Thermal postbuckling analysis of anisotropic laminated beams with different boundary conditions resting on two-parameter elastic foundations

European Journal of Mechanics A/Solids 54 (2015) 30e43

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Thermal postbuckling analysis of anisotropic laminated beams with different boundary conditions resting on two-parameter elastic foundations Zhi-Min Li a, c, *, Pizhong Qiao b, c a

State Key Laboratory of Mechanical System and Vibration, Shanghai Key Lab of Digital Manufacture for Thin-Walled Structures, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China c Department of Civil and Environmental Engineering, Washington State University, Pullman, WA 99164-2910, USA b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 28 June 2014 Accepted 1 June 2015 Available online 23 June 2015

Thermal postbuckling analysis of shear deformable anisotropic laminated composite beams with temperature-dependent material properties subjected to uniform temperature distribution through the thickness and resting on a two-parameter elastic foundation is presented. The material of each layer of the beam is assumed to be linearly elastic and fiber-reinforced. The governing equations are based on
rma
n-type of kinematic nonlinearity. Reddy's high order shear deformation beam theory with a von Ka Composite beams with clampedeclamped, clampedehinged, and hingedehinged boundary conditions are considered. A numerical solution for the nonlinear partial-integral differential form in terms of the transverse deflection using Galerkin's method is employed to determine the buckling temperatures and postbuckling equilibrium paths of anisotropic laminated beams with uniform temperature distribution through the thickness. The numerical illustrations on the thermal postbuckling response of laminated beams with different types of boundary conditions, ply arrangements (lay-ups), geometric and physical properties are also presented, and the results reveal that the geometric and physical properties, temperature dependent properties, boundary conditions, and elastic foundation all have a significant effect on thermal postbuckling behavior of anisotropic laminated composite beams. © 2015 Elsevier Masson SAS. All rights reserved.

Keywords: Anisotropic laminated composite beams Higher-order shear deformation beam theory Thermal postbuckling

1. Introduction Composite structures, like beams, are widely used in various engineering applications such as airplane wings, helicopter blades as well as many others in aerospace, mechanical, and civil industries. Anisotropic composites provide more design flexibility than conventional materials. Due to their outstanding engineering properties, such as high strength/stiffness to weight ratios, the laminated composite beams are likely to play a remarkable role in the design of various engineering structures and partially replace the conventional isotropic beam structures. However, their increased amount of design parameters and physical phenomena

* Corresponding author. State Key Laboratory of Mechanical System and Vibration, Shanghai Key Lab of Digital Manufacture for Thin-Walled Structures, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China. Tel.: þ86 21 34206542; fax: þ86 21 34204542. E-mail address: [email protected] (Z.-M. Li). http://dx.doi.org/10.1016/j.euromechsol.2015.06.001 0997-7538/© 2015 Elsevier Masson SAS. All rights reserved.

brings some difficulties in analysis of aerospace vehicles, supersonic/hypersonic flight vehicles, reusable space transportation systems, etc. One of the important problems in engineering structures is the thermal buckling of composite beams. A number of useful elastic beams, plates and shells theories have been proposed for analysis of composite structures (Levy, 1877; Reissner, 1975; Levinson, 1981; Murty, 1984; Reddy, 1984; Stein, 1986; Looss and Joseph, 1990; Touratier, 1991; Wu and Chen, 2008; Kargani et al., 2013; Shen, 2011, 2013; Esfahani et al., 2013; Shen and Wang, 2014) due to the rapidly increasing use of advanced composite materials in various industries. The importance and potential benefits of composite beams in engineering practice have inspired continuing research interest. As it is well known that the buckling problems are the geometrically nonlinear ones, the temperature rise causes compressive forces in the beams with immovable ends, therefore leading to the occurrence of buckling phenomenon. A series of research work in the thermal postbuckling analysis of composite beams have been

Z.-M. Li, P. Qiao / European Journal of Mechanics A/Solids 54 (2015) 30e43

conducted in the past decades. Kaju and Kao (1984) investigated the thermal postbuckling of columns. Gauss and Antman (1984) presented the global descriptions of the properties of buckled states of nonlinear thermoelastic beams and plates when heated at their ends and edges. Jekot (1996) evaluated the thermal postbuckling of a beam made of physically nonlinear thermoelastic
rma
n strainmaterial using the geometric equations in the von Ka displacement approximation. Sapountzakis and Tsiatas (2007) studied the flexural buckling of composite Euler-Bernoulli beams of arbitrary cross sections, in which the resulting boundary-value problems were solved using the boundary element method. Nayfeh and Emam (2008) investigated the postbuckling behavior of beams composed of isotropic materials under the axial load with various boundary conditions. Lee and Choi (1999) studied the thermal buckling and postbuckling behaviors of a composite beam with embedded shape memory alloy (SMA) wires using the finite element model in which the SMA wire actuators are modeled with the beam elements and the composite laminates with the shell elements. Asadi et al. (2013, 2014) investigated the large amplitude vibration and nonlinear buckling of SMA fiberreinforced hybrid composite beams with symmetric and asymmetric lay-up using the Euler-Bernoulli beam theory and the nonlinear von-Karman strain field. When the EulereBernoulli beam theory is used for the buckling analysis of laminated beams, the buckling loads are overestimated, due to the consequence of neglecting the transverse shear strain and assuming that the plane normal to the beam axis still remains normal even after the deformation. Due to the high ratio of the extensional modulus to the transverse shear modulus of composite beams, the shear deformation cannot be ignored even for reasonably large slenderness ratio. The first-order shear deformation beam theory (FSDBT), such as Timoshenko beam theory, is then used to describe the kinematics of deformation of laminated beams accurately because the transverse shear stresses are accounted for. Abramovich (1994) investigated the thermal buckling of cross-ply symmetric and nonsymmetric composite beams to solve Timoshenko type equations. Bert (1973) and Dharmarajan and McCutchen (1973) obtained the shear correction coefficients for orthotropic beams. Vosoughi et al. (2012) investigated the thermal buckling and postbuckling responses of symmetric laminated composite beams with temperature-dependent material properties, in which the governing equations are based on the FSDBT
n assumptions. They obtained the critical with the von K
arma temperature as well as the nonlinear equilibrium path (postbuckling behavior) of symmetric laminated beams using a direct iterative method. In fact, the variation in material properties along the thickness of unsymmetrically laminated plates or beams results in quite a different behavior compared to that of symmetrically laminated or single material plates or beams. For example, the bifurcation buckling cannot generally occur for unsymmetrically laminated plates or beams with simply supported edges due to in-plane loadings, i.e., a transverse deflection is initiated, regardless of the magnitude of the loading, which is often the case with laminated composite materials (Leissa, 1986, 1987; Qatu and Leissa, 1993). Khdeir and Reddy (1997) studied the buckling behaviors of cross-ply laminated beams with arbitrary boundary conditions based on the classical, first-, second- and third-order beam theories. Khdeir (2001) evaluated the thermal buckling behavior of cross-ply laminated beams subjected to a uniform temperature rise for various boundary conditions and obtained the exact solution for the critical buckling temperature. Akba and Kocatürk (2011) presented the postbuckling analysis of a simply supported beam subjected to a uniform thermal loading using a total Lagrangian finite element model of two dimensional continuum for an eight-node quadratic element.

31

Patel et al. (1999) studied the free vibration and post-buckling analysis of laminated orthotropic beams resting on a two parameters elastic foundation (Pasternak type) using a three-node shear flexible beam element. Based on shear deformable beam theory, Aydogdu (2007) conducted the thermal buckling analysis of crossply laminated beams subjected to different sets of boundary conditions by the Ritz method. Ghugal and Shimpi (2001) presented a number of beam theories in the literature as well as a review of displacement- and stress-based refined theories for isotropic and anisotropic laminated beams. They solved and evaluated the nonlinear governing equation for the thermal postbuckling of the composite beam using the nonlinear finite element formulation. In spite of the availability of FEM and powerful computer programs, the second- or higher-order analysis of a composite beam is still an impractical task to most structural designers due to the limitation of the number of degrees of freedom (DOF) required to achieve a desired level of precision and efficiency. Recently, comparing the modern approaches to the classical beam theories, including well-known classical results related to Euler-Bernoulli and Timoshenko beam theories, Carrera and Giunta (2010), Carrera and Petrolo (2011), Carrera et al. (2010a,b, 2011a,b) and Giunta et al. (2010, 2011) established the Carrera Unified Formulation (CUF) which has hierarchical properties and is capable of dealing with most typical engineering challenges; in particular, the error can be reduced by increasing the number of the unknown variables. It overcomes the problem of classical formulae that require different formulas for tension, bending, shear and torsion. More important, it can be applied to any beam geometries and loading conditions, reaching a high level of accuracy with low computational cost, and it can tackle problems which in most cases are solved by employing the plate/shell and 3D formulations. As expected, their work provided an effective means in applications related to bridge structures, aircraft wings, helicopters and propeller blades. Vaz et al., 2007 examined a perturbation solution for the initial postbuckling of beams that were supported on an elastic foundation under uniform thermal load. Emama and Nayfeh (2009) obtained a closed-form solution for the postbuckling configurations of composite beams with various boundary conditions based on the classical beam theory and expressed these configurations as functions of the applied axial load. Gupta et al. (2010) studied the thermal postbuckling of columns with axially immovable ends with the RayleigheRitz method. Kim et al. (2013) developed the refined and accurate laminated composite beam element based on the eigenvalue problem for the flexural and torsional analyses. Recently, Ma and Lee (2011, 2012) presented an excellent result of geometrically nonlinear static responses of functionally graded materials (FGM) beams subjected to a uniform in-plane thermal loading using the first-order shear deformation beam theory. They stated that the FGM beams with different boundary conditions have some inherent characteristics due to the inhomogeneous boundary conditions. To the authors' best knowledge, there is no work available in the literature on the thermal postbuckling analysis of shear deformable anisotropic laminated composite beams with different types of boundary conditions resting on twoparameter nonlinear elastic foundation. The present work focuses on the thermal postbuckling analysis of anisotropic laminated composite beams with temperaturedependent material properties subjected to uniform temperature distribution through the thickness and resting on two-parameter (Pasternak-type or Vlasov-type) elastic foundation. The governing equations are based on Reddy's high order shear deformation beam
rm
theory (Reddy, 2004) with a von Ka an-type of kinematic nonlinearity including the beamefoundation interaction. The analysis uses a Galerkin's method to determine the thermal buckling temperatures and postbuckling equilibrium paths of a beam

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with three types of boundary conditions. The numerical results are presented to illustrate the effect of different types of geometric parameter and ply arrangements (lay-up) on the thermal postbuckling behavior of anisotropic laminated composite beams.

and



QX RX



 ¼ 

2. Theoretical development

¼ Consider a laminated composite beam with width b, length L and thickness h, which consists of N plies of any kind. The beam resting on a two-parameter elastic foundation is assumed to be moderately thick, and it is subjected to a uniform temperature rise DT. The axial and transverse displacement fields are expressed as

U 1 ðX; Y; ZÞ ¼ UðXÞ þ ZJðXÞ  c1 Z 3 J þ

vW vX

! (1)

U 2 ðX; Y; ZÞ ¼ 0

(2)

U 3 ðX; Y; ZÞ ¼ WðXÞ

(3)

where U 1 and U 3 are, respectively, the displacements in X- and Zdirections at any material point in the (X, Z) plane. U and W are the longitudinal and transverse displacements along the beam reference plane (X,Y), respectively, and J is the rotation of the normal to the cross-section about Y-axis at the reference plane. The strains can be written as

f εX g ¼

n

f gZX g ¼

ð0Þ

εX

n

o

ð0Þ gZX

n o n o þ Z εXð1Þ þ Z 3 εXð3Þ o

(4)

n o ð2Þ þ Z 2 gZX

(

vU 1 vW þ vX 2 vX

!2 ) ;

n

 o n o  vJ εð1Þ ¼ εXð1Þ ¼ vX (6)

  n o n o vJ v2 W εð3Þ ¼ εð3Þ ¼ c þ 1 X vX vX 2  o n o  n ð0Þ ¼ J þ vW ; gð0Þ ¼ gZX vX !) ( vW ¼ 3c1 Jþ vX

(7)

n o n o ð2Þ gð2Þ ¼ gZX

ð0Þ

ð1Þ

ð3Þ

ð1Þ εX ;

ð3Þ εX Þ,

ð0Þ

ð1Þ

ð3Þ

ð0Þ ðεX ;

B11 D11 F 11



Jþ

9 > > > > =

vW vX

D55 ! F55 > vW > > > > > > ; : c2 J þ vX >

A55 D55

(10)

where NX and M X are the stretching force resultant and bending moment resultant, respectively; P X represents the high-order bending moment resultant, Q X and RX are the transverse shear force resultants. Imposing the traction boundary conditions at the top and bottom free-surfaces ðZ ¼ ±h=2Þ, the transverse shear stress will vanish, i.e., tZX ¼ 0, and the coefficient c1 in the displacement field of Eq. (1) is c1 ¼ 4=3h2 . Let us now consider the influence of a temperature field on the behavior of the laminated beams. It is assumed that one value of the temperature is imposed on the upper surface and the other value on the lower surface. This one-dimensional temperature field is assumed to be constant along the length of the beam. In this case, the temperature distribution along the thickness can be obtained by solving a simple steady state heat transfer equation through the thickness of the beam. For the thermal postbuckling problem, the temperature field can be assumed that the temperature of the top surface is Tt, to the bottom surface temperature Tb, the heat conduction equation for a laminate takes the form

  d dT kðZ; TÞ ¼0 dZ dZ

(11)

can

where the buckling temperature difference DT ¼ TbeTt and hi is the temperature exponent (0  hi < ∞). Note that the value of hi equal to unity represents a linear temperature change across the thickness. While the value of hi excluding unity represents a non-linear temperature change through the thickness. In this paper, uniform temperature distribution through the thickness for shear deformable anisotropic laminated composite beams is valid and satisfies the one-dimensional heat conduction equation of a media in the absence of heat generation. Assumed that the value of hi is zero. T T In the above equations, the thermal force NX , moment MX , and T higher order moment P X are defined by

k¼1 t

(8)

be

solved

in

terms

of

resulting in the following relation

8 9 2 < NX = A11 MX ¼ 4 B11 : ; PX E11

8 > > > > <

gð0Þ gð2Þ

t N Zk  T X  T T NX ; MX ; P X ¼ ðAX Þk 1; Z; Z 3 ðT  Tt ÞdZ

For a one-dimensional composite laminated beam, it is assumed that the Y-direction is free of stresses, i.e., NY ¼ N XY ¼ M Y ¼ M XY ¼ P Y ¼ P XY ¼ 0; while the mid-plane strains, and bending and twisting curvatures corresponding to Ydirection are assumed to be nonzero. The strains and curvatures ðεY ; εY ; εY ; gXY ; gXY ; gXY Þ



(5)

where

n o n o εð0Þ ¼ εXð0Þ ¼

D55 F55

A55 D55

9 8 9 > T > 38 = < NX > E11 < εð0Þ = > T F 11 5 εð1Þ  MX > : ð3Þ ; > > ; : T > H11 ε P X

¼

k1

t N Zk X k¼1 t

 ðAX Þk 1; Z; Z 3 ci DTdZ

(12)

k1

and T

T

SX ¼ MX 

4 T P 3h2 X

(13)

where

(9)

 ~ ðAX Þk ¼ Q ða11 Þk ; 11 k

 ci ¼

 h=2; ði ¼ 0; 1; 2; …Þ

Z 1 þ h 2

hi

 h=2  Z (14)

Z.-M. Li, P. Qiao / European Journal of Mechanics A/Solids 54 (2015) 30e43

33

and E11, E22, G12, G13, G23, n12 and n21 have their usual meanings, and

Q 12 Q 26  Q 16 Q 22 12 Q 66 ~ ¼ Q þ Q 16 Q 26  Q Q 2 Q 12 þ 2 Q 16 ; a11   11 11 Q 22 Q 66  Q 26 Q 22 Q 66  Q 26  

 a11 ¼ s2 c2 a22 (15) and a11 and a22 are the thermal expansion coefficients measured in the fiber and transverse directions, respectively, and the effective stiffness coefficients A11 , B11 , D11 , etc., can be expressed by

2

A11 4B 11 E11

B11 D11 F 11

3 2 3 2 A11 B11 E11 A12 E11 F 11 5 ¼ 4 B11 D11 F11 5  4 B12 E11 F11 H11 E12 H11 2 A22 A26 B22 B26 6 A26 A66 B26 B66 6 6 B22 B26 D22 D26 6 6 B26 B66 D26 D66 6 4 E22 E26 F22 F26 E26 E66 F26 F66

A16 B16 E16 E22 E26 F22 F26 H22 H26

N

X Aij ; Bij ; Dij ; Eij ; Fij ; Hij ¼ k¼1

  1; Z; Z 2 ; Z 3 ; Z 4 ; Z 6 dZ Q ij k

hk1

 ði; j ¼ 1; 2; 6Þ

k¼1

Zhk 

Q ij

 1; Z 2 ; Z 4 dZ k

U¼

b 2

3 2 c4 Q 11 c2 s2 6Q 7 6 4 6 12 7 6 6 6Q 7 6 s 6 22 7 ¼ 6 c3 s 6Q 7 6 6 16 7 6 3 4 Q 5 4 cs 26 Q 66 c2 s2

ZL 

(18)

(16)

N X εð0Þ þ M X εð1Þ þ P X εð3Þ þ Q X gð0Þ þ RX gð2Þ dX

(23)

0

The total work V done by the external force of laminated composite beam is given by

ZL V ¼

ði; j ¼ 5Þ

2 4K 1 W 2 þ K 2 2 2

0

vW vX

!2 3 5dX

(24)

Applying Hamilton's principle,

hk1

where Q ij are the transformed elastic constants, defined by

2

(22)

where q is the angle between the fiber direction and the beam axis, and the reduced stiffness constants Q11, Q22, Q12 and Q66 can be obtained in terms of the engineering constants. The total strain energy U of laminated composite beam shown in Fig. 1 is given by

(17) N

X Aij ; Dij ; Fij ¼

s ¼ sinq

3 B12 B16 E12 E16 D12 D16 F12 F16 5 F12 F16 H12 H16 31 2 3 A12 B12 E12 E26 6 7 E66 7 7 6 A16 B16 E16 7 6 B12 D12 F12 7 F26 7 7 6 7 6 7 F66 7 7 6 B16 D16 F16 7 H26 5 4 E12 F12 H12 5 H66 E16 F16 H16

The cross-sectional stiffness coefficients Aij, Bij, Dij, etc., (i,j ¼ 1,2,6) are defined by

Zhk

c ¼ cosq;

2c2 s2 c4 þ s4 2c2 s2 cs3  c3 s c3 s  cs3

s4 c s c4 cs3 c3 s

2c2 s2

c2 s2

2 2

dU þ dV ¼ 0

3 4c2 s2 2 2 3 4c s 72 7 Q11 2 2 7 4c s

76 Q12 7 7 2cs c2  s2 76 74 Q22 5 2

2 7 2cs c  s  2 5 Q66 2 2 c s

(25)

where d represents the first variation, dU is the virtual strain energy, dV is the virtual work done by external forces.

(19) and

h

i

Q 55 ¼ s2

c2

   Q44 Q55

(20)

where

E11 E22 ; Q22 ¼ ; Q12 ð1  n12 n21 Þ ð1  n12 n21 Þ n21 E11 Q44 ¼ G23 Q55 ¼ G13 ; Q66 ¼ G12 ¼ ð1  n12 n21 Þ

Q11 ¼

(21) Fig. 1. Configuration of anisotropic laminated composite rectangular beam.

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Z.-M. Li, P. Qiao / European Journal of Mechanics A/Solids 54 (2015) 30e43

The governing equations of the problem under investigation include those of equilibrium of the beam, the constitutive law and the strain-displacement compatibility condition. By ignoring the higher order terms are ignored without affecting the accuracy, and the governing equilibrium equations are obtained as

C2 ¼

C1 ¼



B11  c1 E11 A11 1 2L

ZL 0

vNX ¼0 vX

(26)

vQ X vR v2 P v vW  3c1 X þ c1 2X þ NX vX vX vX vX vX

!

v2 W  K1W  K2 2 vX

! ¼0 (27)

vMX vP  QX þ 3c1 RX  c1 X ¼ 0 vX vX



LA11

!2

dX þ

!2

(34)

B11  c1 E11 vJ c1 E11 v2 W þ þ C1 vX A11 A11 vX 2 !2

B11  c1 E11 vJ c1 E11 v2 W 1 vW þ ¼  2 vX vX A11 A11 vX 2 !2

ZL 

B11  c1 E11  1 vW þ dX þ J X¼L  JX¼0 2L vX LA11 0 ! T c1 E11 vW  vW  NX    vX X¼0 LA11 vX X¼L A11

vU 1 vW ¼ vX 2 vX

(28)

where

2 !2 3 2

vU 1 vW 5 þ D11  c1 F 11 vJ  c1 F 11 v W  MT M X ¼ B11 4 þ X vX 2 vX vX vX 2





(35)

(29)

Substituting Eq. (35) into Eqs. (27) and (28), the following governing equations are obtained

and

"

(33)





B11  c1 E11  J X¼L  JX¼0 LA11 ! T vW  vW  NX   vX X¼L vX X¼0 A11

vW vX

c1 E11

 c E vW  JX¼0  1 11 A11 vX X¼0

" !# !

T T vJ v2 W

B11  c1 E11 c1 E11 v3 J bc1 E11 v2 N X v2 P X þ b A55  2c2 D55 þ  bc1 2 þ b c1 F 11  c1 H 11  þ vX vX 2 vX 3 vX A11 vX 2 A11 2 ! ! # ! #

2 ZL  

c1 E11 vW   c21 E11 B  c E v4 W A vW vW v2 W T 11 1 11 11 2   þ  c1 H11 dX þ JX¼L  JX¼0   þ b4  NX X¼L X¼0 4 vX vX vX 2L L L vX vX 2 A11 0 ! v2 W  K1W  K 2 2 ¼ 0 vX 

c22 F55

(36)

2 vU 1 vW P X ¼ E11 4 þ vX 2 vX

!2 3 2

5 þ F 11  c1 H11 vJ  c1 H 11 v W  P T X vX vX 2 (30)

T

T

T

SX ¼ M X  c1 P X

(31)

Substituting Eqs. (10) and (18) into Eq. (26), we can obtained

8 2 !2 3

v < vU 1 vW 5 þ b B11  c1 E11 vJ bA 4 þ vX : 11 vX 2 vX vX ) v2 W T  bc1 E11 2  bN X ¼ 0 vX

(" b



B11  c1 E11

2 #

v2 J vX 2 A11 # " )

 v3 W c1 E11 B11  c1 E11 2 þ  c1 F 11  c1 H11 vX 3 A11 !  vW  b A55  2c2 D55 þ c22 F55 J þ ¼0 vX D11  2c1 F 11 þ c21 H 11 

(37)

The boundary conditions can be expressed as

X ¼ 0; L : W¼ (32)

In this study, the zero distributed axial force is assumed, and the simply supported and immovable end conditions are considered.   Applying the immovable end conditions of W X¼0 ¼ 0, W X¼L ¼ 0,  U X¼0 ¼ 0, U X¼L ¼ 0, Eq. (32) yields

vW ¼ J ¼ 0; vX

W ¼ 0; ¼0

ðClampedeclampedÞ

vW  ¼ 0; vX X¼0 ðClampedehingedÞ  JX¼0 ¼ 0

v2 W vJ ¼ 0; W¼ ¼ vX vX 2

v2 W  ¼ 0; vX 2 X¼L

(38) vJ  vX X¼L (39)

ðHingedehingedÞ

(40)

Z.-M. Li, P. Qiao / European Journal of Mechanics A/Solids 54 (2015) 30e43

Note that Eqs. (36) and (37) are valid for the case of beams with the immovable end conditions, and they are adopted in the following nonlinear analysis. 3. Analytical method and solutions To solve Eqs. 36 and 37 with Eqs. 38e40, it is convenient to first define the following dimensionless quantities,

x ¼ X=L;

 u ¼ U L;



W ¼ W r;

*

W ¼W

*

r;

J ¼ J;

z7

35

v3 w v2 J vw vST  ¼0 þ z6 2  z8 J  z4 3 vx vx vx vx

(44)

where w* ðxÞ is an initial deflection due to the initial thermal bending moment, and wðxÞ is an additional deflection. J* ðxÞ is the mid-plane rotation corresponding to w* ðxÞ. JðxÞ is defined analogously to J* ðxÞ, but it is for wðxÞ. Note that w* ðxÞ ¼ J* ðxÞ ¼ 0 when DT ¼ 0.

sffiffiffiffi I3 ; r¼ I1



B11  c1 E11 rL A11 r 2 rc E z1 ¼ ; z2 ¼ z3 ¼ 1 *11 lT ¼ DT; D*1 D1 D*1 A55  6c1 D55 þ 9c21 F55 L3 A55  6c1 D55 þ 9c21 F55 L2 z4 ¼ ; z5 ¼ ; rD*1 D*1 " #



c1 E11 B11  c1 E11  c1 F 11  c1 H11 L A55  6c1 D55 þ 9c21 F55 L4 A11 L2 z6 ¼ 2 ; z7 ¼ ; z8 ¼ ; rD*1 r r 2 D*1 ! 2 c21 E11  c21 H11

2 A11 B11  c1 E11 L3 c21 E11 L3 L4 L3 ; z16 ¼ ; z17 ¼ ; z18 ¼ ; z19 ¼ ; z9 ¼ * r r D1 rA11 rA11 ! L2  T  DT L2 T T ðzT1 ; zT3 ; zT6 Þ ¼ * ATX ; X ; c1 FXT ; NXT ; MXT ; PXT ¼ * NX ; M X ; c1 P X ; h D1 D1

(41)

B  c E 2  11 1 11 D*1 ¼ D11  2c1 F 11 þ c21 H11  A11

where



The boundary conditions of Eqs. 38e40 become

ATX ; DTX ; FXT



¼

t N Zk X k¼1 t

 ðAX Þk 1; Z; Z 3 ci dZ

(42)

w ¼ J ¼ 0;

k1

The nonlinear Eqs. 36 and 37 may then be written in the dimensionless form as 4

3

w ¼ 0;

2

v w v J vJ v w þ z5 2 z9 4  z7 3 þ z4 vx vx vx vx 3 2   Z1  2     z vw vw vw   5 þ4 1 dx þ z2 ðJx¼1  Jx¼0 Þ  z3  vx vx x¼1 vx x¼0 2 0

x ¼ 0; 1:

2

! # Z1 " * 2 v2 w v2 w* z1 vw vw vw* 4  þ þ2 þ dx vx vx 2 vx vx2 vx2

w¼

ðClampedeclampedÞ

 Jx¼0 ¼ 0;

vJ ¼ 0; vx

vJ  ¼0 vx x¼1

ðHingedehingedÞ

(45) ðClampedehingedÞ

(46)

(47)

In the above equation, l2 denotes the critical buckling load and is given as

2

Z1  2 1þ3mþm2 z1 vw l ¼ð1þmÞzT1 lT  4 dx vx 2 0     vw  vw  þð1þ2mÞz2 ðJx¼1 Jx¼0 Þð1þ2mÞz3  vx x¼1 vx x¼0 2

0

3  *  2  * vw  vw*  *  5v w  þ z2 J x¼1  J x¼0  z3 vx x¼1 vx x¼0 vx2 ! ! v2 w v2 w* v2 w  zT1 lT þ w  K  K 1 2 vx2 vx2 vx2

(48)

v2 N T v2 P T þ z16 2X  z17 2X ¼ 0 vx vx (43)

In order to predict the postbuckling behavior of laminated beams, deflections and angle of twist reduces in establishing the components w and J having continuous derivatives up to the fourth order with respect to x and the axial displacement u having continuous derivatives up to the second order with respect to x satisfying the governing differential Eqs 26e28 inside the beam

36

Z.-M. Li, P. Qiao / European Journal of Mechanics A/Solids 54 (2015) 30e43

and the boundary conditions (45)e(47) at the beam ends x ¼ 0, L. To predict the postbuckling behavior of the beam, displacement functions that satisfy the first buckling modes, calculated from the linear buckling analysis, are employed. Let us assume that the first buckling mode is approximated by a polynomial of the sixth order, which satisfies the geometrical boundary conditions. According to previous research efforts (Sapountzakis and Dourakopoulos, 2010; Esfahani et al., 2013; Gupta et al., 2010; Asadi et al., 2014) the buckled shape should satisfy the geometrical (kinematic) boundary conditions so that the calculated buckling load is close to the true buckling load. However, in an ideal solution the buckled shape described by any chosen function should also satisfy the physical (static) boundary conditions. In most cases the assumed buckled shape does not satisfy the static boundary conditions, which in fact are local expressions of the equilibrium equations. Therefore, the differential equilibrium equations are not generally satisfied by the assumed shape. In order to decrease the error induced by the approximation of the true buckled shape, the employed polynomial in the proposed method will also satisfy the physical linear boundary conditions. In this case, the closed-form solution of two coupled equations for the buckled configurations of the laminated beams gets the following form

wðxÞ ¼ A1 sin gx þ A2 cos gx þ A3 x þ A4

(49)

JðxÞ ¼ C1 sin gx þ C2 cos gx þ C3 x þ C4

(50)

clampedeclamped, clampedehinged, and hingedehinged boundary conditions are, respectively, written as

wðxÞ ¼ A1 ½sinðgxÞ þ f2 cosðgxÞ þ f1 x  f2 ; JðxÞ ¼ A1 ½k1 f2 sinðgxÞ  k1 cosðgxÞ þ k1 

(55)

wðxÞ ¼ A1 ½sinðgxÞ  tan g cosðgxÞ  x tan g þ tan g; JðxÞ ¼ A1 ½  k1 tan g sinðgxÞ  k1 cosðgxÞ þ k2 tan g

(56)

wðxÞ ¼ A1 sinðgxÞ;

(57)

JðxÞ ¼ k1 A1 cosðgxÞ

and the initial deflection caused by the thermal bending are assumed to have similar forms for the associated boundary conditions as

w* ðxÞ ¼ mA1 ½sinðgxÞ þ f2 cosðgxÞ þ f1 x  f2 

(58)

w* ðxÞ ¼ mA1 ½sinðgxÞ  tan g cosðgxÞ  x tan g þ tan g

(59)

w* ðxÞ ¼ mA1 sinðgxÞ

(60)

where

k1 ¼

z7 g3 þ z4 g ; z6 g2 þ z8

k2 ¼

z4 ; z8

f1 ¼ k1

z8 ; z4

f2 ¼

cos g  1 sin g (61)

where A1eA4 and C1eC4 are the constants of integration, and the initial deflection caused by the thermal bending is assumed to have a similar form as

By using the boundary conditions of Eqs. 55e57, the characteristic equation of shear deformable anisotropic laminated beams with the clampedeclamped, clampedehinged, and hingedehinged boundary conditions are then, respectively, given by

w* ðxÞ ¼ m½A1 sin gx þ A2 cos gx þ A3 x þ A4 

2  2 cos g 

(51)

where m is the imperfection parameter. The solution of nonlinear differential equations is obtained after substituting the components w and J from Eqs. (49) and (50) in the equilibrium Eqs (43) And (44), employing the Galerkin's approximation method and deriving the equilibrium equations with respect to the displacement amplitudes w. For the clampedeclamped boundary conditions, the following equations are considered,

wð0Þ ¼ A2 þ A4 ¼ 0; wð1Þ ¼ A1 sin g þ A2 cos g þ A3 þ A4 ¼ 0; Jð0Þ ¼ C2 þ C4 ¼ 0; Jð1Þ ¼ C1 sin g þ C2 cos g þ C3 þ C4 ¼ 0 (52) While for the clampedehinged boundary conditions, the following equations are resulted,

wð0Þ ¼ A2 þ A4 ¼ 0; wð1Þ ¼ A1 sin g þ A2 cos g þ A3 þ A4 ¼ 0; vJ  Jð0Þ ¼ C2 þ C4 ¼ 0; ¼ C1 g cos g  C2 g sin g þ C3 ¼ 0 vx x¼1 (53) Finally, for the hingedehinged boundary conditions, the following equations are considered

wð0Þ ¼ A2 þ A4 ¼ 0 wð1Þ ¼ A1 sin g þ A2 cos g þ A3 þ A4 ¼ 0; vJ  vJ  ¼ C1 g þ C3 ¼ 0; ¼ C1 g cos g  C2 g sin g þ C3 ¼ 0 x¼0 vx vx x¼1 (54) After imposing the associated boundary conditions, the closed form solutions for the laminated beams with the

sin g 

k1 sin g ¼ 0 k2

k1 cos g ¼ 0 k2

sin g ¼ 0

ðClampedeclampedÞ

ðClampedehingedÞ

ðHingedehingedÞ

(62)

(63) (64)

To solve Eqs. 62e64, Newton's iterative method is implemented, and the values of g in the implicit form are obtained from the characteristic equation of shear deformable anisotropic laminated beams with three different end conditions. Substituting Eqs. 62e64 into Eq. (43) and combining with Eq. (44), the critical buckling load l2 is then derived as,

l2 ¼ z5  z9 g2 þ



2 z7 g2 þ z4 K1

þ K  2 g2 z6 g2 þ z8

(65)

It is noted that the equation and the corresponding boundary conditions for a clampedeclamped laminated beam lead to a differential eigenvalue problem; however, an eigenvalue problem does not occur due to the inhomogeneous boundary conditions for a hingedehinged laminated beam, which results in a remarkable discrepancy between clamped and simply supported laminated or FGM beams (Ma and Lee, 2011, 2012; Shen and Wang, 2014). In order to obtain a general solution that can be used for a perfect or imperfect beam in its buckling modes, the postbuckling equilibrium paths can be obtained by substituting Eq. (65) into Eq. (43),

# "

1 þ 3m þ m2 z1 1 2 2 S2 A 1 þ / lT ¼ l þ S1 A 1 þ zT1 ð1 þ mÞ 2

(66)

Z.-M. Li, P. Qiao / European Journal of Mechanics A/Solids 54 (2015) 30e43

In Eq. (66), A1 is taken as the secondary conversion parameter relating to the dimensionless maximum deflection. If the maximum deflection is assumed to be at the point x ¼ xm, it can be then obtained by solving the nonlinear equation vwðxÞ=vx ¼ 0 for different buckling modes of beams with three different types of end conditions,



 A1 ¼ f W m r

Table 2 Comparison of the non-dimensional critical temperature (Tcr L2 a11 =h2 ) results for a three layers (0/90/0) laminated composite beams with different thermal expansion coefficients. E11/E22

a22/a11

Aydogdu (2007)

Vosoughi et al. (2012)

Present

20

3 10 3 10 3 10 3 10 3 10 3 10

0.791 0.640 0.727 0.626 0.666 0.593 0.991 1.403 0.845 1.041 0.745 0.865

0.7957 0.6445 0.7325 0.6303 0.6711 0.5971 0.9960 1.4102 0.8507 1.0481 0.7509 0.8717

0.7891 0.6391 0.7260 0.6247 0.6650 0.5917 0.9877 1.3984 0.8432 1.0389 0.7440 0.8637

(67)

30

The expressions used in Eq. (66) are described in detail for three different types of end conditions in Appendix A, and the postbuckling equilibrium paths can be rewritten as

40

2 !

1 þ 3m þ m2 z1 1 Wm 4 w0 þ ð1 þ 2mÞw1 þ lT ¼ zT1 ð1 þ mÞ r 2 3 !2 Wm þ /5  w2 r

30

20

40

(68)

otherwise stated. It can be seen from Table 1 that the present results agree well with, but are lower than, those of HOBT, FOBT and CBT given by Khdeir (2001). Furthermore, the non-dimensional critical temperature ðTcr L2 a11 =h2 Þ for perfect (0/90/0) cross-ply

a

165 Hinged-hinged beams (0/90)S, L/h = 100 150 T-ID 135

o

T ( C)

In Eq. (68), w0 , w1 and w2 are also described in detail in Appendix A. Eq. (68) can be employed to obtain the numerical results for the full nonlinear buckling load-deflection curves of shear deformable anisotropic laminated beams subjected to uniform temperature distribution. The initial buckling load of a perfect beam can readily * be obtained numerically by setting W =r ¼ 0 (orm ¼ 0), while tak* ing W =r ¼ 0 (note that Wm s 0). In this case, the minimum thermal buckling load is determined by considering Eqs. 62e64 for various values of g.

37

Thermal postbuckling problem of laminated beams is solved using the Galerkin's method and iterative solution technique, and the numerical results are presented in this section. As part of the validation of the present method, the nondimensional critical temperature ðTcr a11 ðL=hÞ2 Þ values of a threelayered cross-ply (0/90/0) laminated beam for different length to thickness ratios, modulus ratios and thermal expansion coefficients ratios subjected to uniform temperature rise are compared among various beam theories (including Classical Beam Theory (CBT), First-Order Beam Theory (FOBT) and Third-Order Beam Theory (HOBT)) (Table 1) and along with the results given in Khdeir (2001) using the material properties (graphiteeepoxy) E11/E22 ¼ 20, G12/ E22 ¼ G13/E22 ¼ 0.6, G23/E22 ¼ 0.2, n12 ¼ 0.25, and a22/a11 ¼ 3, unless

Theory

L/h ¼ 5

10

20

50

100

HOBT (Khdeir, 2001) FOBT (Khdeir, 2001) CBT (Khdeir, 2001) Present E11/E22 HOBT (Khdeir, 2001) FOBT (Khdeir, 2001) CBT (Khdeir, 2001) Present a22/a11 HOBT (Khdeir, 2001) FOBT (Khdeir, 2001) CBT (Khdeir, 2001) Present

0.4678 0.4715 e 0.44908 3 0.7612 0.7625 0.8022 0.62508 3 0.8229 0.8281 1.1072 0.78912

0.8229 0.8281 e 0.78912 10 0.8832 0.8868 1.0370 0.81683 10 0.7077 0.7121 0.9522 0.63915

1.0190 1.0212 e 0.97666 20 0.8229 0.8281 1.1072 0.78912 20 0.5898 0.5935 0.7935 0.50268

1.0921 1.0925 1.1072 1.04656 30 0.7471 0.7528 1.1329 0.72608 50 0.3932 0.3956 0.5290 0.30640

e e e 1.05739 40 0.6796 0.6853 1.1462 0.66506 100 0.2528 0.2543 0.3401 0.18561

Present Vosoughi et al. (2012)

105 0.0

0.1

0.2

0.3

0.4

0.5

Wm/h

b

540 Hinged-hinged beams (0/90)S, L/h = 50 480

o

Table 1 Comparison of the non-dimensional critical temperature (Tcr a11 ðL=hÞ2 ) of a threelayered cross-ply (0/90/0) laminated beam for different length to thickness ratio, modulus ratio and thermal expansion coefficient ratio subjected to uniform temperature rise, based on various beam theories.

T-D 120

T ( C)

4. Numerical results and discussions

T-ID

420 T-D 360 Present Vosoughi et al. (2012) 300 0.0

0.1

0.2

0.3

0.4

0.5

Wm/h Fig. 2. Comparisons of postbuckling load-deflection curves of symmetric cross-ply laminated beams under uniform temperature rise.

38

Z.-M. Li, P. Qiao / European Journal of Mechanics A/Solids 54 (2015) 30e43

laminated beam (L/h ¼ 10) subjected to uniform temperature rise are compared in Table 2 with the results in Vosoughi et al. (2012) and Aydogdu (2007), using th the material properties E11/ E22 ¼ 20, 30, 40, G12/E22 ¼ G13/E22 ¼ 0.6, G23/E22 ¼ 0.2, n12 ¼ 0.25, and a22/a11 ¼ ±3, ±10. The comparisons show that the nondimensional critical temperature values predicted by the present study are in good agreement with those of Vosoughi et al. (2012) and Aydogdu (2007). In addition, the comparison of thermal postbuckling equilibrium paths (i.e., the load vs. deflection curves) of (0/90)S cross-ply laminated beam (L/h ¼ 50 and 100) is shown in Fig. 2 with results given by Vosoughi et al. (2012). In calculation, the material properties of graphite/epoxy composites are independent of the temperature rise DT in the laminated beams (Vosoughi et al., 2012): E11/E22 ¼ 40, G12/E22 ¼ G13/E22 ¼ 0.6, G23/E22 ¼ 0.2, n12 ¼ 0.25, a11 ¼ 1.0106 / C, a22 ¼ 10.0106 / C; while in this study, the material properties are in a linear function of temperature in the laminated beams, the material properties adopted, as given in are:

(except for Fig. 8a and b), adopted as (Wang et al., 2002; Oh et al., 2000): E11 ¼ E01 ¼ 150.0 GPa, E22 ¼ E02 ¼ 9.0 GPa, G12 ¼ G13 ¼ G012 ¼ 7.1 GPa, G23 ¼ G023 ¼ 2.5 GPa, n12 ¼ 0.3, a11 ¼ a01 ¼ 1.1106 / C, a22 ¼ a02 ¼ 25.2106 / C. It is assumed that E01, E02, G012, G023, a01 and a02 are the elastic constants and thermal expansion coefficients (at T0 ¼ 30  C), respectively. Table 3 gives buckling temperature DT ( C) for (0)8T (0/90)2S, (90/0)2S symmetric cross-ply, (±45)2S symmetric angle-ply, and (30/60)2S laminated beams with different values of beam length-tothickness ratio (L/h ¼ 20, 50 and 100) under the hingedehinged boundary conditions subjected to a uniform temperature rise for two cases of thermoelastic material properties, i.e. T-ID and T-D. Here, T-D represents material properties for graphite/epoxy orthotropic layers that are temperature-dependent, whereas T-ID represents material properties for graphite/epoxy orthotropic layers that are temperature-independent. The beams rest on twoparameter (Pasternak-type or Vlasov-type) elastic foundations, and three sets of foundation stiffness are considered. The stiffness

  E11 ¼ E01 1  0:5  103 DT GPa; E22 ¼ E02 1  0:2  103 DT GPa;   G12 ¼ G13 ¼ G012 1  0:2  103 DT GPa; G23 ¼ G023 1  0:2  103 DT GPa;   . .   a11 ¼ 1 þ 0:5  103 DT C; a22 ¼ a02 1 þ 0:5  103 DT C:

where E01, E02, G012, G023, a01 and a02 are the elastic constants and thermal expansion coefficients, i.e., at T0 ¼ 30  C, respectively. To obtain the numerical results, it is necessary to solve Eq. (68) using an iterative method with the following step-by-step procedures: (1) Beginning with W m =r ¼ 0: (2) Assuming that the material properties including the elastic constants and thermal expansion coefficients are constant, e.g., at DT ¼ 0. The thermal buckling load for the beam of temperature-independent material (T-ID) is then obtained. (3) Using the temperature determined in the previous step, the temperature-dependent material properties (T-D) are decided from Eq. (68) and the thermal buckling temperature is obtained again. (4) Repeating Step (3) until the thermal buckling temperature converges. (5) Specifying the new value of W m =r, and repeating Steps (2)e(4) until the thermal postbuckling temperature converges. It can be seen that the temperature reduces the buckling loads of laminated composite beams when the temperature dependency is taken into consideration. Through the close correlation observed between the present model and the earlier works, there exist some small discrepancies due to the fact that different beam theories and analytical approaches are employed in these existing studies. A parametric study is conducted to illustrate the thermal postbuckling behavior of anisotropic laminated beams resting on elastic foundations. Typical results are shown in Figs. 3e8. It should be appreciated that in all of these figures W m =r and DT denote the dimensionless central deflection of the beam and temperature rise, respectively. The graphite/epoxy material is selected for the laminated composite beams in these examples. For these examples, all plies are of equal thickness and the total thickness of the beam is b ¼ h ¼ 8.0 mm, and the material properties of graphite/epoxy

(69)

are (k1, k2) ¼ (10,000, 1000) for the Pasternak elastic foundation, (k1, k2) ¼ (10,000, 0) for the Winkler elastic foundation, and (k1, k2) ¼ (0, 0) for the beam without an elastic foundation. It can be seen that the beams without an elastic foundation support have almost the same buckling temperature as those with the Winkler elastic foundation. The results also show that the temperature reduces the buckling loads when the temperature dependency is put into consideration. Fig. 3 presents the thermal postbuckling load-deflection curves for (0)8T, (0/90)2S, (±45)2S, (30/60)4T, (35/15/0/90/60/45/50/ 25)T and (152/602/302/452)T laminated beams, and it intends to shed light on the material anisotropy (or laminate layup) on the thermal postbuckling behavior. It can be seen that the deflection amplitude of the beam increases remarkably as the temperature rise of laminated beams goes up. The results confirm that the thermal postbuckling load-deflection curves of (0)8T beam is higher than the others, whereas the load-deflection curves of (±45)2S beam is lowest among the considered laminated beams. This means that the material anisotropy or laminate layup has a significant effect on the thermal postbuckling characteristics of laminated beams. It is noted that the thermal postbuckling loaddeflection curves of unsymmetric (152/602/302/452)T, (35/ 15/0/90/60/45/50/25)T and (30/60)4T beams are no longer of the bifurcation type because a transverse deflection is initiated due to the existence of the extensionebending coupling. Fig. 4 shows the thermal postbuckling load-deflection curves for (0/90)2S and (152/602/302/452)T laminated beams with three different boundary conditions (i.e., clampedeclamped, clampedehinged, and hingedehinged). As shown in Fig. 4, the critical temperature for the (0/90)2S symmetric cross-ply laminated beams with the hingedehinged boundary conditions become remarkably lower. In contrast, the symmetric cross-ply laminated beams of clampedeclamped boundary conditions become the highest among the three types of boundary conditions. It can be also seen that the thermal postbuckling load-deflection curves of unsymmetric (152/602/302/452)T beams are no longer of the

Z.-M. Li, P. Qiao / European Journal of Mechanics A/Solids 54 (2015) 30e43

a

a

400

300

Hinged-hinged beams L/h = 50, h = 8.0 mm 1: (0)8T 2: (0/90)2S 3: (152/602/-302/-452)T

39

1000 (0/90)2S L/h = 50, h = 8.0 mm 800 1: Clamped-clamped 2: Clamped-hinged 3: Hinged-hinged

1

1

100

o

200

0 0.0

ΔT ( C)

o

ΔT ( C)

600

2

3

0.5

1.0

400

2 3

200

1.5

2.0

0 0.0

2.5

0.5

1.0

Wm/r

b

150

Hinged-hinged beams L/h = 50, h = 8.0 mm 1: (+45)2S 120 2: (35/-15/0/90/60/45/-50/-25)T 3: (30/60)4T 2 90

60

2.0

2.5

2.0

2.5

500 (152/602/-302/-452)T L/h = 50, h = 8.0 mm 400 1: Clamped-clamped 2: Clamped-hinged 3: Hinged-hinged 1

o

ΔT ( C)

300

o

ΔT ( C)

b

1.5

Wm/r

200

3

2

1

100

3

30

0 0.0

0 0.0

0.5

1.0

1.5

2.0

2.5

Wm/r Fig. 3. Effect of material anisotropy on the thermal postbuckling behavior of laminated beams with the hingedehinged boundary conditions under T-D case.

bifurcation type for the simply supported (hingedehinged) boundary conditions because the initial deflection is not zero anymore due to the existence of extensionebending couplings under the thermal bending moments. Thus, no buckling loads exist, and the beam will buckle at the onset of edge compression when under heat conduction. It also shows that the increase of the coupling stiffness yields an increase of postbuckling strength. Fig. 5 illustrates the thermal postbuckling load-deflection curves for (0/90)S, (0/90)2S, (± 45)S, (± 45)2S, (30/60)S, (30/60)2S, (0/90)2T, (0/90)4T, (± 45)2T, (± 45)4T, (30/60)2T and (30/60)4T laminated beams and shows the effect of the number of plies and their layups on the thermal postbuckling behavior. It is found that the critical temperature values of composite beams, especially for (0/90)S and (0/90)2S symmetric cross-ply laminated ones, with the same total thickness (h ¼ 8.0 mm) become slightly lower as the total number of plies are increased. In contrast, (0/90)2T and (0/90)4T antisymmetric cross-ply laminated beams with the same total thickness (h ¼ 8.0 mm) become slightly higher as the total number of plies are increased at the beginning of increasing deflections. It can be seen

0.5

1.0

1.5

Wm/r Fig. 4. Effect of boundary conditions on the thermal postbuckling behavior of laminated beams under T-D case.

that the critical temperature for (± 45)S, (± 45)2S, (30/60)S, (30/ 60)2S, (± 45)2T, (± 45)4T, (30/60)2T and (30/60)4T laminated beams have fairly small discrepancies as the number of layers increases. Fig. 6 demonstrates the effect of length-to-thickness ratios (L/ h ¼ 20, 50 and 100) on the thermal postbuckling behavior of the (0/ 90)2S, (152/602/302/452)T, (±45)2S and (30/60)2S laminated beams. It can be noted that the length-to-thickness ratio is a crucial parameter in determining the critical temperatures, and it also has a significant effect on the thermal postbuckling characteristics. The result confirms that the thermal postbuckling load-deflection curves of (152/602/302/452)T beams are also no longer of the bifurcation type due to the extensionebending couplings even under the uniform temperature field. Fig. 7 shows the effect of foundation stiffness on the thermal postbuckling behavior of the (0/90)2S, (152/602/302/452)T, (0/ 90)4T and (±45)2S laminated beams. The beams rest on twoparameter (Pasternak-type or Vlasov-type) elastic foundations, and three sets of foundation stiffness are considered. The stiffness are (k1, k2) ¼ (10,000, 1000) for the Pasternak elastic foundation, (k1, k2) ¼ (10,000, 0) for the Winkler elastic foundation, and (k1, k2) ¼ (0,

40

Z.-M. Li, P. Qiao / European Journal of Mechanics A/Solids 54 (2015) 30e43

a

300

Hinged-hinged beams L/h = 50, h = 8.0 mm 250 I&1:(0/90)S, II&1:(0/90)2S I&2:(30/60)S, II&2:(30/60)2S I&3:(+45)S, II&3:(+45)2S 200

1000

I&1 800

Hinged-hinged beams I: (0/90)2S II: (152/602/-302/-452)T

1: L/h = 20 2: L/h = 50 3: L/h = 100

II&1 600

I&1

o

ΔT ( C)

o

ΔT ( C)

a

150 100

II&2

II&1 400

I&2

I&2 200

50 I&3 0 0.0

0.5

1.0

1.5

2.0

II&2

II&3 0 0.0

2.5

0.5

1.0

1.5

Hinged-hinged beams L/h = 50, h = 8.0 mm I&1:(0/90)2T, II&1:(0/90)4T I&2:(30/60)2T, II&2:(30/60)4T 150 I&3:(+45)2T, II&3:(+45)4T

o

II&1

100

I&1 II&2 I&2

50

0 0.0

Hinged-hinged beams I: (+45)2S II&1 250 II: (30/60)4T I&1

150

II&2

II&3

0.5

1.0

1.5

1: L/h = 20 2: L/h = 50 3: L/h = 100

100 I&2

50

I&3

2.0

2.5

Wm/r Fig. 5. Effect of total number of plies on the thermal postbuckling behavior of laminated beams with the hingedehinged boundary conditions under T-D case.

0) for the shell without an elastic foundation. It can be seen that the beams with the foundation support have much higher critical temperature, and the thermal postbuckling equilibrium path curves become much lower except for (±45)2S laminated beams when the foundation stiffness decreases. It is also noted that the initial deflection is not zero and an initial extension occurs for the (152/602/302/452)T and (0/90)4T laminated beams when the heat conduction is taken into consideration. The result confirms that the shape of the load-deflection curves for the perfect beams appears similar to that of beams with an initial deflection due to thermal bending stress. Fig. 8 shows the effect of orthotropic ratio on the thermal postbuckling behavior of the (±45)2S, (0/90)4T, (152/602/302/ 452)T, (0/90)2S, (±30)2S and (±60)2S laminated beams. It should be mentioned that only the value of E11 is varied and the other material properties are kept invariant. The material properties of graphite/epoxy adopted are (Wang et al., 2002; Oh et al., 2000): E11a ¼ 150.0 GPa, E11b ¼ 181.0 GPa, E22 ¼ 9.0 GPa, G12 ¼ G13 ¼ 7.1 GPa, G23 ¼ 2.5 GPa, v12 ¼ 0.3, a11 ¼ 1.1106 / C,

2.5

300

200

ΔT ( C)

o

ΔT ( C)

b

200

2.0

Wm/r

Wm/r

b

I&3 II&3

0 0.0

I&3 0.5

1.0

1.5

2.0

II&3

2.5

Wm/r Fig. 6. Effect of length-to-thickness ratio on the thermal postbuckling behavior of laminated beams with the hingedehinged boundary conditions under T-D case.

a22 ¼ 25.2106 / C. The result indicates that the effect of the increasing extensional modulus E11 results in the increased critical temperatures of the (0/90)2S laminated beams when the deflection goes up. For the symmetric angle-ply, it is worthy noting that the postbuckling characteristics of the beams are not particularly affected by the change of the E11/E22 ratio. In summary, to investigate the significance of the shear deformation on the thermal postbuckling behavior for anisotropic laminated beams, the postbuckling paths (i.e., the load vs. deflection curves) with different length-to-thickness ratios, lay-ups and elastic foundations are plotted in Figs. 3e8. It is noted that the initial deflection of unsymmetric beams is not zero anymore, and an initial extension occurs when the heat conduction is taken into consideration. Due to the existence of the extensionebending couplings, the critical buckling loads for these unsymmetric beams do not exist, and the shape of the load-deflection curves of these beams with a perfect configuration appear similar to those of the beams with an initial deflection or imperfection when subjected to the thermal stress. It can also be seen from these figures (Figs. 3e8) that the lay-ups, length-to-thickness ratio, and elastic foundation

Z.-M. Li, P. Qiao / European Journal of Mechanics A/Solids 54 (2015) 30e43

a

400 Hinged-hinged beams 1: (k , k ) = (0, 0) L/h = 50, h = 8.0 mm 2: (k1, k2) = (10000, 0) 1 2 I: (0/90)2S 3: (k1, k2) = (10000, 1000) 300 II: (152/602/-302/-452)T I&3 I&2 I&1

250

Hinged-hinged beams I: E = 150.0 GPa 11a L/h = 50, h = 8 mm II: E = 181.0 GPa 11b 1: (+45)2S 200 2: (0/90)4T 3: (152/602/-302/-452)T

200

II&2 100 I&3 II&3

100 II&1 0 0.0

0.5

1.0

II&2

1.5

2.0

0 0.0

2.5

b

Hinged-hinged beams L/h = 50, h = 8.0 mm I: (30/60)4T 80 II: (+45)2S

I&3

ΔT ( C)

I&2

0.5

1.0

1.5

2.0

2.5

300

Hinged-hinged beams I: E = 150.0 GPa L/h = 50, h = 8 mm II: E11a = 181.0 GPa 11b 250 1: (0/90)2S 2: (+30)2S II&1 3: (+60)2S 200

o

60

o

ΔT ( C)

I&1

Wm/r

100

I&1

II&1

50

II&3

Wm/r

b

I&2

150

o

ΔT ( C)

o

ΔT ( C)

a

41

II&3 40

II&1

I&1

150

II&2

I&2 100

1: (k1, k2) = (0, 0) 2: (k1, k2) = (10000, 0) 3: (k1, k2) = (10000, 1000)

20

0 0.0

0.5

1.0

1.5

2.0

2.5

Wm/r Fig. 7. Effect of elastic foundation on the thermal postbuckling behavior of laminated beams with the hingedehinged boundary conditions under T-D case.

parameters play a crucial role in determining the thermal postbuckling behavior of shear deformable anisotropic laminated beams.

5. Concluding remarks Thermal postbuckling analysis is presented for the shear deformable anisotropic laminated composite beams with clampedeclamped, clampedehinged, and hingedehinged boundary conditions resting on different types of elastic foundations. The formulations are based on Reddy's high order shear deformation
rma
n nonlinear strain beam theory incorporating the von Ka displacement relations. A closed-form solution for the thermal postbuckling of anisotropic laminated composite beams with and without initial thermal deflection obtained as a function of the applied thermal load is employed to investigate thermal postbuckling behaviors of anisotropic laminated composite beams with different types of boundary conditions. The comparisons and parametric study are performed to validate the present solution

II&2 50 0 0.0

I&3 II&3

0.5

1.0

1.5

2.0

2.5

Wm/r Fig. 8. Effect of orthotropic ratio on the thermal postbuckling behavior of laminated beams with the hingedehinged boundary conditions under T-D case.

and also illustrate the effect of geometry parameters and temperature dependence (e.g., the material properties, ply-up orientation of fibers, the boundary conditions, total number of plies, elastic foundations, and orthotropic ratio) on the postbuckling behavior. Based on the numerical results, it is shown that an increase in the length-to-thickness ratio or decrease in the foundation stiffness greatly lowers the postbuckling paths (load vs. deflection curves). Either due to the constraint at the edges of beam or the decreased length-to-thickness ratio, the temperature dependency reduces the overall stiffness of the beam and cannot be neglected especially for anisotropic laminated beams with the clampedeclamped end conditions. The numerical results also reveal that the end boundary conditions, foundation stiffness, and initial deflections under thermal bending moments have a great effect on the postbuckling behavior of shear deformable anisotropic laminated composite beams. The closed form solution and extensive numerical results presented shed light on the thermal postbuckling behavior with different boundary conditions, and they can be used in design

42

Z.-M. Li, P. Qiao / European Journal of Mechanics A/Solids 54 (2015) 30e43

Table 3 Comparison of the buckling temperature DT ( C) for (0)8T, (0/90)2S, (90/0)2S, (±45)2S, and (30/60)2S laminated beams with the hingedehinged boundary conditions resting on elastic foundation (k1, k2) ¼ (0, 0), (10,000, 0), (10,000, 1000). (E11 ¼ E01 ¼ 150.0 GPa, E22 ¼ E02 ¼ 9.0 GPa, G12 ¼ G13 ¼ G012 ¼ 7.1 GPa, G23 ¼ G023 ¼ 2.5 GPa, n12 ¼ 0.3, a11 ¼ a01 ¼ 1.1106 / C, a22 ¼ a02 ¼ 25.2106 / C, h ¼ 8 mm). Lay-up

L=h ¼ 20.0 T-ID

(k1, k2) ¼ (0, 0) (0)8T 1251.2 (0/90)2S 887.7 (90/0)2S 458.0 (±45)2S 190.5 226.8 (30/60)2S (k1, k2) ¼ (10,000, 0) 1251.6 (0)8T (0/90)2S 888.1 (90/0)2S 458.5 (±45)2S 190.9 (30/60)2S 227.3 (k1, k2) ¼ (10,000, 1000) (0)8T 1255.8 (0/90)2S 892.0 (90/0)2S 462.9 (±45)2S 195.3 (30/60)2S 231.4

50.0

A1 ¼

 Wm r sinðgxm Þ

w0 ¼ z5  z9 g2 þ

100.0

T-D

T-ID

T-D

T-ID

T-D

806.8 596.9 363.4 177.0 207.3

208.9 148.8 74.99 30.77 36.72

187.6 135.8 71.57 30.37 36.12

52.54 37.47 18.81 7.704 9.197

51.00 36.54 18.56 7.677 9.156

807.2 597.3 363.8 177.4 207.7

208.9 148.9 75.07 30.84 36.79

187.7 141.3 71.64 30.43 36.19

52.56 37.49 18.83 7.722 9.214

51.02 36.55 18.60 7.695 9.173

809.7 599.5 367.2 181.4 211.4

209.6 149.5 75.15 31.54 37.46

188.2 146.7 71.71 31.11 36.84

52.73 37.64 19.02 7.895 9.381

51.12 36.71 18.79 7.868 9.341

Acknowledgments The work described in this paper is supported in part by the grants from the National Natural Science Foundation of China (No. 51279222) and National Key Basic Research Program of China (No. 2014CB046600). The first author (Z-M Li) is grateful for the support provided by the China Scholarship Council (CSC) (No. 2011831148) which enables him to conduct research at Washington State University.

(A.2)



2 z g2 þ z4 K1 7

þ K  2 g2 z6 g2 þ z8

For the case of clampedeclamped end conditions

w1 ¼

w2 ¼

z2 ½k2 f2 sin g  k1 ðcos g  1Þ ; ½sinðgxm Þ þ f2 cosðgxm Þ þ f1 xm  f2  1 ½sinðgxm Þ þ f2 cosðgxm Þ þ f1 xm  f2 2  1 2 1 1 1 g þ g sin 2 g þ f22 g2  f22 g sin 2 g þ f12  2 4 2 4   f2 g sin2 g þ 2f1 sin g þ 2f1 f22 sin g

for the case of clampedehinged end conditions

w1 ¼

k1 z2 ðcos g  1  tan g sin gÞ ½sinðgxm Þ  tan g cosðgxm Þ  xm tan g þ tan g 

analysis of shear deformation anisotropic laminated composites with temperature dependent properties.

ðHingedehingedÞ

w2 ¼

z3 gðcos g  1  tan g sin gÞ ; ½sinðgxm Þ  tan g cosðgxm Þ  xm tan g þ tan g 1

½sinðgxm Þ  tan g cosðgxm Þ  xm tan g þ tan g2  1 2 1 1 1 g þ g sin 2 g þ g2 tan2 g  g tan2 g sin 2 g  2 4 2 4  2 2  tan g þ g tan g sin g

and for the case of hingedehinged end conditions

ðk z þ z3 gÞðcos g  1Þ ; w1 ¼  1 2 sinðgxm Þ

w2 ¼

g2 2 sin2 ðgxm Þ

(A.3)

Appendix A References In Eqs. 52e68

  vwðxÞ ðcos g  1Þ ¼ A1 g cosðgxÞ  sinðgxÞ þ f1 ¼ 0 vx sin g ðClampedeclampedÞ vwðxÞ ¼ A1 ½g cosðgxÞ  g tan g sinðgxÞ  tan g ¼ 0 vx ðClampedehingedÞ vwðxÞ ¼ A1 g cosðgxÞ ¼ 0 vx A1 ¼

ðHingedehingedÞ

(A.1)

 Wm r ðClampedeclampedÞ ½sinðgxm Þ þ f2 cosðgxm Þ þ f1 xm  f2 

 Wm r ½sinðgxm Þ  tan g cosðgxm Þ  xm tan g þ tan g ðClampedehingedÞ A1 ¼

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