where P0 is the corresponding material property at the room temperature T0 = 300 K and Pi (i = 1,2,3) are the coefficients of temperature-dependent material properties which are

presented in Table 4. For the resulting composites, it is assumed that G13 = G12 [38]. Also, it is assumed that the pre-twist angle of beam cross section varies along the beam axis as, x L

s

φ (x ) = φ 0  

(28)

13

Without loss of generality and accuracy, in preparing the numerical results, equal numbers of terms are used in the series expansion of the field variables (i.e., N u = N v = N w = N ϕ = N θ = N in Eq. (22)). It should be mentioned that the free vibration results of the FG-CNTRC beams with FG-V and FG- Λ distribution of CNTs are similar and hence, the results corresponding to one of these distributions are reported in this section.

3.1. Convergence and accuracy study Since there is no available result for the free vibration analysis of the pre-twisted FG-CRNTC beams, the results are validated by carrying out the convergence study, and also by performing the comparison studies in the limit cases with those of the other available solutions in open literature. As a first example, in Tables 5 and 6 the fast rate of convergence of the first four frequency parameters of the moderately thick FG-CNTRC pre-twisted beams with a relatively large pretwist angle under thermal environment are shown. In Table 5, the results for the fully clamped FG-CNTRC pre-twisted beams (C-C) with uniform and functionally graded (FG-X) CNTs distributions are displayed. In Table 6, the influences of two different set of boundary conditions on the convergence behavior of the method in analyzing the FG-CNTRC pre-twisted beams with simply supported edges (S-S) and clamped-free edges (C-F) are presented. In all cases under consideration, the fast rate of convergence and numerical stability of the present approach is fairly apparent. Also, one can observe that the acceptable results for the first four frequency parameters of the CNTRC pre-twisted beams with both uniform and non-uniform CNTs distributions can be achieved by using only five terms in the series expansion of the field

14

variables (i.e., Eq. (22)). However, in order to obtain more accurate results, ten terms in the series expansion of the field variables (i.e., N = 10 ) is used to generate the numerical results. In order to verify the accuracy of the method, in Table 7 the fundamental frequency parameter of the FG-CNTRC uniform beams (beams without pre-twist angle) are compared with those obtained by Lin and Xiang [18]. They used the p-Ritz method to obtain the eigenfrequency equations for free vibration of the FG-CNTRC beam based on the first-order and third-order shear deformation beam theories. This comparison study is conducted for the FG-CNTRC beams with different CNTs distributions subjected to simply supported and clamped boundary conditions. The obtained results based on both the FSDT and TSDT are reported in this table. Excellent agreements between the results of the two approaches in all cases are quite evident. As another example to show the correctness and effectiveness of the present approach for the free vibration analysis of pre-twisted beams, the first three frequency parameters of the pretwisted homogeneous beams under three different set of boundary conditions are compared with those obtained by Leung [42] in Table 8. In Ref. [42], the results have been extracted based on the classical theory (CT) of beams. The results are compared for different values of the pre-twist φ angle rate  0 L

  . In addition, the results of both the FSDT and TSDT are presented in this table. 

One can see that close agreements exist between the results of the present work and those of Ref. [42].

3.2. Parametric studies The effects of pre-twist angle ( φ0 ) on the first four frequency parameters of the clamped-free and fully clamped per-twisted FG-CNTRC beams with different CNTs distribution are studied in Figs. 4 and 5, respectively. It is interesting to note that when increasing the pre-twist angle, the

15

first and third frequency parameters of the clamped-free beams increase but, their second and fourth frequency parameters decrease. For the fully clamped per-twisted FG-CNTRC beams, only the fourth natural frequency parameter decreases but the first three frequency parameters increase by increasing the pre-twist angle. This phenomena show that the effect of the pre-twist angle depends on the mode number as well as beam edges boundary conditions. In Table 9, the influences of the pre-twist angle and the rate of pre-twist angle, which depends on the parameter s (see Eq. (28)), on the first three frequency parameters of the fully clamped pre-twisted FG-

CNTRC beams are shown. The results for CNTRC beams with both uniform and FG-X distributions of CNTs are presented in this table. It can be seen that increasing the rate of pretwist angle (i.e., the parameter s), the frequency parameters increase in all cases under consideration. In Fig. 6, the effect of pre-twist angle on the fundamental frequency parameter of the FGCNTRC beams with three different length-to-thickness ratio (L/h) and subjected to four different set of boundary conditions are displayed. It can be observed that for all three values of the length-to-thickness ratio and irrespective of the beam boundary conditions, the fundamental frequency parameter increases by increasing the pre-twist angle. The variation of the first four frequency parameters of the fully clamped per-twisted FGCNTRC beams verses the length-to-thickness ratio in combination with the temperature rise ( ∆T ) are shown in Fig. 7. It is obvious that by increasing the geometrical parameter L/h, the frequency parameters decrease monotonically. In addition, one can observe that by increasing the temperature rise, the frequency parameters decrease, but, their trends of variations against the length-to-thickness ratio are not significantly affected by increasing ∆T .

16

In Figs. 8 and 9, the impact of temperature dependence of material properties on the fundamental frequency parameters of the fully clamped and clamped-simply supported FGCNTRC beams are shown, respectively. The results are prepared for CNTRC beams with uniform and functionally graded distribution of CNTs. One can see that by considering the temperature dependence of material properties, the frequency parameter decreases. This is due to the reduction in the stiffness of the beam which is a result of material stiffness softening. In addition, as one can expected, the influences of the temperature dependence of material properties are more pronounced at high temperature rise. The other interesting finding from these figures is the critical temperature rise of the pre-twisted FG-CNTRC beams under investigation. The temperature at which the fundamental frequency of the beam becomes zero is the beam critical temperature. One can see that this parameter can easily be extracted from these figures and also it is obvious that it highly depends on the temperature dependence of the beam material properties. The influences of twist angle together with the thickness-to-width ratio (i.e., β = h / b ) on the fundamental frequency parameter of the pre-twisted FG-CNTRC beams under different boundary conditions are investigated in Table 10. The results are provided for three different CNTs distribution. It is clear that by increasing the thickness-to-width ratio, the frequency parameter reduces considerably. On the other hand, by increasing the pre-twist angle, the fundamental frequency parameters increase for the all types of considered boundary conditions. The impacts of length-to-thickness (L/h) and CNTs volume fraction on the first four frequency parameters of the fully clamped pre-twisted FG-CNTRC beams in thermal environment are studied in Table 11. For this purpose, the CNTRC beams with uniform and FGX distribution of CNTs are analyzed. It can be observed that for thin as well as moderately thick

17

pre-twisted FG-CNTRC beams under investigation by increasing the CNTs volume fraction, the frequency parameters increase and this material parameter has significant effect on the vibrational characteristics of the FG-CNTRC beams.

4. Conclusion For the first time, the free vibration behavior of the pre-twisted FG-CNTRC beams in thermal environment was investigated. The transverse shear deformation and rotary inertia were included based on the third-order shear deformation theory, which does not require the shear correction factor. The Chebyshev-Ritz method as a computationally efficient and powerful semi-analytical method was employed to derive the eigenfrequency of the FG-CNTRC beams subjected to different boundary conditions. The formulation includes the influences of temperature dependence and the initial thermal stresses. The effects of pre-twist angle together with the different CNTs distributions, geometrical parameters and temperature rise on the frequency parameters of the pre-twisted FG-CNTRC beams were investigated. From the presented results one can concluded that, • By increasing the pre-twist angle, irrespective of the beam boundary conditions, the

fundamental frequency parameter increases. But, its influence on the higher-order frequencies significantly depends on the mode number and the beam boundary conditions, • The temperature dependence of the material properties causes the reduction of the

frequency parameters, especially the reduction is more pronounced at high temperature (i.e., near the critical temperature of the FG-CNTRC beams), • By increasing the thickness-to-width ratio, the frequency parameters reduce,

18

• The increase of CNTs volume fraction increases the frequency parameters, and this

material parameter has significant effect on the natural frequencies, • The presented results clearly show the influences of different material and geometrical

parameters on the vibrational behavior of the pre-twisted FG-CNTRC beams and can be used as benchmark solution for the future research works in the field.

Appendix A. The resultant forces and moments The stress resultants appeared in Eq. (15) are defined as, respectively N xx = ∫

h/ 2

∫

b/2

− h / 2 −b / 2

+

M xy = ∫

 ∂θ ∂ 2 w  4C1Q 4C1 ∂v  4C   B − 2  − 2 B12  ϕ −  + 21 B03 30  2 3h b ∂x  3b   ∂x ∂x 

h/ 2

∫

b/2

− h / 2 −b / 2

+

h/2

∫

b/ 2

−h / 2 −b / 2

z Pxzy =

 ∂ϕ ∂ 2v   − 2  ,  ∂x ∂x 

(A.1)

∂ϕ  ∂w   ∂u    ∂θ  4C Q  − Qϕ  + 12 B22 θ −  − B02  Qθ +  − B11   ∂x  h ∂x   ∂x    ∂x  

σ xx y dy dz = B01 

 ∂θ ∂ 2 w  4C1Q  ∂ϕ ∂ 2 v  4C1 ∂v  4C1     ϕ B − − B − + B − 2  ,   31  13 04  2  3h 2 b2 ∂x  3b 2   ∂x ∂x   ∂x ∂x 

M xz = ∫

+

∂ϕ  ∂w   ∂u    ∂θ  4C Q  − Qϕ  + 12 B21  θ −  − B01  Qθ +  − B10   ∂x  h ∂x   ∂x    ∂x  

σ xx dy dz = B00 

(A.2)

∂ϕ  ∂w   ∂u    ∂θ  4C Q  − Qϕ  + 12 B31 θ −  − B11  Qθ +  − B20   ∂x  h ∂x   ∂x    ∂x  

σ xx z dy dz = B10 

 ∂θ ∂ 2 w  4C1Q 4C1 ∂v  4C   B − 2  − 2 B22  ϕ −  + 21 B13 40  2 3h b ∂x  3b   ∂x ∂x 

 ∂ϕ ∂ 2v   − 2  , ∂ x ∂x  

(A.3)

4C1 Q h / 2 b / 2 4C Q   ∂u  ∂ϕ    ∂θ  4C Q σ xx z 2 y dy dz = 12  B21   − B22  Qθ + − B31  − Qϕ  + 12 B42  2 ∫ ∫ − h / 2 −b / 2 h h   ∂x  ∂x  h   ∂x 

 ∂θ ∂ 2 w  4C1Q ∂w  4C1 ∂v  4C   × θ − − 2  − 2 B33  ϕ −  + 21 B24  + 2 B51  ∂x  3h b ∂x  3b    ∂x ∂x 

19

 ∂ϕ ∂ 2v   − 2  ,  ∂x ∂x 

(A.4)

4C1 h / 2 b / 2 4C   ∂u  ∂ϕ    ∂θ  4C Q σ z 3 dy dz = 21  B30   − B31  Qθ +  − B40  − Qϕ  + 12 B51 2 ∫− h / 2 ∫− b / 2 xx 3h 3h   ∂x  ∂x  h   ∂x 

M xzz =

 ∂θ ∂ 2 w  4C1Q  ∂ϕ ∂ 2v  ∂w  4C1 ∂v  4C   × θ − − 2  − 2 B42  ϕ −  + 21 B33  − 2  ,  + 2 B60  ∂x  3h b ∂x  3b    ∂x ∂x   ∂x ∂x 

Pxyzy =

4C1 Q h / 2 b / 2 4C Q   ∂u  ∂ϕ    ∂θ  4C Q σ xx y 2 z dy dz = 12  B12   − B13  Qθ +  − B22  − Qϕ  + 12 B33 2 ∫ ∫ − h / 2 −b / 2 b b   ∂x  ∂x  h   ∂x 

 ∂θ ∂ 2 w  4C1Q ∂w  4C1 ∂v  4C   × θ − − 2  − 2 B24  ϕ −  + 21 B15  + 2 B42  ∂x  3h b ∂x  3b    ∂x ∂x 

Pzyy =

N xy = ∫

h/2

∫

b/2

−h / 2 −b / 2

 

σ xy dzdy =  D00 −

 ∂ϕ ∂ 2v   − 2  ,  ∂x ∂x 

4C1   ∂v  D02   − ϕ  , 2 b   ∂x 

2C1 h / 2 b / 2 2C  4C   ∂v  σ y 2 dzdy = 21  D02 − 21 D04   − ϕ  , 2 ∫− h / 2 ∫− b / 2 xy b b  b   ∂x 

N xz = ∫

h/ 2

∫

b/ 2

− h / 2 −b / 2

M xzz =

 ∂ϕ ∂ 2v    − 2   , ∂ x ∂x   

(A.6)

4C1 h / 2 b / 2 4C   ∂u  ∂ϕ    ∂θ  4C Q σ y3 dy dz = 21  B03   − B04  Qθ +  − B13  − Qϕ  + 12 B24 2 ∫− h / 2 ∫− b / 2 xx 3b 3b   ∂x  ∂x  h   ∂x 

 ∂θ ∂ 2 w  4C1Q ∂w  4C1 ∂v  4C   × θ − − 2  − 2 B15  ϕ −  + 21 B06  + 2 B33  ∂x  3h b ∂x  3b    ∂x ∂x 

M xyy =

(A.5)

 

σ xz dzdy =  D00 −

4C1   ∂w  D20   −θ  , 2 h   ∂x 

2C1 h / 2 b / 2 2C  4C   ∂w  σ z 2 dzdy = 21  D20 − 21 D40   − θ  , 2 ∫− h / 2 ∫− b / 2 xz h h  h   ∂x 

(A.7)

(A.8)

(A.9)

(A.10)

(A.11)

where the stiffness components are, Bij = ∫

h/2

∫

b/2

− h / 2 −b / 2

Q11 ( z ) z i y j dydz , Dij = ∫

h/ 2

∫

b/2

− h / 2 −b / 2

G12 ( z) z i y j dydz

Appendix B. The elements of stiffness and mass matrices

20

(A.12a, b)

The elements of the stiffness and mass sub-matrices for pre-twisted FG-CNTRC beams appear in Eqs. (25a,b) are as follows, respectively

Kiaai = b00 Di1i,1u u ,

(B.1)

4   Kijab = −0.5b01QLDi1,j0u θ − b10 Di1,j1u θ + C1  2 Q L b21 Di1,j0u θ + b30 Di1,j1u θ  , 3  

(B.2)

  4b K ikac = −b01 Di1k,1u ϕ + 0.5 b10QLDi1k, 0u ϕ + C1  β 2  03 Di1k,1u ϕ − 2 Q L b12 Di1k, 0u ϕ   3

  , 

(B.3)

2   Kilad = 4C1 β 2  Q h b12 Di1l,1u v − b03 λˆ Di1l, 2u v  , 3  

(B.4)

2   Kimae = −4C1  Q h b21Di1m,1u w + b30 λˆ Di1l, 2u v  , 3  

(B.5)

(

)

2 2 0,0 1, 0 0,1 1,1 K bb j j = 0.25 b02 Q L D j j θ θ + 0.5 b11 Q L D j j θ θ + D j j θ θ + b20 D j j θ θ +

 + C1  D0j ,j0θ θ 

2  2 2 2Q L (2b42 − b22 ) + λˆ2 (2d 40 − d 20 ) 

d 00 0 , 0 D j jθ θ 4λˆ2

  8Q L  1, 0 0 ,1  − D j j θ θ + D j j θ θ  3 (b31 − b51 )   

(

)

8  − D1j ,1j θ θ (b40 − 2b60 ) , 3 

(B.6)

K bcjk = 0.5 b02 Q L D0j k,1θ ϕ − 0.25 b11 Q2 L2 D0j k, 0θ ϕ + b11 D1j,k1θ ϕ − 0.5 b20 Q L D1j,k0θ ϕ  β 2  + C1 D 0j k, 0θ ϕ Q 2 L2 (b13 + β 2 b31 − 4β 2 b33 ) − 2 D0j k,1θ ϕ Q L  (b04 − 4b24 ) + b22   3      2 4 2  + D1j,k0θ ϕ Q L (3β 2 b22 + b40 − 4 β 2 b42 )− D1j,k1θ ϕ  β 2  b13 − b33  + b31   3 3 3    

[

]

 0,1 2 2  1 0,2 b42   2 1,1  K bd  jl = C1 2 D j l ϕ v β Q h L(4b33 − b13 ) + 4 β Q h  D j l ϕ v (b04 − 4b24 ) − 4 D j l ϕ v  b22 − 3    3  8  + D1j ,l2ϕ v β 2 λˆ (3 b13 − 4 b33 ) , 9 

21

(B.7)

(B.8)

K bejm = −

d 00 0,1  2   D j m ϕ w + C1  2 D 0j ,m1 ϕ w Q 2 h L (b22 − 4b42 ) + (d 20 − 2d 40 ) + 2 λˆ λˆ   

4 0,2 4 8  D j m ϕ w Q L (b31 − 4b51 ) + Q h D1j,m1 ϕ w (3b31 − 4b51 )+ λˆ D1j,m2 ϕ w (3b40 − 4b60 ) , 3 3 9  d K kcck = b02 Dk1,k1 ϕ ϕ − 0.5 b11 Q L Dk0 k,1ϕ ϕ + Dk1,k0ϕ ϕ + 0.25 b20 Q 2 L2 Dk0 ,k0ϕ ϕ + 002 Dk0 ,k0ϕ ϕ 4λˆ

(

(B.9)

)

32 8 2  + C1  β 2 Q L Dk0,k1ϕ ϕ (b13 − β 2b15 ) + β 2 Dk1,k1 ϕ ϕ  β 2b06 − b04  − 2 Dk0,k0ϕ ϕ β 2 Q2 L2 3 3 3  32  × (b22 − 2β 2 b24 ) + λ2 (d 22 − 2β 2 d 04 ) + Dk1,k0ϕ ϕ β 2 Q L (b13 − β 2 b15 ) , (B.10) 3   d 4  8 K klcd = − 00 Dk0l,1ϕ v + C1 4 Dk1,l1ϕ v β 2Q h β 2b15 − b13  + Dk1,l2ϕ v β 2 λˆ 3b04 − 4β 2b06 2λˆ 3  9  2   4  + 2 Dk0l,1ϕ v β 2 Q 2 h L (b22 − 4 β 2b24 ) + (d 02 − 2β 2d 04 ) + Dk0l, 2ϕ v β 2 Q h (4 β 2b15 − 3 b13 ) , (B.11) λ   9  8 4 ce K km = C1  Dk1,m1 ϕ w Q h 3b22 − 4β 2b 24 + Dk1,m2 ϕ w λˆ 3b31 − 4β 2b 33 + 2 Dk0,m1 ϕ w Q 2 h L 4 β 2b 33 −b31 9 3 4  + Dk0,m2ϕ w Q h (4 β 2 b42 − b40 ) , (B.12) 3  32  Kldd = d 00 Dl1l,1v v + C1 8 Dl1l,1v v β 2 β 2Q2 h 2b24 − d 02 + 2β 2d04 − β 4 Q h b15 λˆ Dl2l,1v v + Dl1l, 2v v l 3  64  + Dl2l, 2v v β 2 λˆ2 b06  + N Th Dl1l,1v v , (B.13) 9  32 64   Klmde = C1  − 16 Dl1m,1 v w β 2 Q2 h 2 b33 + β 2Q h λˆ (b24 Dl2m,1v w − b42 Dl1m, 2v w ) + Dl2m, 2v w β 2 λˆ2b33  , (B.14) 3 9   32  K meem = d 00 Dm1,1m w w + C1 8 Dm1,1m w w 2b42Q 2 h 2 − d 20 + 2d 40 + Q h λˆ b51 Dm1, 2m w w + Dm2,m1 w w 3  64  + Dm2,m2 w wb60λˆ2  + N Th Dm1,1m w w , (B.15) 9 

[

]

(

(

)

(

)

)

(

(

(

)

(

)

)

(

)

)

C C     M iaa = 0.25I 00 Di0i,u0 u , M ijab =  − 0.25 I10 + 1 I 30  Di0j, 0u θ , M ikac =  − 0.25I 01 + 1 β 2 I 03  Di0k, 0u ϕ , i 3 3    

M ilad = −

2C1 2C 4C1 2C   I 03 β 2 λˆ Di0l,u1 v , M imae = − 1 I 30 λˆ Di0m,1u w , M bb I 60 − 1 I 40  D 0j ,j0θ θ , jj = 0.25 I 20 + 3 3 9 3  

 C  4 M bcjk =  0.25 I11 − 1  β 2 I13 + I 31 + β 2 I 33 3  3 

 2C 2  0 ,1  0 , 0 bd  D j k θ ϕ , M jl =  1 β λˆ (I13 − 4 I 33 ) D j l θ v ,  3  

22

4C 2C  2C    M bejm =  1 λˆ (I 40 − 4 I 60 ) D 0j ,m1 θ w , M kcck =  0.25 I 02 + 1 β 4 I 06 − 1 I 04  Dk0k, 0ϕ ϕ , 9 3  3     2C M klcd =  1 β 2 λˆ I 04 − 4 β 2 I 06  3

(

M ldd = 0.25I 00 Dl0l,0v v + l

) D 

0 ,1 kl ϕv

 2C  ce =  1 λˆ (I 31 − 4 I 33 ) Dk0m,1 ϕ w , , M km  3 

16C1 4 ˆ2 16C1 2 ˆ2 β λ I 06 Dl1l,1v v , M lmde = β λ I 33 Dl1m,1 v w , 9 9

M meem = 0.25I 00 Dm0,m0 w w +

16C1 ˆ2 λ I 60 Dm1,1m w w 9

[

(B.16-30)

]

s s 1   d [Fϑ (ξ ) Pα (ξ )] d FΞ (ξ ) Pβ (ξ )  Dαs ,βs ϑ Ξ = ∫   dξ , α , β = i, j , k , l , m, i , j, k , l , m ; −1 dξ s dξ s  

h ϑ, Ξ = u,θ ,ϕ , v, w , λˆ =

(B.31-34)

L

References [1] Lau KT, Hui D. The revolutionary creation of new advanced materials-carbon nanotube composites. Compos Part B-Eng 2002;33:263-77. [2] Sun CH, Li F, Cheng HM, Lu GQ. Axial Young's modulus prediction of single walled carbon nanotubes arrays with diameters from nanometer to meter scales. Appl Phys Lett 2005;87:193101-4. [3] Griebel M, Hamaekers J. Molecular dynamics simulations of the elastic moduli of polymer-carbon nanotube composites. Comput Method Appl Mech Eng 2004;193:1773-88. [4] Liew KM, Lei ZX, Zhang LW. Mechanical analysis of functionally graded carbon nanotube reinforced composites: A review. Compos Struct 2015;120:90-7. [5] Meguid SA, Sun Y. On the tensile and shear strength of nano-reinforced composite interfaces. Mater Design 2004;25:289-96.

23

[6] Shen HS. Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments. Compos Struct 2009;91:9-19. [7] Wang ZX, Shen HS. Nonlinear vibration of nanotube-reinforced composite plates in thermal environments. Comput Mater Sci 2011;50:2319-2330 [8] Shen HS. Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite cylindrical shells. Compos Part B-Eng 2012;43:1030-1038. [9] Shen HS, Xiang Y. Nonlinear vibration of nanotube-reinforced composite cylindrical shells in thermal environments. Comput Method Appl Mech Eng 2012;213:196-205. [10] Lei ZX, Liew KM, Yu JL. Free vibration analysis of functionally graded carbon nanotubereinforced composite plates using the element-free kp-Ritz method in thermal environment. Compos Struct 2013;106:128-138. [11] Shen HS, Xiang Y. Nonlinear analysis of nanotube-reinforced composite beams resting on elastic foundations in thermal environments. Eng Struct 2013;56:698-708. [12] Shen HS, Xiang Y. Nonlinear vibration of nanotube-reinforced composite cylindrical panels resting on elastic foundations in thermal environments. Compos Struct 2014;111:291-300. [13] Shen HS, Xiang Y. Postbuckling of axially compressed nanotube-reinforced composite cylindrical panels resting on elastic foundations in thermal environments. Compos Part BEng 2014;67:50-61. [14] Jooybar N, Malekzadeh P, Fiouz AR. Vibration of functionally graded carbon nanotubes reinforced composite truncated conical panels with elastically restrained against rotation edges in thermal environment. Compos Part B-Eng 2016;106:242-261.

24

[15] Deepak BP, Ganguli R, Gopalakrishnan S. Dynamics of rotating composite beams: A comparative study between CNT reinforced polymer composite beams and laminated composite beams using spectral finite elements. Int J Mech Sci 2012;64:110–126. [16] Wattanasakulpong N, Ungbhakorn V. Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation. Comput Mater Scie 2103;71:201-208. [17] Lin F, Xiang Y. Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories. Appl Math Model 2014;38:3741-3754. [18] Lin F, Xiang Y. Numerical analysis on nonlinear free vibration of carbon nanotube reinforced composite beams. Int J Struct Stab Dy 2014;14: 1350056 (21 pages). [19] Ansari R, Faghih Shojaei M, Mohammadi V, Gholami R, Sadeghi F. Nonlinear forced vibration analysis of functionally graded

carbon nanotube-reinforced

composite

Timoshenko beams. Compos Struct 2014;113:316-327. [20] Jam JE, Kiani Y. Low velocity impact response of functionally graded carbon nanotube reinforced composite beams in thermal environment. Compos Struct 2015;132:35-43. [21] Chaudhari VK, Lal A. Nonlinear free vibration analysis of elastically supported nanotubereinforced composite beam in thermal environment. Procedia Eng 2016;144:928-935. [22] Rafiee M, Nitzsche F, Labrosse M. Rotating nanocomposite thin-walled beams undergoing large deformation. Compos Struct 2016;150:191-199. [23] Wu HL, Yang J, Kitipornchai S. Nonlinear vibration of functionally graded carbon nanotube-reinforced composite beams with geometric imperfections. Compos Part B-Eng 2016;90:86-96.

25

[24] Nejati M, Eslampanah A, Najafizadeh MH. Buckling and vibration analysis of functionally graded carbon nanotube-reinforced beam under axial load. Int J Appl Mech 2016;08: 1650008 (19 pages). [25] Wattanasakulpong N, Mao Q. Stability and vibration analyses of carbon nanotubereinforced composite beams with elastic boundary conditions: Chebyshev collocation method. Mech Adv Mater Struct 2017;24:260-270. [26] Mohammadimehr M, Shahedi S. High-order buckling and free vibration analysis of two types sandwich beam including AL or PVC-foam flexible core and CNTs reinforced nanocomposite face sheets using GDQM. Compos Part B-Eng 2017;108:91-107. [27] Mohammadimehr M, Mostafavifar M. Free vibration analysis of sandwich plate with a transversely flexible core and FG-CNTs reinforced nanocomposite face sheets subjected to magnetic field and temperature-dependent material properties using SGT. Compos Part BEng 2016;94:253-270. [28] Malekzadeh P, Dehbozorgi M. Low velocity impact analysis of functionally graded carbon nanotubes reinforced composite skew plates. Compos Struct 2016:140:728-748. [29] Zhang LW, Zhang Y, Zou GL, Liew KM. Free vibration analysis of triangular CNTreinforced composite plates subjected to in-plane stresses using FSDT element-free method. Compos Struct 2016;149:247-260. [30] Thomas B, Roy T. Vibration analysis of functionally graded carbon nanotube-reinforced composite shell structures. Acta Mech 2016;227:581-599. [31] Tornabene F, Fantuzzi N, Bacciocchi M, Viola E. Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells. Compos Part B-Eng 2016;89:187-218.

26

[32] Lei ZX, Zhang LW, Liew KM. Parametric analysis of frequency of rotating laminated CNT reinforced functionally graded cylindrical panels. Compos Part B-Eng 2016;90:251-266. [33] Kiani Y. Free vibration of FG-CNT reinforced composite spherical shell panels using Gram-Schmidt shape functions. Compos Struct 2017;159:368-381. [34] Zhou D, Lo SH, Au FTK, Cheung YK, Liu WQ. 3-D vibration analysis of skew thick plates using Chebyshev–Ritz method. Int J Mech Sci 2006;48:1481-1493. [35] Malekzadeh P, Bahranifard F, Ziaee S. Three-dimensional free vibration analysis of functionally graded cylindrical panels with cut-out using Chebyshev–Ritz method. Compos Struct 2013;105:1-13. [36] Ghorbani Shenas A, Ziaee S, Malekzadeh P. Vibrational behavior of rotating pre-twisted functionally graded microbeams in thermal environment. Compos Struct 2016;157:222235. [37] Ghorbani Shenas A, Malekzadeh P. Free vibration of functionally graded quadrilateral microplates in thermal environment. Thin-Wall Struct 2016;106:294-315. [38] Han Y, Elliott J. Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Comput Mater Sci 2007;39:315-323. [39] Hyer MW. Stress Analysis of Fiber-Reinforced Composite Materials. 1998, McGraw-Hill,

Singapore. [40] Shen HS, Wang ZX. Assessment of Voigt and Mori–Tanaka models for vibration analysis of functionally graded plates. Compos Struct 2012;94:2197-2208. [41] Kim YW. Temperature dependent vibration analysis of functionally graded rectangular plates. J Sound Vib 2005;284:531-549.

27

[42] Leung AYT. Dynamics and buckling of thin pre-twisted beams under axial load and torque. Int J Struct Stab Dy 2010;10:957-81. Table 1. The boundary exponents for different edge boundary conditions. Su S v S w Sθ Boundary conditions Ru Rv Rw Rθ Rϕ C-C 1 2 2 1 1 1 2 2 1 S-S 1 1 1 0 0 1 1 1 0 C-S 1 2 2 1 1 1 1 1 0 C-F 1 2 2 1 1 0 0 0 0

Sϕ 1 0 0 0

Table 2. CNTs efficiency parameters [7]. * VCNT

η1

η2

η3

0.12

0.137

1.022

0.715

0.17

0.142

1.626

1.138

0.28

0.141

1.585

1.109

Table 3. Temperature-depended material properties of single walled carbon

nanotubes [7]. T (K) E11CNT (TPa ) E 22CNT (TPa ) G12CNT (TPa ) α11CNT (× 10 −6 / K ) α 22CNT (× 10 −6 / K ) 300

5.6466

7.0800

1.9445

3.4584

5.1682

500

5.5308

6.9348

1.9643

4.5361

5.0189

700

5.4744

6.8641

1.9644

4.6677

4.8943

1000

5.2814

6.6220

1.9451

4.2800

4.7532

Table 4.

Coefficients of temperature-dependent material properties for

single walled carbon nanotubes [14]. P0

P1

P2

P3

CNT E11CNT / E011

1

− 1.5849 × 10 −4

3.539 × 10 −7

− 3.707 × 10 −10

CNT CNT E 22 / E022

1

− 1.5852 × 10 −4

3.5408 × 10 −7

− 3.709 × 10 −10

CNT G12CNT / G012

1

8.3093 × 10 −5

− 1.7803 × 10 −7

8.5651 × 10 −11

CNT α 11CNT / α 011

1

2.5039 × 10 −3

− 5.3839 × 10 −6

3.2738 × 10 −9

CNT α 22CNT / α 022

1

− 1.5646 × 10 −4

6.0307 × 10 −8

− 9.4442 × 10 −13

28

Table 5. The convergence behavior of the first four dimensionless natural frequencies

of the fully clamped pre-twisted FG-CNTRC beams in thermal environment * = 0.28 ). ( L / h = 10, β = 1, s = 1, φ0 = π / 2, ∆T = 100 K , VCNT UD FG-X Λ3 Λ1 Λ2 Λ4 Λ1 Λ2 Λ3 N

Λ4

2

2.4505

2.8912

7.0460

7.6985

4.1099

5.4464

7.7738

8.9541

3

2.3469

2.7344

5.3585

6.0687

3.8824

4.3206

6.2358

7.3359

4

2.3289

2.7104

5.2335

5.9109

3.5627

4.1001

6.1034

7.1620

5

2.3140

2.6787

5.1802

5.7543

3.2381

3.9807

6.0803

7.0338

6

2.3134

2.6757

4.9289

5.2905

3.1170

3.4799

6.0630

6.8816

10

2.3028

2.5539

4.6201

5.1398

3.0789

3.3945

6.0523

6.8359

12

2.3028

2.5539

4.6201

5.1398

3.0789

3.3945

6.0523

6.8359

15

2.3028

2.5539

4.6201

5.1398

3.0789

3.3945

6.0523

6.8359

Table 6. The convergence behavior of the first four frequency parameters of the pre-

twisted FG-CNTRC beams with FG-V distribution of CNTs ( L / h = 10, β = 1, s = 1, * = 0.28 ). φ0 = π / 2, ∆T = 100 K , VCNT

S-S

C-F 0.6616

Λ3 3.2772

3.6850

0.5575

0.5847

2.0359

2.3667

3.3036

0.5011

0.5787

1.7077

2.0854

2.9047

3.1448

0.4982

0.5761

1.6812

2.0569

1.4009

2.8997

3.1384

0.4964

0.5744

1.6637

2.0388

1.2208

1.4009

2.8967

3.1345

0.4935

0.5718

1.6341

2.0084

12

1.2208

1.4009

2.8967

3.1345

0.4935

0.5718

1.6341

2.0084

15

1.2208

1.4009

2.8967

3.1345

0.4935

0.5718

1.6341

2.0084

N

Λ1

Λ2

Λ3

Λ4

Λ1

Λ2

2

1.4845

1.6866

3.7437

4.1010

0.5838

3

1.3128

1.5011

3.2251

3.5107

4

1.2271

1.4077

3.0198

5

1.2215

1.4015

6

1.2209

10

29

Λ4

Table 7. Comparison of the fundamental frequency parameter ( Ω = ωL2

ρm Em h 2

) of the

uniform (i.e., φ0 = 0 ) FG-CNTRC beams with simply supported and clamped edges * ( L / h = 10, VCNT = 0.12 ).

S-S

C-C

TSDT

1

FSDT

1

FSDT1

TSDT

Present

[18]

Present

[18]

Present

[18]

Present

[18]

FG- Λ FG-X

11.1603

11.1601

11.1298

11.1295

14.8064

14.8064

14.0624

14.0623

12.3854

12.3850

12.3053

12.3048

15.9808

15.9807

14.8683

14.8681

UD

11.3735

11.3732

11.3299

11.3295

15.3295

15.3294

14.4786

14.4784

Shear correction coefficient ks=5/6 has been used.

 ω 2 ρ A L4 Table 8. Comparison of the first three frequency parameters  λi = 4 i  EI xx 

  of  

homogenous pre-twisted Euler-Bernoulli beam for different values of the pre-twist angle rate (ν = 0.3, I xx = 2 I yy = 1 , AL2 = 10 4 , s=1).

λ1

φ0 C-F

S-S

C-C

1

λ3

λ2

L

TSDT

FSDT1 CT [42]

TSDT

FSDT1 CT [42]

TSDT

FSDT1

CT [42]

0

1.5846

1.5783

1.5768

1.8817

1.8781

1.8751

3.9673

3.9550

3.9472

1

1.5891

1.5829

1.5814

1.8705

1.8669

1.8640

4.0071

3.9947

3.9868

2

1.6019

1.5955

1.5940

1.8426

1.8391

1.8362

4.1099

4.0972

4.0891

3

1.6192

1.6128

1.6112

1.8096

1.8061

1.8033

4.2160

4.2030

4.1947

0

2.6539

2.6441

2.6418

3.1573

3.1450

3.1416

5.3088

5.2903

5.2835

1

2.6642

2.6543

2.6520

3.1310

3.1189

3.1155

5.3751

5.3564

5.3495

2

2.6933

2.6834

2.6810

3.0668

3.0549

3.0516

5.5486

5.5292

5.5221

3

2.7356

2.7255

2.7231

2.9919

2.9803

2.9771 5.7657

5.7456

5.7382

0

3.9981

3.9818

3.9775

4.7531

4.7375

4.7300

6.6367

6.6162

6.6037

1

4.0828

4.0661

4.0617

4.6503

4.6351

4.6277

6.7626

6.7417

6.7290

2

4.2703

4.2529

4.2483

4.4429

4.4283

4.4213

7.0374

7.0157

7.0024

3

4.2932

4.2756

4.2710

4.4225

4.4080

4.4010 7.1645

7.1424

7.1289

A shear correction factor of ks=5/6 has been used.

30

Table 9. The influence of twist angle on the first four frequency parameters of the fully

clamped FG-CNTRC beams in thermal environment ( L / h = 10, β = 1, ∆T = 100 K , * VCNT = 0.17 ).

s

φ0

1

2

3

UD

FG-X 2.3652

Λ3 4.3624

4.7913

1.9763

2.3878

4.4049

4.7678

3.8157

1.9952

2.4091

4.4479

4.7446

3.4948

3.7784

2.0520

2.4338

4.5132

4.6980

1.9173

3.5427

3.7517

2.1000

2.4472

4.5815

4.6621

1.5488

1.8911

3.4443

3.9069

1.9741

2.4105

4.4475

4.8586

45°

1.5633

1.9090

3.4776

3.8878

1.9929

2.4336

4.4908

4.8347

60°

1.5779

1.9271

3.5114

3.8690

2.0118

2.4569

4.5348

4.8112

90°

1.6222

1.9454

3.5626

3.8312

2.0691

2.4805

4.6013

4.7640

120°

1.6531

1.9538

3.6118

3.8039

2.1179

2.4993

4.6713

4.7273

30°

1.5631

1.9001

3.4610

3.9483

1.9926

2.4221

4.4693

4.9103

45°

1.5777

1.9180

3.4946

3.9290

2.0115

2.4452

4.5129

4.8862

60°

1.5924

1.9362

3.5285

3.8099

2.0305

2.4686

4.5570

4.8623

90°

1.6372

1.9546

3.5799

3.8718

2.0885

2.4924

4.6238

4.8147

120°

1.6687

1.9789

3.6295

3.8441

2.1380

2.5266

4.6943

4.7776

Λ1

Λ2

Λ3

Λ4

Λ1

Λ2

30°

1.5362

1.8560

3.3788

3.8531

1.9578

45°

1.5505

1.8735

3.4115

3.8343

60°

1.5651

1.8913

3.4446

90°

1.6090

1.9092

120°

1.6393

30°

31

Λ4

Table 10. The influence of twist angle and thickness-to-width ratio on the fundamental

frequency parameter of the pre-twisted FG-CNTRC beams under different boundary * = 0.28 ). conditions (L/h=10, s=1, ∆T = 0 K, VCNT C-S S-S

C-F

φ0

β

UD

FG-O

FG-X

UD

FG-O

FG-X

UD

FG-O

FG-X

45°

0.5

1.9674

1.3354

2.4238

1.4842

0.9133

1.8418

0.7651

0.4164

0.9698

1

1.8383

0.9477

2.3039

1.4592

0.6092

1.7529

0.7423

0.2642

0.9309

1.5

1.6117

0.6232

2.0588

1.1921

0.3650

1.5525

0.5600

0.1531

0.7424

2

1.4169

0.4384

1.8320

0.9846

0.2501

1.3023

0.4416

0.1039

0.5922

0.5

2.0295

1.3861

2.4918

1.6673

1.0068

2.0747

0.7786

0.4282

0.9847

1

1.8953

0.9801

2.3708

1.6125

0.6645

2.0500

0.7575

0.2729

0.9573

1.5

1.6663

0.6289

2.1265

1.3062

0.3956

1.7035

0.5771

0.1586

0.7644

2

1.4612

0.4393

1.8915

1.0732

0.2705

1.4213

0.4563

0.1077

0.6116

0.5

2.0675

1.4190

2.5359

1.7849

1.0641

2.2254

0.8018

0.4488

1.0104

1

1.9322

1.0001

2.4144

1.7065

0.6975

2.1714

0.7827

0.2880

0.9873

1.5

1.7006

0.6319

2.1689

1.3750

0.4137

1.7949

0.6062

0.1682

0.8017

2

1.4891

0.4425

1.9292

1.1261

0.2825

1.4925

0.4817

0.1143

0.6449

90°

135°

32

Table 11. The influence of length-to-thickness (L/h) together with CNTs volume fraction on

the first four frequency parameters of the fully clamped FG-CNTRC beams in thermal environment ( β = 1, s=1, φ0 = π / 2 , ∆T = 100 K ). * VCNT

L/h

0.12

0.17

0.28

UD

FG-X 1.9668

Λ3 3.3697

3.8235

1.4518

1.6429

2.9636

3.3135

2.3484

1.3221

1.4689

2.6358

3.0233

1.9304

2.1738

1.2403

1.3973

2.4288

2.7911

1.0061

1.8128

2.0355

1.1557

1.2978

2.2747

2.6072

1.6090

1.9092

3.4948

3.7784

2.0520

2.4338

4.5132

4.6980

15

1.4139

1.6123

3.2412

3.4529

1.9006

2.1407

4.3362

4.1376

20

1.2639

1.4355

2.8585

3.0132

1.7066

1.9126

3.8187

3.4198

25

1.1525

1.3259

2.6742

2.8213

1.5625

1.7713

3.5673

3.1646

30

1.0662

1.2266

2.4403

2.5987

1.4508

1.6432

3.2708

2.8685

10

2.3028

2.5539

4.6201

5.1398

3.0789

3.3945

6.0523

6.8359

15

2.0031

2.2427

4.1224

4.4765

2.5912

2.8930

5.4003

5.9537

20

1.8352

1.9969

3.6498

4.0604

2.3740

2.5760

4.7812

5.4003

25

1.6953

1.8883

3.4987

3.8034

2.1930

2.4359

4.5832

5.0585

30

1.5889

1.7609

3.1874

3.5454

2.0554

2.2715

4.1754

4.7153

Λ1

Λ2

Λ3

Λ4

Λ1

Λ2

10

1.2935

1.4627

2.6273

2.9500

1.7232

15

1.1223

1.2736

2.3616

2.5666

20

1.0221

1.1387

2.0732

25

0.9588

1.0832

30

0.8934

10

Fig. 1. The geometry and global coordinate system of the pre-twisted FGCNTRC beams

33

Λ4

Fig. 2. The global and local coordinate system together with the rotation components of the pre-twisted beams

(a)

(b)

(c)

(d)

Fig. 3 (a)-(d): The different CNTs distributions in the thickness direction of the pre-twisted FGCNTRC beams (a) UD, (b) FG-V, (c) FG-X, (d) FG-O.

34

(a)

(b)

(c)

(d)

Fig. 4 (a)-(d): The effects of pre-twist angle ( φ0 ) on the first four frequency parameters of the

clamped-free per-twisted FG-CNTRC beams with different CNTs distribution ( β = 1, s=1, * VCNT = 0.28, ∆T = 0 K, L=10h).

35

(a)

(b)

(c)

(d)

Fig. 5 (a)-(d): The effects of pre-twist angle ( φ0 ) on the first four frequency parameters of

the fully clamped per-twisted FG-CNTRC beams with different CNTs distribution ( β = 1, * s=1, VCNT = 0.28, ∆T = 100 K , L=10h).

36

(a)

(b)

(c)

(d)

Fig. 6 (a)-(d): The effects of twist angle ( φ0 ) on the fundamental frequency parameters of

the per-twisted beams with FG-X distribution of CNTs subjected to different boundary * conditions (a) C-C, (b) C-S (c), S-S, (d) C-F ( β = 1, s=1, VCNT = 0.28, ∆T = 0 ).

37

(a)

(b)

(c)

(d)

Fig. 7 (a)-(d): The effects of length-to-thickness (L/h) in combination with the temperature rise

( ∆T ) on the frequency parameters of the fully clamped per-twisted FG-CNTRC beams with * FG-X distribution of CNTs ( β = 1, φ0 = 3π / 4, s=1, VCNT = 0.12 ).

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(a)

(b)

(c)

(d)

Fig. 8 (a)-(d): The effects of temperature dependence of material properties on the

fundamental frequency parameter of the fully clamped per-twisted FG-CNTRC beams with different distribution of CNTs (a) UD, (b) FG-V, (c) FG-O, (d) FG-X ( β = 1 , φ0 = 2π / 3, s=1, * L=10h, VCNT = 0.28 ).

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(a)

(b)

(c)

(d)

Fig. 9 (a)-(d): The effects of temperature dependence of material properties on the

fundamental frequency parameter of the clamped-simply supported per-twisted FGCNTRC beams with different distribution of CNTs (a) UD, (b) FG-V, (c) FG-O, (d) FG* X ( β = 1 , φ0 = π , s=1, L=10h, VCNT = 0.28 ).

40