Buckling and postbuckling behavior of shear deformable anisotropic laminated beams with initial geometric imperfections subjected to axial compression

Engineering Structures 85 (2015) 277–292

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Buckling and postbuckling behavior of shear deformable anisotropic laminated beams with initial geometric imperfections subjected to axial compression 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 b State Key Laboratory of Ocean Engineering and 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

a r t i c l e

i n f o

Article history: Received 10 May 2014 Revised 13 December 2014 Accepted 15 December 2014

Keywords: Anisotropic laminated composite beams Higher-order shear deformation beam theory Buckling Postbuckling Axial compression Imperfection

a b s t r a c t Buckling and postbuckling behavior of shear deformable anisotropic laminated composite beams with initial imperfection subjected to axial compression is presented. The material in each layer of beams is assumed to be linearly elastic, anisotropic and fiber-reinforced. The governing equations are based on the higher order shear deformation beam theory with a von Kármán-type of kinematic nonlinearity. Composite beams with the fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are considered. A generic imperfection function for one-dimensional composite beams is adopted to model various possible initial geometric (e.g., sine, local, and global type) imperfections. The nonlinear prebuckling deformation and initial geometric imperfection of the beam are both taken into account. A numerical solution of nonlinear partial–integral differential form in terms of the transverse deflection is employed to determine the buckling load and postbuckling equilibrium path of composite beams. The results obtained by combining the Newton’s iterative method with the Galerkin’s method are theoretically exact from the transverse and longitudinal displacements for anisotropic laminated beams under the axial compressive loads using the secondary parameter conversion technique, and they are validated by comparing with those available in the literature. The numerical illustrations are presented for the postbuckling response of laminated beams with different types of boundary conditions, ply arrangements (layups), geometric and physical properties. The results reveal that the geometric and physical properties and boundary conditions have a significant effect on postbuckling behavior of anisotropic laminated composite beams. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Composite structures, like beams and plates, are broadly used in various engineering applications, such as airplane wings, helicopter blades as well as many others in the aerospace, mechanical, and civil industries. Due to the 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 type structures and partially replace the conventional isotropic beam structures. Interest in the structural buckling and postbuckling analysis of anisotropic composite

⇑ Corresponding author at: Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, PR China. E-mail addresses: [email protected], [email protected] (P. Qiao). http://dx.doi.org/10.1016/j.engstruct.2014.12.028 0141-0296/Ó 2014 Elsevier Ltd. All rights reserved.

laminated beams has led to a need for more accurate analysis especially in the case of critical structures. Many studies have observed the buckling and postbuckling behavior of beam-type structures, and numerous attempts have been made to predict such phenomenon for isotropic or orthotropic beams. Theories of beams involve basically the reduction of a three dimensional problem of elasticity theory to a one-dimensional problem. Since the thickness dimension is much smaller than the longitudinal dimension, it is possible to approximate the distribution of the displacement, strain and stress components in the thickness dimension. Based on the assumption that the axial line of the beam is inextensible, Timoshenko and Gere [1] examined the postbuckling of compressed beam clamped at one end and free at the other end and investigated it using the exact expression for the curvature in the differential equation of the deflection curve. Comer and Levy [2] proved that the inflatable

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beams can be considered as the usual Euler–Bernoulli beams. The theory is applicable to slender beams and should not be applied to thick or deep beams since it is based on the assumption that the plane sections perpendicular to the neutral layer before bending remain plane and are perpendicular to the neutral layer after bending, implying that the transverse shear and transverse normal strains are zero. Wang [3] dealt with the buckling of the axial compressive beams with the pinned–fixed ends using a shooting method as well as a perturbation method. Torkamani et al. [4] and Hori and Sasagawa [5] used the approximate second-order analysis, such as the finite element method with large deflections and with or without large strains. Li and Batra [6] investigated the analytical relations between the critical buckling load of a functionally graded material (FGM) Timoshenko beam and that of the corresponding homogeneous Euler–Bernoulli beam subjected to axial compressive load with simply supported, clamped and clamped–free boundaries. Vo and Thai [7] developed a one-dimensional displacement-based finite element method to accurately predict the critical buckling loads of rectangular composite beams with the corresponding mode shapes for various configurations. Furthermore, Chia et al. [8] reported the structural responses of generally-laminated composite columns subjected to uni-axial compression and transverse load, and the closed-form expressions were developed and presented to analyze the buckling and bending responses of generally-laminated composite beams with various boundary supports based on the Euler–Bernoulli beam and classical lamination theories. Emam and Nayfeh [9] and Nayfeh and Emam [10] obtained an exact solution for the postbuckling behavior of the composite beams with fixed–fixed, fixed–hinged and hinged–hinged boundary conditions based on the Euler–Bernoulli beam theory. Emam [11] studied the static and dynamic behavior of geometrically-imperfect laminated composite beams, and as a result, its lateral deflection is obtained as a function of the applied axial load, a parameter designating the laminate, and imperfection. Using the Rayleigh–Ritz method, Gupta et al. [12] predicted the postbuckling behavior of composite beams and compared their solution with the results obtained from the finite element analysis for general lay-up. Khdeir and Reddy [13] studied the buckling behavior of cross-ply laminated beams with arbitrary boundary conditions. The classical, first-, second- and third-order shear deformation theories were used in the analysis. Khamlichi et al. [14] presented the different formulations to the solution of the large-deflection problem of a hinged–hinged elastic bar with outside sway under axial compressive load, and they discussed the effects of the axial strains and shear deformations using the asymptotic expansion technique on the postbuckling behavior. Matsunaga [15] analyzed the buckling stresses of the laminated composite beams by taking into account the complete effects of transverse shear, normal stress and rotary inertia. Aydogdu [16] performed the buckling analysis of cross-ply laminated beams subjected to different sets of boundary conditions by using the Ritz method. The analysis was based on a three-degree-of-freedom shear deformable beam theory. It is found that the postbuckling response increases as the shear deformation becomes more significant. The elasticity solutions for plates of rectangular cross sections were given by Pagano [17,18] by comparing the solutions of several specific boundary value problems in his theory to the corresponding ones of elasticity solutions. In general, it is found that the conventional plate theory leads to a very poor description of laminate response at low span-to-depth ratios, but it converges to the exact solution as this ratio increases. While the cylindrical bending provides a convenient tool for performing a one-dimensional analysis of laminated plates, a theory for anisotropic laminated beams is also important. The difference between the cylindrical bending and beam bending is analogous to the difference between the plane strain and plane stress in classical theory

of elasticity. The major difference is in the term of bending stiffness. Recently, Foraboschi [19,20] developed the analytical modeling within the framework of one-dimensional elasticity. This new approach found the exact solution for the laminated composite beams and laminated glass columns. Moreover, the analytical modeling was recently developed within the framework of twodimensional elasticity [21–24]. The classical and first-order shear deformation theories underestimate the amplitude of buckling while the considered higher-order theories yield very close results. Murthy et al. [25] developed a refined 2-node, 4 degree-of-freedom node beam element based on a higher-order shear deformation theory for the axial flexural shear coupled deformation in asymmetrically-stacked composite beams. In spite of the availability of finite element method 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. The use of elasticity theory is practically unfeasible due to mathematical difficulties and the complexity of laminated systems. This led to the development of refined shear deformation theories for beams which approximate the two dimensional solutions with reasonable accuracy. Most recently, Carrera et al. [26-30] and Giunta et al. [31,32] established the Carrera Unified Formulation (CUF) which has hierarchical properties and is capable of dealing with most typical engineering challenges, i.e., the error can be reduced by increasing the number of the unknown variables. It overcomes the problem of classical formulae that require different equations for tension, bending, shear, and torsion, and it can be applied to any beam geometries and loading conditions, reaching a high level of accuracy with low computational cost. It can tackle problems that in most cases are solved by employing the plate/shell and 3D formulations. The comparison of the analytical results with the experimental results shows good correlation in general. Challamel et al. [33] investigated the buckling behavior of generic higher-order shear deformable beam models in a unified framework. Buckling solutions were presented for usual archetypal boundary conditions, such as pinned–pinned, clamped–free, clamped–hinge, and clamped–clamped boundary conditions. The results were then extended to general boundary conditions based on the generalized linear elastic connection law including the vertical and rotational stiffness boundary conditions. In addition, Emam [34] studied the postbuckling behavior of symmetricallylaminated and simply-supported beams by solving the nonlinear governing equations for which the critical buckling load is obtained by solving their linear counterpart. The results showed that the shear deformation of moderately-thick beams or beams made up of highly anisotropic materials has a significant effect on the postbuckling behavior of laminated composite beams. Moreover, Ghugal and Shimpi [35] presented a review of refined shear deformation theories for the structural analysis of shear deformable isotropic and laminated beams and on the recent advances in the modeling and analysis of laminated beams. Mullapudi and Ayoub [36] analyzed the reinforced concrete columns subjected to combined axial, flexure, shear, and torsional loads. Wang et al. [37] illustrated how the shear deformation theories provide accurate solutions when compared to the classical theory for both the beams and plates. Waas [38] presented an asymptotic initial postbuckling analysis of pinned–pinned and clamped laminated beams, incorporating the first-order shear deformation effects. The imperfection sensitivity of the beams has been extensively studied, and the maximum load carrying capacity is usually calculated as a function of normalized imperfection amplitude. In fact, the effect of initial geometric imperfection may play a great role in the postbuckling behavior of moderately-thick anisotropic

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composite beams. To have accurate buckling load predictions, the analytic models to be developed should be as close to reality as possible, i.e., they have to include any possible geometric imperfections and defects. The initial geometric imperfections are inherently and randomly distributed in many real structures. In practice, however, these structures can possess globally or locally distributed, small and unavoidable initial geometric imperfections (namely, deviations between the actual and intended shapes) during the fabrication process. Introducing the geometric imperfections into the model is inconvenient because their real shape is not known. It is not even possible to determine the most unfavorable shape of those because, as concluded by Schneider et al. [39], there is a dependency between the buckling load and the imperfection shape and amplitude. Simitses and Anastasiadist [40] developed a higher order theory involving initial geometric imperfections for a laminated cylindrical configuration under different mechanical loads. The kinematic relations, constitutive relations, transverse shear effects and boundary conditions were also discussed. However, most of these studies were based on the simplified assumption that the initial geometrical imperfection has a similar form to the deformed shape of the beam, and for the same value of amplitude, the local geometric imperfection has a small effect on the buckling load as well as postbuckling response of beams and plates than a modal imperfection does. The investigations dealing with general geometrical imperfections are limited in number, and this is partly due to the lack of sufficient information about the exact size and shape of the actual imperfections to be discussed. Thus, the significant differences of the mechanical responses of composite structures and their characteristic failure mechanisms when compared with isotropic materials make it extremely complex to establish a reliable and accurate correlational criterion. In addition, the contribution of the imperfection (or damage) to the postbuckling behavior is not fully understood. The present work focuses attention on the buckling and postbuckling behavior of anisotropic laminated composite beams with generic initial geometric imperfections subjected to axial compression. A practical and accurate method for the nonlinear large deflection analysis of anisotropic laminated composite beams is presented. The initial geometric imperfection of laminated beams is taken into account, and its form is assumed to be the products of trigonometric functions and hyperbolic functions in the plane of the beam. The governing equations are based on Reddy’s higher order shear deformation beam theory with a von Kármán-type of kinematic nonlinearity. The nonlinear prebuckling deformation and initial geometric imperfection of the beam are both taken into account. Composite beams with the fixed–fixed (F–F), fixed–hinged (F–H), and hinged–hinged (H–H) boundary conditions are considered. A closed-form solution of nonlinear partial–integral differential equations for the postbuckling deformation is obtained as a function of the applied axial load, which is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations are provided to demonstrate the postbuckling behavior of perfect and imperfect, anisotropic laminated composite beams with different types of boundary conditions, geometric parameters and material properties of shear deformable anisotropic laminated composite beams.

U 1 ðX; Y; ZÞ ¼ UðXÞ þ Z WðXÞ 

4Z 3 3h

2

Wþ

@W @X

! ð1aÞ

U 2 ðX; Y; ZÞ ¼ 0

ð1bÞ

U 3 ðX; Y; ZÞ ¼ WðXÞ

ð1cÞ

where U 1 and U 3 are the displacements in X- and Z-directions, respectively, at any material point in the (X, Z) plane. U and W are the longitudinal and transverse displacements, respectively, along the beam reference plane (X, Y) with the initial deflection denoted by W  , and W is the rotation of the normal to the cross-section about the Y-axis at the reference plane. Z is the depth of the material point measured from the beam reference plane along the positive Z-axis. The strains can be written as

feX g ¼

n

o

n

o

n

ð3Þ þ Z 3 eX eXð0Þ þ Z eð1Þ X

o

ð2Þ

n o n o ð0Þ ð2Þ fcZX g ¼ cZX þ Z 2 cZX

ð3Þ

where

8 9 ! <@U 1 @W 2 @W @W  = e ¼ eð0Þ þ þ ; ¼ X : @X 2 @X @X @X ;    ð1Þ  n ð1Þ o @W e ¼ eX ¼ @X 



n

 ð0Þ

ð3Þ

e



¼

o

n

ð3Þ X

e

o

¼

(

4 2

3h

@W @2W þ @X @X 2

ð4aÞ

!) ð4bÞ

) @W ; c ¼ c ¼ Wþ @X ( !)  ð2Þ  n ð2Þ o 4 @W c ¼ cZX ¼  2 W þ @X h 

ð0Þ



n

ð0Þ ZX

(

o

ð4cÞ

The laminated plate constitutive equations based on the higherorder shear deformation theory can be expressed as 8 9 2 A11 NX > > > > > > > > 6A > NY > 6 12 > > > > > > > > 6 6 A16 > N XY > > > 6 > > > > > > > > 6 B11 > < MX > = 6 6 MY ¼ 6 6 B12 > > > > 6 > > M XY > > 6 B16 > > > > 6 > > > > 6 E11 > > P X > > > > 6 > > 6 > > > > P > Y > > 4 E12 > : ; E16 P XY

A12 A22 A26 B12 B22 B26 E12 E22 E26

A16 A26 A66 B16 B26 B66 E16 E26 E66

B11 B12 B16 D11 D12 D16 F 11 F 12 F 16

B12 B22 B26 D12 D22 D26 F 12 F 22 F 26

B16 B26 B66 D16 D26 D66 F 16 F 26 F 66

E11 E12 E16 F 11 F 12 F 16 H11 H12 H16

E12 E22 E26 F 12 F 22 F 26 H12 H22 H26

8 9 3> eð0Þ > > X > > E16 > > > > ð0Þ > > > 7 e > > Y > > E26 7> > > > 7> ð0Þ > > > c > > E66 7 XY > > 7> > ð1Þ > 7> > > eX > > F 16 7> > 7< ð1Þ = F 26 7 e 7> Y > 7> ð1Þ > F 66 7> >c > > XY > > > 7> > > ð3Þ > > > H16 7 > > 7> e X > > > 7> > > H26 5> ð3Þ > > > > > e > Y > > > ð3Þ > > H66 : ;

ð5Þ

cXY

2. Theoretical formulations Consider a laminated composite beam with width b, length L and thickness h, which consists of N plies of any kind. The beam is assumed to be relatively thick, geometrically imperfect, and it is subjected to axial compressive loading P (see Fig. 1). The axial and transverse displacement fields are expressed as

Fig. 1. Configuration of an anisotropic laminated composite rectangular beam.

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where NX and MX 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 shear force resultants. Imposing the traction boundary conditions at the top and bottom free-surfaces (Z = ±h/2), the transverse shear stress will vanish, i.e., sZX = 0. For a one-dimensional composite laminated beam, it is assumed that the Y-direction is free of stresses, i.e., N Y ¼ N XY ¼ M Y ¼ M XY ¼ PY ¼ PXY ¼ 0, while the mid-plane strains, and bending and twisting curvatures corresponding to the Y-direction are assumed to be nonzero. The strains and curvatures   ð1Þ ð3Þ ð0Þ ð1Þ ð3Þ can be solved in terms of eð0Þ Y ; eY ; eY ; cXY ; cXY ; cXY   ð0Þ ð1Þ ð3Þ eX ; eX ; eX , resulting in the following relation

8 9 2 > A11 < NX > = 6 M X ¼ 4 B11 > > : ; PX E11

38 ð0Þ 9 >e > E11 < = 7 F 11 5 eð1Þ > : ð3Þ > ; e H11

B11 D11 F 11

ð6aÞ

where

2

A11

B11

6 6 B11 4 E11

E11

3

F 11

7 F 11 7 5 H11

A11

B11

E11

A12

A16

B12

B16

E12

E16

6 ¼ 4 B11

D11

7 6 F 11 5  4 B12

B16

D12

D16

F 12

7 F 16 5

D11 2

3

2

E11 F 11 H11 2 A22 A26 B22 6 6 A26 A66 B26 6 6 6 B22 B26 D22 6 6 6 B26 B66 D26 6 6E 4 22 E26 F 22 E26

E66

E12

F 26

E16

3

B26

E22

B66

E26

D26

F 22

D66

F 26

F 26

H22

F 12 F 16 H12 H16 3 2 3 E26 1 A12 B12 E12 7 6 7 E66 7 6 A16 B16 E16 7 7 6 7 7 6 7 F 26 7 6 B12 D12 F 12 7 7 6 7 7 6 7 F 66 7 6 B16 D16 F 16 7 7 6 7 7 6 H26 7 5 4 E12 F 12 H12 5

F 66

H26

H66

E16

F 16

and

h

i Q 55 ¼ s2

c2



Q 44

ð8dÞ

Q 55

where

Q 11 ¼

E11 ; ð1  m12 m21 Þ

Q 44 ¼ G23 ;

Q 22 ¼

Q 55 ¼ G13 ;

E22 ; ð1  m12 m21 Þ

E11, E22, G12, G13, G23, m12 and m21 have their usual meanings, and

c ¼ cos h;

s ¼ sin h

ð8fÞ

where h 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. Applying Hamilton’s principle, the governing equations of the problem under investigation are the equations of equilibrium of the beam, the constitutive law, and the strain–displacement equations. The governing equilibrium equations are obtained, and they can be expressed as

@NX ¼0 @X

ð9aÞ

" !#  @W @W  @Q X @RX @ 2 PX @

¼0  3c1 þ c1 þ NX  P þ @X @X @X @X @X 2 @X

ð9bÞ

@MX @P X  Q X þ 3c1 RX  c1 ¼0 @X @X

ð9cÞ

8 9 2 3 !2 

 @W @ < @U 1 @W @W @W 5 @2W = 4 þ b B11  c1 E11 bA11 þ þ  bc1 E11 ¼0 @X @X @X @X : @X 2 @X @X 2 ;

H16

ð10aÞ

QX

¼

RX

A55

D55

D55

F 55

(

cð0Þ cð2Þ

)

 ¼

A55 D55

8 9 = W þ @W D55 < @X   4 @W : F 55  h2 W þ @X ;

@X

 Aij ; Bij ; Dij ; Eij ; F ij ; Hij ¼

hk

k¼1



N  X Aij ; Dij ; F ij ¼ k¼1

ð8aÞ Z

hk

  Q ij ð1; Z 2 ; Z 4 ÞdZ

hk1

k

ði; j ¼ 5Þ

ð8bÞ

where Q ij are the transformed elastic constants, defined by

2

Q 11

3

2

c4

7 6 6 7 6 2 2 6 6 Q 12 7 6 c s 7 6 6 7 6 4 6 6 Q 22 7 6 s 7¼6 6 7 6 6 6 Q 16 7 6 c3 s 7 6 6 7 6 6 6 Q 26 7 6 cs3 5 4 4 Q 66 c2 s2

2c2 s2

s4

4c2 s2

c 4 þ s4

c 2 s2

4c2 s2

3

72 7 Q 3 7 11 76 7 7 2 2 4 2 2 6 2c s c 4c s 76 Q 12 7 7 76 7 7 cs3  c3 s cs3 2csðc2  s2 Þ 76 Q 22 7 5 4 7 7 c3 s  cs3 c3 s 2csðc2  s2 Þ 7 Q 66 5 2 2c2 s2 c 2 s2 ðc2  s2 Þ

U¼

Z 0

þ

k

ði; j ¼ 1; 2; 6Þ

¼

ðB11  c1 E11 Þ @ 2 W A11

@X

2

þ

c1 E11 @ 3 W A11 @X 3

Integrating Eq. (10) twice, one obtains

  Q ij ð1; Z; Z 2 ; Z 3 ; Z 4 ; Z 6 ÞdZ

hk1

!

ð10bÞ

The cross-sectional stiffness coefficients Aij, Bij, Dij (i, j = 1, 2, 6) are defined by N Z X

@W @ 2 W @W @ 2 W þ2 2 @X @X 2 @X @X

þ 2

ð7Þ





@2U 

ð8eÞ

where the coefficient c1 in Eqs. (9b) and (9c) is c1 = 4/3h2. Substituting Eqs. (6a) and (7) into Eqs. (9a)–(9c), Eq. (9a) can be written as

and

)

m21 E11 ð1  m12 m21 Þ

Q 66 ¼ G12

ð6bÞ (

Q 12 ¼

2 3 !2  @W @W 5 ðB11  c1 E11 Þ 41 @W dX  þ W 2 @X @X @X A11

c1 E11 @W þ C1X þ C2 A11 @X

ð11Þ

For the buckling and postbuckling behavior of laminated composite beams, the beam ends should be restrained before the laminated beams are subjected to axial compressive loads, because the midplane stretching is significant [9,10]. The boundary conditions for the axial displacement are assumed as follows: UjX¼0 ¼ 0, UjX¼L ¼ 0. Applying these boundary conditions, Eq. (11) yields

2 3 !2  1 @W @W @W 4 5dX þ 2 @X @X @X 0   ðB11  c1 E11 Þ  c E @W  þ W  1 11   L L @X 

A11 C1 ¼ L ð8cÞ

X

Z

L

X¼L

C2 ¼ 

ðB11  c1 E11 Þ A11

   W 

X¼0

ð12aÞ

X¼L

 c1 E11 @W  þ  A11 @X 

X¼0

ð12bÞ

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Z.-M. Li, P. Qiao / Engineering Structures 85 (2015) 277–292

Substituting the above relation into the second and third governing equations, i.e., Eqs. (9b) and (9c), one reaches the following governing equations as 



c1 ðF 11  c1 H11 Þ  2

E11

!

@4W

@2W

þ

@X 2 @W @X

@2W

þ ðA55  6c1 D55 þ 9c21 F 55 Þ



"

@W @ W þ @X @X 2

 ð1 þ lÞ kP 

3



þ2

@W @W 5 dX @X @X

3

 f7

! f6

¼0

ð13aÞ 



ðD11  2c1 F 11 þ c21 H11 Þ  



 c1 ðF 11  c1 H11 Þ 

! ðB11  c1 E11 Þ2 @ 2 W

A11

 A55  6c1 D55 þ 9c21 F 55



@W Wþ @X

!

@X 3

ð14aÞ



 @ W ¼ 0 or MX  c1 PX X¼0 ¼ 0;  @X X¼L

¼ 0 ðFixed—hingedÞ

ð14bÞ

X¼L

ðHinged—hingedÞ

ð14cÞ

Z

h=2

dA;

I1 ¼

ðB11  c1 E11 ÞL f2 ¼ ; D1 f4 ¼

Z

h=2

w ¼ lw;

Z 2 dA;

w¼

ðFixed—fixed ðF—FÞÞ

Wjx¼0 ¼ 0;

@W ¼ 0; @x

 @ W ¼ 0 ðFixed—hinged ðF—HÞÞ @x x¼1

ð17aÞ ð17bÞ

ðHinged—hinged ðH—HÞÞ

ð1 þ 2lÞf1 2

ð17cÞ

Z 0

1

 2 # @w dx @x

ð18Þ

The closed-form solution for the buckled configurations of laminated beams has the following deformed shapes

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

ð19aÞ

WðxÞ ¼ C 1 sin cx þ C 2 cos cx þ C 3 x þ C 4

ð19bÞ

w ðxÞ ¼ l½A1 sin cx þ A2 cos cx þ A3 x þ A4 

kP ¼

 W ¼ W r;

PL2  ; bD1

f1 ¼

r¼

I1 ; I0

A11 r 2 ; D1

wð0Þ ¼ A2 þ A4 ¼ 0;

wð1Þ ¼ A1 sin c þ A2 cos c þ A3 þ A4 ¼ 0;

Wð0Þ ¼ C 2 þ C 4 ¼ 0;

Wð1Þ ¼ C 1 sin c þ C 2 cos c þ C 3 þ C 4 ¼ 0

f5 ¼

while for the fixed–hinged (F–H) boundary conditions, the following conditions are met

wð0Þ ¼ A2 þ A4 ¼ 0;

wð1Þ ¼ A1 sin c þ A2 cos c þ A3 þ A4 ¼ 0;

Wð0Þ ¼ C 2 þ C 4 ¼ 0;

 @ W ¼ C 1 c cos c  C 2 c sin c þ C 3 ¼ 0 @x x¼1 ð22Þ

ðA55  6c1 D55 þ 9c21 F 55 ÞL2 ; D1

h  i   c1 F 11  c1 H11  E11 ðB11A c1 E11 Þ L

L2 f6 ¼ 2 ; f7 ¼  r rD1

 4 A55  6c1 D55 þ 9c21 F 55 L ; f8 ¼ r 2 D1

ð20Þ

ð21Þ

c1 E11 L f3 ¼ ; D1

ðA55  6c1 D55 þ 9c21 F 55 ÞL3 ; rD1

ð16bÞ

For the fixed–fixed (F–F) boundary conditions, the following equations are satisfied,

Eqs. (9a)–(9c) can be solved with the boundary condition in Eq. (14). Before proceeding, it is convenient to define the following dimensionless quantities, in which l is defined as the imperfection parameter, sffiffiffiffi

I0 ¼

ð16aÞ

where A1–A4 and C1–C4 are the constants of integration, and the initial geometric imperfection is assumed to have a similar form as

3. Analytical method and solutions

x ¼ X=L;

 2 # 2 @w @ w @W @2w dx þ f4 þ f5 2 2 @x @x @x @x

The boundary conditions of Eqs. (14a)–(14c) become

"

ðFixed—fixedÞ

 ðw; w Þ ¼ W; W  r;

0

1

@ W @ w þ f9 4 ¼ 0 @x @x3

k2 ¼ kP 

@W ¼ 0 or M X ¼ PX ¼ 0; @X

Z

In the above equation, k2 denotes the critical buckling load given as

X ¼ 0; L :

W¼

ð1 þ 2lÞf1 2

@2W @3w @w þ f7 3  f8 W  f4 ¼0 @x @x @x2

w ¼ 0;

The boundary conditions can be expressed as

WjX¼0 ¼ 0;

A11

4

w ¼ W ¼ 0;

ð13bÞ

W ¼ 0;  @Q X   @X 

ðB11  c1 E11 Þ2

x ¼ 0; 1 :

¼0

@MX W ¼ W ¼ 0 or ¼ 0; @X



@X 2

A11

! E11 ðB11  c1 E11 Þ @ 3 W





The nonlinear Eqs. (13a) and (13b) may then be written in the dimensionless form as

!

@X 2

!2



; D1 ¼ ðD11  2c1 F 11 þ c21 H11 Þ 

ð15Þ

@X 3

A11

2



11

D1

! E11 ðB11  c1 E11 Þ @ 3 W

P  c21 H11   4 b A11 @X 2 ! Z L  A11 @ 2 W @ 2 W 4 þ þ 2L @X 2 @X 2 0 

f9 ¼ 

  2 c21 H11  AE11

11

;

and for the hinged–hinged (H–H) boundary conditions, they are

wð0Þ ¼ A2 þ A4 ¼ 0;

wð1Þ ¼ A1 sin c þ A2 cos c þ A3 þ A4 ¼ 0;

 @ W ¼ C 1 c þ C 3 ¼ 0; @x x¼0

 @ W ¼ C 1 c cos c  C 2 c sin c þ C 3 ¼ 0 @x x¼1 ð23Þ

After imposing the associated boundary conditions, the closed form solutions for laminated beams with the fixed–fixed, fixed–hinged,

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and hinged–hinged boundary conditions are, respectively, obtained as



h w ðxÞ ¼ l sech½dðx  #1 Þ cos l1 pðx  #1 Þ r

wðxÞ ¼ A1 ½sinðcxÞ þ f 2 cosðcxÞ þ f 1 x  f 2 ;

where l1 is the half-wave number of the imperfection in the X-axis. This model (Eq. (33)) is capable of modeling a wide range of initial imperfection modes, which include the sine type imperfection when d = 0, l1 = 1, #1 = 0.5; the local type imperfection when d – 0; and the global type imperfection when d = 0, l1 – 1, where l is the imperfection parameter. Applying the relationships in Eqs. (15), (19), and (33) to the differential equation Eq. (16) and using the Galerkin’s method, the buckling and postbuckling behavior of generally-imperfect beams can be solved for the different types of initial imperfections. While Eq. (32) can be employed to obtain the numerical results for the full nonlinear buckling load–deflection curves of shear deformable anisotropic laminated beams subjected to axial compression. The initial buckling load of a perfect beam can readily be obtained numerically by setting W  =r ¼ 0 (or l = 0), while taking W m =r ¼ 0 (note that Wm – 0). In this case, the minimum buckling load is determined by considering Eq. (28) for various values of c.

WðxÞ ¼ A1 ½k1 f 2 sinðcxÞ  k1 cosðcxÞ þ k1 

ð24Þ

wðxÞ ¼ A1 ½sinðcxÞ  tan c cosðcxÞ  x tan c þ tan c; WðxÞ ¼ A1 ½k1 tan c sinðcxÞ  k1 cosðcxÞ þ k2 tan c

ð25Þ

wðxÞ ¼ A1 sinðcxÞ;

ð26Þ

WðxÞ ¼ k1 A1 cosðcxÞ

where

k1 ¼

f7 c3 þ f4 c ; f6 c2 þ f8

k2 ¼

f4 ; f8

f 1 ¼ k1

f8 ; f4

f2 ¼

cos c  1 sin c

ð27Þ

By using Eqs. (21)–(23) of the boundary conditions, the characteristic equation of shear deformable anisotropic laminated beams with the fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are, respectively, given by

2  2 cos c 

sin c 

k1 sin c ¼ 0 ðFixed—fixed ðF—FÞÞ k2

k1 cos c ¼ 0 ðFixed—hinged ðF—HÞÞ k2

sin c ¼ 0 ðHinged—hinged ðH—HÞÞ

ð28aÞ 4. Numerical results and discussions

ð28bÞ ð28cÞ

To solve Eq. (28), the Newton’s iterative method is used, from which the values of c in the implicit solutions for the characteristic equation of shear deformable anisotropic laminated beams with three different end conditions can be obtained. By substituting Eq. (28) into Eq. (16b) and then combining it with Eq. (18), the characteristic equation is thus obtained with the Galerkin’s method,

" # 2 1 ðf7 c2 þ f4 Þ 2 k ¼ f  f9 c  ð1 þ lÞ 5 ðf6 c2 þ f8 Þ 2

ð1 þ 2lÞf1 S1 A21 þ    2

4.1. Comparison studies

ð30Þ

In Eq. (30), 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 = xm which may be obtained to solve the nonlinear equation ow(x)/ox = 0 for different buckling modes of beams with three types of boundary conditions, then

 A1 ¼ f W m =r

ð31Þ

The expression used in Eq. (31) is described for three different types of boundary conditions, and it is presented in detail in Appendix A. The postbuckling equilibrium paths can be rewritten as

ð1 þ 2lÞf1 Wm kP ¼ #0 þ #1 2 r

!2 þ 

Based on the present iterative method and combined with Galerkin method, the computer program packages are made by using the FORTRAN computer language, and this study provides the comprehensive ever-known results which are helpful in better understanding the buckling and postbuckling behavior of axially loaded anisotropic laminated beams with generic geometrical imperfection. Numerical results are presented in this section for perfect and imperfect, shear deformable anisotropic laminated beams subjected to axial compressive loads.

ð29Þ

In order to obtain a general solution that can be used for a perfect or imperfect beam in its buckling modes, Eq. (29) is substituted into Eq. (18) and the postbuckling equilibrium paths can be written as

kP ¼ k2 þ

ð33Þ

ð32Þ

In Eq. (32), #0, #1 and #2 are also described in Appendix A. Instead of assuming the imperfection mode to be the same as the deformed shape, a variety of sine type, local type, and global type imperfections were considered by employing a generic imperfection function from the one-dimensional imperfection model for struts by Wadee [41]. The initial geometric imperfection w⁄ for laminated beam structures is assumed to have a general form of

The methodology presented for buckling analysis of generallylaminated composite beams with various boundary supports is first compared and validated with the existing literature data. Chai et al. [8] reported the buckling responses of generally-laminated composite beams subjected to uni-axial compression and transverse load. The closed-form expressions were developed and presented in their contribution to analyze the buckling behaviors using a combination of the Euler–Bernoulli beam and the classical lamination theory. The buckling loads (N) from the present study are calculated and compared in Table 1 with the experimental and theoretical results of Chai et al. [8]. In calculation, the material properties of glass fiber-reinforced epoxy adopted are: E11 = 46.0 GPa, E22 = 15.0 GPa, G12 = 4.17 GPa, v12 = 0.276, and four types of specimen dimensions length  width  thickness (120  60  1, 240  60  1, 280  20  4 and 320  20  3) for (0)8T, (±45)4S, (45)8T, (90)8T, (0)32T and (0)24T laminated beams with the hinged–hinged or fixed–fixed boundary conditions are tested and compared. It can be seen that the present results agree well with the experimental results of Chai et al. [8]. The comparison shows that the influence of coupling responses in the buckling of laminated columns is more severe in the short wide specimens than those in the long wide specimens. In fact, the transverse shear deformation and coupling coefficients for different lay-up arrangement significantly affect the transverse displacement and the buckling loads of laminated composite beams. The elementary theory of beam, which is based on the Bernoulli–Euler assumption that planes initially normal to the mid-plane remain plane and is normal to the mid-surface after bending, leads to high percentages of error in the analysis of anisotropic beams due to the neglect of transverse shear deformations. It is interesting to note that the present theory provides a more reasonable correlation with the

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Z.-M. Li, P. Qiao / Engineering Structures 85 (2015) 277–292 Table 1 Comparison between experimental and theoretical buckling loads (N). Specimen designation

Experiment [8]

Theory 1a [8]

Theory 2b [8]

Present

CS-0-1c CS-0-2 CS-0-3 CS-45S-1 CS-45S-2 CS-45S-3 CS-45-1 CS-45-2 CS-45-3 CS-90-1 CS-90-2 CS-90-3 CS-0-1 CS-0-2 CS-0-3 CS-45S-1 CS-45S-2 CS-45S-3 CS-45-1 CS-45-2 CS-45-3 CS-90-1 CS-90-2 CS-90-3 SS-0-1 SS-0-2 SS-0-3 CL-0-1 CL-0-2 CL-0-3

561 565 478 205 229 213 179 213 187 154 157 139 180 142 198 107 120 134 115 115 136 108 116 106 529 565 587 179 185 186

630.6

646.6

628.67

Table 3 Comparison of the dimensionless critical buckling loads PL2/E22bh3 obtained by various beam theories and of antisymmetric lay-up (0/90). Theories

183.8

300.6

180.79

184.7

300.6

173.26

205.6

210.9

205.40

157.6

161.7

157.52

46.0

75.2

45.43

46.2

75.2

43.34

51.4

52.7

51.39

617.7

633.4

616.30

199.5

205.6

199.33

a

Theory 1: Euler–Bernoulli beam theory. Theory 2: traditional classical lamination theory. c The first letter in the designation indicates boundary conditions, S-hinged, C-fixed; the second letter refers to the relative specimen length, S-short, L-long. b

experimental results in the longer specimens with severe coupling effects which would remarkably affect the critical buckling load of anisotropic laminated beam-type structures. The buckling behavior of laminated composite beams with different geometric parameters and types of boundary conditions subjected to axial compression is then considered and compared. The dimensionless critical buckling loads PL2/E22hb3 are calculated and compared in Tables 2-5 with the theoretical results of the

L/h

Theories

Hinged–hinged

Fixed–hinged

Fixed–fixed

5

TOBT [13] SOBTa [13] SOBTb [13] FOBT [13] CBT [13] PSDBT [16] RSDT [7] Present

8.613 9.797 8.606 8.606 31.760 8.613 8.609 8.6093

9.814 10.940 9.412 9.412 64.974 – – 8.8875

11.652 12.745 10.802 10.802 127.041 – 11.648 11.177

TOBT [13] SOBTa [13] SOBTb [13] FOBT [13] CBT [13] RSDT [7] Present

18.832 20.353 18.989 18.989 31.760 18.814 18.816

25.857 28.810 25.940 25.940 64.974 – 24.807

34.453 39.187 34.426 34.426 127.041 34.437 34.437

PSDBT [16] RSDT [7] Present

27.084 27.050 27.055

– – 46.367

–

10

20

a b

Shear correction coefficient K2 = 1.0 has been used. Shear correction coefficient K2 = 5/6 has been used.

75.257 75.264

10

20

Hinged–hinged

PSDBT [16] RSDT [7] Present

3.906 3.903 3.9054

– 4.936 4.9399

5.296 5.290 5.2945

Fixed–fixed

RSDT [7] Present

8.670 8.6723

15.619 15.6216

19.757 19.7597

Table 4 Critical buckling loads (PL2/E22bh3) for three-layer (0/90/0) symmetric cross-ply beams with different theories (E11/E22 = 10). Theory

Fixed–hinged

Hinged–hinged

Material I

Material II

L/h = 5 PSDBTdsa [16] PSDBTcs [16] ESDBTds [16] ESDBTcs [16] FSDBTds [16] FSDBTcs [16] HSDBTds [16] HSDBTcs [16] Present

6.480 6.189 6.514 6.245 7.236 6.779 6.480 6.187 6.2049

L/h = 20 PSDBTds [16] PSDBTcs [16] ESDBTds [16] ESDBTcs [16] FSDBTds [16] FSDBTcs [16] HSDBTds [16] HSDBTcs [16] Present

14.910 14.797 14.910 14.807 15.166 15.025 14.910 14.797 14.7957

Material I

Material II

4.688 5.658 4.595 5.479 5.878 6.912 4.693 5.667 4.3965

4.726 4.573 4.733 4.592 5.112 4.893 4.727 4.572 4.7104

3.728 4.241 3.652 4.138 4.444 4.892 3.731 4.246 3.7176

14.081 14.797 13.994 14.807 14.716 15.025 14.085 14.797 13.9536

7.666 7.639 7.666 7.641 7.727 7.695 7.666 7.639 7.6246

7.459 7.570 7.437 7.547 7.619 7.688 7.461 7.571 7.4204

a The subscript ‘ds’ indices correspond to discontinuous interlaminar stresses and those with indices ‘cs’ to continuous interlaminar stresses.

Table 5 Critical buckling loads (PL2/E22bh3) for two-layer (0/90) anti-symmetric cross-ply beams with different theories (E11/E22 = 10). Theory

Table 2 Comparison of the dimensionless critical buckling loads PL2/E22bh3 obtained by various beam theories and of symmetric lay-up (0/90/0).

L/h 5

Fixed–hinged

Hinged–hinged

Material I

Material II

Material I

Material II

L/h = 5 PSDBTds [16] ESDBTds [16] FSDBTds [16] HSDBTds [16] Present

3.305 3.337 3.385 3.303 3.2542

2.846 2.887 2.933 2.845 2.7820

1.919 1.928 1.945 1.919 1.9129

1.765 1.778 1.800 1.765 1.7604

L/h = 20 PSDBTds [16] ESDBTds [16] FSDBTds [16] HSDBTds [16] Present

4.521 4.525 4.532 4.521 4.5002

4.458 4.463 4.474 4.458 4.4357

2.241 2.241 2.243 2.241 2.2320

2.226 2.228 2.230 2.226 2.2178

Euler–Bernoulli classical beam theory (CBT) [13], the first-order beam theory (FOBT) [13], the second-order beam theory (SOBT) [13], the third-order beam theory (TOBT) [13], the refined shear deformation theory (RSDT) [7], the corresponding parabolic shear deformation beam theory (PSDBT) [16], the hyperbolic shear deformation beam theory (HSDBT) [16], the first-order shear deformation beam theory (FSDBT) [16], and the exponential shear deformation beam theory (ESDBT) [16]. In Tables 2 and 3, the

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Material II: G12 = G13 = 0.5E22, G23 = 0.2E22, v12 = 0.25. The comparisons show that the present results are in good agreement with those presented in Aydogdu [16], but slightly lower than those of the refined shear deformation theory (RSDT) by Vo and Thai [7].

following material properties are used in the analysis, E11/E22 = 40, G12 = G13 = 0.6E22, G23 = 0.5E22, v12 = 0.25. While in Tables 4 and 5, the following material properties are used in the analysis, E11/ E22 = 10, Material I: G12 = G13 = 0.6E22, G23 = 0.5E22, v12 = 0.25;

Table 6 Comparison of values of coupling coefficients for different symmetric lay-up orientations of the laminated beam. Beam theory

Bending stiffness

a

A11

B11

D11

Unidirectional Emam and Nayfeh [10] Gunda et al. [12] Present

0.1298 0.1298 0.1298

1 1 1

– 155748100.24 155747790.48

– 0 0

– 12.9791 12.9790

(90/90/0)S Emam and Nayfeh [10] Gunda et al. [12] Present

0.0146 0.0146 0.01456

3.43 3.43 3.441

– 60021633.47 60030108.43

– 0 0

– 1.4564 1.45632

(0/90/90)S Emam and Nayfeh [10] Gunda et al. [12] Present

0.0943 0.0943 0.0943

0.53 0.53 0.531

– 60021633.47 60030108.43

– 0 0

– 9.4336 9.43463

Cross-ply Emam and Nayfeh [10] Gunda et al. [12] Present

0.0101 0.0101 0.0101

1 1 1

– 12,120,000 12,158,376

– 0 0

– 1.01 1.013

a

a

The subscript S indicates a symmetric lay-up.

Table 7 Comparison of values of coupling coefficients for different lay-up orientations of the laminated beams. Beam theory

Bending stiffness

A11

B11

D11

Cross-ply (90/0)3T Emam and Nayfeh [10] Gunda et al. [12] Present

0.065697605 0.065697605 0.065613614

83953250.16 83953250.16 83823160.23

5982.916139 5982.916139 5976.501092

6.996132165 6.996132165 6.986406231

Balanced (45/30/30/60/45/60)T Emam and Nayfeh [10] Gunda et al. [12] Present

0.032991958 0.032991958 0.032989395

53895771.27 53895771.27 53893444.04

4735.283957 4735.283957 4735.034478

3.715238007 3.715238007 3.714955929

General (73/30/50/16/45/20)T Emam and Nayfeh [10] Gunda et al. [12] Present

0.051155393 0.051155393 0.051151854

74850091.28 74850091.28 74849624.19

7193.53743 7193.53743 7195.08256

5.806880841 5.806880841 5.806828468

a

a

The subscript T indicates a total lay-up.

Table 8 The first five buckling loads (in N) for different laminates of fixed–fixed, fixed–hinged, and hinged–hinged beams. Lay-up

Unidirectional Emam and Nayfeh [10]

(0/90/90)S

(90/90/0)S

Cross-ply

Present

Emam and Nayfeh [10]

Present

Emam and Nayfeh [10]

Present

Emam and Nayfeh [10]

Present

Fixed–fixed beams b1 81.9823 b2 167.715 b3 327.929 b4 495.731 b5 737.841

81.4048 166.0954 323.4747 486.5276 719.8822

59.5876 121.901 238.35 360.314 536.288

59.2904 121.0619 236.1309 355.7662 527.4797

9.19926 18.8194 36.797 55.6261 82.7934

9.164371 18.74142 36.62490 55.33096 82.29121

6.39991 13.0926 25.5996 38.699 57.5992

6.369708 13.02662 25.45980 38.47228 57.23429

Fixed–hinged beams b1 41.9288 b2 123.933 b3 246.912 b4 410.879 b5 615.836

41.67134 122.9049 244.0522 404.3391 605.2574

30.4753 90.0785 179.464 298.641 447.61

30.34252 89.55481 178.0298 295.3999 441.1584

4.70484 13.9065 27.7061 46.1048 69.1031

4.686268 13.85059 27.58471 45.87545 68.71134

3.27315 9.67474 19.2751 32.0751 48.0749

3.258604 9.626068 19.17251 31.89482 47.78230

Hinged–hinged beams b1 20.4956 b2 81.9823 b3 184.46 b4 327.929 b5 512.39

20.3837 81.4048 182.6582 323.4747 502.9343

14.8969 59.5876 134.072 238.35 372.422

14.8392 59.2904 133.1616 236.1309 367.7573

2.29982 9.19926 20.6983 36.797 57.4954

2.29058 9.16437 20.61115 36.6249 57.1920

1.59998 6.39991 14.3998 25.5996 39.9995

1.593636 6.369708 14.32536 25.45979 39.76482

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It should be noted that the numerical results given in [7,13,16] were obtained from different shear deformation beam theories. Through the close correlation observed between the present model and the earlier works, the accuracy and adequacy of the present model is again established. The influence of coupling coefficients for different lay-up arrangement (ply angle orientations) of laminated beams with the simply supported boundary conditions are calculated and compared in Tables 6 and 7, including the theoretical results of the Euler–Bernoulli beam theory (EBT) by Emam and Nayfeh [10] and Gunda et al. [12]. In the calculation, the material properties

0.25 2

1

0.20

3

L/h = 250,h = 1.0 mm 1: Fixed-fixed 2: Fixed-hinged 3: Hinged-hinged Present F-F F-H H-H Gupta et al. [12] F-F (R-R) F-H (R-R) H-H (R-R) F-F (FEM) F-H (FEM) H-H (FEM)

X (m)

0.15

0.10

0.05

0.00 -0.5

0.0

0.5

1.0

1.5

2.0

2.5

W (m)

3.0 [10-4

Fig. 2. Comparison of deflection-longitudinal coordinate curves for beams with different types of boundary conditions.

adopted are: E11 = 155.0 GPa, E22 = 12.1 GPa, G12 = G13 = 4.4 GPa, G23 = 2.1 GPa and v12 = 0.248. It can be seen that the present results agree reasonably well with those of Gunda et al. [12]. Then, the comparison of first five buckling loads (N) for different laminates of fixed–fixed, fixed–hinged, and hinged–hinged beams are given in Table 8, in which the present results based on the higher-order beam theory (HBT) and the analytical results of Emam and Nayfeh [10] based on the Euler–Bernoulli beam theory (EBT) are given for the direct comparison. The comparisons show that the present results agree well with those of Emam and Nayfeh [10], and they can be used as a benchmark for other numerical studies. Furthermore, the comparison of deflection-longitudinal coordinate curves for beams with different types of boundary conditions is presented in Fig. 2. The comparisons illustrate that the present results agree well with those of Gunda et al. [12] derived by using the Rayleigh–Ritz method and finite element method. In addition, the curves of postbuckling equilibrium paths for beams with the hinged–hinged and fixed–hinged boundary conditions for different laminates (i.e., unidirectional, (90/90/0)S, (90/90/0)S and cross-ply lay-ups) subjected to axial load are compared in Fig. 3 with results from the closed form solution of Emam and Nayfeh [10] and Gunda et al. [12]. It can be seen that the present results are in reasonable agreements with those of Emam and Nayfeh [10] and Gunda et al. [12]. It should be pointed out that Gunda et al. [12] revised the incorrect expression derived for c by Emam and Nayfeh [10] and obtained the exactly matched results with their formulations for all the lay-ups of symmetrically-laminated composite beams.

Table 9 Imperfection modes (except the wave shape-type imperfection). Sine type

Case G1

a 10 Hinged-hinged

Emam & Nayfeh [10] Gupta et al. [12] L/h = 250, h = 1.0 mm 1 1

Wmid (mm)

1: unidirectional 8 2: (0/90/90) S 3: (90/90/0)S 4: cross-ply

6

4

2 3 4 Present 1 2 3 4

2 3 4 d = 0, l1 = 1, #1 = 0.5 4

d = 0, l1 = 3, #1 = 0.5 Case G3

Case G2

3

2

2 1

0 0

20

40

60

80

100

120

140

160

b

8

W mid (mm)

Axial load (N)

4

d = 0, l1 = 5, #1 = 0.5

Fixed-hinged Emam & Nayfeh [10]Gupta et al. [12] L/h = 250, h = 1.0 mm 1 1 1: unidirectional 2 2 2: (0/90/90)S 3 3 6 4 4 3: (90/90/0)S 4: cross-ply Present

d = 0, l1 = 7, #1 = 0.5 Case L2

Case L1

4

1 2 3 4

d = 15, l1 = 2, #1 = 0.25 3

2

d = 15, l1 = 2, #1 = 0.5 Case L4

Case L3

2 1

0 0

20

40

60

80

100

120

140

160

Axial load (N) Fig. 3. Comparison of postbuckling equilibrium paths for the (a) hinged–hinged and (b) fixed–hinged beams for different laminates subjected to axial load.

d = 15, l1 = 5, #1 = 0.5

d = 15, l1 = 7, #1 = 0.5

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Table 10 Buckling loads (kN) for fixed–fixed (0)8T, (0/90)4T, (0/90)2S, (90/0)2S, (±45)4T, (±45)2S, (302/602)2T, (30/60)2S and (152/602/302/452)T laminated beams with different buckling modes (n = 1, 2, 3 and 4) and length-to-thickness ratio L/h (= 10, 20, 50 and 100). (E11 = 181 GPa, E22 = 10.3 GPa, G12 = G13 = 7.17 GPa, G23 = 2.87 GPa, m12 = 0.25 and h = 8 mm). Buckling loads (kN) Lay-up

n=1

2

3

4

A16

A26

D26

A11

L/h = 10 A B C D E F G H I

192.00 112.46 132.36 88.468 43.856 40.450 45.987 48.689 65.637

234.61 146.76 158.45 125.17 73.825 68.816 76.589 79.626 103.11

317.20 202.25 216.13 175.57 118.01 110.82 121.31 124.55 157.40

365.63 239.72 247.11 216.48 141.82 134.37 144.59 146.34 179.98

0.0 0.0 0.0 0.0 0.1719 16.172 2.967e8 2.967e8 7.706e7

0.0 0.0 0.0 0.0 0.1719 16.172 2.967e8 2.967e8 8.59e6

0.0 0.0 0.0 0.0 2.530e5 685.237 1582.49 1856.38 171.47

1.448e9 7.674e8 7.674e8 7.674e8 1.972e8 2.003e8 2.055e8 2.246e8 3.622e8

L/h = 20 A B C D E F G H I

76.317 41.075 53.383 29.728 12.497 11.438 13.218 14.194 19.787

123.65 69.830 85.517 52.965 24.099 22.139 25.388 27.075 37.126

192.00 112.46 132.36 88.468 43.856 40.450 45.987 48.689 65.637

224.09 136.25 152.84 111.72 60.568 56.143 63.177 66.308 87.645

0.0 0.0 0.0 0.0 0.1719 16.172 2.967e8 2.967e8 7.706e7

0.0 0.0 0.0 0.0 0.1719 16.172 2.967e8 2.967e8 8.59e6

0.0 0.0 0.0 0.0 2.530e5 685.237 1582.49 1856.38 171.47

1.448e9 7.674e8 7.674e8 7.674e8 1.972e8 2.003e8 2.055e8 2.246e8 3.622e8

L/h = 50 A B C D E F G H I

14.660 7.5548 10.339 5.2649 2.0813 1.8999 2.2077 2.3821 3.3608

28.565 14.902 20.090 10.496 4.2151 3.8504 4.4680 4.8149 6.7717

52.604 27.838 36.911 19.847 8.1352 7.4374 8.1650 9.2694 12.986

73.680 39.681 51.510 28.741 12.086 11.062 12.783 13.726 19.132

0.0 0.0 0.0 0.0 0.1719 16.172 2.967e8 2.967e8 7.706e7

0.0 0.0 0.0 0.0 0.1719 16.172 2.967e8 2.967e8 8.59e6

0.0 0.0 0.0 0.0 2.530e5 685.237 1582.49 1856.38 171.47

1.448e9 7.674e8 7.674e8 7.674e8 1.972e8 2.003e8 2.055e8 2.246e8 3.622e8

L/h = 100 A B C D E F G H I

3.7734 1.9300 2.6651 1.3366 0.52338 0.47759 0.55541 0.59973 0.84766

7.6216 3.9113 5.3790 2.7164 1.0680 0.97472 1.1331 1.2231 1.7274

14.660 7.5548 10.339 5.2649 2.0813 1.8999 2.2077 2.3821 3.3608

21.683 11.236 15.274 7.8674 3.1323 2.8601 3.3215 3.5819 5.0464

0.0 0.0 0.0 0.0 0.1719 16.172 2.967e8 2.967e8 7.706e7

0.0 0.0 0.0 0.0 0.1719 16.172 2.967e8 2.967e8 8.59e6

0.0 0.0 0.0 0.0 2.530e5 685.237 1582.49 1856.38 171.47

1.448e9 7.674e8 7.674e8 7.674e8 1.972e8 2.003e8 2.055e8 2.246e8 3.622e8

A–I indicate (0)8T, (0/90)4T, (0/90)2S, (90/0)2S, (±45)4T, (±45)2S, (302/602)2T, (30/60)2S and (152/602/302/452)T.

Table 11 Buckling load (kN) of anisotropic laminated beams with different boundary conditions and length-to-thickness ratio (material I-Graphite-epoxy: E11 = 138 GPa, E22 = 14.5 GPa, G12 = G13 = 5.86 GPa, G23 = 3.52 GPa, m12 = 0.21). Buckling load (kN)

a b c

Lay-up

1a

2

3

4

5

6

7

8

L/h = 10 Fixed–fixed Fixed–hinged Hinged–hinged

151.37 96.660 58.972

96.013 58.550 33.718

114.73 72.165 43.376

36.579 19.942 10.270

34.704 18.875 9.6990

56.952 32.048 16.994

45.991 25.532 13.360

46.669 25.992 13.643

L/h = 20 Fixed–fixed Fixed–hinged Hinged–hinged

58.972 36.142 17.163

33.718 18.298 9.3854

43.376 23.920 12.455

L/h = 50 Fixed–fixed Fixed–hinged Hinged–hinged

11.206 5.8197 2.8789

6.0896 3.1473 1.5510

L/h = 100 Fixed–fixed Fixed–hinged Hinged–hinged

2.8789 1.4781 0.7247

1.5510 0.7953 0.3896

10.270 5.3456 2.6490

9.6990 5.0450 2.4986

16.994 8.9368 4.4656

13.360 6.9940 3.4814

13.643 7.1493 3.5618

8.1162 4.2085 2.0795

1.7019 0.8729 0.4276

1.6049 0.8231 0.4032

2.8756 1.4776 0.7249

2.2395 1.1498 0.5637

2.2918 1.1769 0.5770

2.0795 1.0673 0.5231

0.4276 0.2189 0.1070

0.4032 0.2063 0.1009

0.7249 0.3712 0.1816

0.5637 0.2886 0.1412

0.5770 0.2954 0.1445

Number indicates a stack sequence, in which 1b: (0)8T, 2: (0/90)4T, 3c: (0/90)2S, 4: (±45)4T, 5: (±45)2S, 6: (152/602/302/452)T, 7: (302/602)2T and 8: (30/60)2S. The subscript S indicates a symmetric lay-up. The subscript T indicates a total lay-up.

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Table 12 Buckling load (kN) of anisotropic laminated beams with different boundary conditions and length-to-thickness ratio (material II-Glass fiber reinforced epoxy resin: E11 = 45.6 GPa, E22 = 16.2 GPa, G12 = G13 = 5.83 GPa, G23 = 5.78 GPa, v12 = 0.278). Buckling load (kN)

a

Lay-up

1a

2

3

4

5

6

7

8

L/h = 10 Fixed–fixed Fixed–hinged Hinged–hinged

73.423 41.690 22.284

53.553 29.554 15.385

62.164 34.591 18.154

32.452 17.373 8.8086

33.123 17.745 9.0024

43.889 23.855 12.255

36.730 19.784 10.082

38.420 20.728 10.579

L/h = 20 Fixed–fixed Fixed–hinged Hinged–hinged

22.284 11.754 5.8871

15.385 8.0385 3.9955

18.154 9.5111 4.7383

8.8086 4.5605 2.6490

9.0024 4.6618 2.3008

12.255 6.3731 3.1561

10.082 5.2292 2.5840

10.579 5.4896 2.7138

L/h = 50 Fixed–fixed Fixed–hinged Hinged–hinged

3.7936 1.9503 0.9571

2.5691 1.3186 0.6463

3.0487 1.5655 0.7676

1.4441 0.7400 0.3623

1.4765 0.7567 0.3705

2.0272 1.0396 0.5093

1.6588 0.8503 0.4164

1.7423 0.8932 0.4374

L/h = 100 Fixed–fixed Fixed–hinged Hinged–hinged

0.9571 0.4901 0.2398

0.6463 0.3308 0.1618

0.7676 0.3930 0.1922

0.3623 0.1854 0.0906

0.3705 0.1896 0.0927

0.5093 0.2606 0.1275

0.4164 0.2131 0.1042

0.4374 0.2238 0.1094

Number indicates a stack sequence, in which 1: (0)8T, 2: (0/90)4T, 3: (0/90)2S, 4: (±45)4T, 5: (±45)2S, 6: (152/602/302/452)T, 7: (302/602)2T and 8: (30/60)2S.

All the above comparisons demonstrate that the present results agree well with the existing studies. It is noted that in the above examples the extension/shear, extension/twist, and flexural/twist couplings denoted by the stiffness matrices A16, B16, D16 and the initial geometric imperfections of the beam are not included. 4.2. Parametric studies In this section, a parametric study is undertaken to investigate the postbuckling response of laminated beams with an initial imperfection and subjected to axial compression. To this end, nine different imperfection modes are involved in the numerical illustrations with the following parameters: For the sine-type imperfection,

L1-mode : d ¼ 15;

l1 ¼ 2; #1 ¼ 0:25

L2-mode : d ¼ 15;

l1 ¼ 2; #1 ¼ 0:5

L3-mode : d ¼ 15;

l1 ¼ 5; #1 ¼ 0:5

L4-mode : d ¼ 15;

l1 ¼ 7; #1 ¼ 0:5

a 120

L/h = 20, (0/90)2S 1: Fixed-fixed 2: Fixed-hinged 90 3: Hinged-hinged

l1 ¼ 1; #1 ¼ 0:5 P (kN)

d ¼ 0;

For the local-type imperfections.

For the global-type imperfections,

G1-mode : d ¼ 0;

l1 ¼ 3; #1 ¼ 0:5

G2-mode : d ¼ 0;

l1 ¼ 5 ; #1 ¼ 0:5

G3-mode : d ¼ 0;

l1 ¼ 7; #1 ¼ 0:5

1

60

2

30

3

0 0.0

0.5

1.0

* W /r = 0.0 * W /r = 0.1

1.5

2.0

2.5

Wm/r

P (kN)

40

30

Hinged-hinged, L/h = 20 1: (0) 8T 2: (0/90) 2S 3: (+45) 2S 4: (15 2/60 2/-30 2/-45 2)T 5: (30 2/602 )2T

b 40 *

W /r = 0.0 * W /r = 0.1

30

L/h = 20, (152/602/-302/-452)T 1: Fixed-fixed 2: Fixed-hinged 3: Hinged-hinged

1 1

20

P (kN)

50

2

20 2

10

10

0 0.0

3

5 4 3

0.5

1.0

1.5

*

W /r = 0.0 * W /r = 0.1

2.0

2.5

Wm/r Fig. 4. Anisotropic effect on the postbuckling behavior of laminated beams under axial compressions.

0 0.0

0.5

1.0

1.5

2.0

2.5

W m/r Fig. 5. Boundary condition effect on the postbuckling behavior of laminated beams under axial compressions: (a) (0/90)2S and (b) (152/602/302/452)T.

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A list of the imperfection modes, except for the wave shapetype imperfection, is given in Table 9 where Cases G1, G2, and G3 are the global imperfection modes; while Cases L1, L2, L3, and L4 are the localized imperfection modes. In fact, initial imperfection is usually randomly and uncertainly distributed, thus leading to that the buckling load is uncertain. In the process of structural design, the shape of the initial imperfections cannot be known in advance; therefore, according to the distribution characteristics of initial imperfections related to the manufacturing process, the method of stochastic analysis can be effectively adopted to estimate the initial imperfection. Considering the imperfections of bounded random in the early stage, the Fourier coefficient of initial imperfections can be used to describe the bounded uncertain variables, calculate the buckling load of the upper and lower limits, obtain the buckling load limit range, and eventually establish the lower limit of corresponding design criterion. Such a process can be considered as a reasonable method for predicting the buckling loads of imperfect beams. The parametric study is presented to provide information on the postbuckling response of anisotropic laminated beams subjected to axial compression. Typical results are shown in Tables 10–12 and Figs. 4–9. Both the graphite/epoxy and glass/epoxy (Vetrotex E glass fibers, My 750 epoxy resin) composite materials are 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 = h = 8.0 mm, and the material properties of graphite/epoxy [42] (except for Tables 10 and 12 and Fig. 8) adopted are: E11 = 138 GPa, E22 = 14.5 GPa, G12 = G13 = 5.86 GPa, G23 = 3.52 GPa, m12 = 0.21. It should be noticed that in all of

a

the figures (Figs. 4–9) W  =h denote the dimensionless maximum initial geometric imperfection of the beam. Table 10 gives the buckling loads (kN) for the fixed–fixed (0)8T, (0/90)4T, (0/90)2S, (90/0)2S, (±45)4T, (±45)2S, (302/602)2T, (30/60)2S and (152/602/302/452)T laminated beams with different values of length to thickness ratio L/h (= 10, 20, 50 and 100) subjected to axial compression, respectively. In Table 10, the material properties of graphite/epoxy resin [43] adopted are: E11 = 181 GPa, E22 = 10.3 GPa, G12 = G13 = 7.17 GPa, G23 = 2.87 GPa, v12 = 0.25. It can be seen that the buckling loads remarkably change for different values of length-to-thickness ratio L/h (= 10, 20, 50 and 100), and the buckling loads for the (0)8T laminated beams is the largest among these lay-ups because of the axial stiffness being off the charts relative to the other lay-ups. On the other hand, the extension/shear, extension/twist and flexural/twist couplings stiffness coefficients A16, B16, and D16 are approximately zero-values for (0)8T, (0/90)4T, (0/90)2S and (90/0)2S beams, but these beams have relatively higher buckling loads than others due to the high axial stiffness coefficient A11 . Tables 11 and 12 show the buckling loads (kN) for (0)8T, (0/90)4T, (0/90)2S, (±45)4T, (±45)2S, (152/602/302/452)T, (302/602)2T and (30/60)2S laminated beams with different values of lengthto-thickness ratio L/h (= 10, 20, 50 and 100) and different types of boundary conditions subjected to axial compression, respectively. For Table 12, the material properties of glass fiber reinforced epoxy resin [44] adopted are: E11 = 45.6 GPa, E22 = 16.2 GPa, G12 = G13 = 5.83 GPa, G23 = 5.78 GPa, and v12 = 0.278. It can be seen that the buckling loads significantly changes among different types of end boundary conditions, and as expected, the buckling loads for laminated beams with the fixed–fixed end conditions is the largest

30 Hinged-hinged, L/h = 20, h = 8.0 mm I&1: (0/90)S, II&1: (0/90)2S

a

25 I&2: (±45) , II&2: (±45) S 2S

I&3: (30/60)S, II&3: (30/60)2S

I&1

II&1

Hinged-hinged, (0/90)2S 1: L/h = 20 I: Euler-Bernoulli beam theory 25 2: L/h = 50 II: Present (Higher-order theory) 3: L/h = 100 *

15 *

W /r = 0.0 * W /r = 0.1

10

I&1

W /r = 0.0 * W /r = 0.1

20

P (kN)

P (kN)

20

30

II&1

15 10

I&3 II&3

I&2 II&2

5 0 0.0

5

I&2 II&2

0.5

1.0

1.5

I&3 II&3

2.0

2.5

0 0.0

W m/r

1.0

25 Hinged-hinged, L/h = 20, h = 8.0 mm I&1: (0/90)2T, II&1: (0/90)4T 20 I&2: (±45)2T, II&2: ( ±45)4T I&3: (30/60)2T, II&3: (30/60)4T

15

1.5

2.0

2.5

Wm/r

b

12 Hinged-hinged, (152/602/-302/-452)T 1: L/h = 20 I: Euler-Bernoulli beam theory

10 2: L/h = 50 II: Present (Higher-order theory)

* W /r = 0.0 * W /r = 0.1 II&1 I&1

3: L/h = 100

8

P (kN)

P (kN)

b

0.5

II&3 10

I&3

II&1

*

W /r = 0.0 * W /r = 0.1

I&1

6 4

5

II&2 I&2

0 0.0

0.5

1.0

1.5

2.0

I&2 II&2

2 2.5

Wm/r Fig. 6. Effect of total number of plies on the postbuckling behavior of laminated beams under axial compressions: (a) symmetric laminates and (b) anti-symmetric laminates.

0 0.0

II&3I&3

0.5

1.0

1.5

2.0

2.5

Wm/r Fig. 7. Effect of length to thickness ratio on the postbuckling behavior of laminated beams under axial compressions: (a) (0/90)2S and (b) (152/602/302/452)T.

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c 25

40

Hinged-hinged, L/h = 20 I: E11a = 232.3 GPa II: E11b = 144.8 GPa 30 1: (0/90)2S 2: (0/90)4T I&1

Hinged-hinged, L/h = 20 I: E11a = 232.3 GPa II: E11b = 144.8 GPa 20 1: (±45)2S 2: (90/0)2S

II&1

P (kN)

P (kN)

a

20

I&2

II&2

15

10

II&2

10

I&2

I&1 II&1

5

*

W /r = 0.0 * W /r = 0.1

0 0.0

0.5

1.0

1.5

2.0

*

0 0.0

2.5

W /r = 0.0 * W /r = 0.1

0.5

1.0

W m/r 10

Hinged-hinged, L/h = 20 I: E11a = 232.3 GPa II: E11b = 144.8 GPa 8 1: (± 45)4T 2: (152/602/-302/-452)T

6

d I&2

II&2

P (kN)

P (kN)

b

I&1

4 II&1

2

0 0.0

1.5

2.0

8

Hinged-hinged, L/h = 20 I: E11a = 232.3 GPa II: E11b = 144.8 GPa 6 1: (30/60)2S 2: (60/30)2S

I&1 II&1 I&2 II&2

4

2 *

W /r = 0.0 * W /r = 0.1

* W /r = 0.0 * W /r = 0.1

0.5

1.0

2.5

W m/r

1.5

2.0

2.5

W m/r

0 0.0

0.5

1.0

1.5

2.0

2.5

Wm/r

Fig. 8. Effect of material orthotropy ratio on the postbuckling behavior of laminated beams under axial compressions: (a) (0/90)2S and (0/90)4T, (b) (+/45)4T and (152/602/ 302/452)T, (c) (+/45)2S and (0/90)2S, and (d) (30/60)2S and (60/30)2S.

among the three end conditions. It is obvious that the buckling load for the (0)8T graphite/epoxy beam (L/h = 10) with the fixed– fixed end conditions is the largest among all these laminated beams in both the tables due to the fact that the axial stiffness of (0)8T graphite/epoxy beam is the largest. Fig. 4 presents the postbuckling load–deflection curves for (0)8T, (0/90)2S, (±45)2S, (152/602/302/452)T and (302/602)2T laminated beams with the hinged–hinged boundary condition subjected to axial compression. It can be seen that the postbuckling equilibrium path first goes up slightly and smoothly and then rises sharply when the deflection of the beam is sufficiently larger. The results confirm that the buckling load for the (0)8T beam is the largest among these laminated beams. It is found that an increase in the axial compressive loads is usually required to obtain an increase in deformation and the postbuckling equilibrium path is stable for the range of small scale deformation. Fig. 5 shows the postbuckling load–deflection curves for the (0/90)2S and (152/602/302/452)T laminated beams with different boundary conditions subjected to axial compression. The results show that the (0/90)2S and (152/602/302/452)T beams with the fixed–fixed boundary condition have higher buckling load and postbuckling equilibrium path than these with other boundary conditions. Fig. 6 demonstrates the postbuckling load–deflection curves for the (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 with the hinged–hinged boundary condition subjected to axial compression. It is found that the buckling loads and postbuckling equilibrium paths for symmetric and antisymmetric laminated beams with the same total thickness (h = 8.0 mm) become higher

as the total number of plies are increased. On contrast, for the (0/90)S symmetric cross-ply laminated beams, they have the remarkably lower buckling loads and postbuckling equilibrium paths when the total number of plies are increased. Fig. 7 illustrates the comparisons on the postbuckling load– deflection curves for the (0/90)2S and (152/602/302/452)T laminated beams with the hinged–hinged boundary condition subjected to axial compression for the length-to-thickness ratio varying from 20 to 100 using the Euler–Bernoulli or classical beam theory (CBT) and higher-order shear deformation theory, respectively. It is observed that the critical buckling loads decrease rapidly as the length-to-thickness ratio increases. The length-tothickness ratio is a crucial parameter in determining the critical buckling load, and it also has a significant effect on the postbuckling response. These load vs. deflection curves shown in Fig. 7 indicate that the classical beam theory (CBT) always underestimates the amplitude of buckling loads when compared with the higherorder theories and the shear-deformation effect can be neglected for a relatively larger length to thickness ratio, e.g., when L/h > 50. The effect of material orthotropy ratio on the postbuckling behavior of eight laminated (i.e., (0/90)2S, (0/90)4T, (±45)4T, (152/ 602/302/452)T, (±45)2S, (90/0)2S, (30/60)2S and (60/30)2S) beams with the hinged–hinged end condition under compression load is shown in Fig. 8. It should be mentioned that only the value of E11 in the laminates is varied, and the other material properties keep invariant. The material properties of graphite/epoxy adopted are [45]: E11a = 232.3 GPa, E11b = 144.80 GPa, E22 = 9.65 GPa, G12 = G13 = 4.14 GPa, G23 = 3.45 GPa, v12 = 0.3. The effect of the increasing extensional modulus E11 corresponds to the increased buckling loads of the beam, and it affects considerably the

Z.-M. Li, P. Qiao / Engineering Structures 85 (2015) 277–292

40 Fixed-fixed, L/h = 20 (15 2/60 2/-30 2/-45 2)T

P (kN)

30 1

2

20

*

perfect (W m /r = 0.0) *

10

imperfect (W m /r = 0.1) 1: Mode shape 2: Sine type 3: G1-mode 4: G2-mode 5: G3-mode

3 4 5

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

W m/r

b

40 Fixed-fixed, L/h = 20 (15 2/60 2/-30 2/-45 2)T

P (kN)

30

20

25 *

perfect ( Wm /r = 0.0) *

10

0 0.0

4

3

imperfect ( Wm /r = 0.1) 1: Mode shape 2: L1-mode 3: L2-mode 4: L3-mode 5: L4-mode

1

1.0

1.5

2.0

2.5

3.0

W m/r Fig. 9. Effects of imperfection modes on the postbuckling behavior of laminated beams under axial compressions (a) global imperfection modes and (b) localized imperfection modes.

postbuckling equilibrium paths except the (±45)4T and (±45)2S laminated beams. As a consequence, the smaller the axial stiffness coefficients A11 , the lower the resulting buckling load parameters. For the antisymmetric and symmetric angle-ply laminates, it is of particular interest to note that the buckling characteristics of the beams are not particularly affected by the change of the E11/E22 ratio. Fig. 9 shows the effect of different types of global initial imperfection on the postbuckling behavior of (152/602/302/452)T laminated beams with the fixed–fixed end conditions subjected to axial compression. Fig. 9a examines the postbuckling behavior of (152/602/302/452)T laminated beams with the sine-type and global-type geometric imperfections (i.e., G1-, G2- and G3-modes) which are symmetric about the beam center. The half-wave numbers for the G1-, G2- and G3-modes are 3, 5, and 7, respectively. The effect of the half-wave number of globally-distributed imperfections on the postbuckling response of laminated beams is seen to be not very significant. The postbuckling equilibrium path of the imperfect beam with the sine type imperfection is initially higher than the curve of the perfect beam. The effect of the halfwave number of the imperfection mode on the postbuckling behavior of a laminated beam is further studied in Fig. 9b through a comparison of its response sensitivity to the local imperfections which are also symmetric with respect to the beam center. For this purpose, L1, L2, L3 and L4 are chosen to be the local imperfection modes in this example, and the half-wave numbers are 2, 5 and 7 in the X-direction. The postbuckling sensitivity of laminated beams to the asymmetrically distributed local imperfections is also investigated. This is undertaken by considering an L1-mode imperfection and the geometric center of which is deviated from the beam center along the X-axis with varying values of #1 used in

Eq. (33). Note that the unsymmetrical L1-mode virtually becomes a symmetric L2-mode when #1 = 0.5. Overall, the postbuckling strength becomes higher as the imperfection half-wave number increases, and the curves of imperfect beams with L3-mode and L4-mode imperfections are close to the curve of the perfect beam. This implies that the postbuckling equilibrium path tends to be much less affected if the local initial geometrical imperfection has a higher half-wave number. Fig. 10 shows the effect of varying the amplitude of the imperfection on the load–deflection curves for the (0/90)2S, (0/90)4T, (±45)2S and (152/602/302/452)T laminated beams with the hinged–hinged end condition subjected to axial compression. Imperfections with varying amplitudes ranging from 0.1r to 0.5r are in the form of the first buckling mode. In fact, the amplitudes are defined somewhere between 0 and 0.5r; however, its exact amplitudes are usually not known by the designer, who may be only able to specify the maximum contribution of imperfection to each mode but not the overall maximum amplitude. It is noted that each load–deflection curve in Fig. 10 follows a similar path, i.e., as the load increases, the corresponding increase in deflection becomes disproportionately larger. In each case, however, there remains some residual load carrying capacity in the postbuckling region. Increasing the imperfection size can be seen, in general, to decrease the prebuckling stiffness as expected, though this effect is minimal for the imperfections which are less than the radius of gyration of the cross section or thickness of the beam. To investigate the significance of the shear deformation on the postbuckling response, the buckling load for anisotropic laminated beams with different parameters (e.g., the length-to-thickness ratios, boundary conditions and lay-ups) can be determined using

a

20

Hinged-hinged, L/h = 20 I: (0/90)2S II: (0/90)4T I&1-4

15 II&1-4

P (kN)

a

10

1: 2: 3: 4:

5

0 0.0

0.5

1.0

1.5

*

W /r = 0.0 * W /r = 0.1 * W /r = 0.2 * W /r = 0.5 2.0

2.5

W m/r

b

10

Hinged-hinged, L/h = 20 I: (±45)2S 8 II: (152/602/-302/-452)T

II&1-4

P (kN)

290

6

4 1: 2: 3: 4:

2 I&1-4

0 0.0

0.5

1.0

1.5

*

W /r = 0.0 * W /r = 0.1 * W /r = 0.2 * W /r = 0.5

2.0

2.5

Wm/r Fig. 10. Effect of the amplitude of the imperfection on the postbuckling behavior of laminated beams under axial compressions: (a) (0/90)2S and (0/90)4T, and (b) (+/ 45)2S and (152/602/302/452)T.

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the present formulations. As it is well known from the literature, the shear deformation has a significant effect on the critical buckling loads. The same is observed from the present study (see the data in the tables and figures presented). This effect is more noticeable for composites with high material orthotropy, i.e., high E11/E22 ratio (see Fig. 8). It can also be seen from Figs. 5 and 7 that the boundary conditions and length-to-thickness ratio are both the crucial factors in determining the critical buckling loads and postbuckling equilibrium paths of shear deformable anisotropic laminated composite beams.

@wðxÞ ¼ A1 ½c cosðcxÞ  c tan c sinðcxÞ  tan c @x ¼ 0 ðFixed—hingedÞ

A1 ¼

W m =r ½sinðcxm Þ þ f 2 cosðcxm Þ þ f 1 xm  f 2 

5. Concluding remarks

A1 ¼

W m =r ½sinðcxm Þ  tan c cosðcxm Þ  xm tan c þ tan c

A1 ¼

W m =r sinðcxm Þ

In this study, the buckling and postbuckling analysis is presented for the shear deformable anisotropic laminated composite beams subjected to axial compression with different boundary conditions. The formulations are based on the higher-order shear deformation beam theory incorporating von Kármán nonlinear strain displacement relations, which leads to the nonlinear integral–differential equations. The obtained closed-form solution for postbuckling of shear deformable anisotropic laminated composite beams with and without initial geometric imperfection obtained as a function of the applied load is employed to investigate the postbuckling behaviors of shear deformable anisotropic laminated composite beams. The numerical solutions of buckling loads and postbuckling equilibrium paths are obtained using the Newton’s iterative method, Galerkin’s method and the secondary parameter conversion technique. A parametric study for different material properties, boundary conditions under different sets of geometry parameter conditions is conducted. The numerical results show that the geometric parameter, boundary conditions and material properties have significant effects on the postbuckling behavior of laminated beams. In particular, an increase in the length-to-thickness ratio or decrease in the orthotropy, i.e., E11/E22 ratio, to a certain extent, lowers the postbuckling curves, but it has an insignificant effect on the imperfection types of the postbuckling response of laminated beams. The effects of a wide range of initial geometrical imperfection modes on the postbuckling response of anisotropic laminated beams are also evaluated, and the results show that the postbuckling strength is relatively insensitive to the sine-mode and local imperfections but highly sensitive to the G2-mode global imperfections which are located at the center of the beam. The effect of a local imperfection becomes much less as its center deviates from the center of the beam. The present study sheds light on the buckling and postbuckling behavior of laminated beams with different (or in other words, more generically) geometric initial imperfections. Acknowledgments The work presented in this paper is supported in part by the grants from the National Natural Science Foundation of China (Nos. 50909059, 51279222 and 51478265). The first author (Z.-M. Li) is grateful for the support provided by the China Scholarship Council (CSC) which enables him to conduct research at Washington State University. Appendix A In Eqs. (30)–(32)

 @wðxÞ ðcos c  1Þ ¼ A1 c cosðcxÞ  sinðcxÞ þ f 1 @x sin c ¼ 0 ðFixed—fixedÞ

@wðxÞ ¼ A1 c cosðcxÞ ¼ 0 ðHinged—hingedÞ @x

ðA:1Þ

ðFixed—fixedÞ

ðFixed—hingedÞ

ðHinged—hingedÞ

ðA:2Þ

" # 2 1 ðf7 c2 þ f4 Þ 2 #0 ¼ f  f9 c  ð1 þ lÞ 5 ðf6 c2 þ f8 Þ For the case of fixed–fixed end conditions

#1 ¼

1 2

½sinðcxm Þ þ f 2 cosðcxm Þ þ f 1 xm  f 2   1 2 1 1 1  c þ c sin 2c þ f 22 c2  f 22 c sin 2c 2 4 2 4  2 2 2 þ f 1  f 2 c sin c þ 2f 1 sin c þ 2f 1 f 2 sin c

for the case of fixed–hinged end conditions

#1 ¼

1 2

½sinðcxm Þ  tan c cosðcxm Þ  xm tan c þ tan c  1 2 1 1 1  c þ c sin 2c þ c2 tan2 c  c tan2 c sin 2c 2 4 2 4  2  tan2 c þ c tan c sin c

and for the case of hinged–hinged end conditions

#1 ¼

c2 2

2 sin ðcxm Þ

ðA:3Þ

References [1] Timoshenko SP, Gere JM. Theory of elastic stability. New York: McGraw-Hill; 1961. [2] Comer RL, Levy S. Deflections of an inflated circular cylindrical cantilever beam. AIAA J 1963;1(7):1652–5. [3] Wang CY. Post-buckling of a clamped simply supported elastica. Int J NonLinear Mech 1997;32:1115–22. [4] Torkamani MAM, Sonmez MM, Cao J. Second-order elastic plane-frame analysis using finite-element method. ASCE J Struct Eng 1997;123:1225–35. [5] Hori A, Sasagawa A. Large deformation of inelastic large space frame I: analysis. ASCE J Struct Eng 2000;126:580–8. [6] Li SR, Batra RC. Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler–Bernoulli beams. Compos Struct 2013;95(1):5–9. [7] Vo TP, Thai HT. Free vibration of axially loaded rectangular composite beams using refined shear deformation theory. Compos Struct 2012;94(11):3379–87. [8] Chai GB, Yap CW, Lim TM. Bending and buckling of a generally laminated composite beam-column. Proc Inst Mech Eng, Part L: J Mater Des Appl 2010;224(1):1–7. [9] Nayfeh AH, Emam SA. Exact solutions and stability of the postbuckling configurations of beams. Nonlinear Dyn 2008;54(4):395–408. [10] Emam SA, Nayfeh AH. Postbuckling and free vibrations of composite beams. Compos Struct 2009;88:636–42. [11] Emam SA. A static and dynamic analysis of the postbuckling of geometrically imperfect composite beams. Compos Struct 2009;90(2):247–53. [12] Gupta RK, Gunda JB, Janardhan GR, Rao GV. Post-buckling analysis of composite beams: simple and accurate closed-form expressions. Compos Struct 2010;92(8):1947–56.

292

Z.-M. Li, P. Qiao / Engineering Structures 85 (2015) 277–292

[13] Khdeir AA, Reddy JN. Buckling of cross-ply laminated beams with arbitrary boundary conditions. Compos Struct 1997;37(1):1–7. [14] Khamlichi A, Bezzazi M, Limam A. Buckling of elastic cylindrical shells considering the effect of localized axisymmetric imperfections. Thin-Wall Struct 2004;42:1035–47. [15] Matsunaga H. Vibration and buckling of multilayered composite beams according to higher order deformation theories. J Sound Vib 2001;246: 47–62. [16] Aydogdu M. Buckling analysis of cross-ply laminated beams with general boundary conditions by Ritz method. Compos Sci Technol 2006;66:1248–55. [17] Pagano NJ. Exact solutions for composite laminates in cylindrical bending. J Compos Mater 1969;3:398–411. [18] Pagano NJ. Exact solutions for rectangular bidirectional composites and sandwich plates. J Compos Mater 1970;4(1):20–34. [19] Foraboschi P. Analytical solution of two-layer beam taking into account nonlinear interlayer slip. J Eng Mech 2009;135(10):1129–46. [20] Foraboschi P. Laminated glass column. Struct Eng 2009;87(18):20–6. [21] Foraboschi P. Three-layered plate: elasticity solution. Compos B Eng 2014;60:764–76. [22] Foraboschi P. Hybrid laminated-glass plate: design and assessment. Compos Struct 2013;106:250–63. [23] Biolzi L, Cattaneo S, Rosati G. Progressive damage and fracture of laminated glass beams. Constr Build Mater 2010;24(4):577–84. [24] Biolzi L, Cagnacci E, Orlando M, Piscitelli L, Rosati G. Long term response of glass-PVB double-lap joints. Compos B Eng 2014;63:41–9. [25] Murthy MVVS, Mahapatra DR, Badarinarayana K, Gopalakrishnan S. A refined higher order finite element for asymmetric composite beams. Compos Struct 2005;67(1):27–35. [26] Carrera E, Giunta G. Refined beam theories based on a unified formulation. Int J Appl Mech 2010;2(1):117–43. [27] Carrera E, Giunta G, Nali P, Petrolo M. Refined beam elements with arbitrary cross-section geometries. Comput Struct 2010;88(5–6):283–93. [28] Carrera E, Petrolo M. On the effectiveness of higher-order terms in refined beam theories. J Appl Mech 2011;78(2):021013. [29] Carrera E, Petrolo M, Nali P. Unified formulation applied to free vibration finite element analysis of beams with arbitrary section. Shock Vib 2011;18(3):485–502. [30] Carrera E, Petrolo M, Varello A. Advanced beam formulations for free vibration analysis of conventional and joined wings. J Aerosp Eng 2011;25(2):282–93.

[31] Giunta G, Belouettar S, Carrera E. Analysis of FGM beams by means of classical and advanced theories. Mech Adv Mater Struct 2010;17(8):622–35. [32] Giunta G, Biscani F, Carrera E, Belouettar S. Analysis of thin-walled beams via a one-dimensional unified formulation. Int J Appl Mech 2011;3(3):407–34. [33] Challamel N, Mechab I, El Meiche N, Houari MSA, Ameur M, Ait Atmane H. Buckling of generic higher-order shear beam/columns with elastic connections: local and nonlocal formulation. J Eng Mech 2013;139(8):1091–109. [34] Emam SA. Analysis of shear-deformable composite beams in post-buckling. Compos Struct 2011;94(1):24–30. [35] Ghugal YM, Shimpi RP. A review of refined shear deformation theories for isotropic and anisotropic laminated beams. J Reinf Plast Compos 2001;20(3):255–72. [36] Mullapudi T, Ayoub A. Analysis of reinforced concrete columns subjected to combined axial, flexure, shear, and torsional loads. J Struct Eng 2013;139(4):561–73. [37] Wang CM, Reddy JN, Lee KH. Shear deformable beams and plates: relationships with classical solutions. Oxford (UK): Elsevier; 2000. [38] Waas AM. Initial post buckling behavior of shear deformable symmetrically laminated beams. Int J Non-Linear Mech 1992;27(5):817–32. [39] Schneider W, Timmel I, Höhn K. The conception of quasi-collapse-affine imperfections: a new approach to unfavourable imperfections of thin-walled shell structures. Thin-Wall Struct 2005;43(8):1202–24. [40] Simitses GJ, Anastasiadist JS. Shear deformable theories for cylindrical laminates-equilibrium and buckling with applications. AIAA J 1992;30(3):826–34. [41] Wadee MA. Effects of periodic and localized imperfections on struts on nonlinear foundations and compression sandwich panels. Int J Solids Struct 2000;37(8):1191–209. [42] Wang SS, Srinivasan S, Hu HT, HajAli R. Effect of material nonlinearity on buckling and postbuckling of fiber composite laminated plates and cylindrical shells. Compos Struct 1995;33(1):7–15. [43] Wu Z, Chen WJ. An assessment of several displacement-based theories for the vibration and stability analysis of laminated composite and sandwich beams. Compos Struct 2008;84:337–49. [44] Messager T, Pyrz M, Gineste B, Chauchot P. Optimal laminations of thin underwater composite cylindrical vessels. Compos Struct 2002;58(4):529–37. [45] Li J, Shi CX, Kong XS, Li XB, Wu WG. Free vibration of axially loaded composite beams with general boundary conditions using hyperbolic shear deformation theory. Compos Struct 2013;97:1–14.