Optimal Design of Laminated Composite Plates for Maximum Post buckling Strength Omprakash Seresta1

Department of Aerospace and Ocean Engineering

Mostafa M. Abdalla2

Department of Aerospace and Ocean Engineering

and

Zafer Gürdal 3

Departments of Aerospace and Ocean Engineering, and Engineering Science and Mechanics

Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 24061

In this paper, we study the optimization of compression loaded laminated composite plates for maximum failure load in post buckling regime. For the purpose of comparison, we also consider the design for maximum buckling capacity. The optimization procedure is to provide lay-up configuration. The design variable is the stacking sequence of laminate with discrete orientation angles chosen from 0, ±45, and 90 degrees. A standard genetic algorithm (GA) is used for the optimization because of its efficiency in handling discrete variables. The plate is modelled using von Kármán plate equations and assumed to be loaded by uniform edge displacements corresponding to biaxial compression. The analysis is performed using a Rayleigh-Ritz method, and the post buckling equilibrium path is traced using normal flow algorithm. Numerical results indicate that designs optimized for the maximization of buckling load fail at load levels significantly lower than designs directly optimized for the maximization of failure load. It is also observed that the designs optimized for maximum failure load have a higher in-plane stiffness than designs optimized for maximum buckling load. Since post buckling analysis is computationally expensive, we propose to use a multi criteria optimization that include both buckling load and in-plane stiffness as an alternative for the maximization of post buckling strength. Numerical results indicate that this approach leads to improved designs at a substantially reduced computational cost.

Nomenclature 0 (subscript) quantity associated with neutral surface

ba, length and width of plate

ijA inplane stiffnesses

B (superscript) quantity associated with parameters at buckling

ijD flexural stiffnesses

21 , EE longitudinal and transverse young’s moduli of the lamina f factor of safety F (superscript) quantity associated with parameters at failure

12G shear modulus of the lamina

ijK tangential stiffness matrix coefficients

1 Research Assistant, Student member, AIAA, oseresta@vt.edu 2 Research Assistant, Student member, AIAA, moabdall@vt.edu 3 Professor, Associate Fellow, AIAA, zgurdal@vt.edu

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46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference18 - 21 April 2005, Austin, Texas

AIAA 2005-2128

Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

gijK geometric stiffness matrix coefficients NLijklK nonlinear stiffness tensor coefficients

n number of half waves of w along x, and y direction

900 , cc nn number of contiguous zero or ninety degree plies

N total number of plies

yavgxavg NN , average in-plane stress resultants computed at the displaced edge t thickness of plies

wvu ,, displacement along x, y, and z direction

xyyx γεε ,, engineering strains along global axis

allε maximum allowed strain

1221 ,, γεε principal strains along material axis vi

ui

wi ΦΦΦ ,, assumed functions for w, u, v

xyyx κκκ ,, curvature terms F

cr λλ , critical buckling factor and failure factor

12ν poisson’s ratio Π total potential of the plate

iθ fiber orientation of ith layer

Θ vector of design variables

I. Introduction AMINATED composite plates are extensively used in the fields of automotive, aerospace, and marine engineering. This is primarily due to high specific strength and stiffness properties that these materials offer.

Optimal design of composite plates is important to fully realize their strength carrying capacity. The use of fiber composites as plate materials allows the designer to use fiber orientation angles of the individual plies as design variables in the optimization formulation, thus tailoring the design for the particular loads expected during service.

L Much work has been carried out to optimally design the stacking sequence (fiber orientation angle of the plies) of the laminate to maximize the buckling capacity. A comprehensive literature on the optimization of composite laminates for maximization of buckling load can be found in [1-6]. However, laminated composite plates possess significant post buckling strength. The post buckling behaviour of laminated composite plate is extensively studied and can be found in [7-9]. A structural component in its life cycle may be designed to carry loads both below and beyond the buckling load. Hence, it is necessary to integrate the post buckling behaviour of the laminated plate in the design optimization procedure. The obvious difficulty lies in the fact that beyond the buckling load or bifurcation point, the stress as well as strain distribution changes substantially and the failure may occur inside the domain due to excessive straining of the plies. A few related design problems considering the post buckling response of composite plates are studied in [10-15]. Perry et al. [10] studied the strain distribution at discrete predetermined points for optimized post buckling behaviour for the uni-axial case. Shin et al. [11] studied the minimum weight design of laminated composite plates for post buckling performance. In both reference [10] and [11], the design variables were the thicknesses of plies with pre-selected fiber orientations. Adali et al. [12] proposed a multi-objective formulation to simultaneously take into account both the buckling and post buckling behaviour. They defined a design index as a weighted sum of pre-buckling, buckling and initial post buckling stiffness. But in their work, they neglected the change in the stress/strain distribution in the domain in the post buckling regime. Diaconu et al. [13] studied the optimum design of infinite length laminated composite plates subjected to compression. In their study, they used maximum normal displacement and strain as objective function. In this paper, we formulate the optimization problem to maximize the failure load of the laminated composite plate with fibre orientation of the layers as design variables. The fiber angles are chosen from a discrete set, and a genetic algorithm (GA) is used as the design optimization platform because of its efficiency in handling discrete

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variables [3, 5]. For optimization purpose, it is important to select an efficient post buckling analysis procedure since the post buckling response will be evaluated for numerous designs. Although the finite element method is a powerful tool for structural analysis problems, the nonlinear nature of the post buckling response makes the use of a finite element code undesirable due to excessive computational time required. In this paper, we approximate the plate problem using the Rayleigh-Ritz method, and the post buckling response is traced using normal flow algorithm [17]. The maximum strain criterion is used to predict the laminate failure in the post buckling regime [18]. A number of case studies is presented for different loading conditions. The results are compared with that of buckling optimization problem. The rest of the paper is organized as follows: problem formulation, analysis formulation, optimization formulation, numerical results, followed by discussion and conclusions.

II. Problem formulation A simply supported laminated plate of dimension a × b (Figure 1)

subjected to compressive edge displacements λu0, and λv0 is considered. The buckling load crλ is the critical value of λ for which the laminate buckles, and Fλ is the value of λ in the post buckling regime when the maximum strain reaches permissible limits or the laminate fails. The laminate is made up of N plies with orientations restricted to 0, ±45, and 90 degrees. Each ply has a constant thickness t and plies are made up of the same material. Thus the total thickness of laminate is h = N t. The laminate is assumed to be symmetric and balanced. Hence, only the half-laminate stacking sequence is used as the design variable vector. The vector of design variables is denoted as Θ . The laminate is optimized to maximize Fλ . In the next section, the analysis model is developed for moderately large deformation using the Rayleigh-Ritz method applied to the von Kármán plate equations.

Figure 1: Panel

III. Analysis formulation Assuming that the plate is thin, such that the Kirchhoff hypothesis is valid, the strain in terms of mid plane strain

(superscript 0) is given as 0 0 0, ,x x x y y y xy xy xyz z .zε ε κ ε ε κ γ γ κ= + = + = + (1)

In the moderately large rotation case, according to the von Kármán model, the mid-plane strains and curvature terms are given in terms of mid plane displacements u, v, w along x, y, and z axes respectively as,

0 2 0 2 0, , , , , , ,

, , ,

1 1; ;2 2; ; 2 ,

x x x x y y xy y x x

x xx y yy xy xy

u w v w u v w w

w w w

ε ε γ

κ κ κ

= + = + = + +

= − = − = −

, y (2)

where a comma (“,”) in subscript indicates derivative with respect to the variable following it. The total potential, using the above expressions of strain, of a symmetric and balanced laminate [16] is given by

( ) ( ) ( ) ( )2 2 20 0 0 0 0 2 2 211 12 22 66 11 12 22 66

0 0

1, , 2 22

a b

x x y y xy x x y y xyu v w A A A A D D D D dxdyε ε ε ε γ κ κ κ κ κ⎡ ⎤Π = + + + + + + +⎢ ⎥⎣ ⎦∫ ∫ (3)

where the in-plane stiffness coefficients - A11, A12, A22, and A66, and flexural stiffness coefficients - D11, D12, D22, and D66 of the laminate are given in terms of engineering constants of the material (see Appendix), and the lamination sequence of the plate. Following the Rayleigh-Ritz procedure [16], the displacement functions are assumed of the form,

( )1

,n

wi i

i

w x y a=

= Φ∑ ,

( )2

0

1

,n

ui i

i

u xu x y b

aλ

=

= − + Φ∑ , and (4)

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( )2

0

1

,n

vi i

i

v yv x y c

bλ

=

= − + Φ∑ , where λ is the displacement load scaling factor.

Depending on the choice of functions , different boundary conditions can be modeled. In the current work, the following boundary conditions are assumed along the edges of the plate:

, ,w ui i andΦ Φ Φv

i

1. At x = 0, ,( ), 0u x y = ( ) 0,v y

v x yb

λ= − , and ( ), 0w x y =

2. At x = a, ,( ) 0,u x y u= ( ) 0,v y

v x yb

λ= − , and ( ), 0w x y =

3. At y = 0, ( ) 0,u x

u x ya

λ= − , , and ( ),v x y = 0 ( ), 0w x y =

4. At y = b, ( ) 0,u x

u x ya

λ= − , , and ( ) 0,v x y v= ( ), 0w x y =

The displacement field is assumed of the form,

( )1 1

, sin sinn n

iji j

i x j yw x y aa bπ π

= =

⎛ ⎞ ⎛ ⎞⎟⎠

= ⎜ ⎟ ⎜⎝ ⎠ ⎝∑∑ ( ),

2 20

1 1

, sin sinn n

iji j

u x i x j yy ba a b

λ π π

= =

⎛ ⎞ ⎛ ⎞= − + ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑∑u x , and

( )2 2

0

1 1

, sinn n

iji j

v y i x j yv x y cb a

λ π π

= =

⎛ ⎞ ⎛= − + ⎜ ⎟ ⎜⎝ ⎠ ⎝∑∑ sin

b⎞⎟⎠

.

The number of assumed modes for in-plane displacements u, and v is twice that of lateral displacement in w. This is done to ensure that the in-plane equilibrium is adequately satisfied. By using the stationarity conditions of total potential and minimizing with respect to Ritz coefficients ai, bi, and ci, we obtain the general equilibrium equation for a symmetric and balanced laminated composite plate (see Appendix for details),

0

0

0

u ub uc uaal il i il i ijl i j

v vb vc vaal il i il i ijl i j

g wba wca waaail i i ikl i k ikl i k ijkl i j kil

g K b K c K a a

g K b K c K a a

K a K a K b a K c a K a a a

λ

λ

λ

− + + + =

− + + + =

− + + + =

=

(5)

The first two equations in (5) are linear in the Ritz coefficients corresponding to in-plane displacements. Thus, the above three equations can be reduced to a single nonlinear equation, by eliminating the bi and ci coefficients from the third equation using the first two equations. For the details of this condensation process the reader is referred to the Appendix. The final set of nonlinear equations governing the plate behavior reduces to, (6) 0g NL

il i ijkl i j kilK K a K a a aλ⎡ ⎤− +⎣ ⎦

A. Buckling analysis The linearised buckling equation can be obtained from Equation (6) by neglecting the non-linear part, leading to

the standard eigenvalue problem, 0g

il iilK K aλ⎡ ⎤− =⎣ ⎦ .

The lowest eigenvalue is the buckling factor crλ , and the corresponding eigenvector is the buckling mode.

B. Post buckling analysis The post buckling equilibrium path is traced using a standard normal flow algorithm [17]. We can write

Equation (6) as, ( ), 0g NL

l i il i i ijkl i j kilF a K a K a K a a aλ λ≡ − + = . This set of nonlinear equations defines an equilibrium load-deflection path. Different methods are available for the tracing of nonlinear equilibrium paths [17]. In this paper, we use the normal flow algorithm because of its robustness and efficiency [17].

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C. Failure analysis The failure load is F λ is defined as the load for which failure occurs. For the purpose of predicting failure, we

use the maximum strain failure criterion, and consider first ply failure. According to the maximum strain criterion, failure is assumed to occur when any of the strains referred to the material axes exceeds its allowable value. For the purpose of simplicity, we assume that the allowable values are the same for all strains. As such failure is assumed to occur when,

( )max max max1 2 12

1 1.0max , ,

all

fε

ε ε γ≥ ,

where f is the factor of safety, and is the maximum values of strain in the plate. The maximum strains in terms of

max max max1 2 12, ,ε ε γ

,x y ,ε ε and xyγ are calculated as,

.

( )( )

( )( )

max 2 21

max 2 22

max 2 212

max cos sin 2cos sin 1,

max sin cos 2cos sin 1,

max cos sin cos sin cos sin 1,

x i y i i i xy

x i y i i i xy

x i i y i i i i xy

i N

i N

i N

ε ε θ ε θ θ θ γ

ε ε θ ε θ θ θ γ

γ ε θ θ ε θ θ θ θ γ

= + + ∀ =

= + − ∀ =

= − + + − ∀ =

In the post buckling regime, the plate has a finite curvature and the strains are no longer constant over the panel. The strains , ,x yε ε and xyγ in the ith layer are computed at the top and bottom surface of each layer of the laminate taking into account the contribution of bending curvature using Equation (1). To locate the maximum strain over the panel, a Simplex search method (fminsearch) implemented in MATLAB [19] is used. To avoid getting stuck at local maxima, the search is repeated with different initial points. We used nine different initial points, one at the center, four corner nodes and four midpoints of the edges of the plate. The choice of step size used for tracing the nonlinear response is important. If we use too large a step size then it is difficult to compute the exact failure load since the strains calculated at the failure load may well exceed the allowable limit by considerable margin. On the other hand, if we use too small a step size then it is computationally expensive. Instead we use a variable step size. We start with a large step size and the step size is decreased by a factor of two with successive iterations. However, to avoid the step size becoming too small, a lower limit on step size is specified.

IV. Optimization formulation In this paper, for the purpose of comparison we consider two objectives – maximization of buckling load and

maximization of failure load. A number of practical engineering constraints are imposed to obtain realistic designs. First, we restrict the lamination to symmetric balanced laminates. The symmetry condition of laminate is taken care of implicitly by using only half of the laminate stacking sequence as design variables. The balanced condition of the laminate is imposed explicitly by penalizing designs violating the constraint. The second practical condition is the ply contiguity constraint. When the numbers of contiguous plies of the same orientation are large composite laminates are known to experience matrix cracking. Therefore, ply contiguity constraint [1] is imposed so that no more than a certain number of plies of same fiber orientation angles are allowed successively. In the current formulation, no more than four plies of 0 (nc0) or 90 (nc90) degree fiber orientation angles are allowed successively. For ±45 degree plies, the ply contiguity constraint is satisfied automatically.

Depending on the optimization objective, we have the following two optimization problems,

Maximization of buckling load The optimization problem is formulated as Maximize ( )crλ Θ

subject to: 1. 0 4cn ≤ 2. 90 4cn ≤ 3. the laminate is symmetric and balanced

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Maximization of failure load The laminate is loaded beyond the critical or first bifurcation point. The objective of the design optimization

problem is to maximize the failure load. Maximize ( )F λ Θ

subject to 1. 0 4cn ≤ 2. 90 4cn ≤ 3. the laminate is symmetric and balanced

Detailed optimization for post-buckling strength, even with simplified analyses as proposed, is quite

computationally intensive. As the results presented indicate, designs optimized for post buckling strength have high values of inplane stiffness. It is interesting to investigate whether considering the inplane stiffness of the panel in the buckling optimization formulation can lead to improved designs from strength perspective. To this end, we reformulate the maximization of buckling factor to include the inplane stiffness properties in an attempt to reproduce the failure loads designs without incurring the computational cost of nonlinear post buckling analysis. Hence, the buckling optimization problem is reformulated as,

Maximize ( )crλ Θ

subject to 1. 0 4cn ≤ 2. 90 4cn ≤ 3. laminate is symmetric and balanced 4. ( ) allyx EEE ≥,min

Ex and Ey are the effective inplane stiffness along x and y direction respectively, and Eall, is the minimum allowed effective inplane stiffness. Several different optimization algorithms can be applied to above formulations. Among them, integer programming and genetic algorithms have been widely applied to solve discrete design variable problem. Previous works by the authors indicated the suitability of genetic algorithm for stacking sequence design of composite laminate [1,3-6]. In this work, a standard genetic algorithm is used as an optimizer. Genetic algorithms are probabilistic search algorithms based on natural selection to guide the exploration of design space toward a global optimum. The common features of a standard GA are population initialization, parent selection, crossover, mutation, and the selection of successive generations. Each element has many variations, modified to suit a particular problem.

V. Results Results are obtained for 10"×10" graphite-epoxy: T300/N5208 composite plate. The laminates are assumed to

have 16 plies in total. The material properties and the GA parameters are given in Table 1 and 2 respectively.

Table 1. Material properties

Material properties Values Young’s modulus in direction 1, E1 18.5 ×106 lb/in2

Young’s modulus in direction 1, E2 1.6 ×106 lb/in2

Shear modulus, G12 0.832 ×106 lb/in2

Poisson’s ratio, ν12 0.35 Max. permissible strain 0.004 Ply Thickness 0.005 in

Table 2. GA parameters

GA parameters Values Initial population 20 Probability of crossover 1.0 Probability of mutation 0.05 No. of elites retained 1

Since, the post buckling analysis is computationally expensive, all the results presented here are for n = 2. The non-linear analysis is verified with that of commercial finite element package NASTRAN [20] using 100 CQUAD4 elements. The stacking sequence of the test problem is [(0 45 0 -45)2]s. The plate is edge displacement loaded such

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that u0=0.001, and v0=0.0. Since, the stress resultants are not constant in the post buckling regime, average stress resultants are computed at the edges as

0

1( , ) ( , )b

xavg xN a y N a y db

= ∫ y , and 0

1( , ) ( , )a

yavg yN a y N x b da

= ∫ x .

Figure 2 shows the average axial load, Nx vs edge displacement parameter, λ . The analysis model developed agrees well with the finite element analysis. The maximum error is within roughly 10 percent, which makes it adequate for preliminary design purposes.

Figure 2. Plot of average axial load, Nx,avg and edge displacement, λ.

Next, the optimum designs for four different cases of applied edge displacement loading are tabulated and their performance is evaluated. All the designs reported are obtained after 50 GA iterations. The effective inplane stiffnesses Ex, and Ey, are normalized with respect to E1 [21] (see Appendix) of the optimum designs are also reported. The Nx, and Ny reported are the averaged values computed along the displaced edges. The superscript “B” refers to buckling and “F” refers to failure.

Table 3: Optimum designs obtained for different loading ratio.

Obj. 0u (in)

0v (in) crλ

BxN

(lb/in)

ByN

(lb/in) Fλ

FxN

(lb/in)

FyN

(lb/in) Optimum Designs xE yE

Buckling 0.001 0.0 2.78 87.13 56.84 19.75 449.05 -22.90 [(45 90 -45 90)2]s 0.183 0.583 Failure 0.001 0.0 1.22 98.82 5.5 25.41 1378.7 -571.4 [(0 90)4]s 0.547 0.547

Buckling 0.001 0.001 1.0 85.55 85.55 14.28 431.66 431.66 [(±45)4]s 0.156 0.156 Failure 0.001 0.001 0.61 52.16 52.16 20.38 620.30 620.30 [(0 90)4]s 0.547 0.547

Buckling 0.0015 0.001 0.82 73.41 96.63 11.39 430.80 409.65 [(±45)3 (90)2]s 0.200 0.370 Failure 0.0015 0.001 0.49 61.50 42.83 16.17 924.38 303.86 [(0 90)4]s 0.547 0.547

Buckling 0.002 0.001 0.71 77.21 92.84 8.58 430.13 322.55 [(±45)3 (90)2]s 0.200 0.370 Failure 0.002 0.001 0.40 67.72 36.61 12.89 1082.4 93.51 [(0 90)4]s 0.547 0.547

The critical buckling designs for all loading ratio has lower Fλ compared to the failure load designs. The optimum failure design converges at cross-ply laminate. This correlates well with the results reported by [11] for infinite laminate subjected to compression. It is clear that designs optimized for maximum buckling load are not optimal from ultimate load carrying capacity point of view. A close look at Table 3 reveals that the maximization of post buckling strength leads to optimum design with high values of inplane stiffness. Next, we study the tradeoff between effective inplane stiffness and buckling load of laminated panels as formulated in Sec. IV in an attempt to replicate the designs obtained for maximum failure load. By varying the value of the minimum allowable inplane stiffness, Eall, a pareto front describing the optimal compromise between buckling load and inplane stiffness is generated.

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The pareto front for the loading conditions u0=v0=0.001 is shown in Figure 3. For the example problem solved, we start with a minimum allowable stiffness corresponding to design obtained for maximum buckling load and increase the minimum allowable inplane stiffness until the optimizer is no longer able to find a feasible solution. Table 4 shows the failure analysis of the optimum designs obtained in the pareto front. From Table 4, it is seen that as we increase the effective inplane stiffness of the laminate, the failure load of the laminate increases and the buckling capacity decreases. The optimum designs along the pareto front are reported in Table 4. It is seen that the two extremes of the pareto front indicates the designs obtained for maximum buckling load and maximum failure load, respectively. For Eall = 0.5, the optimum design obtained is cross-ply laminate which is the same design obtained for maximum failure load. The pareto curve is initially flat, which suggests that the post buckling strength of a panel can be increased without sacrificing buckling capacity of the panel. For the example problem solved, it is seen that for Eall less than or equal to 0.35, buckling load has changed very little. From Table 4 it can be seen that for Eall = 0.35, the failure load is 8.8 pc higher compared to design obtained for maximum buckling load while the buckling load dropped by 5 pc.

Figure 3: Pareto curve demonstrating the tradeoff between buckling load maximization and inplane stiffness

for u0=v0=0.001.

Table 4: Failure analysis of optimum designs obtained in the pareto plot.

Eall Designs crλ BxN (lb/in) B

yN (lb/in) Fλ FxN (lb/in) F

yN (lb/in) xE yE 0.15 [(±45)4]s 1.0 85.55 85.55 14.28 431.66 431.66 0.156 0.156 0.25 [(±45)3 0 90]s 0.991 85.03 85.03 15.16 459.54 459.54 0.294 0.294 0.35 [(±45)2 90 02 90]s 0.948 81.37 81.37 15.31 469.25 469.25 0.399 0.399 0.45 [±45 02 90 0 902] 0.833 71.46 71.46 15.51 478.86 478.86 0.481 0.481 0.5 [(0 90)4]s 0.61 52.16 52.16 20.38 620.30 620.30 0.547 0.547

VI. Conclusion In this paper, we formulated the optimal design of composite laminated plates for buckling and post buckling

strength. The results are compared with optimal design for buckling strength. The optimum laminate configuration for post buckling problem is different from the optimum laminate configuration for buckling. The results show that considerable improvement in failure load is achieved. For the current example problems solved, the optimum laminate configuration for post buckling problem converges to cross-ply laminate. It is noted that the designs obtained for maximum failure load has higher inplane stiffness. In an effort to save computational time, a simple buckling optimization framework is presented by including the inplane stiffnesse in the design optimization formulation. The optimum laminate configuration for the maximum failure load in the post buckling regime is successfully reproduced using this new formulation. The results indicate that for preliminary design purposes, the

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use of the proposed buckling optimization formulation with constraint on the minimum allowable inplane stiffness is effective in finding designs with large post buckling strength.

Acknowledgments This research is sponsored by Air Force Office of Scientific Research under grant number F49620-01-1-0178.

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18 Arnold, R. R. and Mayers, J. “Buckling, post buckling, and crippling of materially nonlinear laminated composite plates”, International Journal of Solids & Structures, Vol. 20, 1984, pp. 863-880.

19 The Mathworks Inc., Matlab manual, The Mathworks Inc., 3 Apple Hill drive Natick, MA 01760-2098, USA. 20 MSC.Software, Corp., Nastran manual, MSC.Software Corporation, 2 MacArthur Place, Santa Ana, CA

92707, USA.

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21 Gürdal, Z., Haftka, R. T., and Hajela, P., Design and optimization of laminated composite materials, A Wiley-Interscience Publication, John Wiley & Sons, Inc., 1999.

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Appendix: Laminated plate analysis The reduced stiffnesses of the orthotropic layers are given in terms of engineering constant of the materials as,

111

12 211E

Qν ν

=−

, 21 1 12 212

12 21 12 211 1E E

Qν νν ν ν ν

= =− −

, 222

12 211E

Qν ν

=−

, and 66 12Q G= .

The transformed reduced stiffness of an orthotropic layer at an angle θ is given as, ( )

( ) ( )( )

( ) ( )( )

4 2 2 411 11 12 66 22

2 2 4 412 11 22 66 12

4 2 2 422 11 12 66 22

3 316 11 22 66 12 22 66

326 11 22 66

cos 2 2 sin cos sin ,

4 cos sin cos sin ,

sin 2 2 sin cos cos ,

2 sin cos 2 sin cos

2 sin cos

Q Q Q Q Q

Q Q Q Q Q

Q Q Q Q Q

Q Q Q Q Q Q Q

Q Q Q Q

θ θ θ θ

θ θ θ θ

θ θ θ θ

,θ θ θ

θ θ

= + + +

= + − + +

= + + +

= − − + − +

= − − + ( )θ

( ) ( )3

12 22 66

2 2 4 466 11 22 12 66 66

2 sin cos

2 2 cos sin cos sin .

Q Q Q

Q Q Q Q Q Q

,θ θ

θ θ θ

− +

= + − − + + θ

The inplane and the flexural stiffness are given as,

( ) ( )

( ) ( )

11

3 31

1

,

.

N

ij ij k kkk

N

ij ij k kk k

A Q z z

D Q z z

−=

−=

= −

= −

∑

∑.

The effective normalized (with respect to E1) inplane stiffnesses are calculated as, 2

11 22 12

1 22

1x

A A AE

hE A⎛ ⎞−

= ⎜ ⎟⎝ ⎠

, and 2

11 22 12

1 11

1y

A A AE

hE A⎛ ⎞−

= ⎜ ⎟⎝ ⎠

.

The terms used in equation (4) & (5) are as follows

0 011 12 ,

0 0

a bu ul l

u vxg A A dx

a b⎛ ⎞⎛ ⎞= + Φ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∫ ∫ dy

( )11 , , 12 , ,0 0

a bub u u u uil i x l x i y l yK A A dxdy= Φ Φ + Φ Φ∫ ∫

( )12 , , 66 , ,0 0

a buc v u v uil i y l x i x l yK A A dxdy= Φ Φ + Φ Φ∫ ∫

( )11 , , , 12 , , , 66 , , , 66 , , ,0 0

12

a buaa w w u w w u w w u w w uijl i x j x l x i y j y l x i x j y l y i y j x l yK A A A A= Φ Φ Φ + Φ Φ Φ + Φ Φ Φ + Φ Φ Φ∫ ∫ dxdy

0 012 22 ,

0 0

a bv vl l

u vyg A A dxdy

a b⎛ ⎞⎛ ⎞= + Φ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∫ ∫

( )12 , , 22 , ,0 0

a bvb u v u vil i x l y i y l xK A A= Φ Φ + Φ Φ∫ ∫ dxdy

( )22 , , 66 , ,0 0

a bvc v v v vil i y l y i x l xK A A dxdy= Φ Φ + Φ Φ∫ ∫

( )12 , , , 22 , , , 66 , , , 66 , , ,0 0

12

a bvaa w w v w w v w w v w w vijl i x j x l y i y j y l y i x j y l x i y j x l xK A A A A dxdy= Φ Φ Φ + Φ Φ Φ + Φ Φ Φ + Φ Φ Φ∫ ∫

( )11 , , 12 , , 12 , , 22 , , 66 , ,0 0

4a b

w w w w w w w w w wil i xx l xx i yy l xx i xx l yy i yy l yy i xy l xyK D D D D D= Φ Φ + Φ Φ + Φ Φ + Φ Φ + Φ Φ∫ ∫ dxdy

0 0 0 011 12 , , 12 22 , ,

0 0

a bg w wil i x l x i y l y

u v u v w wK A A A A dxdya b a b

⎛ ⎞⎛ ⎞ ⎛ ⎞= + Φ Φ + + Φ Φ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

∫ ∫

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( )11 , , , 12 , , , 66 , , , 66 , , ,0 0

a bwba u w w u w w u w w u w wikl i x k x l x i x k y l y i y k y l x i y k x l yK A A A A dxdy= Φ Φ Φ + Φ Φ Φ + Φ Φ Φ + Φ Φ Φ∫ ∫

( )12 , , , 22 , , , 66 , , , 66 , , ,0 0

a bwca v w w v w w v w w v w wikl i y k x l x i y k y l y i x k y l x i x k x l yK A A A A dxdy= Φ Φ Φ + Φ Φ Φ + Φ Φ Φ + Φ Φ Φ∫ ∫

11 , , , , 12 , , , , 12 , , , ,

22 , , , , 66 , , , , 66 , , , ,

66 , , , , 66 , , ,

12

w w w w w w w w w w w wi x j x k x l x i y j y k x l x i x j x k y l y

waaa w w w w w w w w w w w wijkl i y j y k y l y i x j y k y l x i x j y k x l y

w w w w w w wi y j x k y l x i y j x k x l

A A A

K A A A

A A

Φ Φ Φ Φ + Φ Φ Φ Φ + Φ Φ Φ Φ +

= Φ Φ Φ Φ + Φ Φ Φ Φ + Φ Φ Φ Φ +

Φ Φ Φ Φ + Φ Φ Φ Φ0 0

,

a b

wy

dxdy

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∫ ∫

{ } { }{ }b

ec

⎧ ⎫⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭

, { }{ }{ }

u

v

gg

g

⎧ ⎫⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭

ub uc

a

vb vc

K KK

K K

⎡ ⎤⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦⎢ ⎥⎡ ⎤ =⎣ ⎦ ⎢ ⎥⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦⎣ ⎦,

1, 2,3, , 22 1, , 4

uaaijlab

ijp vaaijl

K for p nK

K for p n n⎧ =⎪= ⎨ = +⎪⎩

, and

1,2,3, , 22 1, , 4

wbaba iklpkl wca

ikl

for p nKK

for p n nK⎧ =⎪= ⎨ = +⎪⎩

.

( )a a ab a abp pl l ps ijs i j p ps ijs i je K g K K a a h K K aλ λ− − −= − = − a .

Substituting the above expression in the second equation (4.c), we get a single buckling equation,

( ) 0g ba a ab waaail i il i pkl p ps ijs i j k ijkl i j kK a K a K h K K a a a K a a aλ λ −− + − + = .

By rearranging the terms we get the following standard form of buckling equation,

( ) ( ) 0,or

0

g ba waaa ba a abil il pil p i ijkl pkl ps ijs i j k

g NLil il i ijkl i j k

K K K h a K K K K a a a

K K a K a a a

λ

λ

−⎡ ⎤− − + − =⎣ ⎦⎡ ⎤− + =⎣ ⎦

.

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