Effect of Lamination Constitution on Post Buckling Behavior of Symmetrically Laminated Plates with Initial Deflections

Effect of Lamination Constitution on Post-Buckling Behavior of Symmetrically

Laminated Plates with Initial Deflections

*1

Keiichi Nemoto

1

_{, Hisao Kikugawa}

2,*2

_{, Hiroyuki Moriyama}

3

_{and Hirakazu Kasuya}

4 1_{Department of Aerospace Engineering, The Yokohama Rubber Co.,Ltd., Hiratsuka 254–8601, Japan}

2_{Department of Biomedical Engineering, School of Engineering, Tokai University, Hiratsuka 259–1292, Japan} 3_{Department of Prime Mover Engineering, School of Engineering, Tokai University, Hiratsuka 259–1292, Japan} 4_{Professor Emeritus of Tokai University, Hiratsuka 259–1292, Japan}

Advanced fiber-reinforced composite materials are being used as structural members in various fields because of their high strength and high stiffness-to-weight ratios. Hence, analysis of thin laminated structures is important. The post-buckling behaviors of thin laminated plates under uniaxial compression have been discussed by many researchers. However, little research has been performed on the secondary buckling phenomenon for thin laminated plates, which occurs with further increase of load. In this paper, the stability conditions of carbon–epoxy sym-metrically laminated plates with initial deflections under uniaxial compression that are simply supported along four edges are determined using the second variation of total potential energy. The necessity of secondary buckling is proven analytically, and the effects of various factors, such as initial deflection, lamination constitution, and number of layers, are elucidated. [doi:10.2320/matertrans.M2016015]

(Received January 14, 2016; Accepted February 17, 2016; Published April 25, 2016)

Keywords:  structural analysis, composite materials, carbon fiber reinforced plastic (CFRP), secondary buckling, initial deflection, symmetri-cally laminated plates

1.  Introduction

The use of advanced composite materials, such as carbon fiber reinforced plastics, has been rapidly expanding to a wide range of industrial applications because they have excellent properties such as high specific strength and specific stiffness. Composite structures are used in aircraft-wings for the effect of coupling due to bending deformation and twisting defor-mation. In general, symmetrically balanced laminates reduce buckling load for most applications owing to coupling ef-fects1–3)_.

Symmetrically balanced laminates not experience in-plane extension–shear coupling owing to their extensional stiffness, A16 and A26; however, they do experience out-of-plane bend-ing–twisting coupling because of their bending stiffness, D16 and D26. By introducing the non-dimensional anisotropic pa-rameter for a finite number of layers, Nemeth4) _{proved that} lamination constitutions capable of ignoring the effect of the bending–twisting coupling terms, D16 and D26, result in com-pression buckling. Furthermore, Terada, Todoroki, and Shi-mamura5) _{neglected D}

16 and D26 while analyzing the optimal design of symmetrically laminated plates while neglecting subjected to compressions.

Numerous researchers have discussed the post-buckling behavior of thin symmetrically laminated plates under in-plane compression6)_{. However, the equilibrium state after} pri-mary buckling is not stable under continually increasing com-pression, and the structure reconfigures to another equilibrium state of higher order. Uemura and Byon7) _{studied the} second-ary buckling phenomenon, which occurs with further increas-es in load, for thin isotropic platincreas-es. We previously reported the secondary buckling of orthotropic laminated plates8–10)_, but the secondary phenomenon of symmetrically balanced

laminates with initial deflection under compression are still unknown.

In this paper, we examine the buckling problems of sym-metrically laminated square plates with initial deflections that are subjected to axial compression, and we expand the pro-posed analytical techniques from the previous paper10)_{. The} stability of the post-buckling equilibrium states can be dis-cussed using the second variation in the total potential energy. Additionally, the occurrence of secondary buckling was prov-en analytically, and the effects of various factors, especially lamination constitutions4)_{, are discussed.}

2.  Buckling Analysis Method

2.1  Fundamental equations of symmetrically laminated

plates

Figure 1 shows the coordinate system and dimensions of the laminated plates with initial deflection under uniaxial compression Nx. The width and length are given by a and b;

the thickness is h; the terms u, v, and w denote the displace-ment components relative to the center in the x, y, and z direc-tions, respectively; the term w0 denotes the initial deflection. The in-plane nonlinear strain components εx, εy, and γxy and

the changes in curvatures κx, κy, and κxy can be written in

terms of the displacement components:

*1

This Paper was Originally Published in Japanese in J. Japan Inst. Met. Mater. 79 (2015) 49–56.

*2

Corresponding author, E-mail: [email protected]

Fig. 1 Configuration and coordinates of symmetrically laminated plate with small initial deflection under uniaxial compression.

[image:1.595.306.547.670.763.2]

εx= ∂u ∂x+ 1 2 ∂w ∂x 2 + ∂w ∂x ∂w0 ∂x ,

εy=∂v ∂y+ 1 2 ∂w ∂y 2 + ∂w ∂y ∂w0 ∂y ,

γxy= ∂u ∂y+ ∂v ∂x+ ∂w ∂x ∂w ∂y + ∂w ∂x ∂w0 ∂y

+ ∂w0

∂x ∂w ∂y ,          (1)

κx=−∂ 2_w

∂x2, κy=− ∂2w

∂y2, κxy=−2 ∂2w

∂x∂y. (2)

The axial, circumferential, and shear strains of the median surface are given in terms of the median surface stresses from the usual form of Hooke s law for thin plates:

εx=σx/E−υσy/E, εy=σy/E−υσx/E, γxy=τxy/G.

(3)

In this paper, we treat the isotropic modulus in-plane with the composite laminated plates. Therefore, E (Ex =  Ey) is the modulus of elasticity, υ (υx =  υy) is Poisson s ratios, and G (=  Gxy) is the shear modulus. The membrane forces per unit width Nx, Ny, and Nxy are defined as,

Nx=

h/2

−h/2

σxdz= Eh 1₋υ2

∂u ∂x+ 1 2 ∂w ∂x 2 + ∂w ∂x ∂w0 ∂x

+υ ∂v

∂y+ 1 2 ∂w ∂y 2 + ∂w ∂y ∂w0 ∂y ,

Ny=

h/2

−h/2

σydz= Eh 1−υ2

∂v ∂y+ 1 2 ∂w ∂y 2 + ∂w ∂y ∂w0 ∂y

+υ ∂u

∂x+ 1 2 ∂w ∂x 2 + ∂w ∂x ∂w0 ∂x ,

N_xy=

h/2

−h/2

τxydz=Gh ∂u ∂y + ∂v ∂x+ ∂w ∂x ∂w ∂y + ∂w ∂x ∂w0 ∂y + ∂w0 ∂x ∂w ∂y .                  (4) In this paper, the membrane force and bending moment re-sultants for symmetrically balanced laminates at the middle surface are,    Nx Ny Nxy   =   

A11 A12 0 A12 A11 0 0 0 A66

      εx εy γxy   ,    Mx My Mxy   =   

D11 D12 D16 D12 D22 D26 D16 D26 D66

      κx κy κxy   ,        (5)

where Aij (i, j =  1, 2, 6) is the extensional stiffness and Dij (i, j  =  1, 2, 6) is the bending stiffness of the laminated plate. Thus, we can calculate the material constants EL, ET, υL, υT, GLT, of the unidirectional composites and the lamination

an-gle θ11)_{. Subscripts L and T denote the longitudinal and} trans-verse directions, respectively, of the fiber.

However, post-buckling analyses of thin plates considering bending–twisting coupling via D16 and D26 are very difficult. The effect of the bending–twisting coupling can be ignored when the anisotropic parameters γ and δ are smaller than 0.24)_:

r= D16 4_D3

11D22

, δ= D26 4_D

11D322

(6)

The energy Um due to the strain in the middle of the plate

and the energy Ub due to bending are, respectively,

U_m=1 2

a

0 b

0

(Nxεx+Nyεy+Nxyγxy)dxdy, (7)

Ub=1₂ a

0 b

0

(Mxκx+Myκy+Mxyκxy)dxdy. (8)

The total strain energy is the summation of the energy due to bending and energy due to the strain in the middle of plate:

Π =Um+Ub. (9)

By introducing eqs. (3), (5), (6), and (7) into eq. (8), we ob-tain,

Π = 1

2 a 0 b 0 1 h

Nx2

E −

2υ E NxNy+

Ny2

E +

Nxy2

G

+ D11 ∂ 2_w

∂x2 2

+2D12∂ 2_w

∂x2 ∂2w

∂y2

+D22 ∂ 2_w

∂y2 2

+4D66 ∂ 2_w

∂x∂y

2

dxdy. (10)

2.2  Equilibrium and compatibility equations

The equilibrium equations in the x, y, and z directions of a laminated plate are expressed as,

∂N_x ∂x +

∂N_xy ∂y =0, ∂Nxy

∂x + ∂Ny

∂y =0,      (11)

D11∂ 4_w

∂x4 +2(D12+2D66) ∂4w ∂x2_∂y2 +D22

∂4w ∂y4

=N_x ∂ 2_(w+w

0) ∂x2 +Ny

∂2(w+w0) ∂y2

+2Nxy ∂

2_(w+w 0)

∂x∂y . (12)

The Airy function F, which satisfies eq. (11), is defined by,

Nx=h∂

2_F

∂y2, Ny=h ∂2F

∂x2, Nxy=−h ∂2F

∂x∂y, (13) and reduces eq. (12) to the form,

D11∂ 4_w

∂x4 +2(D12+2D66) ∂4w ∂x2_∂y2 +D22

=h ∂ 2_F

∂y2

∂2(w+w0) ∂x2 +

∂2F ∂x2

∂2(w+w0) ∂y2

−2∂2F

∂x∂y

∂2(w+w0)

∂x∂y . (14)

By eliminating u and v from eq. (1), the compatibility equation is obtained:

H11 ∂ 4_F

∂x4 + ∂4F

∂y4 +2(H12+H66) ∂4F ∂x2_∂y2

= 1 h

   ∂

2_w

∂x∂y

2

−∂_∂2_xw₂ ∂_∂2_yw₂  

, (15)

where,

H11=A11A66/H, H12=−A12A66/H

H66=(A112−A122)/H, H=A112A66−A122A66. (16)

The average axial strain in the x-direction, εmx, in the

mid-dle of the plane is defined by using by H11 and H12 in eq. (16):

εmx=−1_a a

0 ∂u ∂xdx

=₋1 a

a

0

  hH11∂

2_F

∂y2 +hH12 ∂2F

∂x2 −

1 2

∂w ∂x

2



dx. (17)

The post-buckling behavior can then be studied through the solution of the nonlinear simultaneous equilibrium and compatibility equations involving stress function F and de-flection w under the given boundary conditions. Approximate solutions of these equations will be sought because obtaining the exact solution is very difficult.

2.3 Analysis of post-buckling behavior

The simply supported out-of-plane boundary conditions are

w=∂ 2_w

∂x2 =0 at x=0,a,

w=∂ 2_w

∂y2 =0 at y=0,b,

    

(18)

and the in-plane conditions are expressed as follows: u=constant along y-axis at x=0,a,

b

0

N_xdy=₋N_xb, N_xy=0,

v=constant along x-axis at y=0,b,

a

0

N_ydx=0, N_xy=0.

      

(19)

Here, w and w0 are expressed by terms that correspond to symmetrical and asymmetrical modes, respectively, along the loading axis:

w=w11sin π_ax sin π_by +w21sin 2_aπx sin π_by ,

w0=c11sin π_ax sin π_by +c21sin 2_aπx sin π_by ,

    

(20)

where w11 is the half-wave number of the buckling half-wave in the x- and y-directions, and w21 is the wave number of the

buckling half-wave in the x-direction and the half-wave num-ber of the buckling half-wave in the y direction; c11 and c21 represent the amplitude of the initial deflection. Substituting eq. (20) into the right-hand side of eq. (15) yields:

F= 1

2hλ2(w11 2_+2w

11c11)

× _16Hλ4 11 cos

2πx

a +

1 16H11cos

2πy b

+ 2

hλ2(w212+2w21c21)

× _256Hλ4 11cos

4πx

a +

1 16H11 cos

2πy b

−_4h1_λ2(w11w21+w11c21+w21c11)

× _Hλ4 11

1 9cos

3πx a −cos

πy a

−_81H λ4cos(3πx/a) cos(2πy/b) 11+36(2H12+H66)λ2+16H11λ4

+ λ

4_{9 cos(πx/a) cos(2πy/b)}

H11+4(2H12+H66)λ2+16H11λ4 − σxy2

2 ,(21)

where λ =  a/b.

When we substitute the deflection w and stress function F in the equilibrium equation in the lateral deflection, i.e., as in eq. (12), the following simultaneous equation is obtained by applying the Galerkin method. Then, w11 and w21 are deter-mined by applying the Newton–Raphson method9)_{. We obtain} the relationship between the average axial compressive stress σx and the average shortening of the middle surface of the

laminated plates εmx:

εmx=−1_a a

0 ∂u ∂x dx

=₋1 a

a

0

 

εx−1₂ ∂

w ∂x

2

−∂_∂w_x∂_∂w_x0   dx

=H11σxh+π

2

8a2(w112+2w11c11+4w212+8w21c21). (22)

2.4  Stability criterion of post-buckling

The equilibrium equations can be obtained by equating the first variation δΠ of the total potential energy Π to zero, i.e., by the stationary potential energy. The stability of the equilib-rium state can be discussed by the following second variation δ2_Π:

δ2Π =1

2 a

0

b

0

1 h

δNx2

E +

δNy2

E −

2υ

EδNxδNy+ δNxy2

G

+ Nx ∂δw

∂x

2

+Ny ∂δw

∂y

2

+2Nxy∂δw

∂x ∂δw

∂y

+ D11 ∂ 2_δw

∂x2 2

+2D12∂ 2_δw

∂x2 ∂2δw

∂y2

+D22 ∂ 2_δw

∂y2 2

+4D66 ∂ 2_δw

∂x∂y

2

where,

δNx= ₁Eh

−υ2 ∂δu

∂x + ∂w ∂x

∂δw ∂x +

∂w0 ∂x

∂δw ∂x

+υ ∂δv ∂y +

∂w ∂y

∂δw ∂y +

∂w0 ∂y

∂δw ∂y ,

δNy=₁Eh

−υ2 ∂δv

∂y + ∂w

∂y ∂δw

∂y + ∂w0

∂y ∂δw

∂y

+υ ∂δu ∂x +

∂w ∂x

∂δw ∂x +

∂w0 ∂x

∂δw ∂x ,

δNxy=Gh ∂δv

∂x + ∂δu

∂y + ∂δw

∂x ∂w ∂y +

∂w ∂x

∂δw ∂y

+ ∂w0 ∂x

∂δw ∂y +

∂δw ∂x

∂w0 ∂y .

             (24)

If δ2Π is positive, the equilibrium state is stable; if it is zero, the equilibrium state is neutral; if it is negative, the equi-librium state is unstable. The assumed virtual displacement poses a problem because the sign of δ2_{Π cannot be judged} independently of the assumed patterns. In this study, a meth-od is proposed to set the appropriate virtual displacements. That is, the variation in δ2_{Π with respect to δu, δv, and δw is} considered essential for obtaining the extremum of δ2_{Π under} the given boundary conditions for a set of assumed virtual displacements. The following equilibrium equations for vir-tual displacements are obtained:

∂δNx

∂x +

∂δNxy

∂y =0,

∂δNxy

∂x +

∂δNy

∂y =0,

     (25)

D11∂ 4_δw

∂x4 +2(D12+2D66) ∂4δw

∂x2_∂y2 +D22 ∂4δw

∂y4

=δNx ∂

2_(w+w 0)

∂x2 +2δNxy

∂2(w+w0) ∂x∂y

+δNy ∂

2_(w+w 0) ∂y2 +Nx

∂2δw

∂x2

+2Nxy∂

2_δw

∂x∂y+Ny

∂2δw

∂y2 . (26)

The virtual stress function δF that satisfies eqs. (25) is,

δN_x=h∂

2_δF

∂y2 , δNy=h ∂2δF

∂x2 , δNxy =−h ∂2δF

∂x∂y. (27) If we then eliminate δu and δv from eq. (24), the following compatibility equation for virtual displacements is obtained:

H11∂ 4_δF

∂x4 +(2H12+H66) ∂4δF

∂x2_∂y2 +H11 ∂4δF

∂y4

= 1 h 2

∂2w

∂x∂y

∂2δw

∂x∂y −

∂2w

∂x2 ∂2δw

∂y2 − ∂2δw

∂x2 ∂2w

∂y2

+2∂

2_w 0 ∂x∂y

∂2δw

∂x∂y−

∂2w0 ∂x2

∂2δw

∂y2 − ∂2δw

∂x2 ∂2w0

∂y2 . (28)

The boundary conditions for the virtual displacements δu, δv, and δw and the virtual membrane forces δNxy are

δu=0, δNxy=0, δw= ∂

2_δw

∂x2 =0 at x=0,a,

δv=0, δNxy=0, δw=∂

2_δw

∂y2 =0 at y=0,b,      (29)

and, for the simply supported condition, δw is assumed to be:

δw=δwpqsin p_aπxsinqπ_by, (30)

where p and q are positive integers. This corresponds to the predominant deflected mode of the post-buckled state, which is excluded because it does not satisfy δu or δv in eq. (29). Substituting w and δw of eqs. (20) and (30), respectively, into eq. (28), δF is determined so as to satisfy eq. (29):

δF= λ

2

4hδwpq(w11+c11)

×

   −

cos (1+p)

a πx cos

(1+q)

b πy (p−q)

2

H11(1+p)4+λ2(2H12+H66)(1+p)2(1+q)2+λ4H11(1+q)4

+

cos (1+p)

a πx cos

(1₋q)

b πy (p+q)

2

H11(1+p)4+λ2(2H12+H66)(1+p)2(1−q)2+λ4H11(1−q)4

+

cos (1−p)

a πx cos

(1+q)

b πy (p+q)2 H11(1−p)4+λ2(2H12+H66)(1−p)2(1+q)2+λ4H11(1+q)4

−

cos (1−p)

a πx cos

(1₋q)

b πy (p−q)

2

H11(1−p)4+λ2(2H12+H66)(1−p)2(1−q)2+λ4H11(1−q)4     +λ 2

4hδwpq(w21+c21)

×

   −

cos (2+p)

a πx cos

(1+q)

b πy (p−2q)

2

H11(2+p)4+λ2(2H12+H66)(2+p)2(1+q)2+λ4H11(1+q)4

+

cos (2+p)

a πx cos

(1₋q)

b πy (p+2q)

2

H11(2+p)4+λ2(2H12+H66)(2+p)2(1−q)2+λ4H11(1−q)4

+

cos (2−p)

a πx cos

(1+q)

b πy (p+2q)2 H₁₁(2₋p)4_+λ2_(2H

12+H66)(2−p)2(1+q)2+λ4H11(1+q)4

−

cos (2−p)

a πx cos

(1₋q)

b πy (p−2q)

2

H11(2−p)4+λ2(2H12+H66)(2−p)2(1−q)2+λ4H11(1−q)4    , (31) where membrane forces Nx, Ny, and Nxy are determined by

3. Numerical Results and Discussion

The solution method outlined above was applied to investi-gate the secondary buckling behavior of symmetrically lami-nated plates. The elastic constants of unidirectional carbon fiber–reinforced epoxy composites having fiber volume con-tents of Vf =  60%, which were obtained using the approximate

average method, were confirmed by experiments12) _{and are as} follows:

E_L=137 (GPa), E_T =8.17 (GPa), G_LT =4.75 (GPa), υL=0.316, υT =0.0189.

(32) The relationship between the non-dimensional average ax-ial compression K =  Nxb2/π 2√D11D22  and the average axial

strain in the x-direction εmx of the symmetrically laminated

square plates (λ =  1) without initial deflections is shown in Fig. 2. Figures 2(a) and (b) show results for lamination con-stitutions [±45/0/90] and [0/90/±45], respectively. The number of layers N =  8 case is shown by the solid line; the number of layers N =  24 case is shown by the dash-dot line; the number of layers N =  48 case is shown by the

dash-dot-dot line. In these figures, Kp is the primary buckling value,

and Ks is the secondary buckling value. The numbers in the

parentheses represent the number of primary buckling half-waves (i.e., secondary buckling three-half-half-waves) in the x- di-rections and the number of primary buckling half-waves (i.e., secondary buckling two-half-waves) in the y-directions.

The relationship between the non-dimensional average ax-ial compression K and average axax-ial strain of the x-axis direc-tion εmx of the symmetrically laminated square plates (λ =  1)

with initial deflections is shown in Fig. 3. Figures 3(a) and (b) show results for the lamination constitutions [±45/0/90] and [0/90/±45], respectively, and include the corresponding number of layers N =  48 case results. Results for c11 =  0.0 and c21 =  0.0 are shown by the solid line; results for c11 =  0.05 and c21 =  0.0 are shown by the dash-dot line; results for c11 =  0.10 and c21 =  0.0 are shown by the dash-dot-dot line. Again, c11 and c21 are the initial deflection amplitudes of the symmetri-cal and asymmetrisymmetri-cal modes, respectively, based on eq. (20). Furthermore, c11 =  0.1, which is the half-wave in the-x direc-tion and the half-wave in the y-direcdirec-tion, indicates that the magnitude is 10% of the plate thickness.

The relationship between the non-dimensional average

ax-Fig. 2 The relationship between axial compressive stress K and average axial shortening εmx without initial deflection at number of layers N =  8, 24 and 48.

(a) Lamination constitution [±45/0/90], (b) Lamination constitution [0/90/±45].

Fig. 3 The relationship between axial compressive stress and average axial shortening with initial deflection c11 =  0.0, 0.05 and 0.10 at number of

layers N =  48.

[image:5.595.303.547.355.732.2] [image:5.595.48.293.355.734.2]

ial compression K and the number of plies 8 M of the sym-metrically laminated square plates with and without initial deflections is shown in Fig. 4. For numbers of plies ranging from 8 to 48, Figs. 4(a) and (b) show results for the case of c11 =  0.0 and c21 =  0.0 (i.e., without initial deflection10)) and c11 =  0.10 and c21 =  0.00, respectively. The secondary buck-ling values for the isotropic plates8) _{are K}

S =  25.99 and KS = 

26.04 in Figs. 4(a) and (b), respectively.

Table 1 lists the variation of the secondary buckling stress-es for symmetrically laminated platstress-es with initial deflections of c11 =  0.0, 0.05, and 0.10 under compression. The analytical method indicates that the second variation of the total poten-tial energy δ2_{Π strongly depends on the number of half-waves} p and q in the virtual displacement; the minimum values are then the secondary buckling stresses.

Tables 2 and 3 list the secondary buckling stresses and number of half-waves of virtual displacements p and q for lamination constitutions [±45/0/90] and [0/90/±45], respec-tively, for 8, 16, 24, 32, 40, or 48 layers, with initial deflec-tions of c11 =  0.0, 0.05, and 0.10 and c21 =  0.0, 0.05, and 0.10. As shown in Table 2 and 3, for all lamination constitutions,

w11 governs the conditions regarding the deflection pattern of post-buckling. The half-wave numbers of the primary

buck-ling in the x- and y-direction are determined. Thus, in this case, w21 is ignored. As seen in the tables, minimum values of Ks are obtained when virtual displacements are considered

regardless of the amplitude of the initial deflection.

From these results, it is clear that the post-buckled equilib-rium state with small-wave modes p and q (in both the x- and

y-directions) cannot be proven unstable using only a virtual displacement. However, it can be proven that the post-buck-led equilibrium state becomes unstable when virtual displace-ments p and q are considered and the inevitable secondary buckling is given analytically.

As shown in Figs. 2 and 3, the secondary buckling stress was several times that of the primary buckling stress; the stresses were obtained for the load-carrying capacity of thin laminated plates after primary buckling. The curves in Figs. 2 and 3 become straight lines with slopes equal to those prior to buckling. After primary buckling, the slopes of these lines may decrease. This indicates a loss of stiffness in the

Fig. 4 The relationship between non-dimensional secondary buckling stresses K and number of plies, 8 M at initial deflection c11 =  0.00 and

0.10.

[image:6.595.50.293.68.443.2]

(a) c11 =  0.00, c21 =  0.00, (b) c11 =  0.10, c21 =  0.00.

Table 1 Variation of secondary buckling stresses with various deflection patterns at number of layers N =  48, lamination constitution [0/90/±45].

q c11 c21 p

1 2 3 4

1

0.00 0.00 ̶

Stable Stable Stable

0.05 0.00 Stable Stable Stable

0.10 0.00 Stable Stable Stable

2

0.00 0.00 Stable 34.97 25.01 29.44

0.05 0.00 Stable 35.05 25.03 29.45

0.10 0.00 Stable 35.14 25.06 29.46

3

0.00 0.00 Stable 139.64 50.59 46.06

0.05 0.00 Stable 139.71 50.61 46.07

[image:6.595.306.550.95.235.2]

0.10 0.00 Stable 139.79 50.93 46.08

Table 2 Secondary buckling stresses of symmetrically laminated plates with variation of amplitude of initial deflection, and number of layers at lamination constitution [±45/0/90].

Lamination

constitution Number of layers c11 c21 p q Ks

[±45/0/90]sym

8

0.00 0.00 3 2 38.50 0.05 0.00 3 2 38.54 0.10 0.00 3 2 38.58

16

0.00 0.00 3 2 31.85 0.05 0.00 3 2 31.88 0.10 0.00 3 2 31.91

24

0.00 0.00 3 2 29.80 0.05 0.00 3 2 29.83 0.10 0.00 3 2 29.86

32

0.00 0.00 3 2 28.82 0.05 0.00 3 2 28.84 0.10 0.00 3 2 28.88

40

0.00 0.00 3 2 28.23 0.05 0.00 3 2 28.26 0.10 0.00 3 2 28.29

48

[image:6.595.307.549.295.547.2]

post-buckling range that is greatly dependent on the number of half-waves of the virtual displacements p and q.

As shown in Fig. 4, the secondary buckling values Ks for the balanced laminated plates are different owing to the stack-ing sequence. Because the flexible stiffness Dij differs be-tween the lamination constitutions [±45/0/90] and [0/90/±45], these stiffness values affect the secondary buck-ling. The secondary buckling values Ks also approach that of the isotropic material as the number of layers increases.

As shown in Tables 2 and 3, the number of half-waves of the virtual displacements p and q perfectly agree for all num-bers of layers. There was little change in the secondary buck-ling stress within the ranges of initial deflection amplitudes

considered here. In addition, the secondary buckling stresses were mostly unaffected by the presence of any initial deflec-tion, and the secondary buckling Ks increased only slightly as the amplitude of the initial deflection increased. In addition, the secondary buckling values Ks decrease significantly when

the lamination constitution [0/90/±45] was 8 plies.

4.  Conclusions

We analytically showed that the post-buckling equilibrium states for simply supported symmetrically laminated plates with initial deflections could be proven unstable. A method based on the secondary variation in the total potential energy was proposed for evaluating the stability of post-buckling equilibrium states, and the inevitable secondary buckling was derived. The effect of lamination constitution, sequence of lamination, lamination angle, and initial deflections on the post-buckling equilibrium states and secondary buckling stresses were discussed.

REFERENCES

1) H. Kasuya and M. Uemura: Trans. Jpn. Soc. Mech. Eng. 51 (1985) 393–401.

2) H. Fukunaga, H. Sekine, M. Sato and A. Iino: Trans. Jpn. Soc. Mech. Eng. 59 (1993) 2343–2349.

3) M. Sato, A. Iino, H. Fukunaga and H. Sekine: Trans. Jpn. Soc. Mech. Eng. 60 (1994) 853–859.

4) M.P. Nemeth: AIAA J. 24 (1986) 1831–1835.

5) Y. Terada, A. Todoroki and Y. Shimamura: Trans. Jpn. Soc. Mech. Eng. 66 (2000) 714–720.

6) G. J. Turvey and I. H. Marshall: Buckling and Postbuckling of

Compos-ite Plates, (Chapman & Hall, London, 1995).

7) O.-I. Byon and M. Uemura: Trans. Jpn. Soc. Mech. Eng. 43 (1977) 2818–2827.

8) H. Kasuya, A. Minobe and K. Nemoto: Trans. Jpn. Soc. Mech. Eng. 58 (1992) 1544–1549.

9) H. Kasuya and A. Minobe: J. Soc. Mater. Sci. Jpn. 42 (1993) 804–810. 10) H. Kasuya, K. Nemoto, S. Endo and H. Kikugawa: J. Jpn. Soc. Design

Eng. 46 (2011) 454–460.

11) R. M. Jones: Mechanics of Composite Materials, (McGraw-Hills, New-York, 1975) Chap. 4.

[image:7.595.46.292.103.356.2]

12) M. Uemura and N. Yamada: J. Soc. Mater. Sci. 24 (1975) 156–163. Table 3 Secondary buckling stresses of symmetrically laminated plates

with variation of amplitude of initial deflection, and number of layers at lamination constitution [0/90/±45].

Lamination

constitution Number of layers c11 c21 p q Ks

[0/90/±45]sym

8

0.00 0.00 3 2 22.39 0.05 0.00 3 2 22.41 0.10 0.00 3 2 22.43

16

0.00 0.00 3 2 23.51 0.05 0.00 3 2 23.53 0.10 0.00 3 2 23.56

24

0.00 0.00 3 2 24.18 0.05 0.00 3 2 24.21 0.10 0.00 3 2 24.23

32

0.00 0.00 3 2 24.58 0.05 0.00 3 2 24.60 0.10 0.00 3 2 24.63

40

0.00 0.00 3 2 24.83 0.05 0.00 3 2 24.85 0.10 0.00 3 2 24.88

48