Nonlinear vibration of functionally graded fiber reinforced composite laminated beams with piezoelectric fiber reinforced composite actuators in thermal environments

Engineering Structures 90 (2015) 183–192

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Nonlinear vibration of functionally graded fiber reinforced composite laminated beams with piezoelectric fiber reinforced composite actuators in thermal environments Hui-Shen Shen a,⇑, De-Qing Yang b a b

School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 26 December 2014 Revised 4 February 2015 Accepted 5 February 2015 Available online 9 March 2015 Keywords: Hybrid laminated beam Functionally graded materials Vibration Elastic foundation Thermo-piezoelectric effects

a b s t r a c t A large amplitude flexural vibration of a hybrid laminated beam resting on an elastic foundation in thermal environments is investigated. The hybrid laminated beam is consists of fiber reinforced composite (FRC) layers and piezoelectric fiber reinforced composite (PFRC) actuators. The fiber reinforcements are assumed to be distributed either uniformly (UD) or functionally graded (FG) of piece-wise type along the thickness of the beam. The motion equations are based on a higher order shear deformation theory and von Kármán strain displacement relationships. The beam-foundation interaction and thermopiezoelectric effects are also included. The material properties of both FRCs and PFRCs are estimated through a micromechanical model and are assumed to be temperature dependent. A two-step perturbation approach is employed to determine the nonlinear to linear frequency ratios of hybrid laminated beams. Detailed parametric studies are carried out to investigate effects of material property gradient, temperature variation, applied voltage, stacking sequence as well as the foundation stiffness on the linear and nonlinear vibration characteristics of the hybrid laminated beams. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Fiber reinforced composites (FRCs) possess better material properties than the conventional isotropic materials. Owing to their high strength-to-weight and stiffness-to-weight ratios, FRCs have been widely used in the aerospace, automobile, mechanical, and civil engineering fields. Piezoelectric fiber reinforced composites (PFRCs) have presented as the new class of smart materials. The important feature of PFRCs is that the monolithic piezoelectric fibers are longitudinally reinforced in the polymer matrix. PFRCs may be used as sensors and actuators for a new generation of smart adaptive structures. PFRCs provide the structures with selfmonitoring and self-controlling capabilities. Consequently, hybrid laminated structures where a FRC substrate is coupled with surface bonded or embedded PFRC actuator and/or sensor layers are becoming increasingly important. Many studies have been made on linear vibration and vibration control of functionally graded and laminated beams with integrated or patched piezoelectric sensors and actuators [1–8]. However, relatively few works have been done on the nonlinear vibration of ⇑ Corresponding author. E-mail address: [email protected] (H.-S. Shen). http://dx.doi.org/10.1016/j.engstruct.2015.02.005 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.

FRC laminated beams containing piezoelectric layers. Azrar et al. [9] and Przybylski [10] studied large amplitude free vibration of an isotropic beam with piezoelectric actuators bonded to its top and bottom surfaces based on the classical Euler–Bernoulli beam theory. Fu et al. [11] presented nonlinear free vibration and dynamic stability analyses of functionally graded ceramic–metal beams with surface bonded piezoelectric actuators by using the Galerkin method based on the classical Euler–Bernoulli beam theory. Also, Rafiee et al. [12] studied large amplitude free vibration of functionally graded carbon nanotube reinforced composite beams with surface bonded piezoelectric layers by using the Galerkin method based on the classical Euler–Bernoulli beam theory. Mareishi et al. [13] studied large static deflection, large amplitude free vibration and postbuckling of laminated beams with surface bonded PFRC layers. In their analysis, the formulations are based on the classical Euler–Bernoulli beam theory, and one end of the beam is assumed to be movable. Moreover, Youzera et al. [14] studied the nonlinear forced vibration of three-layered, symmetric laminated beams with PFRC core. In their analytical formulations, both normal and shear deformations are considered in the core by using the higher order zig-zag theory. In the above studies, the host laminates are usually assumed to be cross-ply type. For cross-ply laminated beams, the results obtained by different researchers seem to be a

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reasonable agreement. However, the analysis of anisotropic laminated beams is a complex task due to the bending-extension, flexural-twist and extension-twist couplings. There are unresolved discrepancies between the results obtained by different researchers for laminated beams having arbitrary lay-up [15–17]. There are two ways used in the literature to involve the bending-extension, flexural-twist and extension-twist couplings in one-dimensional beam theory. One way uses refined higher order shear deformation theories [18–22]. Another way uses refined beam stiffnesses [16,21,23,24]. Kapania and Racitij [25] and Singh et al. [26] presented nonlinear vibration analyses for unsymmetrically laminated beams by using FEM based on the refined classical lamination theory, first order shear deformation theory and higher order shear deformation theory, respectively. Gunda et al. [27] studied a large amplitude vibration of symmetric and asymmetric laminated composite beams with axially immovable ends by using the Ritz method based on the classical lamination theory. Li and Qiao [28] studied a large amplitude vibration of anisotropic laminated beams resting on a two-parameter elastic foundation based on a refined higher order shear deformation theory. In the aforementioned studies, however, the fiber reinforcements are usually assumed to be distributed uniformly in each ply and the fiber volume fraction does not vary spatially at the macroscopic level. Functionally graded materials (FGMs) are a new generation of composite materials in which the microstructural details are spatially varied through non-uniform distribution of the reinforcement phase. Two kinds of FGMs are designed to improve mechanical behavior of plate/beam structures. One is functionally graded unidirectional fibers reinforced composites [29–32]. Another one, like functionally graded ceramic–metal materials, is functionally graded particles reinforced composites [33–37]. The concept of functionally graded material can be utilized for the laminates by non-homogeneous distribution of fiber reinforcements into the matrix with a piece-wise type so that the mechanical behavior of laminated beams can be improved. In the present work, we focus our attention on the large amplitude vibration of FRC laminated beams with PFRC actuators resting on an elastic foundation in thermal environments. We choose two kinds of FRC laminated beams, i.e. uniformly distributed (UD) and functionally graded (FG) FRC laminated beams, in the current study. The motion equations are based on a higher order shear deformation theory and von Kármán-type nonlinear strain–displacement relationships and including the extension-twist, extension-flexural and flexural-twist couplings. The beam-foundation interaction and thermo-piezoelectric effects are also included, and the material properties of both FRCs and PFRCs are estimated through a micromechanical model and are assumed to be temperature dependent. Two ends of the beam are assumed to be simply supported and in-plane boundary conditions are assumed to be immovable. The effects of the fiber reinforcement phase, temperature variation, applied voltage, as well as foundation stiffness on the nonlinear vibration characteristics of hybrid FG-FRC laminated beams are discussed in detail. 2. Theoretical development Consider a FRC laminated beam with two pinned ends, which consists of N plies and one of which may be PFRC. Each ply may have different value of fiber volume fraction. The fiber reinforcement distribution is functionally graded of piece-wise type in the thickness direction, as reported in [30–32]. The beam has length L, width b, and thickness h. The beam is exposed to elevated temperature and is subjected to a transverse dynamic load Q combined with electric loads. The beam is referred to a coordinate system (X, Y, Z) in which X and Y are in the length and width directions of the beam and Z is in the direction of the downward normal to the

middle surface. Let U be the displacement in the longitudinal direction, and W be the deflection of the beam. Wx is the mid-plane rotation of the normal about the Y axis. The beam rests on a twoparameter elastic foundation. As is customary [28,34], the foundation is assumed to be a compliant foundation, which means that no part of the beam lifts off the foundation in the large amplitude vibration region. The load–displacement relationship of the foun2

2

dation is assumed to be p ¼ K 1 W  K 2 ðd W=dX Þ, where p is the force per unit area, K 1 is the Winkler foundation stiffness and K 2 is the shearing layer stiffness of the foundation. Several higher order shear deformation beam theories have been developed, and the major difference of these higher order shear deformation beam theories lies in that they use different shape functions in the displacement field. Levinson [38] and Reddy [39] assumed that the displacement field can be expressed by

! 4 3 @W   U 1 ¼ UðX; tÞ þ Z Wx ðX; tÞ  2 Z Wx þ ; @X 3h U 2 ¼ V ¼ 0; U 3 ¼ WðX; tÞ;

ð1Þ

where t represents time, U, W and Wx are unknowns. Based on this higher order shear deformation beam theory and von Kármán-type nonlinear strain–displacement relationships, the motion equations of a hybrid functionally graded FRC laminated beam, which includes the beam-foundation interaction and thermo-piezoelectric effect, can be derived and can be expressed by

S11

@4W

þ S12

@ 3 Wx

S21

@3W @X

3

¼ eI 3

1 Nx ¼ L

@X

þ S22

3

þ

B11 @ 2 NP

@ 2 Wx @X

2

2

þ

@ 2 MP

 S23 Wx þ

2

þ Nx

@2W

A11 @X @X 2 @X ! @2W @2W @ 3 Wx 4 @4W ¼ I1 2 þ bI 5 þ Q  K1W  K2  2 bI 7 2 2 2  @X@ t 3h @X @t2 @t @X @X

4

@W @X

!  S26

@ Wx 4 @3W  2 eI 5 2  @t @X@t2 3h

ð2Þ

@NP @SP þ @X @X

2

ð3Þ

82 9 !2 !3 Z L< = 2 A @W @ W 4 @ W @ W 11 x x P 4 5  N dX þ B11  2 E11 þ 2 ; 2 @X @X 3h @X @X 0 : ð4Þ

in which

S11 ¼ 

4 2

3h

F 11 

B11 E11 A11

! ;

  4 B11  2 E11 ; A11 3h 3h "  # 4 4 E11 4 S21 ¼  2 F 11  2 H11  B11  2 E11 ; A11 3h 3h 3h      2 4 4 4 1 4 S22 ¼ D11  2 F 11  2 F 11  2 H11  B11  2 E11 ; A11 3h 3h 3h 3h     4 4 4 B11 4 E11 S23 ¼ A55  2 D55  2 D55  2 F 55 ; S26 ¼  ; ð5Þ A11 3h2 A11 h h h S12 ¼ D11 

4

2

F 11 

B11

where A11 , B11 , etc., are the beam reduced stiffnesses and can be expressed by [16,21]

ðA11 ; B11 ; D11 ; E11 ; F 11 ; H11 Þ ¼ b

N Z X k¼1

hk

hk1

e 11 Þ ð1; Z; Z 2 ; Z 3 ; Z 4 ; Z 6 ÞdZ ðQ k ð6Þ

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H.-S. Shen, D.-Q. Yang / Engineering Structures 90 (2015) 183–192

ðA55 ; D55 ; F 55 Þ ¼ b

N Z X k¼1

hk hk1

e 55 Þ ð1; Z 2 ; Z 4 ÞdZ ðQ k

ð7Þ

e 11 and Q e 55 are the refined transformed elastic constants, in which Q defined by 2

2

e 11 ¼ Q 11 þ ðQ 16 Þ Q 22  2Q 12 Q 16 Q 26 þ ðQ 12 Þ Q 66 ; Q 2 ðQ 26 Þ  Q 22 Q 66

where DT = T  T0 is the temperature rise from the reference temperature T0 at which there are no thermal strains. For beam-like piezoelectric material, only the transverse electric field component EZ is dominant, and EZ is defined as EZ = U,Z, where U is the potential field. If the voltage applied to the actuator is in the thickness only, then [40]

EZ ¼

Vk hp

ð17Þ

2

e 55 ¼ Q 55  ðQ 45 Þ ; Q Q 44

ð8Þ

e 11 ¼ Q 11 and Q e 55 ¼ Q 55 when only 0- and 90It is noted that Q plies are considered, and Q ij are the transformed elastic constants, defined by

2

Q 11

3

2

c4

7 6 6 6 Q 12 7 6 c2 s2 7 6 6 6 Q 7 6 s4 6 22 7 6 7¼6 6 6 Q 16 7 6 c3 s 7 6 6 7 6 3 6 4 Q 26 5 4 cs Q 66 c 2 s2 2

Q 44

3

2

2c2 s2

s4

4c2 s2

c 4 þ s4

c 2 s2

4c2 s2

3

72 3 7 Q 11 7 76 Q 7 2c2 s2 c4 4c2 s2 76 12 7 7 3 3 3 2 2 76 cs  c s cs 2csðc  s Þ 74 Q 22 5 7 c3 s  cs3 c3 s 2csðc2  s2 Þ 7 5 Q 66 2 2c2 s2 c 2 s2 ðc2  s2 Þ

ð9Þ

E11 ; ð1  m12 m21 Þ m21 E11 ¼ ; ð1  m12 m21 Þ

Q 12

Q 22 ¼

ð10Þ

Q 55 ¼ G13 ;

Q 66 ¼ G12

ð11Þ

and c = cos h, s = sinh where h is the lamination angle with respect to the beam X-axis. In Eqs. (2) and (3), the inertias Ii (i = 1, 2, 3, 4, 5, 7) are defined by

ðI1 ; I2 ; I3 ; I4 ; I5 ; I7 Þ ¼

N Z X

hk

hk1

k¼1

qk ð1; Z; Z 2 ; Z 3 ; Z 4 ; Z 6 ÞdZ

ð12Þ

and

4

8

16

I ; I3 ¼ I3  2 I5 þ 4 I7 ; eI 3 2 4 3h 3h 9h I2 I2 4 I2 I4 e ¼ I3  ; I5 ¼ I5  2 I7 ; I 5 ¼ I5  ; bI 5 I1 I1 3h 4 I4 I4 4 ¼ eI 3 þ 2 eI 5 ; eI 7 ¼ I7  ; bI 7 ¼ eI 5 þ 2 eI 7 ; I 1 3h 3h

I2 ¼ I2 

N Z X k¼1

4 3h

2

tk tk1

½Bx k ð1; Z; Z 3 Þ

Vk dZ hp

ð18Þ

PE

ð19Þ

In Eqs. (15) and (18)

Ax ¼ Q 11 ðc2 a11 þ s2 a22 Þ þ Q 12 ðs2 a11 þ c2 a22 Þ þ Q 16 2csða11  a22 Þ

ð21Þ where a11 and a22 are the thermal expansion coefficients in the X and Y directions for kth ply, d31 and d32 are the piezoelectric strain constants of kth ply, and can be obtained by [40]

2

E22 ; ð1  m12 m21 Þ

Q 44 ¼ G23 ;

PE  ¼

ð20Þ

where

Q 11 ¼

ME

SE ¼ M E 

s2

6 7 6 4 Q 45 5 ¼ 4 cs s2 Q 55

½ NE

Bx ¼ Q 11 ðc2 d31 þ s2 d32 Þ þ Q 12 ðs2 d31 þ c2 d32 Þ þ Q 16 2csðd31  d32 Þ

3   7 Q 44 cs 5 Q 55 c2

c2

where Vk is the applied voltage across the kth ply, and hp is the thickness of the PFRC layer, and

3 2 32 0 0 e31 Q 11 0 0 d31 6 7 6 76 0 0 e ¼ 4 4 0 0 d32 5 4 Q 12 32 5 0 0 0 k 0 0 0 k 0

Q 12 Q 22 0

0

3

7 0 5 Q 66 k

ð22Þ

The effective material properties for both FRCs and PFRCs can be expressed in terms of a micromechanical model by the rule of mixture, such that [41] f E11 ¼ V f E11 þ V m Em

ð23Þ

f f m2f Em =E22 þ m2m E22 =Em  2m f mm Vf 1 Vm ¼ f þ m  Vf Vm f E22 E22 E V f E22 þ V m Em

ð24Þ

Vf Vm 1 ¼ þ Gij G f Gm ij

ð25Þ

ðij ¼ 12; 13 and 23Þ

m12 ¼ V f m f þ V m mm

ð26Þ

In Eqs. (2)–(4) the equivalent thermo-piezoelectric loads are defined by

q ¼ V f q f þ V m qm

ð27Þ

2

f f f f f where E11 , E22 , G12 , G13 , G23 , mf and qf are the Young’s moduli, shear moduli, Poisson’s ratio and mass density, respectively, of the fiber, while Em, Gm, mm and qm are the corresponding properties for the matrix, respectively. Vf and Vm are the fiber and matrix volume fractions and must satisfy the unity condition of Vf + Vm = 1. Similarly, the effective thermal expansion coefficients in the longitudinal and transverse directions may be written by

NP

3

2

NT

3

2

NE

3

6 MP 7 6 MT 7 6 ME 7 6 7 6 7 6 7 6 P 7¼6 T 7þ6 E 7 4P 5 4P 5 4P 5 SP

ST

ð13Þ

ð14Þ

SE

The beam is considered to be at an isothermal state and the temperature field can be assumed uniformly distributed in the beam. The forces, moments and higher order moments caused by elevated temperature are defined by

½ NT

M

T

T

P ¼

N Z X k¼1

T

T

S ¼M 

4 3h

2

P

T

tk

t k1

½Ax k ð1; Z; Z 3 ÞDT dZ

ð15Þ

ð16Þ

a11 ¼

f f V f E11 a11 þ V m Em am f V f E11 þ V m Em

f a22 ¼ ð1 þ m f ÞV f a22 þ ð1 þ mm ÞV m am  m12 a11

ð28Þ

ð29Þ

f f where a11 , a22 and am are thermal expansion coefficients of the fiber and the matrix respectively, and the effective piezoelectric moduli e31 and e32 can be expressed by [42]

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H.-S. Shen, D.-Q. Yang / Engineering Structures 90 (2015) 183–192

f f m f f e31 ¼ V f e31  ðV m V f =HÞfðC 13  Cm 13 Þ½ðV m C 22 þ V f C 22 Þe33



f ðV m C 23

þ

f V f Cm 23 Þe31 

þ

f ðC 12



f Cm 12 Þ½ðV m C 33

k¼1

f m f f þ V f Cm 33 Þe31  ðV m C 23 þ V f C 23 Þe33 g

ð30Þ

f f f f f e32 ¼ e31 þ ðV m =HÞfC 22 ½ðV m C 23 þ V f Cm 23 Þe33  ðV m C 33 f f m f f f þ V f Cm 33 Þe31   C 23 ½ðV m C 22 þ V f C 22 Þe33  ðV m C 23 f þ V f Cm 23 Þe31 g

ð31Þ

in which 2

f f m f m H ¼ ðV m C 22 þ V f Cm 22 ÞðV m C 33 þ V f C 33 Þ  ðV m C 23 þ V f C 23 Þ

ð32Þ

f f and e31 and e33 are the piezoelectric coefficients of the fiber, and C ijf

tively. The relation between

(i, j = 1–6) and

f E11 ,

f E22 ,

f G12 ,

f G13

and

f G23 can be found in Reddy [39] and other textbooks. It is assumed that the material property of matrix C m ij (i, j = 1–6) is a function of temperature, so that all effective material properties of FRCs and PFRCs are temperature dependant. It is worth noting that Eq. (4) is only valid in the case of the two ends of the beam are immovable, that is, UðX ¼ 0Þ ¼ UðX ¼ LÞ ¼ 0. Such end condition is acceptable in the nonlinear bending or nonlinear vibration analysis of beams, but is unacceptable for the compressive postbuckling analysis, even though the two ends of the beam are assumed to be clamped, as previously mentioned in Shen [43].

3. Solution procedure Before carrying out the solution process, it is convenient to first define the following dimensionless quantities for such beams

X W 1 L2 N x x ¼ p ; W ¼ ; Wx ¼ Wx ; Nx ¼ ; L L p p2 D11 !   L2 4 L4 L4 ðMx ; P x Þ ¼ M x ; 2 P x ; ðK 1 ;k1 Þ ¼ K 1 ; ; p2 hD11 p4 D11 E0 I 3h ! L2 L2 1 ; ðS11 ;S12 ;S21 ;S22 Þ; ; ðc11 ; c12 ; c21 ; c22 Þ ¼ ðK 2 ; k2 Þ ¼ K 2 p2 D11 E0 I D11 L2 A11 L 4 4 ; ðc ; c Þ ¼ ðB11  2 E11 ; 2 E11 Þ; p p2 D11 14 15 p2 D11 3h 3h   1 4 bI1 E0 L2 B11 ;B11  2 E11 ; c17 ¼  ; ðc16 ; c26 Þ ¼ p2 q0 D11 A11 L 3h   4 4 bE0 ; ðc18 ; c19 ; c28 ; c29 Þ ¼  bI 5 ; 2 bI 7 ; eI 3 ;  2 eI 5 q0 D11 3h 3h sffiffiffiffiffiffi rffiffiffiffiffiffi pt E0 L q0 L2 ATx ; ; xL ¼ XL t¼ ; cT1 ¼ L q0 p E0 p2 D11   L2 4 L2 BEx DTx ; 2 F Tx ; cp1 ¼ ; ðcT3 ; cT6 Þ ¼ 2 p hD11 p2 D11 3h   L2 4 QL3 DEx ; 2 F Ex ; kq ¼ 4 ; ð33Þ ðcp3 ; cp6 Þ ¼ 2 p EI p hD11 3h

c23 ¼

L2

2D 11

S23 ; c13 ¼

in which the alternative forms k1 and k2 are not needed until the numerical examples are considered. E0 and q0 are the reference values of Em and qm at the room temperature (T0 = 25 °C), and ATx , BEx , etc., are defined by

ðATx ; DTx ; F Tx ÞDT ¼

N Z X k¼1

tk

t k1

½Ax k ð1; Z; Z 3 ÞDT dZ

ð34Þ

tk

t k1

½Bx k ð1; Z; Z 3 Þ

Vk dZ hp

ð35Þ

where Ax and Bx are given in detail in Eqs. (20) and (21). The Eqs. (2)–(4) may then be rewritten in the following dimensionless form (Z " # ) p @4W @3W c13 @W 2 @ Wx @2W @2W c11 4  c12 3 x  p ð Þ þ c14  c15 2 dx @x 2 @x @x @x @x @x2 0 ! @2W @ 2 NP @ 2 MP @2W þ C 1 2  c16  C2 ¼ kq  K 1 W  K 2 2 @x2 @x2 @x @x

þ c17

and C m ij are the elastic constants of the fiber and the matrix, respecC ijf

N Z X

ðBEx ; DEx ; F Ex ÞDV ¼

c21

@2W @ 3 Wx @4W þ c18 þ c19 2 2 2 2 @x@t @t @x @t

ð36Þ

  @3W @ 2 Wx @W @NP @SP  c26  c þ c W þ  C3 x 22 23 3 2 @x @x @x @x @x

¼ c28

@ 2 Wx @3W þ c 29 @x@t 2 @t2

ð37Þ

where

C 1 ¼ cT1 DT þ cp1 DV; C 2 ¼ cT3 DT þ cp3 DV; C 3 ¼ ðcT3  cT6 ÞDT þ ðcp3  cp6 ÞDV;

ð38Þ

Eqs. (36) and (37) are the governing equations for hybrid FG-FRC laminated beams with immovable end conditions, and are adopted in the following nonlinear vibration analysis. It is worth noting that Eq. (36) is a weakly nonlinear equation that does not contain large rotation effect and, therefore, is inadequate for nonlinear analysis of cantilever beams. For nonlinear vibration problem, we seek for the relationship between the vibration amplitude and the frequency. We assume that the solutions of Eqs. (36) and (37) can be expressed as

f ðx; tÞ ¼ W  ðxÞ þ Wðx; tÞ; W

e x ðx; tÞ ¼ W ðxÞ þ Wx ðx; tÞ; W x

ð39Þ

where W⁄(x) is an initial deflection due to initial thermo-piezoelectric bending moment, and W(x, t) is an additional deflection. Wx ðxÞ is the mid-plane rotation corresponding to W⁄(x). Wx(x, t) is defined analogously to Wx ðxÞ, but is for W(x, t). Note that W⁄(x) = Wx ðxÞ ¼ 0 when DT = DV = 0. Substituting Eqs. (39) into (36) and (37), we obtain two sets of equations that can be solved in sequence. The first set of equations yields the particular solution of static deflection due to thermo-piezoelectric bending stresses, and the second set of equations gives the homogeneous solution of vibration characteristics on the initial deflected beam. This homogeneous solution can be determined by means of a two-step perturbation technique, for which the small perturbation parameter has no physical meaning at the first step, and is then replaced by a dimensionless vibration amplitude at the second step. In the present case, we assume that

Wðx; s; eÞ ¼

X

e j wj ðx; sÞ;

j¼1

Wx ðx; s; eÞ ¼

X

X

j¼1

j¼1

e j Wxj ðx; sÞ; kq ðx; s; eÞ ¼

e j kj ðx; sÞ;

ð40Þ

where e is a small perturbation parameter, and s ¼ et is introduced to improve perturbation procedure for solving nonlinear vibration problem. In such a case the motion equations may be re-written as

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H.-S. Shen, D.-Q. Yang / Engineering Structures 90 (2015) 183–192

@4W @ 3 Wx  c12 4 @x3 @x( # ) !  Z p"  c13 @W 2 @ Wx @2W @2 W @2 W  p þ c14  c15 2 dx þ @x @x2 2 @x @x @x2 0 (Z " ! # )     2 2 p c13 @W @W @W @ Wx @ W @2W 2 þ c14 dx þ  c15 p 2 2 @x @x @x @x @x @x2 0 ! ! @2W @2W @ 3 Wx @4 W ð41Þ ¼ kq  K 1 W  K 2 2 þ e2 c17 þ c18 þ c19 2 2 @x@ s2 @x @ s2 @x @ s

c11



@3W @ 2 Wx @W  c22 þ c23 Wx þ @x3 @x @x2 ! 2 3 @ Wx @ W þ c29 ¼ e2 c28 @x@ s2 @ s2

c21



ð42Þ

ð43Þ

All of the major steps of the solution methodology will be described below and more details may be found in Shen [44]. Substituting Eqs. (40) into (41) and (42), collecting the terms of the same order of e, a set of perturbation equations is obtained. It is assumed that ð1Þ

w1 ðx; sÞ ¼ A10 ðsÞ sin mx

ð44Þ

and the initial deflection is assumed to be

W  ðxÞ ¼ A10 sin x

ð45Þ

By using Eqs. (44) and (45) to solve Eqs. (41) and (42) step by step, we obtain asymptotic solutions ð1Þ

Wðx; t; eÞ ¼ eA10 ðtÞ sin mx þ Oðe4 Þ

ð46Þ

3 ð3Þ 4 Wx ðx; t; eÞ ¼ eBð1Þ 10 ðtÞ cos mx þ e B10 ðtÞ cos mx þ Oðe Þ

ð47Þ

€ ð1Þ ðtÞg þ eAð1Þ ðtÞg  sin mx þ ½ðeAð1Þ ðtÞÞ2 g kq ðx; t; eÞ ¼ ½eA 30 31 321 10 10 10 ð1Þ

3

þ ðeA10 ÞðeA10 Þg 322  sin mx þ ½ðeA10 ðtÞÞ g 33   sin mx þ Oðe4 Þ

ð48Þ

Note that in Eqs. (46)–(48) s is replaced back by t. For free vibration problem kq ¼ 0. Applying Galerkin procedure to Eq. (48), one has 2

g 30

ð1Þ

d ðeA10 Þ dt

2

ð1Þ

ð1Þ 2

  2   3 k G20 k k   C 33 þ  G10 G10 G10 G10

ð51Þ

where k ¼

4

p





cT3  ðcT3  cT6 Þ







c11  c12



c12 c12 DT þ cp3  ðcp3  cp6 Þ DV ; c22 þ c23 c22 þ c23



c21  c23 þ ðK 1 þ K 2 Þ  ðcT1 DT þ cp1 DVÞ; c22 þ c23    2 c  c23 p2 G30 G20 G20 ¼ 2p c15  c14 21 2 ; G30 ¼ c13 ; C 33 ¼ c22 þ c23 4 G10 G10 G10 ¼

2

  @W  @ Wx  ¼ 0; Wx js¼0 ¼ ¼ 0;  @ s s¼0 @ s s¼0

ð1Þ



ð52Þ

The solution of Eq. (49) may be expressed by

The boundary conditions are assumed to be simply supported and the initial conditions are assumed to be

Wjs¼0 ¼

U ¼ eA10 ¼

ð1Þ 3

þ g 31 ðeA10 Þ þ g 32 ðeA10 Þ þ g 33 ðeA10 Þ ¼ 0

ð49Þ

where

  c m2  c23 g 30 ¼ c17 þ m2 c19 þ c18 21 2 c22 m þ c23   4 c21 m c21 m2  c23 c þ c ; þ 29 28 c22 m2 þ c23 c22 m2 þ c23   c m2  c23 þ ðK 1 þ K 2 m2 Þ g 31 ¼ m4 c11  c12 21 2 c22 m þ c23     c m2  c23 c c U þ 2pm2 c15  c14 21 23 U; þ 2pm c15  c14 21 2 c22 m þ c23 c22 þ c23   c21 m2  c23 p2 p2 3 ð50Þ  C 4 c13 U; g 33 ¼ m4 c13 ; g 32 ¼ 2pm c15  c14 2 c22 m þ c23 4 4 in which C4 = 3 (for m = 1) and C4 = m2 (for m – 1), and

xNL

9g g  10g 232 W m ¼ xL 41 þ 31 33 2 L 12g 31

!2 31=2 5

ð53Þ

where xL = [g31/g30]1/2 is the dimensionless linear frequency, and W m =L is the dimensionless amplitude of the beam. According to Eq. (33), the corresponding linear frequency can be expressed as XL = xL(p/L)(E0/q0)1/2, where E0 and q0 are defined as in Eq. (33). 4. Numerical results and discussion Numerical results are presented in this section for (0/90)2S symmetric cross-ply, (0/90)4T unsymmetric cross-ply, (45/45)2S symmetric angle-ply and (45/45)4T antisymmetric angle-ply laminated beams with fully covered or embedded PFRC actuators resting on elastic foundations. Four types of host FG-FRC laminated beams are considered. For Type V, the fiber volume fractions are assumed to have graded distribution [0.8/0.75/0.7/0.65/0.55/0.5/ 0.45/0.4] for eight plies, referred to as FG-V. For Type K the distribution of fiber reinforcements is inversed, i.e. [0.4/0.45/0.5/0.55/ 0.65/0.7/0.75/0.8], referred to as FG-K. For Type X, a mid-plane symmetric graded distribution of fiber reinforcements is achieved, i.e. [0.8/0.7/0.5/0.4/0.4/0.5/0.7/0.8], and for type O the fiber volume fractions are assumed to have [0.4/0.5/0.7/0.8/0.8/0.7/0.5/0.4], referred to as FG-X and FG-O, respectively. A uniformly distributed (UD) FRC laminated beam with the same thickness is also considered as a comparator for which the fiber volume fraction of each ply is identical and Vf = 0.6. In such a way, the two cases of UD- and FG-FRC laminated beams will have the same value of total fraction of fiber reinforcements. For the sake of brevity, (0/90)2S symmetric cross-ply, (0/90)4T unsymmetric cross-ply, (45/45)2S symmetric angle-ply and (45/45)4T antisymmetric angle-ply laminated beams with a piezoelectric layer bonded at the top surface or embedded at the middle surface are referred to as (p/(0/90)2/(90/0)2), ((0/90)2/p/(90/0)2), (p/(0/90)4), ((0/90)2/p/(0/ 90)2), (p/(45/45)2/(45/45)2), ((45/45)2/p/(45/45)2), (p/(45/ 45)4) and ((45/45)2/p/(45/45)2), respectively. For all cases discussed below, the host beam is made of FRCs with polymer matrix. The material properties of fibers are assumed f f ¼ 233:05 GPa;E22 ¼ to be anisotropic and are taken to be [45]: E11 f f f ¼ 8:96 GPa, mf = 0.2, a11 ¼ 0:54  106 = C, a22 ¼ 23:1 GPa;G12

10:08  106 = C and qf=1750 kg/m3. Unlike in [1–12], the PFRC actuator is used instead of integrated or patched piezoelectric actuator in the present analysis. PZT-5A is selected for the piezoelectric fiber and the material properties of which f f f f f are [46]: C 11 ¼ C 22 ¼ 121 GPa;C 33 ¼ 111 GPa;C 12 ¼ 75:4 GPa;C 13 ¼ f f f f f f C 23 ¼ 75:2 GPa, C 44 ¼ C 55 ¼ 21:1 GPa;C 66 ¼ 22:6 GPa;e31 ¼ e32 ¼ 5:4 f f f c=m2 , e33 ¼ 15:8 c=m2 ; a11 ¼ a22 ¼ 1:5  106 = C and qpf ¼ 7700 3 kg=m . The fiber volume fraction of FRRC layer is taken to be Vf = 0.9 enable the piezoelectric effect being about largest [42].

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It is worth noting that the present solution is also valid for the case of monolithic piezoelectric layer, if only the fiber volume fraction Vf in Eq. (18) is taken to be Vf = 1, as previously reported in Huang and Shen [47]. The material properties of matrix are assumed to be m m in which T = T0 + DT Cm 11 ¼ C 22 ¼ C 33 ¼ ð5:407  0:0047TÞ GPa, m m m and T0 = 25 °C (room temperature), C m 12 ¼ C 13 ¼ C 23 ¼ 0:515C 11 ; m m m m m C 44 ¼ C 55 ¼ C 66 ¼ 0:242C 11 ; qm ¼ 1200 kg=m3 and a = 45.0  (1 + 0.001DT)  106/°C. As part of the validation of the present method, the first four dimensionless natural frequencies of (h/h)S symmetric laminated beams with different lamination angles are calculated in Table 1 and are compared with the theoretical solutions of Chandrashekhara et al. [48] based on the first order shear deformation theory with the shear correction factor taken to be 5/6, the FEM results of Vo and Thai [20] based on a refined shear deformation theory, the Ritz method results of Aydogdu [49] based on a higher order shear deformation theory, and a semi-analytical– numerical method results of Qu et al. [22] based on a higher order shear deformation theory. As a second example, the first four dimensionless natural frequencies of (30/50/50/30)T unsymmetric angle-ply laminated beams are calculated in Table 2 and are compared with the state-space-based differential quadrature method results of Chen et al. [15]. The geometric parameters and material properties adopted in Tables 1 and 2 are: L = 0.381 m, b = h = 0.0254 m, E11 = 144.8 GPa, E22 = 9.65 GPa, G12 = G13 = 4.14 GPa, G23 = 3.45 GPa, m12 = 0.3 and q = 1389.23 kg/m3. The pffiffiffiffiffiffiffiffiffiffiffiffiffi dimensionless frequency is defined by x ¼ XðL2 =hÞ q=E11 . As a third example, the nonlinear to linear frequency ratios xNL/xL at W m =h ¼ 1:0 for a (45/30/30/60/45/60)T unsymmetric angleply laminated beam are calculated in Table 3 and are compared with the Ritz method results of Gunda et al. [27] based on the Euler–Bernoulli beam theory, the approximate analytical solution of Emam [50] based on the Euler–Bernoulli beam theory, and the perturbation method results of Li and Qiao [28] based on a refined shear deformation theory. The geometric parameters and material properties adopted are: L = 0.25 m, b = 0.01 m, h = 0.01 m, E11 = 155 GPa, E22 = 12.1 GPa, G12 = G13 = 4.4 GPa, G23 = 2.1 GPa, m12 = 0.248 and q = 1570 kg/m3. As a last example, the curves of nonlinear to linear frequency ratios xNL/xL as function of dimensionless amplitudes of (45/45)4T and (90/0)2S laminated beams are plotted in Fig. 1 and are compared with the perturbation method results of Li and Qiao [28]. The computing data adopted are: E11 = 37.41 GPa, E22 = 13.67 GPa, G12 = 5.478 GPa, G13 = 6.03 GPa, G23 = 6.666 GPa, m12 = 0.3, q = 1968.9 kg/m3, L/h = 10 and h = 0.008 m. Note that in these four examples the material properties are assumed to be independent of temperature. These comparison studies confirm that there are unresolved discrepancies between the results obtained by different authors for angleply laminated beams by using different methods based on the different theories. It can be found that, for the cases of cross-ply laminated beams, the present results are identical to the existing results, whereas in most cases of angle-ply laminated beams, the present results are compared well with the existing results. A parametric study has been carried out and typical results are shown in Tables 4–7 and Figs. 2–5. For these examples, the

Table 2 Comparisons of nondimensional frequencies x for a (30/50/50/30)T angle-ply laminated beam. Source

x1

x2

x3

x4

Chen et al. [15] Present

0.9790 0.9726

3.8585 3.7617

8.4823 8.0326

14.6309 13.3651

Table 3 Comparisons of nonlinear to linear frequency ratios xNL/xL for a (45/30/30/60/45/ 60)T laminated beam at W m =h ¼ 1:0. Source

xNL/xL

Gunda et al. [27] Emam [50] Li and Qiao [28] Present

1.6561 1.8131 2.4378 1.8562

geometric parameters are taken to be L/h = 10, b = 0.06 m, and the total thickness of the beam h = 0.0509 m, whereas the thickness of piezoelectric layers is 0.1 mm. The dimensionless natural pffiffiffiffiffiffiffiffiffiffiffiffiffi ~ ¼ XðL2 =hÞ q =E0 , where q0 and E0 are frequency is defined by X 0

the reference values of qm and Em at DT = 0. The boundary conditions are pinned–pinned and two ends are immovable. Table 4 presents the first four dimensionless natural frequencies of (45/45)4T, (0/90)4T and (0/90)2S laminated beams with surface-bonded or embedded PFRC layers for four types of FG distribution of fiber reinforcements. The results for the same beam with UD distribution of fiber reinforcements are also listed for direct comparison. It can be seen that, for all the cases, the beam of FG-X type has the largest, whereas the beam of FG-O type has the lowest natural frequencies among the five beams. Table 5 shows the effect of temperature rise along with the foundation stiffness on the fundamental frequencies of (0/90)2S laminated beams with surface bonded or embedded PFRC layers resting on elastic foundations in thermal environments. Three sets of thermal environmental conditions, i.e. DT = 0, 50 and 100 °C, are considered. Two foundation models are considered. The stiffnesses are (k1, k2) = (100, 10) for the Pasternak elastic foundation, (k1, k2) = (100, 0) for the Winkler elastic foundation and (k1, k2) = (0, 0) for the beam without any elastic foundation. Then Table 6 shows the effect of applied voltage along with the foundation stiffness on the fundamental frequencies of (45/45)4T and (0/ 90)4T laminated beams with embedded PFRC layers resting on elastic foundations. Three sets of applied voltage, i.e. DV = 200, 0 and +200 V, are considered. From Tables 5 and 6, it can bee seen that the fundamental frequencies are reduced with increase in temperature. It can also be seen that, for the ((45/45)2/p/(45/45)2) beam, the negative applied voltage increases, whereas the positive applied voltage decreases fundamental frequencies, except for the same beam of FG-V type. In contrast, for the ((0/90)2/p/(0/90)2) beam, the negative applied voltage decreases, whereas the positive applied voltage increases fundamental frequencies, except for the same beam of FG-K type. Table 7 shows foundation stiffness on the nonlinear to linear frequency ratios xNL/xL for (45/45)4T and (0/90)4T laminated

Table 1 Comparisons of nondimensional frequencies x of (h/h)S symmetric laminated beams. Source

0

15

30

45

60

75

90

Chandrashekhara et al. [48] Vo and Thai [20] Qu et al. [22] Aydogdu [49] Present

2.6560 2.6494 2.6297 2.651 2.6484

2.5105 2.4039 1.8773 1.896 1.5795

2.1032 1.5540 1.2251 1.141 0.9988

1.5368 0.9078 0.8745 0.804 0.7960

1.0124 0.7361 0.7583 0.736 0.7313

0.7611 0.7247 0.7325 0.725 0.7248

0.7320 0.7395 0.7302 0.729 0.7297

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H.-S. Shen, D.-Q. Yang / Engineering Structures 90 (2015) 183–192

7 6

1: (45/-45)4T 2: (90/0)2S

5

ωNL/ωL

Table 5 pffiffiffiffiffiffiffiffiffiffiffiffiffi ~ ¼ XðL2 =hÞ q =E0 for (0/90)2S laminated beams with Fundamental frequencies X 0 surface-bonded or embedded PFRC layers resting on elastic foundations in thermal environments (L/h = 10).

L/h=10, h = 0.008 m

Lay-up

4

(p/(0/90)2/(90/0)2)

0 50 100 ((0/90)2/p/(90/0)2) 0 50 100

3

1: Present 1: Li & Qiao [28] 2: Present 2: Li & Qiao [28]

2 1 0

DT (°C) UD

0

1

2

3

(p/(0/90)2/(90/0)2)

4

W/h Fig. 1. Comparisons of frequency–amplitude curves for (45/45)4T and (90/0)2S laminated beams.

0 50 100 0 ((0/90)2/p/(90/0)2) 50 100

(p/(0/90)2/(90/0)2)

Table 4 pffiffiffiffiffiffiffiffiffiffiffiffiffi e ¼ XðL2 =hÞ q =E0 for FRC laminated beams with surfaceNatural frequencies X 0 bonded or embedded PFRC layers (L/h = 10). e1 X

Lay-up (p/(45/45)4)

e2 X

e3 X

e4 X

UD FG-V FG-K FG-X FG-O

3.7675 3.8238 3.8633 4.3125 3.4848

14.0255 14.2694 14.3922 15.6724 13.2139

28.4657 29.0433 29.2344 30.9628 27.4389

44.9701 45.9998 46.2206 47.8378 44.2794

UD FG-V FG-K FG-X FG-O

4.1630 4.3897 4.1502 4.7567 3.8520

15.7684 16.5776 15.7595 17.7719 14.7366

32.6870 34.2341 32.7759 36.2104 30.9606

52.6325 54.9239 52.9495 57.3697 50.5371

(p/(0/90)4)

UD FG-V FG-K FG-X FG-O

9.8779 10.0082 9.5661 10.4155 9.2758

29.5376 30.4770 28.8849 29.8519 29.5244

50.6980 52.6009 49.8106 50.2045 52.4473

72.1030 74.6239 70.9592 70.7430 75.7791

((0/90)2/p/(0/90)2)

UD FG-V FG-K FG-X FG-O

11.1964 11.3211 10.7730 11.9814 10.3208

35.7026 36.6203 34.5861 37.0553 34.3428

63.4558 65.5978 61.7090 64.6662 62.8084

91.6567 94.9785 89.2489 92.5400 92.1074

(p/(0/90)2/(90/0)2)

UD FG-V FG-K FG-X FG-O

11.0438 10.9238 10.9331 11.6555 10.2516

31.2800 31.5412 31.5575 31.4775 31.1742

52.5930 53.2580 53.2791 51.9874 54.1067

74.5488 75.2452 75.2747 73.2486 77.5916

UD FG-V FG-K FG-X FG-O

12.3164 11.9711 12.1996 13.4045 10.9324

37.6617 37.6608 37.5072 39.2813 35.4986

65.4217 66.2534 65.2858 66.7999 63.8965

93.6221 94.9655 93.4111 94.7066 93.0396

((45/45)2/p/(45/45)2)

((0/90)2/p/(90/0)2)

beams with embedded PFRC layers resting on elastic foundations. As expected, the fundamental frequency is increased, but the nonlinear to linear frequency ratio is decreased with increase in foundation stiffness. Fig. 2 presents the frequency–amplitude curves of the four types of FG (p/(45/45)4) beams and compares with its UD comparator. It can be seen that the frequency–amplitude curves of FG-V and FG-K (p/(45/45)4) beams are very close. It can also be seen that the FG-O (p/(45/45)4) beam has the highest, whereas the FG-X (p/(45/45)4) beam has the lowest frequency–amplitude curve among the four FG (p/(45/45)4) beams. Hence, in the

0 50 100 ((0/90)2/p/(90/0)2) 0 50 100

FG-K

FG-X

FG-O

(k1, k2) = (0, 0) 11.0438 10.9238 10.9718 10.1981 10.8950 9.7636 12.3164 11.9711 12.1779 10.3281 11.9769 7.8443

FG-V

10.9331 10.2011 9.7691 12.1996 11.8748 11.3772

11.6555 11.5644 11.4673 13.4045 13.3101 13.1971

10.2516 10.2051 10.1555 10.9324 10.6143 10.4185

(k1, k2) = (100, 0) 11.3343 11.2175 11.2643 10.5545 11.1895 9.9638 12.6134 12.2750 12.4833 10.7917 12.2981 8.5494

11.2265 10.5578 9.9666 12.5010 12.2009 11.7387

11.9312 11.8422 11.7474 13.6765 13.5854 13.4773

10.5639 10.5188 10.4707 11.2678 10.9819 10.6307

(k1, k2) = (100, 10) 11.6139 11.49999 11.5456 10.8904 11.4727 10.2411 12.8999 12.5678 12.7769 11.2169 12.6055 9.1650

11.5087 10.8938 10.2423 12.7914 12.5126 12.0858

12.1972 12.1102 12.0175 13.9398 13.8515 13.7479

10.8634 10.8195 10.7728 11.5893 11.3299 10.9457

Table 6 pffiffiffiffiffiffiffiffiffiffiffiffiffi ~ ¼ XðL2 =hÞ q =E0 for (45/45)4T and (0/90)4T laminated Fundamental frequencies X 0 beams with embedded PFRC layers resting on elastic foundations under applied voltage (L/h = 10). Lay-up

DV (V) UD

((45/45)2/p/(45/45)2) 200 0 +200 ((0/90)2/p/(0/90)2) 200 0 +200

FG-V

FG-K

FG-X

FG-O

(k1, k2) = (0, 0) 4.1704 4.3871 4.1682 4.7618 3.8606 4.1630 4.3897 4.1502 4.7567 3.8520 4.1557 4.3923 4.1321 4.7516 3.8433 11.1954 11.3189 10.7733 11.9804 10.3198 11.1964 11.3211 10.7730 11.9814 10.3208 11.1974 11.3233 10.7727 11.9823 10.3217

(k1, k2) = (100, 0) ((45/45)2/p/(45/45)2) 200 4.9776 5.1593 0 4.9732 5.1609 +200 4.9689 5.1626 ((0/90)2/p/(0/90)2) 200 11.5214 11.6400 0 11.5223 11.6420 +200 11.5232 11.6440 (k1, k2) = (100, 10) ((45/45)2/p/(45/45)2) 200 5.6632 5.8217 0 5.6603 5.8228 +200 5.6574 5.8239 ((0/90)2/p/(0/90)2) 200 11.8343 11.9483 0 11.8352 11.9502 +200 11.8360 11.9520

4.9768 5.4794 4.7250 4.9663 5.4761 4.7203 4.9558 5.4728 4.7156 11.1133 12.2840 10.6745 11.1130 12.2849 10.6754 11.1127 12.2858 10.6762 5.6646 6.1059 5.4462 5.6575 6.1035 5.4431 5.6504 6.1011 5.4401 11.4390 12.5764 11.0134 11.4387 12.5772 11.0141 11.4384 12.5781 11.0149

following examples only FG-O and UD FRC laminated beams are considered. Fig. 3 shows the effect of foundation stiffness on the frequency– amplitude curves of FG-O and UD ((45/45)2/p/(45/45)2) beams resting on elastic foundations. The results confirm that the fundamental frequencies are increased, but the frequency–amplitude curves are decreased with increase in foundation stiffness under the same environmental condition. Fig. 4 shows temperature rise on the nonlinear vibration characteristics of FG-O and UD (p/(0/90)2/(90/0)2) beams. Three sets of thermal environmental conditions, i.e. DT = 0, 100 and 200 °C, are considered. From Table 5 and Fig. 3, it can be seen that the fundamental frequency is decreased, but the nonlinear to linear

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Table 7 Nonlinear to linear frequency ratios xNL/xL for (45/45)4T and (0/90)4T laminated beams with embedded PFRC layers resting on elastic foundations (L/h = 10). e X

Lay-up

W m =h 0.2

UD FG-V FG-K FG-X FG-O UD FG-V FG-K FG-X FG-O

4.1630 4.3897 4.1502 4.7567 3.8520 11.1964 11.3211 10.7730 11.9814 10.3208

(k1, k2) = (0, 0) 1.0517 1.0508 1.0444 1.0430 1.0633 1.0540 1.0461 1.0586 1.0466 1.0648

((45/45)2/p/(45/45)2)

UD FG-V FG-K FG-X FG-O

4.9732 5.1609 4.9663 5.4761 4.7203

(k1, k2) = (100, 0) 1.0369 1.0370 1.0336 1.0329 1.0430

((0/90)2/p/(0/90)2)

UD FG-V FG-K FG-X FG-O

11.5223 11.6420 11.1130 12.2849 10.6754

((45/45)2/p/(45/45)2)

UD FG-V FG-K FG-X FG-O UD FG-V FG-K FG-X FG-O

5.6603 5.8228 5.6575 6.1035 5.4431 11.8352 11.9502 11.4387 12.5772 11.0141

((45/45)2/p/(45/45)2)

((0/90)2/p/(0/90)2)

((0/90)2/p/(0/90)2)

1.0512 1.0442 1.0552 1.0445 1.0608 (k1, k2) = (100, 10) 1.0288 1.0292 1.0270 1.0267 1.0327 1.0487 1.0425 1.0522 1.0426 1.0573

2.00

1.25

L/h=10 ΔT = ΔV = 0

1.0

1.1936 1.1901 1.1674 1.1626 1.2337 1.2016 1.1737 1.2177 1.1755 1.2390

1.3983 1.3918 1.3476 1.3384 1.4746 1.4137 1.3600 1.4443 1.3635 1.4847

1.6426 1.6327 1.5655 1.5514 1.7573 1.6659 1.5845 1.7120 1.5898 1.7722

1.9114 1.8981 1.8075 1.7883 2.0646 1.9426 1.8331 2.0042 1.8403 2.0845

1.1406 1.1409 1.1284 1.1257 1.1626

1.2950 1.2957 1.2708 1.2655 1.3384

1.4844 1.4856 1.4469 1.4384 1.5514

1.6972 1.6988 1.6457 1.6342 1.7884

1.1916 1.1669 1.2057 1.1680 1.2252

1.3946 1.3468 1.4216 1.3489 1.4585

1.6371 1.5642 1.6778 1.5675 1.7332

1.9040 1.8057 1.9585 1.8101 2.0325

1.1107 1.1123 1.1040 1.1028 1.1252 1.1828 1.1607 1.1951 1.1612 1.2130

1.2352 1.2385 1.2216 1.2191 1.2644 1.3775 1.3346 1.4013 1.3355 1.4354

1.3909 1.3961 1.3694 1.3654 1.4367 1.6111 1.5455 1.6472 1.5471 1.6985

1.5686 1.5757 1.5387 1.5331 1.6318 1.8690 1.7804 1.9176 1.7825 1.9863

((45/-45)2/p/(45/-45)2) L/h=10

ΔT = ΔV = 0

1: (k1, k2) = (0, 0)

UD FG-V FG-Λ FG-X FG-O

1.5

2: (k1, k2) = (100, 0)

3: (k1, k2) = (100, 10) 1: UD 1: FG-O 2: UD 2: FG-O 3: UD 3: FG-O

1.0

1.00 0.0

0.8

2.0

ωNL/ωL

ωNL/ωL

1.50

0.6

2.5

(p/(45/-45)4)

1.75

0.4

0.2

0.4

0.6

0.8

1.0

W/h Fig. 2. Comparisons of frequency–amplitude curves of five types of (p/(45/45)4) beams.

frequency ratio is increased with increase in temperature rise. The results confirm that the temperature variation only has a very small effect on the frequency–amplitude curves for both FG-O and UD (p/(0/90)2/(90/0)2) beams. Fig. 5 shows applied voltage on the nonlinear vibration characteristics of FG-O and UD ((45/45)2/p/(45/45)2) beams. Three sets of applied voltage, i.e. DV = 200, 0 and +200 V, are considered. It is found that the negative applied voltage increases the fundamental frequencies, but decreases the frequency–amplitude

0.5 0.0

0.2

0.4

0.6

0.8

1.0

W/h Fig. 3. The effect of foundation stiffness on the frequency–amplitude curves of UD and FG-O ((45/45)2/p/(45/45)2) beams.

curves. In contrast, the positive applied voltage decreases the fundamental frequencies, but increases the frequency–amplitude curves. The results confirm that the applied voltage only has a very small effect on the frequency–amplitude curves for both FG-O and UD ((45/45)2/p/(45/45)2) beams. Very high voltages will be able to influence the fundamental frequencies and the frequency–amplitude curves of the hybrid laminated beams. However, such high voltages cannot be applied, because they lead to a breakdown in the material properties.

H.-S. Shen, D.-Q. Yang / Engineering Structures 90 (2015) 183–192

2.00

References

ωNL/ωL

(p/(0/90)2/(90/0)2) L/h=10

1.75

ΔV = 0

1.50

1: ΔT = 0 o 2: ΔT = 100 C o 3: ΔT = 200 C 1: UD 1: FG-O 2: UD 2: FG-O 3: UD 3: FG-O

1.25

1.00 0.0

0.2

0.4

0.6

0.8

1.0

W/h Fig. 4. The effect of temperature rise on the frequency–amplitude curves of UD and FG-O (p/(0/90)2/(90/0)2) beams.

2.00 ((45/-45)2/p/(-45/45)2)

ωNL/ωL

191

1.75

L/h=10 ΔT = 0

1.50

1: ΔV = -200 V 2: ΔV = 0 3: ΔV = +200 V 1: UD 2: UD 3: UD 1: FG-O 2: FG-O 3: FG-O

1.25

1.00 0.0

0.2

0.4

0.6

0.8

1.0

W/h Fig. 5. The effect of applied voltage on the frequency–amplitude curves of UD and FG-O ((45/45)2/p/(45/45)2) beams.

5. Conclusions The large amplitude vibration analysis for FG-FRC laminated beams with fully covered or embedded PFRC actuators have been presented on the basis of a micromechanical model and multi-scale approach. The material properties of both FRCs and PFRCs are assumed to be temperature dependent. A parametric study has been carried out for UD and FG cross-ply and angle-ply laminated beams in different set of thermo-piezoelectric loading conditions. The results show that the temperature variation has a moderately effect on the natural frequencies of the hybrid FRC laminated beam, but only has a small effect on the nonlinear to linear frequency ratios of the same beam. The applied voltage only has a small effect on the natural frequencies of the hybrid FRC laminated beam, but this effect on the nonlinear frequency–amplitude curves of the same beam may be neglected. The numerical results reveal that the foundation stiffness has a significant effect on the nonlinear vibration characteristics of the hybrid FRC laminated beam when supported by an elastic foundation. Acknowledgment The support for this work, provided by the National Natural Science Foundation of China under Grant 51279103 is gratefully acknowledged.

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