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Thermal buckling of hybrid laminated composite plates with a hole
Composite Structures 68 (2005) 247–254 www.elsevier.com/locate/compstruct
Thermal buckling of hybrid laminated composite plates with a hole € Ahmet Avci, Omer Sinan Sahin *, Mesut Uyaner Department of Mechanical Engineering, Selcßuk University, 42075 Konya, Turkey Available online 13 May 2004
Abstract Thermal buckling analysis of symmetric and antisymmetric cross-ply laminated hybrid composite plates with a hole subjected to a uniform temperature rise for different boundary conditions are presented in this paper. The first-order shear deformation theory in conjunction with variational energy method is employed in the mathematical formulation. The eight-node Lagrangian finite element technique is used for finding the thermal buckling temperatures of hybrid laminates. The effects of hole size, lay-up sequences and boundary conditions on the thermal buckling temperatures are investigated. The results are shown in graphical form for various conditions. 2004 Published by Elsevier Ltd. Keywords: Hybrid composite plates; Thermal buckling; Finite element method
1. Introduction The buckling of fiber reinforced plates is an important consideration in the design process in a number of engineering fields. The buckling problem under thermal loadings is one of the practical importances for structures work at elevated temperatures. Gossard et al. [1] was the first to consider the thermal buckling of plates and in their work simply supported rectangular isotropic plate subjected to tent-like temperature distribution was investigated. Chang and Shiao [2] investigated the thermal buckling of isotropic and composite plates with hole by using both closed form solution and finite elements method. Thermal buckling of antisymmetric cross-ply composite laminates was investigated by Mathew et al. [3]. Abramovich[4] investigated the thermal buckling of cross-ply symmetric and nonsymmetrical laminated beams by using firstorder deformation theory. Murphy and Ferreira [5] studied theoretical and experimental approaches to determine the buckling temperature and buckling mode for flat rectangular plates. Huang and Tauchert [6] determined thermal buckling of clamped symmetric angle-ply laminated plates using a Fourier series approach and the finite element method. Prabhu and *
Corresponding author. Tel.: +90-332-223-18-98; fax: +90-332-24106-51. € E-mail address: [email protected] (O.S. Sahin). 0263-8223/$ - see front matter 2004 Published by Elsevier Ltd. doi:10.1016/j.compstruct.2004.03.017
Dhanaraj [7], Chandrashekhara [8], Thangaratnam and Ramaohandran [9], and Chen et al. [10] also evaluated the thermal buckling of the laminates subjected to uniform temperature rise or nonuniform temperature fields using the finite element approach. Mannini [11] investigated the thermal buckling of cross-ply laminates by using first-order shear deformation theory and the Rayleigh–Ritz method. Dawe and Ge [12] presented results for thermally loaded shear deformable composite laminates using the spline finite strip method. Yettram and Brown [13] studied the buckling of square perforated plates under biaxial loadings using a direct matrix method. Chang and Hsu [14] evaluated the buckling and vibration of isotropic plates containing circular or a square opening. The post buckling analysis of isotropic and composite square plates involving circular holes were carried out by Vanden Brink and Kamat [15] by using finite element method. Larson [16] also used a finite element approach and perturbation method for investigating the buckling of orthotropic compressed plates with circular holes. Lin and Kuo [17] investigated the buckling of cross-ply and angle-ply laminated plates with circular holes under in plane static loadings. In their study, they utilized the finite elements method in order to find the critical buckling loads. The present paper aims to determine the buckling temperature and buckling mode shapes for hybrid composite laminates with different circular hole diameters using the finite element method. The thermal
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buckling of symmetric and antisymmetric cross-ply laminates with holes is investigated, based on the firstorder deformation theory in conjunction with the variational energy method. The finite element approach is used for finding the thermal buckling temperatures for aluminum–glass/epoxy, aluminum–boron/epoxy and aluminum–boron/epoxy–glass/epoxy hybrid laminates. The effects of hole diameter, lay-up sequences and boundary conditions on the thermal buckling temperatures are numerically solved. The results are presented in graphical form for various boundary conditions.
2. Mathematical formulation The laminated orthotropic construction of the plate is consistedP of N layers. Each layer is of thickness tk , so N that h ¼ k¼1 tk is the total thickness of the laminate. The longitudinal and lateral dimensions of the laminate are a and b and subjected to uniform temperature difference DT between ambient and laminated plate as shown Fig. 1. The linear stress–strain relation for each layer is expressed with x, y-axes and has the form 8 9 2 38 > Q11 Q12 Q16 > = < rx > < ex ax DT 6 7 ey ay DT ry ¼ 4 Q12 Q22 Q26 5 > > ; : > : sxy k Q16 Q26 Q66 k cxy axy DT # " syz cyz Q44 Q45 ¼ sxz cxz Q45 Q55
9 > = > ;
ð1Þ
k
where rx , ry , sxy , syz and sxz are the stress components, Qij are transformed reduced stiffnesses, which can be expressed in terms of the orientation angle and the engineering constant of the material [20]. DT is tem-
perature increase, ax and ay are the coefficients of thermal expansion in directions of x and y axes, respectively. axy is the apparent coefficient of thermal shear, such as ax ¼ a1 cos2 h þ a2 sin2 h ay ¼ a2 cos2 h þ a1 sin2 h
ð2Þ
axy ¼ 2ða1 a2 Þ sin h cos h a1 and a2 are the thermal expansion coefficients of the lamina along the longitudinal and transverse directions of fibers respectively. In this study first-order shear deformation theory is used. The displacements u, v and w can be written as follows: uðx; y; zÞ ¼ u0 ðx; yÞ þ zwx ðx; yÞ vðx; y; zÞ ¼ v0 ðx; yÞ þ zwy ðx; yÞ
ð3Þ
wðx; y; zÞ ¼ wðx; yÞ where u0 , v0 , w are the displacements at any point of the middle surface, and wx , wy are the bending rotations of normals to the mid plane about the x and y axes, respectively. The bending strains ex , ey and transverse shear strains cxy , cyz , cxz at any point of the laminate are ow 8 9 ou0 x0 e ox < x = ox owy ov0 ; ey ¼ þ z oy : ; ou oy ov 0þ 0 cxy owx þ owy oy ox oy ox ow cyz oy wy ¼ ð4Þ c ow þ w xz
x
ox
The resultant forces Nx , Ny and Nxy , moments Mx , My and Mxy and shearing forces Qx , Qy per unit length of the plate are given as
y 2
Ny Nx
1
y
θ
d b
x
x z h/2
a
h/2
N xy Fig. 1. Geometry of the problem and coordinates.
x
A. Avci et al. / Composite Structures 68 (2005) 247–254
8 9 > = < rx > 6 7 ry ð1; zÞ dz 4 Ny My 5 ¼ h=2 > ; : > Nxy Mxy sxy Z h=2 Qx sxz ¼ dz Qy syz h=2 2
Nx
Mx
3
Z
where
h=2
ð5Þ
oR
Here dA ¼ dx dy, R is the region of a plate excluding the b b hole. N n and N s are in-plane loads applied on the boundary oR. For the equilibrium, the potential energy P must be stationary. The equilibrium equations of the cross-ply laminated plate subjected to temperature change can be derived from the variational principle through use of stress–strain and strain–displacement relations. One may obtain these equations by using dP ¼ 0 [17,18]. 2.1. Finite element formulations
A
and T
½Bg ½Dg ½Bg dA A
ð12Þ
ðA44 ; A55 Þ ¼
Z
h=2
ðQ44 ; Q55 Þ dz
ð13Þ
h=2
A44 and A55 , are the shear correction factors for rectangular cross-section are given by k12 ¼ k22 ¼ 5=6 [19]. The total potential energy principle for the plate satisfies the assembly of the element equations. The element stiffness and the geometric stiffness matrices are assembled. The corresponding eigenvalue problem can be solved using any standard eigenvalue extraction procedures [10,18] 8 9 < ui = ½½K0 kb ½K0g vi ¼ 0 ð14Þ : ; wi where ½K0 ¼ ½Kb þ ½Ks ; kb ½K0g ¼ ½Kg
ð15Þ
The product of kb and the initial guest value DT is the critical buckling temperature Tcr , that is Tcr ¼ kb DT
ð16Þ
3. Numerical result and discussion
In general, a closed form solution is difficult to obtain for buckling problems [17,18]. Therefore numerical methods are usually used for finding an approximate solution. In order to study the buckling of the plate, an eightnode Lagrangian finite element analysis is applied in this study. The stiffness matrix of the plate is obtained by using the minimum potential energy principle. Bending stiffness [Kb ], shear stiffness [Ks ] and geometric stiffness [Kg ] matrices can be expressed as Z ð8Þ ½Kb ¼ ½Bb T ½Db ½Bb dA A Z ð9Þ ½Ks ¼ ½Bs T ½Ds ½Bs dA
Z
ð11Þ
h=2
ð6Þ
where Ub is the strain energy of bending, Us is the strain energy of shear and V represents the potential energy of in-plane loadings due to temperature change. Z Z Z 1 h=2 Ub ¼ ðrx ex þ ry ey þ sxy cxy Þ dA dz 2 h=2 R ð7Þ Z Z Z 1 h=2 Us ¼ ðsxz cxz þ syz cyz Þ dA dz 2 h=2 R Z Z 1 ½N 1 ðow=oxÞ2 þ N 2 ðow=oyÞ2 V ¼ 2 R Z b b þ 2N 12 ðow=oxÞðow=oyÞ dA ðN n u0n þ N s u0s Þ ds
½Kg ¼
" # Aij Bij k12 A44 0 ½Ds ¼ ½Db ¼ Bij Dij 0 k22 A55 " # N 1 N 12 ½Dg ¼ N 12 N 2 Z h=2 Qij ð1; z; z2 Þ dz ði; j ¼ 1; 2; 6Þ ðAij ; Bij ; Dij Þ ¼
The total potential energy P of a laminated plate under thermal loading is equal to P ¼ Ub þ Us þ V
249
ð10Þ
There are many techniques to solve eigenvalue problems. In this study the Newton Raphson method is applied to obtain numerical solutions of the problem. For thermal buckling due to a DT temperature change in the plate, the uniaxial or biaxial in-plane loads are developed along the rectangular edges, while the circular hole edge is free. The cross-ply laminated hybrid composite plates used here have several thicknesses and bonded symmetrically and antisymmetrically. For the computations thermo-elastic properties considered for the aluminum, E-glass/epoxy, and boron/ epoxy composites are given in Table 1. Here, E1 and E2 are elastic moduli in 1 and 2 directions respectively, m12 is Poisson’s ratio and a1 and a2 are thermal expansion coefficients of the materials used in the solution. The effect of a12 is neglected. Stacking sequence of hybrid composite plates have been taken both symmetric and antisymmetric. Some of
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Table 1 Material properties G12 (GPa)
70 6 19
52 3 4.8
23.6 · 10 7.0 · 106 4.14 · 106
23.6 · 106 2.30 · 105 1.91 · 105
Fig. 2. Typical mesh for four edges clamped (CCCC) plate.
1. Four edges simply supported (SSSS): a a At x ¼ ; ; u ¼ w ¼ wy ¼ 0 2 2
CCCC
250 200
v ¼ w ¼ wx ¼ 0
180
CCCC
o C
Tc
3. Four edges clamped (CCCC): a a At x ¼ ; ; u ¼ w ¼ wy ¼ wx ¼ 0 2 2
(a)
150 100 50
SSSS SFSF
SSSS
150
SFSF
CFCF
120
CFCF
2. Two edges simply supported and two edges are free (SFSF): a a At x ¼ ; ; u ¼ w ¼ wy ¼ 0 2 2
b b At y ¼ ; ; 2 2
a2 (C1 )
6
0.33 0.3 0.21
stacking sequences have been represented below. The letters A, B and G represent, aluminum, boron/epoxy and glass/epoxy composites respectively. The sequence of 10 layers symmetric lay up of boron/ epoxy–glass/epoxy is 0G /90B /90B /0G /90B /90B /0G / 90B /90B /0G . The sequence of 6 layers antisymmetric lay up of Aluminum–boron/epoxy–glass/epoxy is 0G /90B /0A / 0G /90B /0A . And sequence of 6 layers symmetric lay up of aluminum–boron/epoxy–glass/epoxy is 0G /90B /0A /0A / 90B /0G . Each layer has 0.25 mm thickness and the length of one edge of plate is 100 mm. h=b ratio, represents the total thickness of composite plate to length of one side of composite plate and d=b ratio, represents the hole size to length of one side of composite plate. A wide range of boundary conditions can be accommodated, but only four kinds of boundary conditions are chosen as defined below.
b b At y ¼ ; ; 2 2
a1 (C1 )
m12
C
70 15 207
E2 (GPa)
o
Aluminum E-glass/epoxy Boron/epoxy
E1 (GPa)
Tc
Material
90 60 30
0 0 0.01 0.015 0.020 0.025 0.01 0.015 0.020 0.025 (b) h/b h/b
Fig. 3. Effect of stacking on buckling temperature for (a) antisymmetric and (b) symmetric boron–epoxy/glass–epoxy hybrid composites without hole.
v ¼ w ¼ wx ¼ wy ¼ 0
4. Two edges clamped and two edges are free (CFCF): a a At x ¼ ; ; u ¼ w ¼ wy ¼ wx ¼ 0 2 2
Fig. 2 shows the meshed plate. Four edges of plate have divided into 10 parts disregarding the hole size. The variation of critical buckling temperature Tc with h=b (plate thickness to length of the one edge of the plate), for boron–epoxy/glass–epoxy hybrid composites
consisting of 4, 6, 8 and 10 layers is shown in Figs. 3 and 4. For this material, the buckling temperatures for both symmetric and antisymmetric plates increase as the h=b ratio increases. Both plates with hole and plates without hole show the same behavior for four different boundary conditions. It is observed that the critical buckling temperature reaches its maximum value for four edge clamped (CCCC) plate. This behavior is same for all hole sizes except for the case d=b ¼ 0:3. The critical
A. Avci et al. / Composite Structures 68 (2005) 247–254
SSSS
400
SFSF
SFSF
CFCF
300 o
Tc
200 100
100
0 0.01 0.015 0.02 0.025
0 0.01 0.015 0.02 0.025 (b)
h/b
h/b
Fig. 4. Effect of stacking on buckling temperature for (a) antisymmetric and (b) symmetric boron–epoxy/glass–epoxy hybrid composites for d=b ¼ 0:3.
CCCC
CCCC SSSS
50
SFSF CFCF
SFSF CFCF
30 20
o
C
40
10 0
(a)
0.1 0.2 0.3 0.4 0.5 d/b
0
(b)
0.1 0.2 0.3 0.4 0.5 d/b
Fig. 5. Effect of hole size on buckling temperature for (a) antisymmetric and (b) symmetric boron–epoxy/glass–epoxy hybrid composites composed of four layers.
CCCC
CCCC SSSS SFSF C
CFCF o
180 160 140 120 100 80 60 40 20 0
Tc
Tc
o C
buckling temperature for the case of d=b ¼ 0:3 reaches its maximum value for two edge clamped (CFCF) plate. In Figs. 3 and 4, the curves present a monotonically increasing Tcr as the size of hole increases. It can be seen that the critical temperatures of clamped cases are always higher than those of the simply supported plates. Fig. 5 represents the variation of critical buckling temperature Tc with d=b for boron/epoxy–glass/epoxy hybrid composites. For symmetric and antisymmetric lay up, it is concluded that, there is no change at critical
140 120
CCCC
100 80 60 40
SFSF
50
CFCF
40
(a)
d/b
40
20
20
0
Fig. 6. Effect of hole size on buckling temperature for (a) antisymmetric and (b) symmetric aluminum/boron–epoxy/glass–epoxy hybrid composites composed of six layers.
(a)
SFSF CFCF
10
10
d/b
50
CFCF
30
0 0.1 0.2 0.3 0.4 0.5 (b)
SSSS
SFSF
30
20 0 0 0.1 0.2 0.3 0.4 0.5
CCCC
SSSS
SSSS
C
0
o
80 70 60 50 40 30 20 10 0
Tc
Tc
o
C
SSSS
Tc
(a)
C
Tc
200
CFCF
buckling temperature with the variation of hole size for SFSF and SSSS plates. For four edges clamped (CCCC) plate, critical temperature gradually increases as the d=b increases. Fig. 6 shows the variation of critical buckling temperature Tc versus d=b for aluminum–boron/epoxy– glass/epoxy six layered hybrid composite plates. It is concluded that, there are no change at the critical temperatures for four edge simple supported (SSSS) and two edge simply supported (SFSF) plates with the variation of hole size for both symmetric and antisymmetric plates. The greatest buckling temperature is reached for four edge clamped plate at antisymmetric lay up, though it can be seen a little decrease at small hole sizes. The behavior of symmetric and antisymmetric single edge supported plates is different. Symmetric cross-ply laminates does not yield the highest buckling resistance as usually expected, because of absence of bending–extension coupling. The buckling temperature of four edges and two edges clamped antisymmetric plate remains constant though at small hole sizes, but, buckling temperatures of four edges and two edges clamped symmetric plate have a gradual increment by d=b ratio. The variation of critical buckling temperature Tc with d=b for aluminum/boron–epoxy hybrid composites is shown in Fig. 7. The critical temperatures for four edges simple supported and two edge simply supported plates are not affected by hole size. A gradual increment is seen at four edge clamped plate. For two edge clamped plate with hole size of d=b ¼ 0–0.1 the buckling temperature increases, but buckling temperature is almost constant for plate with bigger hole sizes. Both symmetric and antisymmetric plates show almost same behavior. In Fig. 8, the variation of critical buckling temperature Tc versus d=b for aluminum/glass–epoxy hybrid composites are shown. It can be observed that the critical temperatures for all boundary conditions except for four edge clamped plates are not affected by hole size. The critical temperature Tc for four edges clamped plate rapidly increases as d=b ratio increases.
o
o
C
C
300
CCCC
SSSS
Tc
400
CCCC
251
0 0.1 0.2 0.3 0.4 0.5 d/b
0 0 0.1 0.2 0.3 0.4 0.5 (b) d/b
Fig. 7. Effect of hole size on buckling temperature for (a) antisymmetric and (b) symmetric aluminum/boron–epoxy hybrid composites composed of four layers.
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40
0 0 0.1 0.2 0.3 0.4 0.5
Tc
10 0 0 0.1 0.2 0.3 0.4 0.5 (b)
d/b
0 Fig. 8. Effect of hole size on buckling temperature for (a) antisymmetric and (b) symmetric aluminum/glass–epoxy hybrid composites with four layers.
Antisym Sym.
60
10 8 6
20
4
10
2 0
0 0 (a)
Sym.
0.1 0.2 0.3 0.4 0.5 d/b
C o
Tc
30
Fig. 10. Relationship between buckling temperature and hole size of boron/epoxy–glass/epoxy composite plates for four edge clamped (a) antisymmetric, (b) symmetric plates.
Antisym
14 12
Tc oC
o C
40
Tc
50
0.1 0.2 0.3 0.4 0.5 d/b
0 (b)
0.1 0.2 0.3 0.4 0.5
180 160 140 120 100 80 60 40 20 0
d/b
A-G
A-B
A-B B-G A-B-G
B-G
120
A-B-G
90 60 30 0
0 0.1 0.2 0.3 0.4 0.5
(a)
A-G
150
Tc
70
(a)
8 Layer 400 10 Layer 350 300 250 200 150 100 50 0 0 0.1 0.2 0.3 0.4 0.5 (b) d/b
10 Layer
C
d/b
6 Layer 8 Layer
450 400 350 300 250 200 150 100 50 0
o C
20
4 Layer 6 Layer
Tc
CFCF
30
o C
CFCF
10
(a)
SFSF
SFSF
4 Layer
o
SSSS
o
Tc
20
CCCC SSSS
o C
o C
30
50
CCCC
Tc
40
d/b
0
(b)
0.1 0.2 0.3 0.4 0.5 d/b
Fig. 9. Relationship between Buckling temperature and hole size of boron–epoxy/glass–epoxy composite plates composed of four layers for (a) two edge clamped and (b) two edge simply supported conditions.
Fig. 11. Relationship between buckling temperature and hole size of various composite plates for four edge clamped plates consisting of six layers (a) antisymmetrical, (b) symmetrical plate.
Fig. 9 depicts the relationship between buckling temperature and hole size of boron–epoxy/glass–epoxy composite plates for two edge clamped (a) and two edge simply supported conditions. As shown in Fig. 9, for both two edge clamped and two edge simply supported plates, higher temperature difference is needed for buckling of antisymmetrical stacked plate than symmetric one. Expected results are seen in Fig. 10. As number of layers increases, the temperature difference for buckling
increases. This behavior is same for both antisymmetric and symmetric lay up. But, temperature difference for buckling of antisymmetrical stacked plate is higher than symmetric stacked plate. Fig. 11 shows the relationship between buckling temperature and hole size of various composite plates for four edges clamped plates consisting of six layers. As seen in Fig. 11, for all materials used in the solution, as the hole size increases, the buckling temperature increases rapidly. For these four composite materials, the
Fig. 12. First buckling mode shapes for (a) four edge simply supported and (b) two edge simply supported aluminum–glass/epoxy plates with four layers stacked antisymmetrically.
A. Avci et al. / Composite Structures 68 (2005) 247–254
253
Fig. 13. First buckling mode shapes for (a) two edge clamped and (b) four edge clamped aluminum–glass/epoxy plates with four layers stacked antisymmetrically.
Fig. 14. (a) First and (b) second buckling mode shapes for two edge clamped aluminum–glass/epoxy plates with four layers stacked antisymmetrically.
highest buckling temperature is reached for aluminum– boron/epoxy–glass/epoxy composites. The first and second buckled mode shapes generated aluminum–glass/epoxy cross-ply laminated plates with four boundary conditions are shown in Figs. 12–14. It is found that critical temperature for four edge clamped plate is 16.33 C, for two edge clamped is 14.32 C, for four edge simply supported plate is 5.86 C and for two edge simply supported plate is 4.15 C, respectively. As shown in Fig. 11 critical temperatures are 12.02 for mode1 and 18.26 for mode 2 for two edge clamped aluminum–glass/epoxy plates with four layers stacked antisymmetrically. The mode shapes presented in Figs. 12–14 show considerable skewing for the laminated plates.
4. Conclusions Thermal buckling behaviors of cross-ply laminated hybrid plates with holes have been examined by employing the first-order shear deformation theory and finite element technique. Both symmetric and antisymetric lay-up sequence are considered and various boundary conditions are taken in to account. For boron/epoxy–glass/epoxy hybrid composites, critical buckling temperature increases as h=b ratio increases.
The boundary condition has a strong impact on the critical buckling temperature. It is evident that, clamped plates have a much higher buckling temperature because of stiffer constraints. Four edges clamped plates have the highest buckling temperature and for this boundary condition, critical buckling temperature is strongly depend on hole size. It is concluded that there are no significant change at the critical buckling temperature of four edges simply supported and two edges simply supported plates. It can be seen that the critical buckling temperatures of clamped cases are always higher than those of the simply supported plates. For this material, there is a little difference between symmetric and antisymmetric lay up. Because of absence of bending– extension coupling, symmetric cross-ply laminates does not yield the highest buckling resistance as usually expected. It is also observed that there are no significant variations at the critical buckling temperatures of aluminum/boron–epoxy/glass–epoxy, aluminum/glass–epoxy and aluminum/boron–epoxy hybrid composites for all boundary conditions used in the solution except for four edges clamped plates. It can be seen that the critical buckling temperatures goes up as the number of layers increases. This behavior is same for both antisymmetric and symmetric lay up. But, buckling temperatures of antisymmetric stacked plates are higher than those of symmetric stacked plates.
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The highest buckling temperature is obtained for aluminum–boron/epoxy–glass/epoxy composites. References [1] Gossard ML, Seide P, Roberts WM. Thermal buckling of plates NACA TN 2771. 1952. [2] Chang JS, Shiao FJ. Thermal buckling analysis of isotropic and composite plates with hole. J Thermal Stresses 1990;13:315–32. [3] Mathew TC, Singh G, Rao GV. Thermal buckling of cross-ply composite laminates. Comput Struct 1992;42(2):281–7. [4] Abramovich H. Thermal buckling of cross-ply laminates using a first-order deformation theory. Comput Struct 1994;28:201–13. [5] Murphy KD, Ferreira D. Thermal buckling of rectangular plates. Int J Solids Struct 2000;38:3979–94. [6] Huang NN, Tauchert TR. Thermal buckling of clamped symmetric laminated plates. J Thin-walled Struct 1992;13:259–73. [7] Prabhu MR, Dhanaraj R. Thermal buckling of laminated composite plates. Comput Struct 1994:1193–204. [8] Chandrashekhara MR. Buckling of Multilayered Composite Plates under Uniform Temperature Field. In: Birman V, Hui D, editors. Thermal Effects on Structures and Materials, Vol. 203, Vol. 110. ASME pub., AMD; 1990. p. 29–33. [9] Thangaratnam KR, Ramaohandran J. Thermal buckling of composite laminated plates. Comput Struct 1989;32:1117–24.
[10] Chen LW, Lin PD, Chen LY. Thermal buckling behavior of thick composite laminated plates under non-uniform temperature distribution. Comput Struct 1991;41:637–45. [11] Mannini A. Shear deformation effects on thermal buckling of cross-ply composite laminates. Composite Struct 1997;39:1–10. [12] Dawe DJ, Ge YS. Thermal buckling of shear-deformable composite laminated plates by the spline finite strip method. Comput Methods, Appl Mech Engrg 1999;185:347–66. [13] Yettram A, Brown CJ. The elastic stability of square perforated plates under biaxial loadings. Comput Struct 1986;22(4):589–94. [14] Chang CL, Hsu DS. Vibration and stability of plates with hole. Proc of the 10th Natl Conf Theo Appl Mech STAM ROC, 1986. p. 235–41. [15] VanDen Brink DJ, Kamat MP. Post-buckling response of isotropic and laminated composites square plates with circular holes. In: Proc Int Conf Composite Mater., San Diego, California, 1985. p. 1393–409. [16] Larsson PL. On buckling of orthotropic compressed plates with circular holes. Composite Struct 1987;7:103–21. [17] Lin CH, Kuo CS. Buckling of laminated plates with holes. J Composite Mater 1989;23:536–53. [18] Bathe KJ. Finite element procedures in engineering analysis. Englewood Cliffs, NJ: Prentice-Hall, Inc.; 1982. [19] Whitney JM. Shear correction factors for orthotropic laminates under static load. J Appl Mech 1973;(March):302–4. [20] Jones RM. Mechanics of composite materials. Tokyo: McGrawHill Kogagusha Ltd; 1975.
Thermal buckling of hybrid laminated composite plates with a hole € Ahmet Avci, Omer Sinan Sahin *, Mesut Uyaner Department of Mechanical Engineering, Selcßuk University, 42075 Konya, Turkey Available online 13 May 2004
Abstract Thermal buckling analysis of symmetric and antisymmetric cross-ply laminated hybrid composite plates with a hole subjected to a uniform temperature rise for different boundary conditions are presented in this paper. The first-order shear deformation theory in conjunction with variational energy method is employed in the mathematical formulation. The eight-node Lagrangian finite element technique is used for finding the thermal buckling temperatures of hybrid laminates. The effects of hole size, lay-up sequences and boundary conditions on the thermal buckling temperatures are investigated. The results are shown in graphical form for various conditions. 2004 Published by Elsevier Ltd. Keywords: Hybrid composite plates; Thermal buckling; Finite element method
1. Introduction The buckling of fiber reinforced plates is an important consideration in the design process in a number of engineering fields. The buckling problem under thermal loadings is one of the practical importances for structures work at elevated temperatures. Gossard et al. [1] was the first to consider the thermal buckling of plates and in their work simply supported rectangular isotropic plate subjected to tent-like temperature distribution was investigated. Chang and Shiao [2] investigated the thermal buckling of isotropic and composite plates with hole by using both closed form solution and finite elements method. Thermal buckling of antisymmetric cross-ply composite laminates was investigated by Mathew et al. [3]. Abramovich[4] investigated the thermal buckling of cross-ply symmetric and nonsymmetrical laminated beams by using firstorder deformation theory. Murphy and Ferreira [5] studied theoretical and experimental approaches to determine the buckling temperature and buckling mode for flat rectangular plates. Huang and Tauchert [6] determined thermal buckling of clamped symmetric angle-ply laminated plates using a Fourier series approach and the finite element method. Prabhu and *
Corresponding author. Tel.: +90-332-223-18-98; fax: +90-332-24106-51. € E-mail address: [email protected] (O.S. Sahin). 0263-8223/$ - see front matter 2004 Published by Elsevier Ltd. doi:10.1016/j.compstruct.2004.03.017
Dhanaraj [7], Chandrashekhara [8], Thangaratnam and Ramaohandran [9], and Chen et al. [10] also evaluated the thermal buckling of the laminates subjected to uniform temperature rise or nonuniform temperature fields using the finite element approach. Mannini [11] investigated the thermal buckling of cross-ply laminates by using first-order shear deformation theory and the Rayleigh–Ritz method. Dawe and Ge [12] presented results for thermally loaded shear deformable composite laminates using the spline finite strip method. Yettram and Brown [13] studied the buckling of square perforated plates under biaxial loadings using a direct matrix method. Chang and Hsu [14] evaluated the buckling and vibration of isotropic plates containing circular or a square opening. The post buckling analysis of isotropic and composite square plates involving circular holes were carried out by Vanden Brink and Kamat [15] by using finite element method. Larson [16] also used a finite element approach and perturbation method for investigating the buckling of orthotropic compressed plates with circular holes. Lin and Kuo [17] investigated the buckling of cross-ply and angle-ply laminated plates with circular holes under in plane static loadings. In their study, they utilized the finite elements method in order to find the critical buckling loads. The present paper aims to determine the buckling temperature and buckling mode shapes for hybrid composite laminates with different circular hole diameters using the finite element method. The thermal
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buckling of symmetric and antisymmetric cross-ply laminates with holes is investigated, based on the firstorder deformation theory in conjunction with the variational energy method. The finite element approach is used for finding the thermal buckling temperatures for aluminum–glass/epoxy, aluminum–boron/epoxy and aluminum–boron/epoxy–glass/epoxy hybrid laminates. The effects of hole diameter, lay-up sequences and boundary conditions on the thermal buckling temperatures are numerically solved. The results are presented in graphical form for various boundary conditions.
2. Mathematical formulation The laminated orthotropic construction of the plate is consistedP of N layers. Each layer is of thickness tk , so N that h ¼ k¼1 tk is the total thickness of the laminate. The longitudinal and lateral dimensions of the laminate are a and b and subjected to uniform temperature difference DT between ambient and laminated plate as shown Fig. 1. The linear stress–strain relation for each layer is expressed with x, y-axes and has the form 8 9 2 38 > Q11 Q12 Q16 > = < rx > < ex ax DT 6 7 ey ay DT ry ¼ 4 Q12 Q22 Q26 5 > > ; : > : sxy k Q16 Q26 Q66 k cxy axy DT # " syz cyz Q44 Q45 ¼ sxz cxz Q45 Q55
9 > = > ;
ð1Þ
k
where rx , ry , sxy , syz and sxz are the stress components, Qij are transformed reduced stiffnesses, which can be expressed in terms of the orientation angle and the engineering constant of the material [20]. DT is tem-
perature increase, ax and ay are the coefficients of thermal expansion in directions of x and y axes, respectively. axy is the apparent coefficient of thermal shear, such as ax ¼ a1 cos2 h þ a2 sin2 h ay ¼ a2 cos2 h þ a1 sin2 h
ð2Þ
axy ¼ 2ða1 a2 Þ sin h cos h a1 and a2 are the thermal expansion coefficients of the lamina along the longitudinal and transverse directions of fibers respectively. In this study first-order shear deformation theory is used. The displacements u, v and w can be written as follows: uðx; y; zÞ ¼ u0 ðx; yÞ þ zwx ðx; yÞ vðx; y; zÞ ¼ v0 ðx; yÞ þ zwy ðx; yÞ
ð3Þ
wðx; y; zÞ ¼ wðx; yÞ where u0 , v0 , w are the displacements at any point of the middle surface, and wx , wy are the bending rotations of normals to the mid plane about the x and y axes, respectively. The bending strains ex , ey and transverse shear strains cxy , cyz , cxz at any point of the laminate are ow 8 9 ou0 x0 e ox < x = ox owy ov0 ; ey ¼ þ z oy : ; ou oy ov 0þ 0 cxy owx þ owy oy ox oy ox ow cyz oy wy ¼ ð4Þ c ow þ w xz
x
ox
The resultant forces Nx , Ny and Nxy , moments Mx , My and Mxy and shearing forces Qx , Qy per unit length of the plate are given as
y 2
Ny Nx
1
y
θ
d b
x
x z h/2
a
h/2
N xy Fig. 1. Geometry of the problem and coordinates.
x
A. Avci et al. / Composite Structures 68 (2005) 247–254
8 9 > = < rx > 6 7 ry ð1; zÞ dz 4 Ny My 5 ¼ h=2 > ; : > Nxy Mxy sxy Z h=2 Qx sxz ¼ dz Qy syz h=2 2
Nx
Mx
3
Z
where
h=2
ð5Þ
oR
Here dA ¼ dx dy, R is the region of a plate excluding the b b hole. N n and N s are in-plane loads applied on the boundary oR. For the equilibrium, the potential energy P must be stationary. The equilibrium equations of the cross-ply laminated plate subjected to temperature change can be derived from the variational principle through use of stress–strain and strain–displacement relations. One may obtain these equations by using dP ¼ 0 [17,18]. 2.1. Finite element formulations
A
and T
½Bg ½Dg ½Bg dA A
ð12Þ
ðA44 ; A55 Þ ¼
Z
h=2
ðQ44 ; Q55 Þ dz
ð13Þ
h=2
A44 and A55 , are the shear correction factors for rectangular cross-section are given by k12 ¼ k22 ¼ 5=6 [19]. The total potential energy principle for the plate satisfies the assembly of the element equations. The element stiffness and the geometric stiffness matrices are assembled. The corresponding eigenvalue problem can be solved using any standard eigenvalue extraction procedures [10,18] 8 9 < ui = ½½K0 kb ½K0g vi ¼ 0 ð14Þ : ; wi where ½K0 ¼ ½Kb þ ½Ks ; kb ½K0g ¼ ½Kg
ð15Þ
The product of kb and the initial guest value DT is the critical buckling temperature Tcr , that is Tcr ¼ kb DT
ð16Þ
3. Numerical result and discussion
In general, a closed form solution is difficult to obtain for buckling problems [17,18]. Therefore numerical methods are usually used for finding an approximate solution. In order to study the buckling of the plate, an eightnode Lagrangian finite element analysis is applied in this study. The stiffness matrix of the plate is obtained by using the minimum potential energy principle. Bending stiffness [Kb ], shear stiffness [Ks ] and geometric stiffness [Kg ] matrices can be expressed as Z ð8Þ ½Kb ¼ ½Bb T ½Db ½Bb dA A Z ð9Þ ½Ks ¼ ½Bs T ½Ds ½Bs dA
Z
ð11Þ
h=2
ð6Þ
where Ub is the strain energy of bending, Us is the strain energy of shear and V represents the potential energy of in-plane loadings due to temperature change. Z Z Z 1 h=2 Ub ¼ ðrx ex þ ry ey þ sxy cxy Þ dA dz 2 h=2 R ð7Þ Z Z Z 1 h=2 Us ¼ ðsxz cxz þ syz cyz Þ dA dz 2 h=2 R Z Z 1 ½N 1 ðow=oxÞ2 þ N 2 ðow=oyÞ2 V ¼ 2 R Z b b þ 2N 12 ðow=oxÞðow=oyÞ dA ðN n u0n þ N s u0s Þ ds
½Kg ¼
" # Aij Bij k12 A44 0 ½Ds ¼ ½Db ¼ Bij Dij 0 k22 A55 " # N 1 N 12 ½Dg ¼ N 12 N 2 Z h=2 Qij ð1; z; z2 Þ dz ði; j ¼ 1; 2; 6Þ ðAij ; Bij ; Dij Þ ¼
The total potential energy P of a laminated plate under thermal loading is equal to P ¼ Ub þ Us þ V
249
ð10Þ
There are many techniques to solve eigenvalue problems. In this study the Newton Raphson method is applied to obtain numerical solutions of the problem. For thermal buckling due to a DT temperature change in the plate, the uniaxial or biaxial in-plane loads are developed along the rectangular edges, while the circular hole edge is free. The cross-ply laminated hybrid composite plates used here have several thicknesses and bonded symmetrically and antisymmetrically. For the computations thermo-elastic properties considered for the aluminum, E-glass/epoxy, and boron/ epoxy composites are given in Table 1. Here, E1 and E2 are elastic moduli in 1 and 2 directions respectively, m12 is Poisson’s ratio and a1 and a2 are thermal expansion coefficients of the materials used in the solution. The effect of a12 is neglected. Stacking sequence of hybrid composite plates have been taken both symmetric and antisymmetric. Some of
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Table 1 Material properties G12 (GPa)
70 6 19
52 3 4.8
23.6 · 10 7.0 · 106 4.14 · 106
23.6 · 106 2.30 · 105 1.91 · 105
Fig. 2. Typical mesh for four edges clamped (CCCC) plate.
1. Four edges simply supported (SSSS): a a At x ¼ ; ; u ¼ w ¼ wy ¼ 0 2 2
CCCC
250 200
v ¼ w ¼ wx ¼ 0
180
CCCC
o C
Tc
3. Four edges clamped (CCCC): a a At x ¼ ; ; u ¼ w ¼ wy ¼ wx ¼ 0 2 2
(a)
150 100 50
SSSS SFSF
SSSS
150
SFSF
CFCF
120
CFCF
2. Two edges simply supported and two edges are free (SFSF): a a At x ¼ ; ; u ¼ w ¼ wy ¼ 0 2 2
b b At y ¼ ; ; 2 2
a2 (C1 )
6
0.33 0.3 0.21
stacking sequences have been represented below. The letters A, B and G represent, aluminum, boron/epoxy and glass/epoxy composites respectively. The sequence of 10 layers symmetric lay up of boron/ epoxy–glass/epoxy is 0G /90B /90B /0G /90B /90B /0G / 90B /90B /0G . The sequence of 6 layers antisymmetric lay up of Aluminum–boron/epoxy–glass/epoxy is 0G /90B /0A / 0G /90B /0A . And sequence of 6 layers symmetric lay up of aluminum–boron/epoxy–glass/epoxy is 0G /90B /0A /0A / 90B /0G . Each layer has 0.25 mm thickness and the length of one edge of plate is 100 mm. h=b ratio, represents the total thickness of composite plate to length of one side of composite plate and d=b ratio, represents the hole size to length of one side of composite plate. A wide range of boundary conditions can be accommodated, but only four kinds of boundary conditions are chosen as defined below.
b b At y ¼ ; ; 2 2
a1 (C1 )
m12
C
70 15 207
E2 (GPa)
o
Aluminum E-glass/epoxy Boron/epoxy
E1 (GPa)
Tc
Material
90 60 30
0 0 0.01 0.015 0.020 0.025 0.01 0.015 0.020 0.025 (b) h/b h/b
Fig. 3. Effect of stacking on buckling temperature for (a) antisymmetric and (b) symmetric boron–epoxy/glass–epoxy hybrid composites without hole.
v ¼ w ¼ wx ¼ wy ¼ 0
4. Two edges clamped and two edges are free (CFCF): a a At x ¼ ; ; u ¼ w ¼ wy ¼ wx ¼ 0 2 2
Fig. 2 shows the meshed plate. Four edges of plate have divided into 10 parts disregarding the hole size. The variation of critical buckling temperature Tc with h=b (plate thickness to length of the one edge of the plate), for boron–epoxy/glass–epoxy hybrid composites
consisting of 4, 6, 8 and 10 layers is shown in Figs. 3 and 4. For this material, the buckling temperatures for both symmetric and antisymmetric plates increase as the h=b ratio increases. Both plates with hole and plates without hole show the same behavior for four different boundary conditions. It is observed that the critical buckling temperature reaches its maximum value for four edge clamped (CCCC) plate. This behavior is same for all hole sizes except for the case d=b ¼ 0:3. The critical
A. Avci et al. / Composite Structures 68 (2005) 247–254
SSSS
400
SFSF
SFSF
CFCF
300 o
Tc
200 100
100
0 0.01 0.015 0.02 0.025
0 0.01 0.015 0.02 0.025 (b)
h/b
h/b
Fig. 4. Effect of stacking on buckling temperature for (a) antisymmetric and (b) symmetric boron–epoxy/glass–epoxy hybrid composites for d=b ¼ 0:3.
CCCC
CCCC SSSS
50
SFSF CFCF
SFSF CFCF
30 20
o
C
40
10 0
(a)
0.1 0.2 0.3 0.4 0.5 d/b
0
(b)
0.1 0.2 0.3 0.4 0.5 d/b
Fig. 5. Effect of hole size on buckling temperature for (a) antisymmetric and (b) symmetric boron–epoxy/glass–epoxy hybrid composites composed of four layers.
CCCC
CCCC SSSS SFSF C
CFCF o
180 160 140 120 100 80 60 40 20 0
Tc
Tc
o C
buckling temperature for the case of d=b ¼ 0:3 reaches its maximum value for two edge clamped (CFCF) plate. In Figs. 3 and 4, the curves present a monotonically increasing Tcr as the size of hole increases. It can be seen that the critical temperatures of clamped cases are always higher than those of the simply supported plates. Fig. 5 represents the variation of critical buckling temperature Tc with d=b for boron/epoxy–glass/epoxy hybrid composites. For symmetric and antisymmetric lay up, it is concluded that, there is no change at critical
140 120
CCCC
100 80 60 40
SFSF
50
CFCF
40
(a)
d/b
40
20
20
0
Fig. 6. Effect of hole size on buckling temperature for (a) antisymmetric and (b) symmetric aluminum/boron–epoxy/glass–epoxy hybrid composites composed of six layers.
(a)
SFSF CFCF
10
10
d/b
50
CFCF
30
0 0.1 0.2 0.3 0.4 0.5 (b)
SSSS
SFSF
30
20 0 0 0.1 0.2 0.3 0.4 0.5
CCCC
SSSS
SSSS
C
0
o
80 70 60 50 40 30 20 10 0
Tc
Tc
o
C
SSSS
Tc
(a)
C
Tc
200
CFCF
buckling temperature with the variation of hole size for SFSF and SSSS plates. For four edges clamped (CCCC) plate, critical temperature gradually increases as the d=b increases. Fig. 6 shows the variation of critical buckling temperature Tc versus d=b for aluminum–boron/epoxy– glass/epoxy six layered hybrid composite plates. It is concluded that, there are no change at the critical temperatures for four edge simple supported (SSSS) and two edge simply supported (SFSF) plates with the variation of hole size for both symmetric and antisymmetric plates. The greatest buckling temperature is reached for four edge clamped plate at antisymmetric lay up, though it can be seen a little decrease at small hole sizes. The behavior of symmetric and antisymmetric single edge supported plates is different. Symmetric cross-ply laminates does not yield the highest buckling resistance as usually expected, because of absence of bending–extension coupling. The buckling temperature of four edges and two edges clamped antisymmetric plate remains constant though at small hole sizes, but, buckling temperatures of four edges and two edges clamped symmetric plate have a gradual increment by d=b ratio. The variation of critical buckling temperature Tc with d=b for aluminum/boron–epoxy hybrid composites is shown in Fig. 7. The critical temperatures for four edges simple supported and two edge simply supported plates are not affected by hole size. A gradual increment is seen at four edge clamped plate. For two edge clamped plate with hole size of d=b ¼ 0–0.1 the buckling temperature increases, but buckling temperature is almost constant for plate with bigger hole sizes. Both symmetric and antisymmetric plates show almost same behavior. In Fig. 8, the variation of critical buckling temperature Tc versus d=b for aluminum/glass–epoxy hybrid composites are shown. It can be observed that the critical temperatures for all boundary conditions except for four edge clamped plates are not affected by hole size. The critical temperature Tc for four edges clamped plate rapidly increases as d=b ratio increases.
o
o
C
C
300
CCCC
SSSS
Tc
400
CCCC
251
0 0.1 0.2 0.3 0.4 0.5 d/b
0 0 0.1 0.2 0.3 0.4 0.5 (b) d/b
Fig. 7. Effect of hole size on buckling temperature for (a) antisymmetric and (b) symmetric aluminum/boron–epoxy hybrid composites composed of four layers.
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40
0 0 0.1 0.2 0.3 0.4 0.5
Tc
10 0 0 0.1 0.2 0.3 0.4 0.5 (b)
d/b
0 Fig. 8. Effect of hole size on buckling temperature for (a) antisymmetric and (b) symmetric aluminum/glass–epoxy hybrid composites with four layers.
Antisym Sym.
60
10 8 6
20
4
10
2 0
0 0 (a)
Sym.
0.1 0.2 0.3 0.4 0.5 d/b
C o
Tc
30
Fig. 10. Relationship between buckling temperature and hole size of boron/epoxy–glass/epoxy composite plates for four edge clamped (a) antisymmetric, (b) symmetric plates.
Antisym
14 12
Tc oC
o C
40
Tc
50
0.1 0.2 0.3 0.4 0.5 d/b
0 (b)
0.1 0.2 0.3 0.4 0.5
180 160 140 120 100 80 60 40 20 0
d/b
A-G
A-B
A-B B-G A-B-G
B-G
120
A-B-G
90 60 30 0
0 0.1 0.2 0.3 0.4 0.5
(a)
A-G
150
Tc
70
(a)
8 Layer 400 10 Layer 350 300 250 200 150 100 50 0 0 0.1 0.2 0.3 0.4 0.5 (b) d/b
10 Layer
C
d/b
6 Layer 8 Layer
450 400 350 300 250 200 150 100 50 0
o C
20
4 Layer 6 Layer
Tc
CFCF
30
o C
CFCF
10
(a)
SFSF
SFSF
4 Layer
o
SSSS
o
Tc
20
CCCC SSSS
o C
o C
30
50
CCCC
Tc
40
d/b
0
(b)
0.1 0.2 0.3 0.4 0.5 d/b
Fig. 9. Relationship between Buckling temperature and hole size of boron–epoxy/glass–epoxy composite plates composed of four layers for (a) two edge clamped and (b) two edge simply supported conditions.
Fig. 11. Relationship between buckling temperature and hole size of various composite plates for four edge clamped plates consisting of six layers (a) antisymmetrical, (b) symmetrical plate.
Fig. 9 depicts the relationship between buckling temperature and hole size of boron–epoxy/glass–epoxy composite plates for two edge clamped (a) and two edge simply supported conditions. As shown in Fig. 9, for both two edge clamped and two edge simply supported plates, higher temperature difference is needed for buckling of antisymmetrical stacked plate than symmetric one. Expected results are seen in Fig. 10. As number of layers increases, the temperature difference for buckling
increases. This behavior is same for both antisymmetric and symmetric lay up. But, temperature difference for buckling of antisymmetrical stacked plate is higher than symmetric stacked plate. Fig. 11 shows the relationship between buckling temperature and hole size of various composite plates for four edges clamped plates consisting of six layers. As seen in Fig. 11, for all materials used in the solution, as the hole size increases, the buckling temperature increases rapidly. For these four composite materials, the
Fig. 12. First buckling mode shapes for (a) four edge simply supported and (b) two edge simply supported aluminum–glass/epoxy plates with four layers stacked antisymmetrically.
A. Avci et al. / Composite Structures 68 (2005) 247–254
253
Fig. 13. First buckling mode shapes for (a) two edge clamped and (b) four edge clamped aluminum–glass/epoxy plates with four layers stacked antisymmetrically.
Fig. 14. (a) First and (b) second buckling mode shapes for two edge clamped aluminum–glass/epoxy plates with four layers stacked antisymmetrically.
highest buckling temperature is reached for aluminum– boron/epoxy–glass/epoxy composites. The first and second buckled mode shapes generated aluminum–glass/epoxy cross-ply laminated plates with four boundary conditions are shown in Figs. 12–14. It is found that critical temperature for four edge clamped plate is 16.33 C, for two edge clamped is 14.32 C, for four edge simply supported plate is 5.86 C and for two edge simply supported plate is 4.15 C, respectively. As shown in Fig. 11 critical temperatures are 12.02 for mode1 and 18.26 for mode 2 for two edge clamped aluminum–glass/epoxy plates with four layers stacked antisymmetrically. The mode shapes presented in Figs. 12–14 show considerable skewing for the laminated plates.
4. Conclusions Thermal buckling behaviors of cross-ply laminated hybrid plates with holes have been examined by employing the first-order shear deformation theory and finite element technique. Both symmetric and antisymetric lay-up sequence are considered and various boundary conditions are taken in to account. For boron/epoxy–glass/epoxy hybrid composites, critical buckling temperature increases as h=b ratio increases.
The boundary condition has a strong impact on the critical buckling temperature. It is evident that, clamped plates have a much higher buckling temperature because of stiffer constraints. Four edges clamped plates have the highest buckling temperature and for this boundary condition, critical buckling temperature is strongly depend on hole size. It is concluded that there are no significant change at the critical buckling temperature of four edges simply supported and two edges simply supported plates. It can be seen that the critical buckling temperatures of clamped cases are always higher than those of the simply supported plates. For this material, there is a little difference between symmetric and antisymmetric lay up. Because of absence of bending– extension coupling, symmetric cross-ply laminates does not yield the highest buckling resistance as usually expected. It is also observed that there are no significant variations at the critical buckling temperatures of aluminum/boron–epoxy/glass–epoxy, aluminum/glass–epoxy and aluminum/boron–epoxy hybrid composites for all boundary conditions used in the solution except for four edges clamped plates. It can be seen that the critical buckling temperatures goes up as the number of layers increases. This behavior is same for both antisymmetric and symmetric lay up. But, buckling temperatures of antisymmetric stacked plates are higher than those of symmetric stacked plates.
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