Buckling and free vibration analysis of laminated composite plates using an eﬃcient and simple higher order shear

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c AFM, EDP Sciences 2016 DOI:10.1051/meca/2015112 www.mechanics-industry.org

M echanics

& I ndustry

Buckling and free vibration analysis of laminated composite plates using an eﬃcient and simple higher order shear

deformation theory

Belkacem Adim

^1,4

, Tahar Hassaine Daouadji

^1,3,a

, Boussad Abbes

²

and A. Rabahi

^1,4

1 Département de génie civil, Université Ibn Khaldoun Tiaret, BP 78 Zaaroura, 14000 Tiaret, Algérie

2 Laboratoire GRESPI, Campus du Moulin de la Housse, BP 1039, 51687 Reims Cedex 2, France

3 Laboratoire de Géomatique et Développement Durable, Université Ibn Khaldoun, Tiaret, Algérie

4 Laboratoire de Technologie Industrielle, Universit´e Ibn Khaldoun, Tiaret, Alg´erie Received 13 February 2015, Accepted 21 November 2015

Abstract – In this paper, the buckling and free vibration analysis of laminated composite plates using an efficient and simple higher order shear deformation theory are examined by using a refined shear deformation theory. This theory is based on the assumption that the transverse displacements consist of bending and shear components where the bending components do not contribute to shear forces, and likewise, the shear components do not contribute to bending moments. The most interesting feature of this theory is that it allows for parabolic distributions of transverse shear stresses across the plate thickness and satisfies the conditions of zero shear stresses at the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns in the present theory is four, as against five in other shear deformation theories. In this analysis, the equations of motion for simply supported thick laminated rectangular plates are derived and obtained through the use of Hamilton’s principle. The closed- form solutions of anti-symmetric cross-ply and angle- ply laminates are obtained using Navier solution.

Numerical results of the present study are compared with three-dimensional elasticity solutions and results of the ﬁrst-order and the other higher-order theories reported in the literature. It can be concluded that the proposed theory is accurate and simple in solving the buckling and free vibration behaviors of laminated composite plates.

Key words: Analytical solutions / laminated composite plates / higher-order shear deformation theory / buckling / free vibration / Navier solution

1 Introduction

Laminated composite plates are widely used in industry and new fields of technology. Due to the high degrees of anisotropy and the low rigidity in transverse shear of the plates, the Kirchhoff hypothesis as a classical theory is no longer adequate. The hypothesis states that the normal to the midplane of a plate remains straight and normal after deformation because of the negligible transverse shear effects. Refined theories without this assumption have been used recently. The classical laminate plate theory CLPT underpredicts deflections and over predicts frequencies as well as buckling loads with moderately thick plates. Many shear deformation theories account- ing for transverse shear effects have been developed to overcome the deficiencies of the CLPT. The first-order

a Corresponding author:daouadjitah@yahoo.fr

shear deformation theories FSDT based on Reissner [1]

and Mindlin [2] account for the transverse shear eﬀects by the way of linear variation of in-plane displacements through the thickness. A number of shear deformation theories have been proposed to date. The ﬁrst such theory for laminated isotropic plates was apparently [3]. This theory was generalized to laminated anisotropic plates in reference [4]. It was shown in references [5–7], the FSDT violates equilibrium conditions at the top and bottom faces of the plate, shear correction factors are re- quired to rectify the unrealistic variation of the shear strain/stress through the thickness. In order to overcome the limitations of FSDT, higher-order shear deformation theories HSDT, since which involve higher-order terms in Taylor’s expansions of the displacements in the thickness coordinate, were developed by Librescu [8], Levinson [9], Bhimaraddi and Stevens [10], Reddy [11], Ren [12], Kant and Pandya [13], and Mohan et al. [14]. A good review

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of these theories for the analysis of laminated composite plates is available in references [15–19]. A reﬁned plate theory using only two unknown functions was developed by Shimpi [20] for isotropic plates, and was extended by Shimpi and Patel [21,22] for orthotropic plates. The most interesting feature of this theory is that it does not require shear correction factors, and has strong similari- ties with the classical plate theory in some aspects such as governing equation, boundary conditions and moment expressions.

In this paper, a reﬁned and simple theory of plates is presented and applied to the investigation of buckling and free vibration behavior of laminated composite plates. This theory is based on the assumption that the in-plane and transverse displacements consist of bending and shear components where the bending components do not contribute to shear forces, and likewise, the shear components do not contribute to bending moments. The most interesting feature of this theory is that it allows for parabolic distributions of transverse shear stresses across the plate thickness and satisﬁes zero shear stress conditions at the top and bottom surfaces of the plate without using shear correction factors. The equations of motion are derived using Hamilton’s principle. The fundamental frequencies are found by solving an eigenvalue equation.

The results obtained by the present method are compared with solutions and results of the ﬁrst-order and the other higher-order theories.

2 Reﬁned plate theory for laminated composite plates

2.1 Basic assumptions

Consider a rectangular plate of total thickness h com- posed of n orthotropic layers with the coordinate system as shown in Figure1. Assumptions of the reﬁned plate’s theory are as follows:

– The displacements are small in comparison with the plate thickness and, therefore, strains involved are inﬁnitesimal.

– The transverse displacementwincludes three components of bendingw_band shearw_s. These components are functions of coordinatesx,y, and time tonly.

w(x, y, t) =w_b(x, y, t) +w_s(x, y, t) (1) – The transverse normal stressσ_z is negligible in com-

parison with in-plane stressesσ_xandσ_y.

– The displacements U in x-direction and V in y- direction consist of extension, bending, and shear components:

U =u+u_b+u_s, V =v+v_b+v_s (2) – The bending components u_b and v_b are assumed to be similar to the displacements given by the

classical plate theory. Therefore, the expression for u_b andv_bcan be given as:

u_b=−z∂w_b

∂x , v_b=−z∂w_b

∂y (3a)

– The shear componentsu_sandv_sgive rise, in con- junction with w_s, to the parabolic variations of shear strains γ_xz, γ_yz and hence to shear stresses σ_xz,σ_yz through the thickness of the plate in such a way that shear stressesσ_xz, σ_yz are zero at the top and bottom faces of the plate. Consequently, the expression foru_sandv_scan be given as:

u_s=f(z)∂w_s

∂x, v_s=f(z)∂w_s

∂y (3b)

2.2 Kinematics

Based on the assumptions made in the preceding section, the displacement ﬁeld can be obtained using Equations (1)–(3) as:

u(x, y, z, t) =u(x, y, t)−z∂w_b

∂x +f(z)∂w_s

∂x v(x, y, z, t) =v(x, y, t)−z∂w_b

∂y +f(z)∂w_s

∂y

w(x, y, z, t) =w_b(x, y, t) +w_s(x, y, t) (4a) where u and v are the mid-plane displacements of the plate in the x and y direction, respectively; w_b and w_s are the bending and shear components of transverse displacement, respectively, whilef(z) represents shape functions determining the distribution of the transverse shear strains and stresses along the thickness and is given as the present model; the function f(z) is an hyperbolic shape function (Hyperbolic Shear Deformation Theory):

f(z) =z

1 + 3π 2 sech²

1 2

−3π 2 htanh

z h

(4b) It should be noted that unlike the ﬁrst-order shear deformation theory, this theory does not require shear correction factors. The strains associated with the displacements in Equation (4) are:

ε_x=ε⁰_x+zk_x^b+fk_x^s ε_y=ε⁰_y+zk^b_y+fk^s_y γ_xy=γ_xy⁰ +zk_xy^b +fk_xy^s γ_yz=gγ^s_yz

γ_xz=gγ^s_xz

ε_z= 0 (5)

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Fig. 1.Coordinate system and layer numbering used for a typical laminated plate.

where:

ε⁰_x=∂u

∂x, k_x^b=−∂²w_b

∂x² , k_x^s =−∂²w_s

∂x² , ε⁰_y =∂v

∂y, k_y^b=−∂²w_b

∂y² , k^s_y=−∂²w_s

∂y² γ_xy⁰ =∂u

∂y + ∂v

∂x, k_xy^b =−2∂²w_b

∂x∂y, k_xy^s =−2∂²w_s

∂x∂y, γ_yz^s = ∂w_s

∂y , γ_xz^s = ∂w_s

∂x (6) g(z) = 1−f(z) andf(z) =^df(z)_dz .

2.3 Constitutive equations

Under the assumption that each layer possesses a plane of elastic symmetry parallel to the x-y plane, the constitutive equations for a layer can be written as

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ σ_x σ_y σ_xy σ_yz σ_xz

⎫⎪

⎪⎪

⎪⎬

⎪⎪

⎭

=

⎡

⎢⎢

⎣

Q₁₁ Q₁₂ 0 0 0 Q₁₂ Q₂₂ 0 0 0

0 0 Q₆₆ 0 0

0 0 0 Q₄₄ 0

0 0 0 0 Q₅₅

⎤

⎥⎥

⎦

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ ε_x ε_y γ_xy γ_yz γ_xz

⎫⎪

⎪⎪

⎪⎬

⎪⎪

⎭ (7)

where Q_ij are the plane stress-reduced stiﬀnesses, and are known in terms of the engineering constants in the material axes of the layer:

Q₁₁= E₁₁

1−ν₁₂ν₂₁, Q₂₂= E₂₂

1−ν₁₂ν₂₁, Q₁₂= ν₁₂E₂₂ 1−ν₁₂ν₂₁, Q₆₆=G₁₂, Q₄₄=G₂₃, Q₅₅=G₁₃ (8)

Since the laminate is made of several orthotropic layers with their material axes oriented arbitrarily with respect to the laminate coordinates, the constitutive equations of each layer must be transformed to the laminate coordinates (x, y, z). The stress-strain relations in the laminate coordinates of the kth layer are given as

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ σ_x σ_y σ_xy σ_yz σ_xz

⎫⎪

⎪⎪

⎪⎬

⎪⎪

⎭

(k)

=

⎡

⎢⎢

⎣

Q¯₁₁ Q¯₁₂ Q¯₁₆ 0 0 Q¯₁₂ Q¯₂₂ Q¯₂₆ 0 0 Q¯₁₆ Q¯₂₆ Q¯₆₆ 0 0 0 0 0 Q¯₄₄ Q¯₄₅ 0 0 0 Q¯₄₅ Q¯₅₅

⎤

⎥⎥

⎦

(k)⎧

⎪⎪

⎨

⎪⎪

⎩ ε_x ε_y γ_xy γ_yz γ_xz

⎫⎪

⎪⎪

⎪⎬

⎪⎪

⎭

(k)

(9)

(4)

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ Nx

Ny

Nxy

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ M_x^b M_y^b Mxy^b

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ M_x^s My^s

M_xy^s

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

=

⎡

⎢⎢

⎢⎣

⎡

⎢⎢

⎢⎣

A11 A12 A16

A12 A22 A26

A16 A26 A66

⎤

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎣

B11 B12 B16

B12 B22 B26

B16 A26 A66

⎤

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎣

B^s₁₁ B^s₁₂ B^s₁₆ B^s12 B^s22 B^s26

B^s₁₆ B^s₂₆ B^s₆₆

⎤

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎣

B11 B12 B16

B12 B22 B26

B16 A26 A66

⎤

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎣

D11 D12 D16

D12 D22 D26

D16 D26 D66

⎤

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎣

D11^s D^s12 D^s16

D₁₂^s D^s₂₂ D^s₂₆ D₁₆^s D^s₂₆ D^s₆₆

⎤

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎣

B^s₁₁ B^s₁₂ B₁₆^s B^s₁₂ B^s₂₂ B₂₆^s B^s₁₆ B^s₂₆ B₆₆^s

⎤

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎣

D₁₁^s D₁₂^s D₁₆^s D₁₂^s D₂₂^s D₂₆^s D₁₆^s D₂₆^s D₆₆^s

⎤

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎣

H₁₁^s H₁₂^s H₁₆^s H₁₂^s H₂₂^s H₂₆^s H₁₆^s H₂₆^s H₆₆^s

⎤

⎥⎥

⎥⎦

⎤

⎥⎥

⎥⎦

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ ε⁰x

ε⁰_y γ_xy⁰

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ k^bx

k^by

k_xy^b

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ kx^s

k_y^s kxy^s

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

(14a)

⎧⎨

⎩ Q^s_yz Q^s_xz

⎫⎬

⎭=

⎡

⎣A^s₄₄A^s₄₅ A^s₄₅A^s₅₅

⎤

⎦

⎧⎨

⎩ γ_yz^s γ_xz^s

⎫⎬

⎭ (14b)

where ¯Q_ijare the transformed material constants given as Q¯₁₁=Q₁₁cos⁴θ+2(Q₁₂+ 2Q₆₆) sin²θcos²θ+Q₂₂sin⁴θ Q¯₁₂= (Q₁₁+Q₂₂−4Q₆₆) sin²θcos²θ+Q₁₂(sin⁴θ+cos⁴θ) Q¯₂₂=Q₁₁sin⁴θ+2(Q₁₂+2Q₆₆) sin²θcos²θ+Q₂₂cos⁴θ Q¯₁₆= (Q₁₁−Q₁₂−2Q₆₆) sinθcos³θ

+ (Q₁₂−Q₂₂+ 2Q₆₆) sin³θcosθ Q¯₂₆= (Q₁₁−Q₁₂−2Q₆₆) sin³θcosθ

+ (Q₁₂−Q₂₂+ 2Q₆₆) sinθcos³θ Q¯₆₆= (Q₁₁+Q₂₂−2Q₁₂−2Q₆₆) sin²θcos²θ

+Q₆₆(sin⁴θ+ cos⁴θ) Q¯₄₄=Q₄₄cos²θ+Q₅₅sin²θ Q¯₄₅= (Q₅₅−Q₄₄) cosθsinθ

Q¯₅₅=Q₅₅cos²θ+Q₄₄sin²θ (10) In whichθis the angle between the globalx-axis and the localx-axis of each lamina.

2.4 Governing equations

The strain energy of the plate can be written as U =1

2

V

σ_ijε_ijdV = 1 2

V

(σ_xε_x+σ_yε_y

+σ_xyγ_xy+σ_yzγ_yz+σ_xzγ_xz)dV (11) Substituting Equations (5) and (9) into Equation (11) and integrating through the thickness of the plate, the strain

energy of the plate can be rewritten as U = 1

2

A

N_xε⁰_x+N_yε⁰_y+N_xyγ_xy⁰ +M_x^bk^b_x+M_y^bk_y^b +M_xy^bk_xy^b +M_x^sk_x^s+M_y^sk_y^s+M_xy^s k_xy^s +Q^s_yzγ_yz^s

+Q^s_xzγ_xz^s }dxdy (12)

where the stress resultantsN,M, and Qare deﬁned by (N_x, N_y, N_xy) =

_h/2

−h/2

(σ_x, σ_y, σ_xy)dz

= N k=1

_z_k+1

zk

(σ_x, σ_y, σ_xy)dz

(M_x^b, M_y^b, M_xy^b) = _h/2

−h/2

(σ_x, σ_y, σ_xy)zdz

= N k=1

_z_k+1

zk

(σ_x, σ_y, σ_xy)zdz

(M_x^s, M_y^s, M_xy^s ) = _h/2

−h/2

(σ_x, σ_y, σ_xy)fdz

= N k=1

_z_k+1

zk

(σ_x, σ_y, σ_xy)fdz

(Q^s_xz, Q^s_yz) = _h/2

−h/2

(σ_xz, σ_yz)gdz

= N k=1

_z_k+1

zk

(σ_xz, σ_yz)gdz (13)

Substituting Equation (9) into Equation (13) and integrating through the thickness of the plate, the stress resultants are given as:

See equations (14a) and (14b) above

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whereA_ij,B_ij, etc., are the plate stiﬀnesses, deﬁned by A_ij, B_ij, D_ij, B_ij^s, D^s_ij, H_ij^s

= h/2

−h/2

Q¯_ij(1, z, z², f(z), zf(z), f²(z))dz,(i, j) = (1,2,6) (15a) A^s_ij =

h/2

−h/2

Q¯_ij[g(z)]²dz,(i, j) = (4,5) (15b)

The work done by applied forces can be written as:

V = 1 2

A

N_x⁰∂²(w_b+w_s)

∂x² +N_y⁰∂²(w_b+w_s)

∂y² +2N_xy⁰ ∂²(w_b+w_s)

∂x∂y

dxdy (16) whereN_x⁰,N_y⁰ andN_xy⁰ are in-plane distributed forces.

The kinetic energy of the plate can be written as T = 1

2

V

ρ¨u_iidV=1 2

A

δu

I₁u¨−I₂∂w¨_b

∂x w−I₄∂w¨_s

∂x

+δv

I₁v¨−I₂∂w¨_b

∂y −I₄∂w¨_s

∂y

+δw_b

I₁( ¨w_b+ ¨w_s) +I₂ ∂u¨

∂x +∂v¨

∂y

−I₃ ∂²w¨_b

∂x² +∂²w¨_b

∂y²

−I₅ ∂²w¨_s

∂x² +∂²w¨_s

∂y²

+δw_s

I₁( ¨w_b+ ¨w_s) +I₄ ∂u¨

∂x +∂v¨

∂y

−I₅ ∂²w¨_b

∂x² +∂²w¨_b

∂y²

−I₆ ∂²w¨_s

∂x² +∂²w¨_s

∂y² dxdy (17) whereρis the mass of density of the plate andI_iare the inertias deﬁned by

(I₁, I₂, I₃, I₄, I₅, I₆) = h/2

−h/2

ρ(1, z, z², f(z), zf(z),[f(z)]²)dz (18) Hamilton’s principle [18] is used herein to derive the equations of motion appropriate to the displacement ﬁeld and the constitutive equation. The principle can be stated in analytical form as

_t

0

δ(U +V −T)dt= 0 (19) whereδindicates a variation with respect toxandy.

Substituting Equations (12), (16) and (17) into Equa- tion (19) and integrating the equation by parts, collecting

the coeﬃcients of δu, δv,δw_b and δw_s, the equations of motion for the laminate plate are obtained as follows:

δu: ∂N_x

∂x +∂N_xy

∂y =I₁u¨−I₂∂w¨_b

∂x −I₄∂w¨_s

∂x δv: ∂N_xy

∂x +∂N_y

∂y =I₁v¨−I₂∂w¨_b

∂y −I₄∂w¨_s

∂y δw_b: ∂²M_x^b

∂x² + 2∂²M_xy^b

∂x∂y +∂²M_y^b

∂y² +N(w)

=I₁( ¨w_b+ ¨w_s) +I₂ ∂¨u

∂x+∂¨v

∂y

−I₃ ∂²w¨_b

∂x² +∂²w¨_b

∂y²

−I₅ ∂²w¨_s

∂x² +∂²w¨_s

∂y²

δw_s: ∂²M_x^s

∂x² +2∂²M_xy^s

∂x∂y +∂²M_y^s

∂y² +∂Q^s_xz

∂x +∂Q^s_yz

∂y +N(w)

=I₁( ¨w_b+ ¨w_s) +I₄ ∂¨u

∂x+∂¨v

∂y

−I₅ ∂²w¨_b

∂x² +∂²w¨_b

∂y²

−I₆ ∂²w¨_s

∂x² +∂²w¨_s

∂y²

(20) where N(w) is deﬁned by

N(w) =N_x⁰∂²(w_b+w_s)

∂x² +N_y⁰∂²(w_b+w_s)

∂y²

+ 2N_xy⁰ ∂²(w_b+w_s)

∂x∂y (21) Equation (20) can be expressed in terms of displacements (u,v,w_b,w_s) by substituting for the stress resultants from Equation (14). For homogeneous laminates, the equations of motion (20) take the form

See equations (22a)–(22d) next page.

3 Analytical solutions

3.1 Analytical solutions for antisymmetric cross-ply laminates

The Navier solutions can be developed for rectangular laminates with two sets of simply supported boundary conditions. For antisymmetric cross-ply laminates, the following plate stiﬀnesses are identically zero:

A₁₆=A₂₆=D₁₆=D₂₆=D₁₆^s =D₂₆^s =H₁₆^s =H₂₆^s = 0 B₁₂=B₂₆=B₁₆=B₆₆=B₁₂^s =B₁₆^s =B₂₆^s =B₆₆^s

=A^s₄₅= 0

B₂₂=−B₁₁, B₂₂^s =−B₁₁^s (23)

(6)

A11∂²u

∂x² + 2A16 ∂²u

∂x∂y+A66∂²u

∂y² +A16∂²v

∂x² + (A12+A66) ∂²v

∂x∂y+A26∂²v

∂y²

−B11∂³wb

∂x³ −3B16 ∂³wb

∂x²∂y−(B12+ 2B66) ∂³wb

∂x∂y² −B26∂³wb

∂y³

−B^s₁₁∂³ws

∂x³ −3B₁₆^s ∂³ws

∂x²∂y−(B₁₂^s + 2B₆₆^s ) ∂³ws

∂x∂y² −B₂₆^s ∂³ws

∂y³ =I1u¨−I2∂w¨b

∂x −I4∂w¨s

∂x (22a)

A16∂²u

∂x² + (A12+A66) ∂²u

∂x∂y+A26∂²u

∂y² +A66∂²v

∂x² + 2A26 ∂²v

∂x∂y+A22∂²v

∂y²

−B16∂³wb

∂x³ −(B12+ 2B66)∂³wb

∂x²∂y−3B26 ∂³wb

∂x∂y²−B22∂³wb

∂y³

−B₁₆^s ∂³ws

∂x³ −(B₁₂^s + 2B₆₆^s ) ∂³ws

∂x²∂y −3B₂₆^s ∂³ws

∂x∂y² −B₂₂^s ∂³ws

∂y³ =I1v¨0−I2∂w¨b

∂y −I4∂w¨s

∂y (22b)

B11∂³u

∂x³ + 3B16 ∂³u

∂x²∂y+ (B12+ 2B66) ∂³u

∂x∂y² +B26∂³u

∂y³ +B16∂³v

∂x³ + (B12+ 2B66) ∂³v

∂x²∂y+ 3B26 ∂³v

∂x∂y² +B22∂³v

∂y³

−D11∂⁴wb

∂x⁴ −4D16 ∂⁴wb

∂x³∂y−2(D12+ 2D66) ∂⁴wb

∂x²∂y² −4D26 ∂⁴wb

∂x∂y³ −D22∂⁴wb

∂y⁴

−D11^s ∂⁴ws

∂x⁴ −4D16^s ∂⁴ws

∂x³∂y−2(D^s12+ 2D66^s ) ∂⁴ws

∂x²∂y² −4D^s26 ∂⁴ws

∂x∂y³ −D^s22∂⁴ws

∂y⁴ +N(w)

=I1( ¨wb+ ¨ws) +I2

∂u¨

∂x+∂v¨

∂y

−I3

∂²w¨b

∂x² +∂²w¨b

∂y²

−I5

∂²w¨s

∂x² +∂²w¨s

∂y²

(22c)

B₁₁^s ∂³u

∂x³ + 3B₁₆^s ∂³u

∂x²∂y + (B^s₁₂+ 2B₆₆^s ) ∂³u

∂x∂y² +B₂₆^s ∂³u

∂y³ +B16^s ∂³v

∂x³ + (B12^s + 2B66^s ) ∂³v

∂x²∂y+ 3B^s26 ∂³v

∂x∂y² +B^s22∂³v

∂y³

−D^s₁₁∂⁴wb

∂x⁴ −4D₁₆^s ∂⁴wb

∂x³∂y−2(D^s₁₂+ 2D^s₆₆) ∂⁴wb

∂x²∂y² −4D₂₆^s ∂⁴wb

∂x∂y³ −D₂₂^s ∂⁴wb

∂y⁴

−H11^s ∂⁴ws

∂x⁴ −4H16^s ∂⁴ws

∂x³∂y −2(H12^s + 2H66^s ) ∂⁴ws

∂x²∂y² −4H26^s ∂⁴ws

∂x∂y³ −H22^s ∂⁴ws

∂y⁴ +A^s₅₅∂²ws

∂x² +A^s₄₄∂²ws

∂y² + 2A^s₄₅∂²ws

∂x∂y+N(w)

=I1( ¨wb+ ¨ws) +I4

∂u¨

∂x+∂¨v

∂y

−I5

∂²w¨b

∂x² +∂²w¨b

∂y²

−I6

∂²w¨s

∂x² +∂²w¨s

∂y²

(22d)

(7)

The following boundary conditions for antisymmetric cross-ply laminates can be written as

v(0, y) =w_b(0, y) =w_s(0, y) = ∂w_b

∂y (0, y)

=∂w_s

∂y (0, y) = 0

v(a, y) =w_b(a, y) =w_s(a, y) =∂w_b

∂y (a, y)

=∂w_s

∂y (a, y) = 0

N_x(0, y) =M_x^b(0, y) =M_x^s(0, y) =N_x(a, y) =M_x^b(a, y)

=M_x^s(a, y) = 0

u(x,0) =w_b(x,0) =w_s(x,0) = ∂w_b

∂x (x,0)

=∂w_s

∂x (x,0) = 0

u(x, b) =w_b(x, b) =w_s(x, b) = ∂w_b

∂x (x, b)

=∂w_s

∂x (x, b) = 0

N_y(x,0) =M_y^b(x,0) =M_y^s(x,0) =N_y(x, b) =M_y^b(x, b)

=M_y^s(x, b) = 0 (24)

The boundary conditions in Equation (24) are satisﬁed by the following expansions

u= ∞ m=1

∞ n=1

U_mne^iωtcos(αx) sin(βy)

v= ∞ m=1

∞ n=1

V_mne^iωtsin(αx) cos(βy)

w_b= ∞ m=1

∞ n=1

W_bmne^iωtsin(αx) sin(βy)

w_s= ∞ m=1

∞ n=1

W_smne^iωtsin(αx) sin(βy) (25)

where U_mn, V_mn, W_bmn and W_smn unknown parameters must be determined,ω is the eigen frequency associated with (m, n) the eigen-mode, andα=^mπ_a andβ= ^nπ_b .

Substituting Equations (23) and (25) into Equa- tion (22), the Navier solution of antisymmetric cross-ply

laminates can be determined from equations

⎡

⎢⎢

⎢⎣

s₁₁ s₁₂ s₁₃ s₁₄ s₁₂ s₂₂ s₂₃ s₂₄ s₁₃ s₂₃ s₃₃+k s₃₄+k s₁₄ s₂₄ s₃₄+k s₄₄+k

⎤

⎥⎥

⎥⎦

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ U_mn V_mn W_bmn W_smn

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

+

⎡

⎢⎢

⎢⎣

m₁₁ 0 0 0

0 m₂₂ 0 0

0 0 m₃₃ m₃₄

0 0 m₃₄ ms₄₄

⎤

⎥⎥

⎥⎦

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ U¨_mn V¨_mn W¨_bmn W¨_smn

⎫⎪

⎪⎪

⎪⎬

⎪⎪

⎭

=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 0 0 0 0

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭ (26)

where

s₁₁=A₁₁α²+A₆₆β², s₁₂=αβ(A₁₂+A₆₆), s₁₃=−B₁₁α³, s₁₄=−B^s₁₁α³

s₂₂=A₆₆α²+A₂₂β², s₂₃=B₁₁β³, s₂₄=B₁₁^s β³ s₃₃=D₁₁α⁴+ 2(D₁₂+ 2D₆₆)α²β²+D₂₂β⁴ s₃₄=D^s₁₁α⁴+ 2(D^s₁₂+ 2D^s₆₆)α²β²+D^s₂₂β⁴ s₄₄=H₁₁^s α⁴+ 2(H₁₂^s + 2H₆₆^s )α²β²+H₂₂^s β⁴

+A^s₅₅α²+A^s₄₄β²

m₁₁=m₂₂=I₁, m₃₃=I₁+I₃(α²+β²)

m₃₄=I₁+I₅(α²+β²), m₄₄=I₁+I₆(α²+β²), k=N_x⁰α²+N_y⁰β² (27)

3.2 Analytical solutions for antisymmetric angle-ply laminates

For antisymmetric angle-ply laminates, the following plate stiﬀnesses are identically zero:

A₁₆=A₂₆=D₁₆=D₂₆=D^s₁₆=D^s₂₆=H₁₆^s

=H₂₆^s = 0

B₁₁=B₁₂=B₂₂=B₆₆=B^s₁₁=B^s₁₂=B^s₂₂

=B₆₆^s =A^s₄₅= 0 (28)