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Thermal buckling of hybrid angle-ply laminated composite plates with a hole
COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 65 (2005) 1780–1790 www.elsevier.com/locate/compscitech
Thermal buckling of hybrid angle-ply laminated composite plates with a hole ¨ mer Sinan S O ß ahin
*
Selc¸uk University, Department of Mechanical Engineering, Konya, Turkey Received 13 October 2004; received in revised form 8 March 2005; accepted 11 March 2005 Available online 19 April 2005
Abstract In this paper, thermal buckling analysis of symmetric and antisymmetric laminated hybrid composite plates with a hole subjected to a uniform temperature rise for different boundary conditions is presented. The first-order shear deformation theory in conjunction with variational energy method is employed in mathematical formulation. The eight-node Lagrangian finite element technique is used for obtaining the thermal buckling temperatures of glass–epoxy/boron–epoxy hybrid laminates. The effects of hole size, layup sequences and boundary conditions on thermal buckling temperatures are investigated. The results are shown in graphical form for various boundary conditions. 2005 Elsevier Ltd. All rights reserved. Keywords: Hybrid composite plates; Thermal buckling; Finite element method
1. Introduction Fiber reinforced composites because of their low weight, high strength and anisotropic properties, which can be tailored through the variation of fiber angle and stacking sequence, are used extensively in aerospace structures and transportation industries. Thermal buckling could be caused when a structure is exposed to a temperature rise. Due to the solar radiation heating and aerodynamically thermal boundary effects the temperature change during aeronautical mission are important. Since, structural components of these high speed machines are usually exposed to non-uniform heat flux and temperature distribution. Considerable work has been done on thermal problems of plates. Gossard et al. [1] used a Rayleigh–Ritz method to find the deflections of plates or initially
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0266-3538/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2005.03.007
imperfect plates exposed to thermal buckling. An extensive overview of the general buckling problems of laminated composite plate was made by Liessa [2]. In this study, some complicated effects were investigated such as shear deformation, hygrothermal factors and postbuckling behavior. Bednarczyk and Rihter [3] investigated the influence of temperature distribution on buckling modes. The thermally induced buckling of antisymmetric angle-ply laminated plates with Levytype boundary conditions was investigated by Chen and Liu [4]. Thermal buckling behavior of composite laminated plates with transverse shear deformation was studied by Sun and Hsu [5]. Chockalingam et al. [6] investigated the thermal buckling of antisymmetric cross-ply hybrid laminates by using a one-dimensional finite element technique based on first-order shear deformation theory. Thangaratnam and Ramachandran [7] and Chen et al. [8] also presented thermal buckling of laminates subjected to uniform temperature rise or non-uniform temperature fields using finite element approach. Thermal
¨ .S. S O ß ahin / Composites Science and Technology 65 (2005) 1780–1790
buckling analysis of cross-ply laminates is also studied by using first-order shear deformation theory and Rayleigh–Ritz method by Mannini [9]. Dawe and Ge [10] presented results for thermally loaded shear deformable composite laminates using the spline finite strip method. Yettram and Brawn [11] predicted the buckling of square perforated plates under biaxial loadings using a direct matrix method. The buckling and vibration of isotropic plates with a circular or a square hole are discussed by Chang and Hsu [12]. The post-buckling analysis of isotropic and composite square plates with circular holes was carried out by VanDen Brink and Kamat [13] by using a finite element technique. Larsson [14] also used a finite element approach and perturbation method for investigating the buckling of orthotropic compressed plates with circular holes. Lin and Kuo [15] investigated the buckling of cross-ply and angle-ply laminated plates with circular holes under in plane static loadings. For finding the critical loads they used a finite element analysis. The present paper is an attempt to determine the buckling temperature and buckling mode for hybrid composite laminates with circular holes using finite element method. Thermal buckling of symmetric and antisymmetric angle-ply laminates with holes is investigated, based on the first-order deformation theory in conjunction with variational energy method. The eight-node Lagrangian finite element approach is used for obtaining the thermal buckling temperatures of boron– epoxy/glass–epoxy hybrid laminates. The effects of hole diameter, lay-up sequences and boundary conditions on thermal buckling temperatures are investigated. The results are shown in graphical form for various boundary conditions.
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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 in Fig. 1. The linear stress–strain relation for each layer is expressed with x, y axes and has the form 9 8 9 2 38 > Q11 Q12 Q16 > = = < rx > < ex ax DT > 6 7 ey ay DT ry ¼ 4 Q12 Q22 Q26 5 ; > > > ; ; : > : c a DT sxy k Q16 Q26 Q66 k xy xy ð1Þ k # " cyz syz Q44 Q45 ¼ ; sxz cxz Q45 Q55 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 [18]. DT is the temperature 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. The displacements u, v and w can be written as follows:
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
¨ .S. S O ß ahin / Composites Science and Technology 65 (2005) 1780–1790
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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, y axes, respectively. The bending strains ex, ey and shear strains cxy, cyz, cxz at any point of the laminate are owx0 8 9 ou0 e ox > > x ox < = ow ov ey ¼ oy0 þ z oyy ; > ow owy ; ou0 ov0 : > cxy oy þ ox xþ ð4Þ oy ox ow cyz oy wy ¼ : 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 8 9 2 3 Nx Mx Z h=2 > = < rx > 6 7 N M ry ð1; zÞ dz; ¼ 4 y y 5 h=2 > ; : > N xy M xy sxy ð5Þ ( ) Z h=2 Qx sxz dz; ¼ Qy syz h=2 In Ref. (5), it was shown that the effect of transverse shear correction coefficients is negligible for the plates with high length-to-thickness ratios. In this study, the numerical solutions was performed on the plates with laminate length (b)-to-thickness of plate (h) ratios varying 40–100 in order not to cause rough errors by neglecting the transverse shear effects. The total potential energy P of a laminated plate under thermal loading is equal to P ¼ Ub þ Us þ V ;
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 h=2 Z Z U b ¼ 1=2 ðrx ex þ ry ey þ sxy cxy Þ dA dz; Z
h=2
h=2 Z Z
R
ðsxz cxz þ syz cyz Þ dA dz;
U s ¼ 1=2 R h=2 Z Z h 2 2 V ¼ 1=2 N 1 ðow=oxÞ þ N 2 ðow=oyÞ R i þ 2N 12 ðow=oxÞðow=oyÞ dA Z b b N n uon þ N s uos ds:
ð7Þ
oR
Here dA = 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 angle-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 = 0 [15,16]. 2.1. Finite element formulations In general, a closed form solution is difficult to obtain for buckling problems [15,16]. 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 T ð8Þ ½K b ¼ ½Bb ½Db ½Bb dA; A
½K s ¼
Z
½Bs T ½Ds ½Bs dA
ð9Þ
A
and ½K g ¼
Z
T
½Bg ½Dg ½Bg dA;
ð10Þ
A
where
" Aij Bij k 21 A44 ½Db ¼ ; ½Ds ¼ Bij Dij 0 " # N 1 N 12 ; ½Dg ¼ N 12 N 2
ðAij ; Bij ; Dij Þ ¼
Z
# 0 ; k 22 A55
ð11Þ
h=2
Qij ð1; z; z2 Þ dz
ði; j ¼ 1; 2; 6Þ;
ð12Þ
h=2
ðA44 ; A55 Þ ¼
Z
h=2
ðQ44 ; Q55 Þ dz:
ð13Þ
h=2
In which the term A45 is neglected in comparison with A44 and A55, and the shear correction factors for rectangular cross-section are given by k 21 ¼ k 22 ¼ 5=6 [17]. 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 [8,16] 8 9 > = < ui > ¼ 0; ð14Þ ½½K 0 kb ½K 0g vi > ; : > wi
¨ .S. S O ß ahin / Composites Science and Technology 65 (2005) 1780–1790
where ½K 0 ¼ ½K b þ ½K s ;
kb ½K 0g ¼ ½K g :
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Table 2 Stacking sequence of hybrid laminates
ð15Þ
Lay-up sequence
The product of kb and the initial guest value DT is the critical buckling temperature Tcr, that is
Glass–epoxy/boron–epoxy (symmetric lay up, 10 layers) 600G = 600B = 600B =600G = 600B = 600B =600G = 600B = 600B =600G
T cr ¼ kb DT :
Glass–epoxy/boron–epoxy (antisymmetric lay up, 10 layers) 600G = 600B =600G = 600B =600G = 600B =600G = 600B =600G = 600B
ð16Þ
3. Numerical result and discussion
3. Four edges clamped (CCCC): At x = a/2, a/2; u = w = wy = wx = 0 At y = b/2, b/2; v = w = wx = wy = 0 4. Two edges clamped (CC): At x = a/2, a/2; u = w = wy = wx = 0 At y = b/2, b/2; w = 0
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 angle-ply laminated hybrid composite plates used here have several thicknesses and bonded symmetrically and antisymmetrically. For the computations thermo-elastic properties considered for the 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 PoissonÕs 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 stacking sequences have been represented in Table 2. The letters A and B represents glass–epoxy and boron–epoxy, respectively. 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.
The variation of critical buckling temperature of plate without hole for (15/15)2 configuration is shown in Fig. 2. As expected, temperatures increase as the h/b ratio increases for both symmetric and antisymmetric cases. But, critical buckling temperatures of antisymmetric laminates are higher than that of symmetric plate. The highest buckling temperatures are obtained for two edge clamped plate and the lowest temperatures obtained for four edges simply supported plate. The relationship between buckling temperature of plate with hole and plate thickness presented in Fig. 3 for (15/15)2 configuration. In Fig. 3, the curves show a monotonically increasing Tcr as the hole diameter increase. It can be seen that the buckling temperatures of clamped cases are always higher than those of simply supported plates. Figs. 3–6 show that as plate thickness increases, the buckling temperature of both antisymmetric and symmetric plates with hole increases. Though, the highest buckling temperatures are obtained for two edge clamped plate without hole, the highest critical buckling temperatures are obtained for four edge clamped plate with hole size of d/b = 0.5. The lowest buckling temperatures are obtained for two edge simply supported plate. This behavior is same for both antisymmetric and symmetric plates. Effects of boundary condition on the buckling behavior are same for all layer configurations. As shown in Figs. 3–6, the highest buckling temperatures are obtained for four edges clamped plate and the lowest buckling temperatures are obtained for two edges simply supported plate.
1. Four edges simply supported (SSSS): At x = a/2, a/2; u = w = wy = 0 At y = b/2, b/2; v = w = wx = 0 2. Two edges simply supported (SS): At x = a/2, a/2; u = w = wy = 0 At y = b/2, b/2; w = 0
Table 1 Material properties for using numerical solutions Material E-glass/epoxy Boron/epoxy
E1 (GPa) 15 207
E2 (GPa) 6 19
G12 (GPa) 3 4.8
m12 0.3 0.21
a1 (1/C) 6
7.0 · 10 4.14 · 106
a2 (1/C) 2.30 · 105 1.91 · 105
¨ .S. S O ß ahin / Composites Science and Technology 65 (2005) 1780–1790
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500
500
400
400
Tc ˚C
300
ssss
200
ss
100
cc
0 0.01 (a)
0.015 0.02 h/b
Tc ˚C
cccc
cccc
300
ssss
200
ss cc
100 0 0.01
0.025 (b)
0.015 0.02 h/b
0.025
Fig. 2. Variation of buckling temperatures with thickness of (a) antisymmetric, (b) symmetric laminate without hole for various boundary conditions for (15/15)2 configuration.
700
600
600 ssss
400
ss
300
cc
Tc ˚C
Tc ˚C
500
cccc
500
cccc
400
ssss
300
ss cc
200
200
100
100 0 0.01 (a)
0 0.015 0.02 h/b
0.025
0.01 (b)
0.015 0.02 h/b
0.025
Fig. 3. Variation of buckling temperatures with thickness of (a) antisymmetric, (b) symmetric laminate with hole size of d/b = 0.5 for various boundary conditions for (15/15)2 configuration.
600
500
500
400 cccc
300 200 100
Tc ˚C
Tc ˚C
400
0.01
cccc
ssss
200
ssss
ss
100
ss cc
cc
0
0 (a)
300
0.015 0.02 h/b
0.025
0.01 (b)
0.015 0.02 h/b
0.025
Fig. 4. Variation of buckling temperatures with thickness of (a) antisymmetric, (b) symmetric laminate with hole size of d/b = 0.5 for various boundary conditions for (30/30)2 configuration.
The buckling temperatures for two edges and four edges clamped plates are relatively close for (60/60)2 configuration. On the other hand the buckling temperatures for two edges and four edges simply supported plates are close as well. But, buckling temperatures for four edges simply supported plate is greater than that of two edges simply supported one. As shown in Fig. 6, for (60/60)2 configuration, the buckling temperatures for four edges and two edges clamped plates are almost equal. At the same time, the buckling temperatures for four edges and two edges simply supported plate are close as well.
Figs. 7 and 8 show the variation of critical buckling temperature and hole diameter (15/15)2 configuration and (30/30)2 configuration, respectively. For two and four edge simply supported plates, as hole diameter increases, the buckling temperature remains almost constant for both symmetric and antisymmetric plates. As the hole diameter increases, the buckling temperature of four edge clamped plate increases. At hole sizes bigger than d/b = 0.2 the buckling temperature increases parabolic, as hole diameter increases. The buckling temperature for two edge clamped plate shows a clear decrease, as the hole size increases. Above-mentioned behaviors
¨ .S. S O ß ahin / Composites Science and Technology 65 (2005) 1780–1790
600
500
400 300
cccc
200
ssss
100
ss
Tc ˚C
500 Tc ˚C
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0.01 (a)
0.015 0.02 h/b
cccc
300
ssss ss
200
cc
100
cc
0
400
0 0.01
0.025
0.015 0.02 h/b
(b)
0.025
Fig. 5. Variation of buckling temperatures with thickness of (a) antisymmetric, (b) symmetric laminate with hole size of d/b = 0.5 for various boundary conditions for (45/45)2 configuration.
500
400
300
ssss
200
ss
ssss 200
0 0.01
0.015
0.02
cc
0 0.01
0.025
h/b
(a)
ss
100
cc
100
cccc
300
cccc
Tc ˚C
Tc ˚C
400
0.015
(b)
0.02 0.025 h/b
250
250
200
200 cccc
150
ssss
100 50
Tc ˚C
Tc ˚C
Fig. 6. Variation of buckling temperatures with thickness of (a) antisymmetric, (b) symmetric laminate with hole size of d/b = 0.5 for various boundary conditions for (60/60)2 configuration.
ssss
ss
100
cc
50
0
cccc
150
ss cc
0 0
0.1
0.2
0.3 0.4
0.5
0
d/b
(a)
0.1
0.2
0.3 0.4
0.5
d/b
(b)
Fig. 7. Variation of buckling temperatures of six layered (a) antisymmetric, (b) symmetric plate with hole for various boundary conditions for (15/15)2 configuration.
250
200 150 cccc
150
ssss
100
(a)
cccc ssss
100
ss
ss
cc
50
cc
50 0
Tc ˚C
Tc ˚C
200
0 0
0.1
0.2
d/b
0.3 0.4
0.5
0 (b)
0.1
0.2
0.3 0.4
0.5
d/b
Fig. 8. Variation of buckling temperatures with hole size for six layered (a) antisymmetric, (b) symmetric plate for various boundary conditions for (30/30)2 configuration.
¨ .S. S O ß ahin / Composites Science and Technology 65 (2005) 1780–1790
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200 160 160 cccc
100 80
ssss ss
60 40
cccc
120
Tc ˚C
Tc ˚C
140 120
cc
ssss
80
ss cc
40
20 0
0 0
0.1
0.2
0.3 0.4 0.5
0
d/b
(a)
0.1
0.2
0.3 0.4
0.5
d/b
(b)
Fig. 9. Variation of buckling temperatures of six layered (a) antisymmetric, (b) symmetric plate with hole size for various boundary conditions for (45/45)2 configuration.
160 140 120 100 80 60 40 20 0
Tc ˚C
cccc ssss ss cc
0 (a)
0.1 0.2 0.3 0.4 0.5 d/b
buckling temperature of four edge clamped plate increases. As seen in Figs. 9 and 10 the buckling temperatures for antisymmetric plate are higher than that of symmetric plate. Though, the buckling temperatures for two edge clamped (45/45)2 plate, show a decreasing tendency, the plate with (60/60)2 configuration shows an increase. Figs. 11–14 show the variation of critical buckling temperatures with hole diameters for various layer configurations.
160 140 120 100 80 60 40 20 0
cccc
Tc ˚C
are same for both antisymmetric and symmetric laminated plates. Variation of critical buckling temperature with hole size are same for both (15/15)2 and (30/ 30)2 configurations. In Figs. 9 and 10 the variation of critical buckling temperature and hole diameter is seen. For this layer configuration, for two and four edge simply supported plates, as hole diameter increases, the buckling temperature remains almost constant for both symmetric and antisymmetric plates. As hole diameter increases, the
(b)
ssss ss cc
0
0.1
0.2 0.3 0.4 0.5 d/b
Fig. 10. Variation of buckling temperatures with hole size for six layered (a) antisymmetric, (b) symmetric plate for various boundary conditions for (60/60)2 configuration.
700
600 500
500
4 Layers
400
6 Layers
300
8 Layers 10 Layers
200
6 Layers
300
8 Layers
200
10 Layers
100
100
0
0
0 0.1 0.2 0.3 0.4 0.5 (a) d/b
4 Layers
400
Tc ˚C
Tc ˚C
600
(b)
0 0.1 0.2 0.3 0.4 0.5 d/b
Fig. 11. Variation of buckling temperatures with hole size for (a) antisymmetric, (b) symmetric four edge clamped plate for various layer numbers for (15/15)2 configuration.
¨ .S. S O ß ahin / Composites Science and Technology 65 (2005) 1780–1790 500
600 500
400 4 Layers 6 Layers
300
8 Layers
6 Layers 8 Layers
200
10 Layers
200
4 Layers
300
Tc ˚C
Tc ˚C
400
10 Layers
100
100
0
0 0 0.1 0.2 0.3 0.4 0.5 d/b
(a)
1787
0 (b)
0.1 0.2 0.3 0.4 0.5 d/b
Fig. 12. Variation of buckling temperatures with hole size for (a) antisymmetric, (b) symmetric four edges clamped plate for various layer numbers for (30/30)2 configuration.
500
600 500
400 4 Layers
4 Layers 6 Layers
300
8 Layers
Tc ˚C
Tc ˚C
400
8 Layers 10 Layers
100
100
0
0 0 0.1 0.2 0.3 0.4 0.5 d/b
(a)
6 Layers
200
10 Layers
200
300
0
(b)
0.1 0.2 0.3 0.4 0.5 d/b
Fig. 13. Variation of buckling temperatures with hole size for (a) antisymmetric, (b) symmetric four edges clamped plate for various layer numbers for (45/45)2 configuration.
500
400 300 4 Layers
300
6 Layers
Tc ˚C
Tc ˚C
400
8 Layers
200
6 Layers
200
8 Layers 10 Layers
10 Layers
100
100
0 0
0 (a)
4 Layers
0 0.1 0.2 0.3 0.4 0.5 d/b
(b)
0.1 0.2 0.3 0.4 0.5 d/b
Fig. 14. Variation of buckling temperatures with hole size for (a) antisymmetric, (b) symmetric four edges clamped plate for various layer numbers for (60/60)2 configuration.
As shown in Figs. 11–14 as the number of layers namely, the thickness of plate increases, the critical buckling temperature increases. This behavior is same for all layer configurations used in this solution. As shown in Figs. 11–14, the critical buckling temperatures of antisymmetric plates are higher than that of symmetric plates.
Symmetric laminates, does not yield the highest buckling resistance as usually expected, because of effect of bending–extension coupling. As shown in Figs. 11–14, buckling temperatures for (15/15)2 configuration are the highest and buckling temperatures for (60/60)2 configuration are the lowest, comparing with other layer configurations.
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The first and second buckled mode shapes generated glass–epoxy/boron–epoxy angle-ply laminated six layered plates with four boundary conditions are shown in Figs. 15–18. It is found that critical temperature for
(15/15)2 configuration plate without hole for four edge clamped plate is 70.95 C, for two edge clamped is 172.88 C, for four edge simply supported plate is 33.23 C and for two edge simply supported plate is
Fig. 15. First buckling mode shapes for (15/15)2 configuration six layered laminate without hole for boundary conditions (a) four edge clamped (CCCC), (b) four edge simply supported (SSSS) plates.
Fig. 16. First buckling mode shapes for (15/15)2 configuration six layered laminate without hole for boundary conditions (a) two edge clamped (CC), (b) two edge simply supported (SS) plates.
Fig. 17. First buckling mode shapes for (30/30)2 configuration six layered laminate with hole for boundary conditions (a) four edge clamped (CCCC), (b) four edge simply supported (SSSS) plates.
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Fig. 18. First buckling mode shapes for (30/30)2 configuration six layered laminate with hole for boundary conditions (a) two edge clamped (CC), (b) two edge simply supported (SS) plates.
94.43 C, respectively. For (30/30)2 configuration plate with hole size of d/b = 0.3 for four edge clamped plate is 107.35 C, for two edge clamped is 123.74 C, for four edge simply supported plate is 28.89 C and for two edge simply supported plate is 29.2 C, respectively. The mode shapes presented in Figs. 15–18 show considerable skewing for the laminated plates.
4. Conclusions Thermal buckling behaviors of angle-ply laminated plates with holes have been examined by employing the first-order shear deformation theory and finite element technique. Both symmetric and antisymmetric lay-up sequence are considered and various boundary conditions are taken into account. Four edges clamped plates with hole have the highest buckling temperature and four edges simply supported plate have the lowest buckling temperature for all layer configurations. The critical buckling temperature is strongly depend on hole size for two edges and four edges clamped plates. For four edge clamped plates, as the hole size increases, the buckling temperature shows an increase. But, as the hole size increases, the buckling temperatures for two edge clamped plate decreases. The critical buckling temperature of four edge simply supported and simply supported plates do not affected so much by hole size and plates have this boundary conditions result the lowest buckling temperature. For this case, there is a little difference between symmetric and antisymmetric lay up. As number of layers namely the plate thickness increases, more temperature difference is needed for buckling. This behavior is same for both antisymmetric and symmetric lay up. But, higher temperature difference is needed for buckling of antisymmetrical stacked plate than the symmetric one.
For four layer configuration used in the solution, the highest buckling temperature difference is reached for (15/15)2 configuration and the lowest buckling temperature difference is reached for (60/60)2 configuration.
References [1] Gossard ML, Seide P, Roberts WM. Thermal buckling of plates. NACA TN 2771; 1952. [2] Liessa AW. A review of laminated composite plate buckling. Appl Mech Rec 1987;40:575–91. [3] Bednarczyk H, Rihter M. Buckling of plated due to selfequilibrated thermal stresses. J Thermal Stresses 1985;8: 139–52. [4] Chen W, Liu WH. Thermal buckling of antisymmetric angle-ply laminated plates – an analytical Levy-type solution. J Thermal Stresses 1993(16):401–19. [5] Sun LX, Hsu TR. Thermal buckling of laminated composite plates with transverse shear deformation. Comput Struct 1990;36(5):883–9. [6] Chockalingam S, Mathew TC, Singh G, Rao GV. Critical temperatures of hybrid laminates using finite elements. Comput Struct 1992;43(5):995–8. [7] Thangaratnam KR, Ramachandran J. Thermal buckling of composite laminated plates. Comput Struct 1989;32:1117–24. [8] 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. [9] Mannini A. Shear deformation effects on thermal buckling of cross-ply composite laminates. Comp Struct 1997;39:1–10. [10] 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. [11] Yettram A, Brawn CJ. The elastic stability of square perforated plates under biaxial loadings. Comput Struct 1986;22(4):589–94. [12] Chang CL, Hsu DS. Vibration and stability of plates with hole. Proc of the 10th Natl Conf on Theo and Appl Mechanics STAM ROC 1986:235–41. [13] VanDen Brink, DJ. Kamat MP. Post-buckling response of isotropic and laminated composites square plates with circular holes. In: Proc Int Conf Composite Material, San Diego, CA; 1985, p. 1393–1409. [14] Larsson PL. On buckling of orthotropic compressed plates with circular holes. Comp Struct 1987;7:103–21.
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[15] Lin CH, Kuo CS. Buckling of laminated plates with holes. J Comp Mater 1989;23:536–53. [16] Bathe KJ. Finite element procedures in engineering analysis. Englewood Cliffs, NJ: Prentice-Hall; 1982.
[17] Whitney JM. Shear correction factors for orthotropic laminates under static load. J Appl Mech 1973(march):302–4. [18] Jhones RM. Mechanics of composite materials. Tokyo: McGrawHill Kogagusha Ltd; 1975.
Thermal buckling of hybrid angle-ply laminated composite plates with a hole ¨ mer Sinan S O ß ahin
*
Selc¸uk University, Department of Mechanical Engineering, Konya, Turkey Received 13 October 2004; received in revised form 8 March 2005; accepted 11 March 2005 Available online 19 April 2005
Abstract In this paper, thermal buckling analysis of symmetric and antisymmetric laminated hybrid composite plates with a hole subjected to a uniform temperature rise for different boundary conditions is presented. The first-order shear deformation theory in conjunction with variational energy method is employed in mathematical formulation. The eight-node Lagrangian finite element technique is used for obtaining the thermal buckling temperatures of glass–epoxy/boron–epoxy hybrid laminates. The effects of hole size, layup sequences and boundary conditions on thermal buckling temperatures are investigated. The results are shown in graphical form for various boundary conditions. 2005 Elsevier Ltd. All rights reserved. Keywords: Hybrid composite plates; Thermal buckling; Finite element method
1. Introduction Fiber reinforced composites because of their low weight, high strength and anisotropic properties, which can be tailored through the variation of fiber angle and stacking sequence, are used extensively in aerospace structures and transportation industries. Thermal buckling could be caused when a structure is exposed to a temperature rise. Due to the solar radiation heating and aerodynamically thermal boundary effects the temperature change during aeronautical mission are important. Since, structural components of these high speed machines are usually exposed to non-uniform heat flux and temperature distribution. Considerable work has been done on thermal problems of plates. Gossard et al. [1] used a Rayleigh–Ritz method to find the deflections of plates or initially
*
Tel.: +90 332 223 18 98; fax: +90 332 241 06 51. E-mail address: [email protected].
0266-3538/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2005.03.007
imperfect plates exposed to thermal buckling. An extensive overview of the general buckling problems of laminated composite plate was made by Liessa [2]. In this study, some complicated effects were investigated such as shear deformation, hygrothermal factors and postbuckling behavior. Bednarczyk and Rihter [3] investigated the influence of temperature distribution on buckling modes. The thermally induced buckling of antisymmetric angle-ply laminated plates with Levytype boundary conditions was investigated by Chen and Liu [4]. Thermal buckling behavior of composite laminated plates with transverse shear deformation was studied by Sun and Hsu [5]. Chockalingam et al. [6] investigated the thermal buckling of antisymmetric cross-ply hybrid laminates by using a one-dimensional finite element technique based on first-order shear deformation theory. Thangaratnam and Ramachandran [7] and Chen et al. [8] also presented thermal buckling of laminates subjected to uniform temperature rise or non-uniform temperature fields using finite element approach. Thermal
¨ .S. S O ß ahin / Composites Science and Technology 65 (2005) 1780–1790
buckling analysis of cross-ply laminates is also studied by using first-order shear deformation theory and Rayleigh–Ritz method by Mannini [9]. Dawe and Ge [10] presented results for thermally loaded shear deformable composite laminates using the spline finite strip method. Yettram and Brawn [11] predicted the buckling of square perforated plates under biaxial loadings using a direct matrix method. The buckling and vibration of isotropic plates with a circular or a square hole are discussed by Chang and Hsu [12]. The post-buckling analysis of isotropic and composite square plates with circular holes was carried out by VanDen Brink and Kamat [13] by using a finite element technique. Larsson [14] also used a finite element approach and perturbation method for investigating the buckling of orthotropic compressed plates with circular holes. Lin and Kuo [15] investigated the buckling of cross-ply and angle-ply laminated plates with circular holes under in plane static loadings. For finding the critical loads they used a finite element analysis. The present paper is an attempt to determine the buckling temperature and buckling mode for hybrid composite laminates with circular holes using finite element method. Thermal buckling of symmetric and antisymmetric angle-ply laminates with holes is investigated, based on the first-order deformation theory in conjunction with variational energy method. The eight-node Lagrangian finite element approach is used for obtaining the thermal buckling temperatures of boron– epoxy/glass–epoxy hybrid laminates. The effects of hole diameter, lay-up sequences and boundary conditions on thermal buckling temperatures are investigated. The results are shown in graphical form for various boundary conditions.
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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 in Fig. 1. The linear stress–strain relation for each layer is expressed with x, y axes and has the form 9 8 9 2 38 > Q11 Q12 Q16 > = = < rx > < ex ax DT > 6 7 ey ay DT ry ¼ 4 Q12 Q22 Q26 5 ; > > > ; ; : > : c a DT sxy k Q16 Q26 Q66 k xy xy ð1Þ k # " cyz syz Q44 Q45 ¼ ; sxz cxz Q45 Q55 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 [18]. DT is the temperature 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. The displacements u, v and w can be written as follows:
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
¨ .S. S O ß ahin / Composites Science and Technology 65 (2005) 1780–1790
1782
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, y axes, respectively. The bending strains ex, ey and shear strains cxy, cyz, cxz at any point of the laminate are owx0 8 9 ou0 e ox > > x ox < = ow ov ey ¼ oy0 þ z oyy ; > ow owy ; ou0 ov0 : > cxy oy þ ox xþ ð4Þ oy ox ow cyz oy wy ¼ : 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 8 9 2 3 Nx Mx Z h=2 > = < rx > 6 7 N M ry ð1; zÞ dz; ¼ 4 y y 5 h=2 > ; : > N xy M xy sxy ð5Þ ( ) Z h=2 Qx sxz dz; ¼ Qy syz h=2 In Ref. (5), it was shown that the effect of transverse shear correction coefficients is negligible for the plates with high length-to-thickness ratios. In this study, the numerical solutions was performed on the plates with laminate length (b)-to-thickness of plate (h) ratios varying 40–100 in order not to cause rough errors by neglecting the transverse shear effects. The total potential energy P of a laminated plate under thermal loading is equal to P ¼ Ub þ Us þ V ;
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 h=2 Z Z U b ¼ 1=2 ðrx ex þ ry ey þ sxy cxy Þ dA dz; Z
h=2
h=2 Z Z
R
ðsxz cxz þ syz cyz Þ dA dz;
U s ¼ 1=2 R h=2 Z Z h 2 2 V ¼ 1=2 N 1 ðow=oxÞ þ N 2 ðow=oyÞ R i þ 2N 12 ðow=oxÞðow=oyÞ dA Z b b N n uon þ N s uos ds:
ð7Þ
oR
Here dA = 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 angle-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 = 0 [15,16]. 2.1. Finite element formulations In general, a closed form solution is difficult to obtain for buckling problems [15,16]. 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 T ð8Þ ½K b ¼ ½Bb ½Db ½Bb dA; A
½K s ¼
Z
½Bs T ½Ds ½Bs dA
ð9Þ
A
and ½K g ¼
Z
T
½Bg ½Dg ½Bg dA;
ð10Þ
A
where
" Aij Bij k 21 A44 ½Db ¼ ; ½Ds ¼ Bij Dij 0 " # N 1 N 12 ; ½Dg ¼ N 12 N 2
ðAij ; Bij ; Dij Þ ¼
Z
# 0 ; k 22 A55
ð11Þ
h=2
Qij ð1; z; z2 Þ dz
ði; j ¼ 1; 2; 6Þ;
ð12Þ
h=2
ðA44 ; A55 Þ ¼
Z
h=2
ðQ44 ; Q55 Þ dz:
ð13Þ
h=2
In which the term A45 is neglected in comparison with A44 and A55, and the shear correction factors for rectangular cross-section are given by k 21 ¼ k 22 ¼ 5=6 [17]. 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 [8,16] 8 9 > = < ui > ¼ 0; ð14Þ ½½K 0 kb ½K 0g vi > ; : > wi
¨ .S. S O ß ahin / Composites Science and Technology 65 (2005) 1780–1790
where ½K 0 ¼ ½K b þ ½K s ;
kb ½K 0g ¼ ½K g :
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Table 2 Stacking sequence of hybrid laminates
ð15Þ
Lay-up sequence
The product of kb and the initial guest value DT is the critical buckling temperature Tcr, that is
Glass–epoxy/boron–epoxy (symmetric lay up, 10 layers) 600G = 600B = 600B =600G = 600B = 600B =600G = 600B = 600B =600G
T cr ¼ kb DT :
Glass–epoxy/boron–epoxy (antisymmetric lay up, 10 layers) 600G = 600B =600G = 600B =600G = 600B =600G = 600B =600G = 600B
ð16Þ
3. Numerical result and discussion
3. Four edges clamped (CCCC): At x = a/2, a/2; u = w = wy = wx = 0 At y = b/2, b/2; v = w = wx = wy = 0 4. Two edges clamped (CC): At x = a/2, a/2; u = w = wy = wx = 0 At y = b/2, b/2; w = 0
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 angle-ply laminated hybrid composite plates used here have several thicknesses and bonded symmetrically and antisymmetrically. For the computations thermo-elastic properties considered for the 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 PoissonÕs 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 stacking sequences have been represented in Table 2. The letters A and B represents glass–epoxy and boron–epoxy, respectively. 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.
The variation of critical buckling temperature of plate without hole for (15/15)2 configuration is shown in Fig. 2. As expected, temperatures increase as the h/b ratio increases for both symmetric and antisymmetric cases. But, critical buckling temperatures of antisymmetric laminates are higher than that of symmetric plate. The highest buckling temperatures are obtained for two edge clamped plate and the lowest temperatures obtained for four edges simply supported plate. The relationship between buckling temperature of plate with hole and plate thickness presented in Fig. 3 for (15/15)2 configuration. In Fig. 3, the curves show a monotonically increasing Tcr as the hole diameter increase. It can be seen that the buckling temperatures of clamped cases are always higher than those of simply supported plates. Figs. 3–6 show that as plate thickness increases, the buckling temperature of both antisymmetric and symmetric plates with hole increases. Though, the highest buckling temperatures are obtained for two edge clamped plate without hole, the highest critical buckling temperatures are obtained for four edge clamped plate with hole size of d/b = 0.5. The lowest buckling temperatures are obtained for two edge simply supported plate. This behavior is same for both antisymmetric and symmetric plates. Effects of boundary condition on the buckling behavior are same for all layer configurations. As shown in Figs. 3–6, the highest buckling temperatures are obtained for four edges clamped plate and the lowest buckling temperatures are obtained for two edges simply supported plate.
1. Four edges simply supported (SSSS): At x = a/2, a/2; u = w = wy = 0 At y = b/2, b/2; v = w = wx = 0 2. Two edges simply supported (SS): At x = a/2, a/2; u = w = wy = 0 At y = b/2, b/2; w = 0
Table 1 Material properties for using numerical solutions Material E-glass/epoxy Boron/epoxy
E1 (GPa) 15 207
E2 (GPa) 6 19
G12 (GPa) 3 4.8
m12 0.3 0.21
a1 (1/C) 6
7.0 · 10 4.14 · 106
a2 (1/C) 2.30 · 105 1.91 · 105
¨ .S. S O ß ahin / Composites Science and Technology 65 (2005) 1780–1790
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500
500
400
400
Tc ˚C
300
ssss
200
ss
100
cc
0 0.01 (a)
0.015 0.02 h/b
Tc ˚C
cccc
cccc
300
ssss
200
ss cc
100 0 0.01
0.025 (b)
0.015 0.02 h/b
0.025
Fig. 2. Variation of buckling temperatures with thickness of (a) antisymmetric, (b) symmetric laminate without hole for various boundary conditions for (15/15)2 configuration.
700
600
600 ssss
400
ss
300
cc
Tc ˚C
Tc ˚C
500
cccc
500
cccc
400
ssss
300
ss cc
200
200
100
100 0 0.01 (a)
0 0.015 0.02 h/b
0.025
0.01 (b)
0.015 0.02 h/b
0.025
Fig. 3. Variation of buckling temperatures with thickness of (a) antisymmetric, (b) symmetric laminate with hole size of d/b = 0.5 for various boundary conditions for (15/15)2 configuration.
600
500
500
400 cccc
300 200 100
Tc ˚C
Tc ˚C
400
0.01
cccc
ssss
200
ssss
ss
100
ss cc
cc
0
0 (a)
300
0.015 0.02 h/b
0.025
0.01 (b)
0.015 0.02 h/b
0.025
Fig. 4. Variation of buckling temperatures with thickness of (a) antisymmetric, (b) symmetric laminate with hole size of d/b = 0.5 for various boundary conditions for (30/30)2 configuration.
The buckling temperatures for two edges and four edges clamped plates are relatively close for (60/60)2 configuration. On the other hand the buckling temperatures for two edges and four edges simply supported plates are close as well. But, buckling temperatures for four edges simply supported plate is greater than that of two edges simply supported one. As shown in Fig. 6, for (60/60)2 configuration, the buckling temperatures for four edges and two edges clamped plates are almost equal. At the same time, the buckling temperatures for four edges and two edges simply supported plate are close as well.
Figs. 7 and 8 show the variation of critical buckling temperature and hole diameter (15/15)2 configuration and (30/30)2 configuration, respectively. For two and four edge simply supported plates, as hole diameter increases, the buckling temperature remains almost constant for both symmetric and antisymmetric plates. As the hole diameter increases, the buckling temperature of four edge clamped plate increases. At hole sizes bigger than d/b = 0.2 the buckling temperature increases parabolic, as hole diameter increases. The buckling temperature for two edge clamped plate shows a clear decrease, as the hole size increases. Above-mentioned behaviors
¨ .S. S O ß ahin / Composites Science and Technology 65 (2005) 1780–1790
600
500
400 300
cccc
200
ssss
100
ss
Tc ˚C
500 Tc ˚C
1785
0.01 (a)
0.015 0.02 h/b
cccc
300
ssss ss
200
cc
100
cc
0
400
0 0.01
0.025
0.015 0.02 h/b
(b)
0.025
Fig. 5. Variation of buckling temperatures with thickness of (a) antisymmetric, (b) symmetric laminate with hole size of d/b = 0.5 for various boundary conditions for (45/45)2 configuration.
500
400
300
ssss
200
ss
ssss 200
0 0.01
0.015
0.02
cc
0 0.01
0.025
h/b
(a)
ss
100
cc
100
cccc
300
cccc
Tc ˚C
Tc ˚C
400
0.015
(b)
0.02 0.025 h/b
250
250
200
200 cccc
150
ssss
100 50
Tc ˚C
Tc ˚C
Fig. 6. Variation of buckling temperatures with thickness of (a) antisymmetric, (b) symmetric laminate with hole size of d/b = 0.5 for various boundary conditions for (60/60)2 configuration.
ssss
ss
100
cc
50
0
cccc
150
ss cc
0 0
0.1
0.2
0.3 0.4
0.5
0
d/b
(a)
0.1
0.2
0.3 0.4
0.5
d/b
(b)
Fig. 7. Variation of buckling temperatures of six layered (a) antisymmetric, (b) symmetric plate with hole for various boundary conditions for (15/15)2 configuration.
250
200 150 cccc
150
ssss
100
(a)
cccc ssss
100
ss
ss
cc
50
cc
50 0
Tc ˚C
Tc ˚C
200
0 0
0.1
0.2
d/b
0.3 0.4
0.5
0 (b)
0.1
0.2
0.3 0.4
0.5
d/b
Fig. 8. Variation of buckling temperatures with hole size for six layered (a) antisymmetric, (b) symmetric plate for various boundary conditions for (30/30)2 configuration.
¨ .S. S O ß ahin / Composites Science and Technology 65 (2005) 1780–1790
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200 160 160 cccc
100 80
ssss ss
60 40
cccc
120
Tc ˚C
Tc ˚C
140 120
cc
ssss
80
ss cc
40
20 0
0 0
0.1
0.2
0.3 0.4 0.5
0
d/b
(a)
0.1
0.2
0.3 0.4
0.5
d/b
(b)
Fig. 9. Variation of buckling temperatures of six layered (a) antisymmetric, (b) symmetric plate with hole size for various boundary conditions for (45/45)2 configuration.
160 140 120 100 80 60 40 20 0
Tc ˚C
cccc ssss ss cc
0 (a)
0.1 0.2 0.3 0.4 0.5 d/b
buckling temperature of four edge clamped plate increases. As seen in Figs. 9 and 10 the buckling temperatures for antisymmetric plate are higher than that of symmetric plate. Though, the buckling temperatures for two edge clamped (45/45)2 plate, show a decreasing tendency, the plate with (60/60)2 configuration shows an increase. Figs. 11–14 show the variation of critical buckling temperatures with hole diameters for various layer configurations.
160 140 120 100 80 60 40 20 0
cccc
Tc ˚C
are same for both antisymmetric and symmetric laminated plates. Variation of critical buckling temperature with hole size are same for both (15/15)2 and (30/ 30)2 configurations. In Figs. 9 and 10 the variation of critical buckling temperature and hole diameter is seen. For this layer configuration, for two and four edge simply supported plates, as hole diameter increases, the buckling temperature remains almost constant for both symmetric and antisymmetric plates. As hole diameter increases, the
(b)
ssss ss cc
0
0.1
0.2 0.3 0.4 0.5 d/b
Fig. 10. Variation of buckling temperatures with hole size for six layered (a) antisymmetric, (b) symmetric plate for various boundary conditions for (60/60)2 configuration.
700
600 500
500
4 Layers
400
6 Layers
300
8 Layers 10 Layers
200
6 Layers
300
8 Layers
200
10 Layers
100
100
0
0
0 0.1 0.2 0.3 0.4 0.5 (a) d/b
4 Layers
400
Tc ˚C
Tc ˚C
600
(b)
0 0.1 0.2 0.3 0.4 0.5 d/b
Fig. 11. Variation of buckling temperatures with hole size for (a) antisymmetric, (b) symmetric four edge clamped plate for various layer numbers for (15/15)2 configuration.
¨ .S. S O ß ahin / Composites Science and Technology 65 (2005) 1780–1790 500
600 500
400 4 Layers 6 Layers
300
8 Layers
6 Layers 8 Layers
200
10 Layers
200
4 Layers
300
Tc ˚C
Tc ˚C
400
10 Layers
100
100
0
0 0 0.1 0.2 0.3 0.4 0.5 d/b
(a)
1787
0 (b)
0.1 0.2 0.3 0.4 0.5 d/b
Fig. 12. Variation of buckling temperatures with hole size for (a) antisymmetric, (b) symmetric four edges clamped plate for various layer numbers for (30/30)2 configuration.
500
600 500
400 4 Layers
4 Layers 6 Layers
300
8 Layers
Tc ˚C
Tc ˚C
400
8 Layers 10 Layers
100
100
0
0 0 0.1 0.2 0.3 0.4 0.5 d/b
(a)
6 Layers
200
10 Layers
200
300
0
(b)
0.1 0.2 0.3 0.4 0.5 d/b
Fig. 13. Variation of buckling temperatures with hole size for (a) antisymmetric, (b) symmetric four edges clamped plate for various layer numbers for (45/45)2 configuration.
500
400 300 4 Layers
300
6 Layers
Tc ˚C
Tc ˚C
400
8 Layers
200
6 Layers
200
8 Layers 10 Layers
10 Layers
100
100
0 0
0 (a)
4 Layers
0 0.1 0.2 0.3 0.4 0.5 d/b
(b)
0.1 0.2 0.3 0.4 0.5 d/b
Fig. 14. Variation of buckling temperatures with hole size for (a) antisymmetric, (b) symmetric four edges clamped plate for various layer numbers for (60/60)2 configuration.
As shown in Figs. 11–14 as the number of layers namely, the thickness of plate increases, the critical buckling temperature increases. This behavior is same for all layer configurations used in this solution. As shown in Figs. 11–14, the critical buckling temperatures of antisymmetric plates are higher than that of symmetric plates.
Symmetric laminates, does not yield the highest buckling resistance as usually expected, because of effect of bending–extension coupling. As shown in Figs. 11–14, buckling temperatures for (15/15)2 configuration are the highest and buckling temperatures for (60/60)2 configuration are the lowest, comparing with other layer configurations.
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The first and second buckled mode shapes generated glass–epoxy/boron–epoxy angle-ply laminated six layered plates with four boundary conditions are shown in Figs. 15–18. It is found that critical temperature for
(15/15)2 configuration plate without hole for four edge clamped plate is 70.95 C, for two edge clamped is 172.88 C, for four edge simply supported plate is 33.23 C and for two edge simply supported plate is
Fig. 15. First buckling mode shapes for (15/15)2 configuration six layered laminate without hole for boundary conditions (a) four edge clamped (CCCC), (b) four edge simply supported (SSSS) plates.
Fig. 16. First buckling mode shapes for (15/15)2 configuration six layered laminate without hole for boundary conditions (a) two edge clamped (CC), (b) two edge simply supported (SS) plates.
Fig. 17. First buckling mode shapes for (30/30)2 configuration six layered laminate with hole for boundary conditions (a) four edge clamped (CCCC), (b) four edge simply supported (SSSS) plates.
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Fig. 18. First buckling mode shapes for (30/30)2 configuration six layered laminate with hole for boundary conditions (a) two edge clamped (CC), (b) two edge simply supported (SS) plates.
94.43 C, respectively. For (30/30)2 configuration plate with hole size of d/b = 0.3 for four edge clamped plate is 107.35 C, for two edge clamped is 123.74 C, for four edge simply supported plate is 28.89 C and for two edge simply supported plate is 29.2 C, respectively. The mode shapes presented in Figs. 15–18 show considerable skewing for the laminated plates.
4. Conclusions Thermal buckling behaviors of angle-ply laminated plates with holes have been examined by employing the first-order shear deformation theory and finite element technique. Both symmetric and antisymmetric lay-up sequence are considered and various boundary conditions are taken into account. Four edges clamped plates with hole have the highest buckling temperature and four edges simply supported plate have the lowest buckling temperature for all layer configurations. The critical buckling temperature is strongly depend on hole size for two edges and four edges clamped plates. For four edge clamped plates, as the hole size increases, the buckling temperature shows an increase. But, as the hole size increases, the buckling temperatures for two edge clamped plate decreases. The critical buckling temperature of four edge simply supported and simply supported plates do not affected so much by hole size and plates have this boundary conditions result the lowest buckling temperature. For this case, there is a little difference between symmetric and antisymmetric lay up. As number of layers namely the plate thickness increases, more temperature difference is needed for buckling. This behavior is same for both antisymmetric and symmetric lay up. But, higher temperature difference is needed for buckling of antisymmetrical stacked plate than the symmetric one.
For four layer configuration used in the solution, the highest buckling temperature difference is reached for (15/15)2 configuration and the lowest buckling temperature difference is reached for (60/60)2 configuration.
References [1] Gossard ML, Seide P, Roberts WM. Thermal buckling of plates. NACA TN 2771; 1952. [2] Liessa AW. A review of laminated composite plate buckling. Appl Mech Rec 1987;40:575–91. [3] Bednarczyk H, Rihter M. Buckling of plated due to selfequilibrated thermal stresses. J Thermal Stresses 1985;8: 139–52. [4] Chen W, Liu WH. Thermal buckling of antisymmetric angle-ply laminated plates – an analytical Levy-type solution. J Thermal Stresses 1993(16):401–19. [5] Sun LX, Hsu TR. Thermal buckling of laminated composite plates with transverse shear deformation. Comput Struct 1990;36(5):883–9. [6] Chockalingam S, Mathew TC, Singh G, Rao GV. Critical temperatures of hybrid laminates using finite elements. Comput Struct 1992;43(5):995–8. [7] Thangaratnam KR, Ramachandran J. Thermal buckling of composite laminated plates. Comput Struct 1989;32:1117–24. [8] 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. [9] Mannini A. Shear deformation effects on thermal buckling of cross-ply composite laminates. Comp Struct 1997;39:1–10. [10] 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. [11] Yettram A, Brawn CJ. The elastic stability of square perforated plates under biaxial loadings. Comput Struct 1986;22(4):589–94. [12] Chang CL, Hsu DS. Vibration and stability of plates with hole. Proc of the 10th Natl Conf on Theo and Appl Mechanics STAM ROC 1986:235–41. [13] VanDen Brink, DJ. Kamat MP. Post-buckling response of isotropic and laminated composites square plates with circular holes. In: Proc Int Conf Composite Material, San Diego, CA; 1985, p. 1393–1409. [14] Larsson PL. On buckling of orthotropic compressed plates with circular holes. Comp Struct 1987;7:103–21.
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¨ .S. S O ß ahin / Composites Science and Technology 65 (2005) 1780–1790
[15] Lin CH, Kuo CS. Buckling of laminated plates with holes. J Comp Mater 1989;23:536–53. [16] Bathe KJ. Finite element procedures in engineering analysis. Englewood Cliffs, NJ: Prentice-Hall; 1982.
[17] Whitney JM. Shear correction factors for orthotropic laminates under static load. J Appl Mech 1973(march):302–4. [18] Jhones RM. Mechanics of composite materials. Tokyo: McGrawHill Kogagusha Ltd; 1975.