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Thermal buckling load optimization of laminated composite plates
ARTICLE IN PRESS
Thin-Walled Structures 46 (2008) 667–675 www.elsevier.com/locate/tws
Thermal buckling load optimization of laminated composite plates U. Topal, U¨. Uzman Department of Civil Engineering, Karadeniz Technical University, 29000 Gu¨mu¨-shane, Turkey Received 15 November 2007; accepted 26 November 2007 Available online 4 January 2008
Abstract In this study, the applicability of the Modified Feasible Direction (MFD) method on the thermal buckling optimization of laminated plates subjected to uniformly distributed temperature load is investigated. The objective function is to maximize the critical temperature capacity of laminated plates and the fiber orientation is considered as design variable. The first-order shear deformation theory is used in the mathematical formulation. For this purpose, a program based on FORTRAN is used for the optimization of laminated plates. Finally, the effect of aspect ratio, antisymmetric lay-up, boundary condition, material anisotropy, ratio of coefficients of thermal expansion, and hybrid laminates on the results is investigated and the results are compared. r 2007 Elsevier Ltd. All rights reserved. Keywords: Laminated composite plates; Thermal buckling; Modified Feasible Direction method; Optimal designs
1. Introduction Advanced fiber reinforced composite materials used in various fields of engineering, have given a promise for better performance in comparison to their isotropic competitors. An increase in the utilization of these materials is due to their physical properties such as higher strength-to-weight ratio, stiffness-to-weight ratio and versatility. The buckling of fiber reinforced plates under thermal loadings is an important consideration in the design process. Structural optimization of laminated plates involving thermal buckling are found in some papers. Autio [1] optimized the behavior of a laminated plate with given boundary temperatures and displacement constraints. The optimization problem was expressed in terms of lamination parameters. Fares et al. [2] presented a multiobjective optimization problem to determine the optimal layer thickness and optimal closed loop control function for a symmetric cross-ply laminate subjected to thermomechanical loadings. The optimization procedure aimed to maximize the critical combination of the applied edges Corresponding author. Tel.: +90 462 377 4017; fax: +90 462 377 2606.
E-mail address: [email protected] (U. Topal). 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.11.005
load and temperature levels and to minimize the laminate dynamic response subject to constraints on the thickness and control energy. Spallino and Thierauf [3] investigated thermal buckling optimization of laminated composite plates subject to thermal loading. The optimal design problem was solved using evolution strategies under strain and ply contiguity constraints. de Faria and de Almeida [4] addressed buckling optimization of composite plates subjected to uncertain thermal and nonuniform mechanical loadings. The loading configuration was assumed to be described by piecewise linear functions along the plate edges. An optimal design of antisymmetric laminates under thermal loads was given by Adali and Duffy [5] for the nonhybrid and hybrid cases. The optimum results were given for simply supported laminates with graphite, boron and glass layers under a uniform temperature change. Walker et al. [6] investigated the optimal designs of laminated plates subject to nonuniform temperature distributions for maximum buckling temperature. Three different temperature loadings were considered and various combinations of simply supported and clamped boundary conditions were studied. Golden section method was used as optimization routine. Lee et al. [7] presented the design of a thick laminated composite plate subjected to a thermal buckling load under a uniform temperature distribution. In
ARTICLE IN PRESS U. Topal, U. Uzman / Thin-Walled Structures 46 (2008) 667–675
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On the other hand, there is no any research in the literature on the thermal buckling load design of laminates using Modified Feasible Direction (MFD) method. Therefore, the main objective of this paper is to demonstrate the effectiveness of the MFD on the thermal buckling optimization of laminated plates subjected to uniformly distributed temperature load. The objective function is to maximize the critical temperature capacity of laminated plates and the fiber orientation is considered as design variable. The first-order shear deformation theory is used in the mathematical formulation. For this purpose, a program based on FORTRAN is used for optimization of laminated plates. Finally, the effect of aspect ratio, antisymmetric lay-up, boundary condition, material anisotropy, ratio of coefficients of thermal expansion and hybrid laminates on the results is investigated and the results are compared.
design procedures of composite laminated plates for a maximum thermal buckling load, golden section method was used as an optimization routine. Singha et al. [8] maximized buckling temperatures of graphite/epoxy laminated composite plates for a given total thickness. The fiber orientations and thicknesses of layers were adopted as design variables. Thermal buckling analysis was carried out using the finite element method with four node shear deformable plate element. Genetic algorithm (GA) was employed to optimize as many as 10 variables for the five layered plates. Chen et al. [9] investigated design optimization for structural thermal buckling. The analysis of heat conduction, structural stress and buckling were considered at the same time in the design optimization procedure. The coupling sensitivity effects of heat conduction on the structural thermal buckling were studied. In the sensitivity analysis, the semianalytical method was employed. The optimization model was constructed and solved by the sequential linear programming or sequential quadratic programming algorithm. Fares et al. [10] presented design and control optimization to minimize the thermal postbuckling dynamic response and to maximize the buckling temperature level of composite laminated plates subjected to thermal distribution varying linearly through the thickness and arbitrarily with respect to the in-plane coordinates. The thickness of layers and the fiber orientation angles were taken as optimization design variables. Numerical examples were presented for angle-ply antisymmetric laminates with simply supported edges. Mohan and Arvind [11] explored the metaheuristic approach called scatter search for lay-up sequence optimization of laminated composite panels. The main objective of this study was to show the ability of the proposed scatter search algorithm for the combinatorial problem like stacking sequence optimization of laminated composite panels.
2. Basic equations Consider a laminated composite plate of uniform thickness h, having a rectangular plan a b as shown in Fig. 1. The individual layers are assumed to be homogeneous and orthotropic. Perfect bonding is assumed between the layers of laminated composite plates. The displacement field based on the first-order shear deformation theory [12] takes the form uðx; y; zÞ ¼ uo ðx; yÞ þ zcx ðx; yÞ vðx; y; zÞ ¼ vo ðx; yÞ þ zcy ðx; yÞ
(1)
wðx; y; zÞ ¼ wðx; yÞ where uo and vo are the displacements of u and v on the mid-plane. Because the vertical line to the mid-plane of the laminates is not necessarily perpendicular to the deformed mid-plane, the terms cx and cy are independent of qw/qx and qw/qy.
2
1
y z
θ
y
x z b
h/2 x
h/2
a Fig. 1. Geometry and coordinate systems of laminated composite plate.
x
ARTICLE IN PRESS U. Topal, U. Uzman / Thin-Walled Structures 46 (2008) 667–675
The displacement–strain relations, taking Eq. (1) into account are quo qcx 8 9 qx qx x > > < = qc qv y o y ¼ z ; qy qy > :g > ; xy quo qvo qcx qcy þ þ qy qy qx qx
qw gyz qy cy . ¼ gxz qw qx cx
principle. Bending stiffness [Kb], shear stiffness [Ks] and geometric stiffness [Kg] can be calculated as below R ½K b ¼ A ½Bb T ½Db ½Bb dA R ½K s ¼ A ½Bs T ½Ds ½Bs dA (8) R T ½K g ¼ A ½Bg ½Dg ½Bg dA where "
(2) The constitutive relations for a laminated plate accounting for thermal effects can be written as 0 1 0¯ ¯ 12 Q ¯ 16 1 0 x ax DT 1 sx Q11 Q Bs C B¯ B C ¯ 22 Q ¯ 26 C Q @ y A ¼ @Q A @ y ay DT A, (3) 12 ¯ 16 Q ¯ 26 Q ¯ 66 xy axy DT sxy Q tyz txz
! ¼ ðkÞ
ðkÞ
¯ 44 Q ¯ 45 Q
¯ 45 Q ¯ 55 Q
! ðkÞ
gyz gxz
! ,
(4)
¯ ij is the transformed reduced stiffness, ax, ay, axy where Q are the coefficients of thermal expansion and DT is the uniform constant temperature difference. The stress resultants {N}, stress couples {M} and transverse shear stress resultants {Q} are 9 9 9 9 8 8 8 8 Mx > Nx > sx > > > > sx > > > > > > > > Z h=2 > = = = Z h=2 > = < < < < Ny sy dz; My sy zdz ¼ ¼ > > > > > > h=2 > h=2 > > > > > ; ; ; ; :N > :t > :M > :t > xy xy xy xy ( ) Z h=2 ( txz ) Qx ¼ dz, ð5Þ Qy h=2 tyz
0
0
Dij
½Db ¼ " ½Dg ¼
2
#
¯1 N
¯ 12 N
¯ 12 N
¯2 N
;
½Ds ¼ 4
k21 A44
A45
A45
k22 A55
#
3 5,
ð9Þ
Aij and Dij can be calculated as follows R h=2 ¯ ij ð1; z2 Þdz ði; j ¼ 1; 2; 6Þ; ðAij ; Dij Þ ¼ h=2 Q R h=2 ¯ ij dz Aij ¼ h=2 Q ði; j ¼ 4; 5Þ:
(10)
In Eq. (9), k21 and k22 are the shear correction factors and, in this study the shear correction factor is assumed 5/6. The discrete eigenvalue equation of the static buckling problem of laminates can be derived as ð½K b þ K s l½K og Þfug ¼ 0.
(11)
The product of l and the initial guest value DT is the critical buckling temperature Tcr, that is T cr ¼ lDT.
(12)
In this study, subspace iteration technique is applied to obtain numerical solutions of the problem. 3. Modified Feasible Direction method
The total potential energy, p of the plates is p ¼ Ub þ Us þ V,
Aij
(6)
where Ub is the strain energy of bending, Us is the strain energy of shear and V is the potential energy due to external loads. 2.1. Finite element formulation In this study, nine-node Lagrangian rectangular plate element with five degrees of freedom is used for the finite element solution of the laminates. The interpolation function of the displacement field is defined as 0 1 u B v C B C X n B C B w C¼ Nidi (7) B C Bc C i¼1 @ xA cy where di and Ni are the nodal variables and the interpolation function, respectively. The stiffness matrix of the plate is obtained by using the minimum potential energy
The MFD method is one of the most powerful methods for optimization problems. This method takes into account not only the gradients of objective function and constraints, but also the search direction in the former iteration. In this study, there is no any constraint. Fig. 2 shows the iterative process within each optimization process [13,14]. MFD 1. q = 0, Xq = Xm 2. q = q+1 Evaluate objective function F(Xq-1) Δ
ðkÞ
669
3. Calculate gradient of the objective function F(Xq-1) 4. Find the usable-feasible direction Sq 5. Perform a one-dimensional search Xq = Xq-1 + αSq 6. Check convergence. If satisfied, go to 7. Otherwise go to 2 7. Xm = Xq Fig. 2. Flow chart of Modified Feasible Direction method.
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The objective function F(Xi) is accurately modeled as a quadratic polynomial approximation around the current iterate Xi as in Eq. (13). F ðX i Þ ¼ ao þ
Nd X
ai X i þ
i¼1
Nd X
bi X 2i
The number of design sets so far exceeds the maximum number of optimization loops. If the initial design is infeasible and the allowed number of consecutive infeasible designs has been exceeded.
The optimization problem is terminated if all of the following conditions are satisfied:
Changes in the objective function F: (a) The difference between the current value and the best design so far is less than the tolerance tF. jF current F best jptF .
(13)
i¼1
where Nd and Xi are number of design variables and ith design variable, respectively. ai and bi are the coefficients of polynomial function determined by a least squares regression. After the objective function is approximated, their gradients with respect to the design variables are calculated by finite differences methods. The solving process is iterated until convergence is achieved. Convergence or termination checks are performed at the end of each optimization loop. The optimization process continues until either convergence or termination occurs. The process may be terminated before convergence in two cases:
The current design is feasible,
Table 1 Convergence study of the present study for a clamped square laminated plate with literature results for Material 1
(b) The difference between the current value and the previous design is less than the tolerance, jF current F current1 jptF .
Changes in the design variables Xi: (a) The difference between the current value of each design variable and the best design so far is less than the respective tolerance ti. jX icurrent X ibest jpti . (b) The difference between the current value of each design variable and the previous design is less than the respective tolerance, jX icurrent X icurrent1 jptF .
We resolved the optimization process to obtain global maximum from different initial points to check if other solutions are possible. The converge tolerance ratio is considered 0.01 for objective function. The optimization problem is the maximization of the temperature buckling load by changing the ply orientations in the stacking sequence of the laminated plate. The optimal design problem can be stated as follows:
Critical temperature
Ref. [15]
Ref. [16]
Present study
find : y ðT cr Þmax ¼ max T cr ðyÞ
Tcr (1C)
129.91
131.55
130.04
0 pyk p90 :
(14)
y
0.35
Thermal critical temperature
0.3 0.25 Material 2 0.2 0.15 0.1 0.05 0 1
2
3
4
5 6 Mode Number
7
8
9
10
Fig. 3. Thermal critical temperature for a clamped laminated composite plate (a/h ¼ 20) for mode numbers for Material 2.
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4. Numerical results and discussion In this study, four layered, symmetric, clamped angle-ply laminated plate subjected to a uniform temperature load is investigated. Each of the lamina is assumed to be of same thickness. The fiber orientations are given for number of layers as follows yk ¼ ð1Þkþ1 y for kp
N þ 1 and 2
N þ 1 k ¼ 1; . . . ; N, 2 where N is the number of layers. yk ¼ ð1Þk y for kX
ð15Þ
4.1. Convergence of the present study To validate the present study, the obtained critical buckling temperatures are compared with literature results. The laminated plate is modeled using 6 6 mesh. The material properties of the lamina are as follows:
(Material 1) E1 ¼ 76 GPa, E2 ¼ 5.5 GPa, G12 ¼ G13 ¼ 2.30 GPa, G23 ¼ 1.5 GPa, n12 ¼ 0.34, a1 ¼ 4 106 1C1, a2 ¼ 79 106 1C1. (Material 2) E1 ¼ 181 GPa, E2 ¼ 10.3 GPa, G12 ¼ G13 ¼ 7.17 GPa, G23 ¼ 2.39 GPa, n12 ¼ 0.28, a1 ¼ 0.02 106 1C1, a2 ¼ 22.5 106 1C1. (Material 3) E1 ¼ E2 ¼ 70 GPa, G12 ¼ G13 ¼ G23 ¼ 52 GPa, n12 ¼ 0.33, a1 ¼ 23.6 106 1C1, a2 ¼ 23.6 106 1C1.
Table 2 Convergence of the optimum fiber orientations for different boundary conditions with literature results for Material 2 Simply supported
Clamped
Ref. [6]
Ref. [8]
Present study
Ref. [6]
Ref. [8]
Present study
45.1
45.0
45.0
54.3
52.9
54.0
671
(Material 4) E1/E2 ¼ 25, G12/E2 ¼ G13/E2 ¼ 0.5, G23/E2 ¼ 0.2, n12 ¼ 0.25, a1 ¼ 0.02 106 1C1, a2 ¼ 22.5 106 1C1.
Example 1. A single thin (a/h ¼ 40) clamped laminated square plate for Material 1 with fiber orientation angle 451 is considered first to compare the present study with literature results. As seen from Table 1, the results obtained for critical buckling temperature are in very close agreement with literature results [15,16]. Example 2. The next example is comprised of the same plate geometry and the same boundary conditions, with four layered (01/901/01/901) lamination sequence for Material 2 (a/h=20). The results are obtained for 10 modes (Fig. 3). It can be seen that, the present study is in very good agreement with the literature results (see Ref. [16]). 4.2. Optimization problem The convergence of the present optimization study is compared with the literature results [6,8] for Material 2. Therefore, the optimum fiber orientations are obtained for maximum thermal critical temperature with simply supported and clamped angle-ply laminated plates (N=4). It may be seen from Table 2 that, the results compare well. 4.2.1. Effect of plate aspect ratios on the optimal design Fig. 4 shows the influence of a/b ratios on the optimal results for clamped laminated plates under uniform temperature load (N ¼ 4, h ¼ 0.01 m, Material 2). It can be noted from this figure that, the critical buckling resistance is higher for square laminated plate as the plate becomes stiffer. In Table 3, optimum fiber orientations are given for a/b ratios. The buckled mode shapes under uniform temperature load are shown in Fig. 5 for a/b ratios for Material 2.
170
Critical temperature
165 160 155 Material 2
150 145 140
1
1.25
1.5 a/b
1.75
Fig. 4. Effect of plate aspect ratio on the optimal designs for laminated plates for Material 2.
2
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4.2.2. Effect of antisymmetric lay-up on the optimal design The effect of antisymmetric lay-up (y/y/y/y) on the optimal results is investigated for a/h ¼ 10, a/h ¼ 20, a/h ¼ 50 and a/h ¼ 100 for four layered square plates for
Table 3 Effect of the plate aspect ratio on the optimum fiber orientations for clamped laminated plates for Material 2 a/b
yopt (1)
1.00 1.25 1.50 1.75 2.00
54.0 54.5 51.7 55.2 53.8
Material 2. For a given antisymmetric lay-up, the optimum fiber orientations are obtained at 451. Comparative results are given with symmetric lay-up in Fig. 6. As seen, because of the absence of bending–extension coupling, symmetric lay-up does not yield the highest thermal buckling load resistance as usually expected. The differences for critical temperatures are 14.27%, 11.29%, 10.09% and 9.92% for b/h ¼ 10, b/h ¼ 20, b/h ¼ 50 and b/h ¼ 100 between antisymmetric and symmetric lay-up, respectively. As seen, the differences between antisymmetric and symmetric lay-up decrease as a/h ratio increases.
4.2.3. Effect of boundary conditions on the optimal design The iteration procedure may be applied to laminated plates with any combinations of simple support (S),
Fig. 5. Buckled mode shapes of laminated plates for a/b ratios (Material 2).
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clamped support (C) and free edge (F). In this study, four different combinations of the boundary conditions as (SSSS), (CSCS), (CCCC) and (CFCF) are considered as defined below.
2. Two edges simply supported and two edges clamped (CSCS) a a At x ¼ ; ; 2 2 b b At x ¼ ; ; 2 2
1. Four edges simply supported (SSSS) a a At x ¼ ; ; 2 2 b b At x ¼ ; ; 2 2
u ¼ w ¼ cy ¼ 0
u ¼ w ¼ cy ¼ cx ¼ 0 u ¼ w ¼ cx ¼ 0.
3. Four edges clamped (CCCC)
u ¼ w ¼ cx ¼ 0.
a a At x ¼ ; ; 2 2 b b At x ¼ ; ; 2 2
12000 Critical temperature
673
Symmetric lay-up Antisymmetric lay-up
u ¼ w ¼ cy ¼ cx ¼ 0 u ¼ w ¼ cx ¼ cy ¼ 0.
4. Two edges clamped and two edges free (CFCF)
8000
4000
0
a a At x ¼ ; ; 2 2
Material 2
0
20
60
40
80
100
a/h
Fig. 6. Effect of antisymmetric lay-up on the optimal designs for square laminated plates for Material 2.
Table 4 Effect of boundary conditions on the optimal designs for square laminated plates Boundary conditions
yopt (1)
Tcr (1C)
(SSSS) (CCCC) (CSCS) (CFCF)
45.0 54.0 36.6 38.1
1.829603 102 1.674376 102 2.550149 102 3.145864 102
u ¼ w ¼ cy ¼ cx ¼ 0.
Table 4 shows the effect of different boundary conditions on the optimal designs for symmetric square laminated plates (N ¼ 4, h ¼ 0.01 m, Material 2). As seen, the maximum thermal critical buckling load occurs at (CFCF) boundary condition, whereas the minimum thermal critical buckling load occurs at (CCCC) boundary condition. The results are contrast with the critical buckling loads under the mechanical loads. This can be explained that the free edges provide more degrees of freedom and allow the laminated plate to buckle higher temperatures. 4.2.4. Effect of material anisotropy on the optimal design The effect of material anisotropy ratio, E1/E2, on the optimum design is studied for four layered square plates for Material 4 (Fig. 7). It can be seen that, the critical temperature increases with increase of E1/E2 ratio. This is due to the increase in stiffness of laminates with increase in
500
Critical temperature
400 300 Material 4 200 100 0
0
10
30
20
40
50
E1/E2 Fig. 7. Effect of E1/E2 ratio on the optimal designs for square laminated plates for Material 4.
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674
E1. On the other hand, the differences for critical temperature decrease as E1/E2 ratio increases. For example, the differences are 28.04%, 21.41% and 17.25% for E1/E2 ¼ 20–30, E1/E2 ¼ 30–40 and E1/E2 ¼ 40–50, respectively. In Table 5, the effect of E1/E2 ratio on the optimum fiber orientations are given. As seen, as E1/E2 ratio increases, the optimum fiber orientation decreases but this decrease diminishes for larger E1/E2 ratios.
clamped square plates (N ¼ 6, h ¼ 0.01 m). The letters K, G and A represent Material 1, Material 2 and Material 3, respectively. For this purpose, six different combinations such as (KGK)s, (KAK)s, (GAG)s, (KAG)s, (AKG)s and (AKG)s are considered. In Table 7, optimum fiber orientations and the critical temperatures are given for hybrid laminates. As seen, the highest critical temperature is reached for (KGK)s composites, while the lowest is (KAK)s composites.
4.2.5. Effect of thermal expansion ratio on the optimal designs The effect of thermal expansion ratio, a2/a1, on the optimum design is studied for four layered square plates for Material 4 (Fig. 8). As seen, the critical temperature increases with increase in the values of a2/a1 ratio. But this increase diminishes as a2/a1 ratio increases. For example, the differences for critical temperature are 15.24%, 9.13% and 6.34% for a2/a1 ¼ 20–30, a2/a1 ¼ 30–40 and a2/a1 ¼ 40–50, respectively. The optimum fiber orientations are given in Table 6 for a2/a1 ratios. As seen, the optimum fiber orientations decrease gradually with increase of a2/a1 ratio.
5. Conclusions
4.2.6. Effect of hybrid laminates on the optimal design In this study, the effect of hybrid laminates on the optimal design is investigated for symmetrically laminated Table 5 Effect of E1/E2 ratio on the optimum fiber orientations for clamped laminated plates for Material 4
This paper presents optimal design of laminated plates subjected to uniformly distributed temperature load. The objective function is to maximize the critical temperature
Table 6 Effect of a2/a1 ratio on the optimum fiber orientations for clamped laminated plates for Material 4 a2/a1
yopt (1)
10 20 30 40 50
66.7 62.7 61.2 60.4 57.5
Table 7 Effect of hybrid laminates on the optimum results Tcr (1C)
yopt (1)
Laminate configuration
yopt (1)
E1/E2 5 10 20 30 40 50
60.1 55.9 52.7 51.4 50.6 50.2
(KGK)s (KAK)s (GAG)s (KAG)s (AKG)s (AKG)s
45.0 90.0 80.1 90.0 57.0 77.4
110.12 2.23 3.55 2.31 3.58 3.40
180
Critical temperature
160 140 Material 4
120 100 80
10
20
30
40
α2/α1 Fig. 8. Effect of a2/a1 ratio on the optimal designs for square laminated plates for Material 4.
50
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capacity of laminated plates and the fiber orientation is considered as design variable. Numerical experiments have been conducted to evaluate the MFD algorithm for laminated composite optimization problems. Initially, preliminary studies have been carried out by solving problems given in the literature for thermal buckling maximization and the results have been compared with the best known results available in the literature. The numerical results clearly demonstrate the effectiveness of MFD algorithm on the thermal buckling optimization for laminates. This methodology provides engineers with a useful tool for designing laminated composite structures. On the other hand, the results presented herein confirm that the characteristics of thermal buckling are significantly influenced by aspect ratio, antisymmetric lay-up, boundary condition, material anisotropy, ratio of coefficients of thermal expansion and hybrid laminates.
[5] [6]
[7]
[8]
[9]
[10]
[11]
References [12] [1] Autio M. Optimization of coupled thermal-structural problems of laminated plates with lamination parameters. Struct Multidiscip Optim 2001;21(1):40–51. [2] Fares ME, Youssif YG, Hafiz MA. Structural and control optimization for maximum thermal buckling and minimum dynamic response of composite laminated plates. Int J Solids Struct 2004;41(3–4): 1005–19. [3] Spallino R, Thierauf G. Thermal buckling optimization of composite laminates by evolution strategies. Comput Struct 2000;78: 691–7. [4] de Faria AR, de Almeida SFM. Buckling optimization of composite plates subjected to uncertain thermal and nonuniform mechanical
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loadings. Inverse problems, design and optimization symposium Rio de Janeiro, Brazil, 2004. Adali S, Duffy KJ. Optimal design of antisymmetric hybrid laminates against thermal buckling. J Therm Stress 1990;13:57–71. Walker M, Reiss T, Adali S, Verijenko VE. Optimal design of symmetrically laminated plates for maximum buckling temperature. J Therm Stress 1997;20:21–33. Lee YS, Lee YW, Yang MS, Park BS. Optimal design of thick laminated composite plates for maximum thermal buckling load. J Therm Stress 1999;22:259–73. Singha MK, Ramachandra LS, Bandyopadhyay JN. Optimum design of laminated composite plates for maximum thermal buckling loads. J Compos Mater 2000;34(23):1982–97. Chen B, Gu Y, Zhao G, Lin W, Chang TYP, Kuang JS. Design optimization for structural thermal buckling. J Therm Stress 2003; 26(5):479–94. Fares ME, Youssif YG, Hafiz MA. Multiobjective design and control optimization for minimum thermal postbuckling dynamic response and maximum buckling temperature of composite laminates. Struct Multidiscip Optim 2005;30(2):89–100. Mohan RAR, Arvind N. A scatter search algorithm for stacking sequence optimisation of laminate composites. Compos Struct 2005; 70(4):383–402. Ochoa OO, Reddy JN. Finite element analysis of composite laminates. Dordrecht: Kluwer Academic Publishers; 1992. Topal U, Uzman U¨. Optimum design of laminated composite plates to maximize fundamental frequency using MFD method. Struct Eng Mech 2006;24(4):479–91. Topal U, Uzman U¨. Optimum design of laminated composite plates to maximize buckling load using MFD method. Thin-Wall Struct 2007;45(7–8):660–9. Huang NN, Tauchert TR. Thermal buckling of clamped symmetric laminated plates. Thin-Wall Struct 1992;13:259–73. Kabir HRH, Askar H, Chaudhuri RA. Thermal buckling response of shear flexible laminated anisotropic plates using a three-node isoparametric element. Compos Struct 2003;59:173–87.
Thin-Walled Structures 46 (2008) 667–675 www.elsevier.com/locate/tws
Thermal buckling load optimization of laminated composite plates U. Topal, U¨. Uzman Department of Civil Engineering, Karadeniz Technical University, 29000 Gu¨mu¨-shane, Turkey Received 15 November 2007; accepted 26 November 2007 Available online 4 January 2008
Abstract In this study, the applicability of the Modified Feasible Direction (MFD) method on the thermal buckling optimization of laminated plates subjected to uniformly distributed temperature load is investigated. The objective function is to maximize the critical temperature capacity of laminated plates and the fiber orientation is considered as design variable. The first-order shear deformation theory is used in the mathematical formulation. For this purpose, a program based on FORTRAN is used for the optimization of laminated plates. Finally, the effect of aspect ratio, antisymmetric lay-up, boundary condition, material anisotropy, ratio of coefficients of thermal expansion, and hybrid laminates on the results is investigated and the results are compared. r 2007 Elsevier Ltd. All rights reserved. Keywords: Laminated composite plates; Thermal buckling; Modified Feasible Direction method; Optimal designs
1. Introduction Advanced fiber reinforced composite materials used in various fields of engineering, have given a promise for better performance in comparison to their isotropic competitors. An increase in the utilization of these materials is due to their physical properties such as higher strength-to-weight ratio, stiffness-to-weight ratio and versatility. The buckling of fiber reinforced plates under thermal loadings is an important consideration in the design process. Structural optimization of laminated plates involving thermal buckling are found in some papers. Autio [1] optimized the behavior of a laminated plate with given boundary temperatures and displacement constraints. The optimization problem was expressed in terms of lamination parameters. Fares et al. [2] presented a multiobjective optimization problem to determine the optimal layer thickness and optimal closed loop control function for a symmetric cross-ply laminate subjected to thermomechanical loadings. The optimization procedure aimed to maximize the critical combination of the applied edges Corresponding author. Tel.: +90 462 377 4017; fax: +90 462 377 2606.
E-mail address: [email protected] (U. Topal). 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.11.005
load and temperature levels and to minimize the laminate dynamic response subject to constraints on the thickness and control energy. Spallino and Thierauf [3] investigated thermal buckling optimization of laminated composite plates subject to thermal loading. The optimal design problem was solved using evolution strategies under strain and ply contiguity constraints. de Faria and de Almeida [4] addressed buckling optimization of composite plates subjected to uncertain thermal and nonuniform mechanical loadings. The loading configuration was assumed to be described by piecewise linear functions along the plate edges. An optimal design of antisymmetric laminates under thermal loads was given by Adali and Duffy [5] for the nonhybrid and hybrid cases. The optimum results were given for simply supported laminates with graphite, boron and glass layers under a uniform temperature change. Walker et al. [6] investigated the optimal designs of laminated plates subject to nonuniform temperature distributions for maximum buckling temperature. Three different temperature loadings were considered and various combinations of simply supported and clamped boundary conditions were studied. Golden section method was used as optimization routine. Lee et al. [7] presented the design of a thick laminated composite plate subjected to a thermal buckling load under a uniform temperature distribution. In
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On the other hand, there is no any research in the literature on the thermal buckling load design of laminates using Modified Feasible Direction (MFD) method. Therefore, the main objective of this paper is to demonstrate the effectiveness of the MFD on the thermal buckling optimization of laminated plates subjected to uniformly distributed temperature load. The objective function is to maximize the critical temperature capacity of laminated plates and the fiber orientation is considered as design variable. The first-order shear deformation theory is used in the mathematical formulation. For this purpose, a program based on FORTRAN is used for optimization of laminated plates. Finally, the effect of aspect ratio, antisymmetric lay-up, boundary condition, material anisotropy, ratio of coefficients of thermal expansion and hybrid laminates on the results is investigated and the results are compared.
design procedures of composite laminated plates for a maximum thermal buckling load, golden section method was used as an optimization routine. Singha et al. [8] maximized buckling temperatures of graphite/epoxy laminated composite plates for a given total thickness. The fiber orientations and thicknesses of layers were adopted as design variables. Thermal buckling analysis was carried out using the finite element method with four node shear deformable plate element. Genetic algorithm (GA) was employed to optimize as many as 10 variables for the five layered plates. Chen et al. [9] investigated design optimization for structural thermal buckling. The analysis of heat conduction, structural stress and buckling were considered at the same time in the design optimization procedure. The coupling sensitivity effects of heat conduction on the structural thermal buckling were studied. In the sensitivity analysis, the semianalytical method was employed. The optimization model was constructed and solved by the sequential linear programming or sequential quadratic programming algorithm. Fares et al. [10] presented design and control optimization to minimize the thermal postbuckling dynamic response and to maximize the buckling temperature level of composite laminated plates subjected to thermal distribution varying linearly through the thickness and arbitrarily with respect to the in-plane coordinates. The thickness of layers and the fiber orientation angles were taken as optimization design variables. Numerical examples were presented for angle-ply antisymmetric laminates with simply supported edges. Mohan and Arvind [11] explored the metaheuristic approach called scatter search for lay-up sequence optimization of laminated composite panels. The main objective of this study was to show the ability of the proposed scatter search algorithm for the combinatorial problem like stacking sequence optimization of laminated composite panels.
2. Basic equations Consider a laminated composite plate of uniform thickness h, having a rectangular plan a b as shown in Fig. 1. The individual layers are assumed to be homogeneous and orthotropic. Perfect bonding is assumed between the layers of laminated composite plates. The displacement field based on the first-order shear deformation theory [12] takes the form uðx; y; zÞ ¼ uo ðx; yÞ þ zcx ðx; yÞ vðx; y; zÞ ¼ vo ðx; yÞ þ zcy ðx; yÞ
(1)
wðx; y; zÞ ¼ wðx; yÞ where uo and vo are the displacements of u and v on the mid-plane. Because the vertical line to the mid-plane of the laminates is not necessarily perpendicular to the deformed mid-plane, the terms cx and cy are independent of qw/qx and qw/qy.
2
1
y z
θ
y
x z b
h/2 x
h/2
a Fig. 1. Geometry and coordinate systems of laminated composite plate.
x
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The displacement–strain relations, taking Eq. (1) into account are quo qcx 8 9 qx qx x > > < = qc qv y o y ¼ z ; qy qy > :g > ; xy quo qvo qcx qcy þ þ qy qy qx qx
qw gyz qy cy . ¼ gxz qw qx cx
principle. Bending stiffness [Kb], shear stiffness [Ks] and geometric stiffness [Kg] can be calculated as below R ½K b ¼ A ½Bb T ½Db ½Bb dA R ½K s ¼ A ½Bs T ½Ds ½Bs dA (8) R T ½K g ¼ A ½Bg ½Dg ½Bg dA where "
(2) The constitutive relations for a laminated plate accounting for thermal effects can be written as 0 1 0¯ ¯ 12 Q ¯ 16 1 0 x ax DT 1 sx Q11 Q Bs C B¯ B C ¯ 22 Q ¯ 26 C Q @ y A ¼ @Q A @ y ay DT A, (3) 12 ¯ 16 Q ¯ 26 Q ¯ 66 xy axy DT sxy Q tyz txz
! ¼ ðkÞ
ðkÞ
¯ 44 Q ¯ 45 Q
¯ 45 Q ¯ 55 Q
! ðkÞ
gyz gxz
! ,
(4)
¯ ij is the transformed reduced stiffness, ax, ay, axy where Q are the coefficients of thermal expansion and DT is the uniform constant temperature difference. The stress resultants {N}, stress couples {M} and transverse shear stress resultants {Q} are 9 9 9 9 8 8 8 8 Mx > Nx > sx > > > > sx > > > > > > > > Z h=2 > = = = Z h=2 > = < < < < Ny sy dz; My sy zdz ¼ ¼ > > > > > > h=2 > h=2 > > > > > ; ; ; ; :N > :t > :M > :t > xy xy xy xy ( ) Z h=2 ( txz ) Qx ¼ dz, ð5Þ Qy h=2 tyz
0
0
Dij
½Db ¼ " ½Dg ¼
2
#
¯1 N
¯ 12 N
¯ 12 N
¯2 N
;
½Ds ¼ 4
k21 A44
A45
A45
k22 A55
#
3 5,
ð9Þ
Aij and Dij can be calculated as follows R h=2 ¯ ij ð1; z2 Þdz ði; j ¼ 1; 2; 6Þ; ðAij ; Dij Þ ¼ h=2 Q R h=2 ¯ ij dz Aij ¼ h=2 Q ði; j ¼ 4; 5Þ:
(10)
In Eq. (9), k21 and k22 are the shear correction factors and, in this study the shear correction factor is assumed 5/6. The discrete eigenvalue equation of the static buckling problem of laminates can be derived as ð½K b þ K s l½K og Þfug ¼ 0.
(11)
The product of l and the initial guest value DT is the critical buckling temperature Tcr, that is T cr ¼ lDT.
(12)
In this study, subspace iteration technique is applied to obtain numerical solutions of the problem. 3. Modified Feasible Direction method
The total potential energy, p of the plates is p ¼ Ub þ Us þ V,
Aij
(6)
where Ub is the strain energy of bending, Us is the strain energy of shear and V is the potential energy due to external loads. 2.1. Finite element formulation In this study, nine-node Lagrangian rectangular plate element with five degrees of freedom is used for the finite element solution of the laminates. The interpolation function of the displacement field is defined as 0 1 u B v C B C X n B C B w C¼ Nidi (7) B C Bc C i¼1 @ xA cy where di and Ni are the nodal variables and the interpolation function, respectively. The stiffness matrix of the plate is obtained by using the minimum potential energy
The MFD method is one of the most powerful methods for optimization problems. This method takes into account not only the gradients of objective function and constraints, but also the search direction in the former iteration. In this study, there is no any constraint. Fig. 2 shows the iterative process within each optimization process [13,14]. MFD 1. q = 0, Xq = Xm 2. q = q+1 Evaluate objective function F(Xq-1) Δ
ðkÞ
669
3. Calculate gradient of the objective function F(Xq-1) 4. Find the usable-feasible direction Sq 5. Perform a one-dimensional search Xq = Xq-1 + αSq 6. Check convergence. If satisfied, go to 7. Otherwise go to 2 7. Xm = Xq Fig. 2. Flow chart of Modified Feasible Direction method.
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The objective function F(Xi) is accurately modeled as a quadratic polynomial approximation around the current iterate Xi as in Eq. (13). F ðX i Þ ¼ ao þ
Nd X
ai X i þ
i¼1
Nd X
bi X 2i
The number of design sets so far exceeds the maximum number of optimization loops. If the initial design is infeasible and the allowed number of consecutive infeasible designs has been exceeded.
The optimization problem is terminated if all of the following conditions are satisfied:
Changes in the objective function F: (a) The difference between the current value and the best design so far is less than the tolerance tF. jF current F best jptF .
(13)
i¼1
where Nd and Xi are number of design variables and ith design variable, respectively. ai and bi are the coefficients of polynomial function determined by a least squares regression. After the objective function is approximated, their gradients with respect to the design variables are calculated by finite differences methods. The solving process is iterated until convergence is achieved. Convergence or termination checks are performed at the end of each optimization loop. The optimization process continues until either convergence or termination occurs. The process may be terminated before convergence in two cases:
The current design is feasible,
Table 1 Convergence study of the present study for a clamped square laminated plate with literature results for Material 1
(b) The difference between the current value and the previous design is less than the tolerance, jF current F current1 jptF .
Changes in the design variables Xi: (a) The difference between the current value of each design variable and the best design so far is less than the respective tolerance ti. jX icurrent X ibest jpti . (b) The difference between the current value of each design variable and the previous design is less than the respective tolerance, jX icurrent X icurrent1 jptF .
We resolved the optimization process to obtain global maximum from different initial points to check if other solutions are possible. The converge tolerance ratio is considered 0.01 for objective function. The optimization problem is the maximization of the temperature buckling load by changing the ply orientations in the stacking sequence of the laminated plate. The optimal design problem can be stated as follows:
Critical temperature
Ref. [15]
Ref. [16]
Present study
find : y ðT cr Þmax ¼ max T cr ðyÞ
Tcr (1C)
129.91
131.55
130.04
0 pyk p90 :
(14)
y
0.35
Thermal critical temperature
0.3 0.25 Material 2 0.2 0.15 0.1 0.05 0 1
2
3
4
5 6 Mode Number
7
8
9
10
Fig. 3. Thermal critical temperature for a clamped laminated composite plate (a/h ¼ 20) for mode numbers for Material 2.
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4. Numerical results and discussion In this study, four layered, symmetric, clamped angle-ply laminated plate subjected to a uniform temperature load is investigated. Each of the lamina is assumed to be of same thickness. The fiber orientations are given for number of layers as follows yk ¼ ð1Þkþ1 y for kp
N þ 1 and 2
N þ 1 k ¼ 1; . . . ; N, 2 where N is the number of layers. yk ¼ ð1Þk y for kX
ð15Þ
4.1. Convergence of the present study To validate the present study, the obtained critical buckling temperatures are compared with literature results. The laminated plate is modeled using 6 6 mesh. The material properties of the lamina are as follows:
(Material 1) E1 ¼ 76 GPa, E2 ¼ 5.5 GPa, G12 ¼ G13 ¼ 2.30 GPa, G23 ¼ 1.5 GPa, n12 ¼ 0.34, a1 ¼ 4 106 1C1, a2 ¼ 79 106 1C1. (Material 2) E1 ¼ 181 GPa, E2 ¼ 10.3 GPa, G12 ¼ G13 ¼ 7.17 GPa, G23 ¼ 2.39 GPa, n12 ¼ 0.28, a1 ¼ 0.02 106 1C1, a2 ¼ 22.5 106 1C1. (Material 3) E1 ¼ E2 ¼ 70 GPa, G12 ¼ G13 ¼ G23 ¼ 52 GPa, n12 ¼ 0.33, a1 ¼ 23.6 106 1C1, a2 ¼ 23.6 106 1C1.
Table 2 Convergence of the optimum fiber orientations for different boundary conditions with literature results for Material 2 Simply supported
Clamped
Ref. [6]
Ref. [8]
Present study
Ref. [6]
Ref. [8]
Present study
45.1
45.0
45.0
54.3
52.9
54.0
671
(Material 4) E1/E2 ¼ 25, G12/E2 ¼ G13/E2 ¼ 0.5, G23/E2 ¼ 0.2, n12 ¼ 0.25, a1 ¼ 0.02 106 1C1, a2 ¼ 22.5 106 1C1.
Example 1. A single thin (a/h ¼ 40) clamped laminated square plate for Material 1 with fiber orientation angle 451 is considered first to compare the present study with literature results. As seen from Table 1, the results obtained for critical buckling temperature are in very close agreement with literature results [15,16]. Example 2. The next example is comprised of the same plate geometry and the same boundary conditions, with four layered (01/901/01/901) lamination sequence for Material 2 (a/h=20). The results are obtained for 10 modes (Fig. 3). It can be seen that, the present study is in very good agreement with the literature results (see Ref. [16]). 4.2. Optimization problem The convergence of the present optimization study is compared with the literature results [6,8] for Material 2. Therefore, the optimum fiber orientations are obtained for maximum thermal critical temperature with simply supported and clamped angle-ply laminated plates (N=4). It may be seen from Table 2 that, the results compare well. 4.2.1. Effect of plate aspect ratios on the optimal design Fig. 4 shows the influence of a/b ratios on the optimal results for clamped laminated plates under uniform temperature load (N ¼ 4, h ¼ 0.01 m, Material 2). It can be noted from this figure that, the critical buckling resistance is higher for square laminated plate as the plate becomes stiffer. In Table 3, optimum fiber orientations are given for a/b ratios. The buckled mode shapes under uniform temperature load are shown in Fig. 5 for a/b ratios for Material 2.
170
Critical temperature
165 160 155 Material 2
150 145 140
1
1.25
1.5 a/b
1.75
Fig. 4. Effect of plate aspect ratio on the optimal designs for laminated plates for Material 2.
2
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4.2.2. Effect of antisymmetric lay-up on the optimal design The effect of antisymmetric lay-up (y/y/y/y) on the optimal results is investigated for a/h ¼ 10, a/h ¼ 20, a/h ¼ 50 and a/h ¼ 100 for four layered square plates for
Table 3 Effect of the plate aspect ratio on the optimum fiber orientations for clamped laminated plates for Material 2 a/b
yopt (1)
1.00 1.25 1.50 1.75 2.00
54.0 54.5 51.7 55.2 53.8
Material 2. For a given antisymmetric lay-up, the optimum fiber orientations are obtained at 451. Comparative results are given with symmetric lay-up in Fig. 6. As seen, because of the absence of bending–extension coupling, symmetric lay-up does not yield the highest thermal buckling load resistance as usually expected. The differences for critical temperatures are 14.27%, 11.29%, 10.09% and 9.92% for b/h ¼ 10, b/h ¼ 20, b/h ¼ 50 and b/h ¼ 100 between antisymmetric and symmetric lay-up, respectively. As seen, the differences between antisymmetric and symmetric lay-up decrease as a/h ratio increases.
4.2.3. Effect of boundary conditions on the optimal design The iteration procedure may be applied to laminated plates with any combinations of simple support (S),
Fig. 5. Buckled mode shapes of laminated plates for a/b ratios (Material 2).
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clamped support (C) and free edge (F). In this study, four different combinations of the boundary conditions as (SSSS), (CSCS), (CCCC) and (CFCF) are considered as defined below.
2. Two edges simply supported and two edges clamped (CSCS) a a At x ¼ ; ; 2 2 b b At x ¼ ; ; 2 2
1. Four edges simply supported (SSSS) a a At x ¼ ; ; 2 2 b b At x ¼ ; ; 2 2
u ¼ w ¼ cy ¼ 0
u ¼ w ¼ cy ¼ cx ¼ 0 u ¼ w ¼ cx ¼ 0.
3. Four edges clamped (CCCC)
u ¼ w ¼ cx ¼ 0.
a a At x ¼ ; ; 2 2 b b At x ¼ ; ; 2 2
12000 Critical temperature
673
Symmetric lay-up Antisymmetric lay-up
u ¼ w ¼ cy ¼ cx ¼ 0 u ¼ w ¼ cx ¼ cy ¼ 0.
4. Two edges clamped and two edges free (CFCF)
8000
4000
0
a a At x ¼ ; ; 2 2
Material 2
0
20
60
40
80
100
a/h
Fig. 6. Effect of antisymmetric lay-up on the optimal designs for square laminated plates for Material 2.
Table 4 Effect of boundary conditions on the optimal designs for square laminated plates Boundary conditions
yopt (1)
Tcr (1C)
(SSSS) (CCCC) (CSCS) (CFCF)
45.0 54.0 36.6 38.1
1.829603 102 1.674376 102 2.550149 102 3.145864 102
u ¼ w ¼ cy ¼ cx ¼ 0.
Table 4 shows the effect of different boundary conditions on the optimal designs for symmetric square laminated plates (N ¼ 4, h ¼ 0.01 m, Material 2). As seen, the maximum thermal critical buckling load occurs at (CFCF) boundary condition, whereas the minimum thermal critical buckling load occurs at (CCCC) boundary condition. The results are contrast with the critical buckling loads under the mechanical loads. This can be explained that the free edges provide more degrees of freedom and allow the laminated plate to buckle higher temperatures. 4.2.4. Effect of material anisotropy on the optimal design The effect of material anisotropy ratio, E1/E2, on the optimum design is studied for four layered square plates for Material 4 (Fig. 7). It can be seen that, the critical temperature increases with increase of E1/E2 ratio. This is due to the increase in stiffness of laminates with increase in
500
Critical temperature
400 300 Material 4 200 100 0
0
10
30
20
40
50
E1/E2 Fig. 7. Effect of E1/E2 ratio on the optimal designs for square laminated plates for Material 4.
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E1. On the other hand, the differences for critical temperature decrease as E1/E2 ratio increases. For example, the differences are 28.04%, 21.41% and 17.25% for E1/E2 ¼ 20–30, E1/E2 ¼ 30–40 and E1/E2 ¼ 40–50, respectively. In Table 5, the effect of E1/E2 ratio on the optimum fiber orientations are given. As seen, as E1/E2 ratio increases, the optimum fiber orientation decreases but this decrease diminishes for larger E1/E2 ratios.
clamped square plates (N ¼ 6, h ¼ 0.01 m). The letters K, G and A represent Material 1, Material 2 and Material 3, respectively. For this purpose, six different combinations such as (KGK)s, (KAK)s, (GAG)s, (KAG)s, (AKG)s and (AKG)s are considered. In Table 7, optimum fiber orientations and the critical temperatures are given for hybrid laminates. As seen, the highest critical temperature is reached for (KGK)s composites, while the lowest is (KAK)s composites.
4.2.5. Effect of thermal expansion ratio on the optimal designs The effect of thermal expansion ratio, a2/a1, on the optimum design is studied for four layered square plates for Material 4 (Fig. 8). As seen, the critical temperature increases with increase in the values of a2/a1 ratio. But this increase diminishes as a2/a1 ratio increases. For example, the differences for critical temperature are 15.24%, 9.13% and 6.34% for a2/a1 ¼ 20–30, a2/a1 ¼ 30–40 and a2/a1 ¼ 40–50, respectively. The optimum fiber orientations are given in Table 6 for a2/a1 ratios. As seen, the optimum fiber orientations decrease gradually with increase of a2/a1 ratio.
5. Conclusions
4.2.6. Effect of hybrid laminates on the optimal design In this study, the effect of hybrid laminates on the optimal design is investigated for symmetrically laminated Table 5 Effect of E1/E2 ratio on the optimum fiber orientations for clamped laminated plates for Material 4
This paper presents optimal design of laminated plates subjected to uniformly distributed temperature load. The objective function is to maximize the critical temperature
Table 6 Effect of a2/a1 ratio on the optimum fiber orientations for clamped laminated plates for Material 4 a2/a1
yopt (1)
10 20 30 40 50
66.7 62.7 61.2 60.4 57.5
Table 7 Effect of hybrid laminates on the optimum results Tcr (1C)
yopt (1)
Laminate configuration
yopt (1)
E1/E2 5 10 20 30 40 50
60.1 55.9 52.7 51.4 50.6 50.2
(KGK)s (KAK)s (GAG)s (KAG)s (AKG)s (AKG)s
45.0 90.0 80.1 90.0 57.0 77.4
110.12 2.23 3.55 2.31 3.58 3.40
180
Critical temperature
160 140 Material 4
120 100 80
10
20
30
40
α2/α1 Fig. 8. Effect of a2/a1 ratio on the optimal designs for square laminated plates for Material 4.
50
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capacity of laminated plates and the fiber orientation is considered as design variable. Numerical experiments have been conducted to evaluate the MFD algorithm for laminated composite optimization problems. Initially, preliminary studies have been carried out by solving problems given in the literature for thermal buckling maximization and the results have been compared with the best known results available in the literature. The numerical results clearly demonstrate the effectiveness of MFD algorithm on the thermal buckling optimization for laminates. This methodology provides engineers with a useful tool for designing laminated composite structures. On the other hand, the results presented herein confirm that the characteristics of thermal buckling are significantly influenced by aspect ratio, antisymmetric lay-up, boundary condition, material anisotropy, ratio of coefficients of thermal expansion and hybrid laminates.
[5] [6]
[7]
[8]
[9]
[10]
[11]
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