Design of variable-stiffness composite panels for maximum buckling load

Design of variable-sti?ness composite panels

for maximum buckling load

Shahriar Setoodeh,Mostafa M.Abdalla,Samuel T.IJsselmuiden,Zafer Gu ¨rdal *

Faculty of Aerospace Engineering,Delft University of Technology,Kluyverweg 1,2629HS Delft,The Netherlands

Available online 26January 2008

Abstract

A generalized reciprocal approximation is presented for design of variable-sti?ness laminated composite panels for maximum buck-ling load.The buckling load is expanded in terms of the inverse of the sti?ness tensor.For discretized panels such an approximation has

the important property of being separable,which allows the maximization to be carried out at each discrete node separate from the oth-ers.This makes the algorithm particularly suited to parallel computations.The sensitivity analysis is performed exactly using an adjoint method,requiring only one back substitution using the already factored inplane sti?ness matrix with di?erent right hand sides to com-pute the sensitivities for all design variables.A conforming CLPT ?nite element is used for the buckling analysis of rectangular plates and the proposed reciprocal approximation is used to update ?ber orientation angles at each ?nite element node.Numerical results obtained for rectangular plates show that signi?cant improvements can be gained in the buckling load by allowing the sti?ness properties to vary spatially.The case of repeated eigenvalues is handled using a dual formulation.ó2008Elsevier Ltd.All rights reserved.

Keywords:Tow-placement;Variable-sti?ness panels;Buckling design;Reciprocal approximation;Adjoint sensitivity analysis

1.Introduction

Buckling design of laminated composite plates is well studied in the literature [8,9,27].Grenestedt [9]used lami-nation parameters with an approximate feasible domain de?nition to design panels against shear buckling.In his formulation,four design variables corresponding to the bending lamination parameters over the entire domain were used to design a panel.His work was later extended by Fukunaga [8]using exact de?nitions for the feasible combination of lamination parameters for panels with and without shear coupling,and with two and four design variables,respectively.Liu et al.[15]showed that the feasi-ble domain of the bending lamination parameters for lay-ups with only discrete 0 ;?45 ,and 90°layers assumes a hexagonal shape inside the Miki Parabola,and then used it to design panels for maximum buckling load.

Although signi?cant increase in the buckling load can be

obtained through tailoring laminate stacking sequence of composite panels with traditional straight ?bers,the poten-tial of ?brous composites is not fully exploited.An approach for the design of panels with cutouts was ?rst introduced in the late eighties [12]in which curvilinear ?bers were suggested to improve structural response instead of straight ?ber paths.The approach has been gen-eralized later by Gu ¨rdal and Olmedo [10]by designing var-iable-sti?ness laminates that use continuous curvilinear ?ber paths.For such variable-sti?ness panels,the sti?ness properties are continuous functions of position.Ideally,by steering the ?ber paths the ?ber orientation angle can be varied,which in turn changes the sti?ness properties at each point in the panel independently.The additional freedom in locally tailoring the sti?ness properties means that the performance of variable-sti?ness panels can be highly improved over constant-sti?ness (straight ?bers)panels.However,this additional freedom comes at the price of having signi?cantly enlarged design space as well as the complexity of maintaining uniform ?bre paths.

0263-8223/$-see front matter ó2008Elsevier Ltd.All rights reserved.doi:10.1016/https://www.360docs.net/doc/c34344001.html,pstruct.2008.01.008

*

Corresponding author.

E-mail address:z.gurdal@tudelft.nl (Z.Gu ¨rdal).

https://www.360docs.net/doc/c34344001.html,/locate/compstruct

Available online at https://www.360docs.net/doc/c34344001.html,

Composite Structures 87(2009)

109–117

Research in variable-sti?ness concept has been stimu-lated by recent developments of advanced tow-placement technology[5,26].Tow-placement machines are computer numerical controlled multi-axis machines that are capable of steering multiple tows(up to32with each tow having a width of about1/8inch)along prescribed paths.Each tow can be placed at a di?erent rate allowing a radius of curvature as low as12inches[17].Moreover,tows can be independently stopped(cut)and started permitting?exible coverage patterns to build composite panels.

The idea of curved?bers or?ber orientation angle dis-tribution has been used earlier to design composite lami-nates for maximum buckling loads.Improvements in buckling resistance of composite plates with a circular cen-ter hole was studied by Hyer and Lee[13]using curvilinear ?ber format.They used?nite di?erence based sensitivity analysis to?nd conditions that maximize buckling perfor-mance of a plate with a central circular hole.To reduce the problem size,they use a quarter model with selected regions of straight?ber orientations.Banichuk et al.[3] used triangular?nite elements to search for optimal angles of orthotropy,which maximizes the critical buckling parameter.They performed exact sensitivity analysis for both simple and multiple eigenvalues.Their numerical results showed a signi?cant increase in the critical buckling load just with reorientation of the orthotropic material in an optimal manner.Tatting and Gu¨rdal[25]used a two parameter based continuous curvilinear?ber path de?ni-tion,in which?ber orientation angles vary linearly along the panel side,to design variable-sti?ness panels with a cen-ter hole for maximum buckling load.Their study showed that up to60%improvement can be gained by optimal design of panels while complying with the existing manu-facturing techniques for curvilinear?bre paths.A so called discrete material optimization approach was used by Lund et al.[16]based on a semi-analytic sensitivity analysis to design rectangular plates for maximum buckling load.

In the present study,the generalized reciprocal approx-imation approach introduced by Abdalla et al.[2]is extended to buckling design of variable-sti?ness panels. In this formulation,the critical load is approximated using a?rst order Taylor series expansion in terms of the point-wise compliance tensors.Such an approximation has the important property of being separable,which is highly suit-able for parallel computing.Therefore,the problem of maximizing the buckling load is reduced to a simple local optimization problem at any discretization point.For buckling analysis,a conforming bilinear?nite element is used while the sensitivity of the buckling load is performed analytically.Practical manufacturing constraint on smooth spatial variation of the?ber orientation angle has been addressed approximately by using an interpolation scheme.

In the following sections,?rst the?nite element discret-ization of the buckling analysis is brie?y discussed,then the generalized reciprocal approximation applied for maxi-mum buckling design of variable-sti?ness panels is pre-sented followed by the sensitivity analysis.Sti?ness smoothing(or rather compliance smoothing)scheme is introduced next,followed by design update rule for the buckling load maximization and the treatment of the mul-timodal design.Finally numerical results are provided to demonstrate the performance of the current formulation along with the potential gains in the buckling load using a variable-sti?ness design compared to a constant-sti?ness design.The e?ect of boundary conditions on the optimal buckling load are also investigated.

2.Buckling analysis

The buckling loads are determined through the eigen-value problem

eK bàk K gTáa?0;e1Twhere K b is the global bending sti?ness matrix,K g is the global geometric sti?ness matrix,a is the mode shape com-prising of deformation degrees of freedom,and k is load multiplier.The mode shapes are normalized such that

a TáK báa?1:e2T

The geometric sti?ness matrix is constructed through an assembly of element geometric matrices.The sti?ness matrix of each element takes the form

K g

e

?àn x K xàn y K yàn xy K xy;e3Twhere n e?en x;n y;n xyTT is the vector of inplane stress resultants averaged over the element,and K x;K y and K xy are constant matrices that depend only on element geometry.

The averaged inplane stress resultants can be expressed as

n e?A eáe e;e4Twhere A is the inplane sti?ness matrix and e is the average strain vector given by

e e?B eáu e;e5Twhere u is the vector o

f inplane displacements,B is the average element strain displacement matrix(see Appendix for the de?nitions),and u e is the vector of the degrees of freedom associated with nodes connected to the e th ele-ment.The inplane displacements can be found from the solution of the inplane equilibrium equations

K máu?f:e6Twhere f is the vector of inplane loads.

3.The generalized reciprocal approximation

In the standard reciprocal approximation,a function is expanded in a Taylor series in terms of reciprocal variables [11].Reciprocal variables are traditionally de?ned as the reciprocals of the design variables.The reciprocal approx-imation is used extensively for truss design,in which the design variables are the cross sectional areas of the mem-

110S.Setoodeh et al./Composite Structures87(2009)109–117

bers.To generalize the reciprocal approximation,it is noted that the cross sectional area for trusses plays the role of the sti?ness of the member.In composite panel buckling design this rule is played by the inplane and bending sti?-ness matrices A and D.The generalized reciprocal approx-imation is,therefore,obtained by expanding the objective function in a Taylor series in terms of the inverse tensor of the sti?ness tensors,commonly known as the compli-ance tensors,which is denoted by R?Aà1and S?Dà1.

The derivatives of the buckling load with respect to the components of the bending compliance tensor,S,for the element e can be written using the chain rule as

o k o S e ?

o k

o D e

á

o D e

o S e

:e7T

Noting that the second term is simply the derivative of the sti?ness tensor with respect to the compliance tensor,it can be shown that

o k o S e ?àD eá

o k

o D e

áD e:e8T

The same formulation applies to the derivative of the buck-ling with respect to the in-plane compliance tensor

o k o R e ?àA eá

o k

o A e

áA e:e9T

4.Sensitivity analysis

The sensitivity of the critical eigenvalue k cr to a change in a design variable which is generically denoted by b is considered here.The variable b is assumed to a?ect only the local sti?ness properties of a single element i.The expression of the sensitivity can be written as[11]

d k d b ?k a Tá

d K b

d b

àk

d K g

d b

áa:e10T

The sensitivity value in(10)is composed of two terms.The ?rst term depending on the derivative of the bending sti?-ness is local in the case when b is linked to the sti?ness of a single element.By local it is meant that this term can be evaluated using information from a single element,viz.

S b1 a Tád K b

d b

áa?a T

i

á

d K b

i

d b

áa i:e11T

The second term in(10)is not necessarily local.This is due to the fact that even when the sti?ness of a single element is altered,the distribution of the inplane loads is altered for all elements and thus the geometric matrices of all element would change.In the following,an e?cient way for the evaluation of this term is described.

Substituting from(3)into(10)and rearranging the terms S b2is de?ned as

S b2 a Tád K g

d b

áa?à

X

e

s T

e

á

d n e

d b

;e12T

where the vector s e can be calculated locally as

s e?a T

e áK xáa e;a T

e

áK yáa e;a T

e

áK xyáa e

àáT

:e13TThe derivative of the inplane stress resultants can be ob-tained by di?erentiation of(4)as

d n e

d b

?

d A e

d b

áe etA eá

d e e

d b

:e14TThus,the sum in(12)can be decomposed into two terms corresponding to the two terms in(14).The?rst term can be evaluated locally since only the inplane sti?ness ma-trix of the i th element depends on b.The second term in-volves the derivative of the average strain of an arbitrary element with respect to the change of sti?ness of the i th ele-ment and is not local

S b2?às T

i

á

d A i

d b

áe ià

X

e

s T

e

áA eá

d e e

d b

:e15T

To evaluate the second term above,which is denoted by S b22in(15)term,(5)is di?erentiated with respect to b to obtain

d e e

d b

?B eá

d u e

d b

:e16TThus the strain term S b22simpli?es to

S b22?àg T

e

á

d u e

d b

;e17Twhere the vector g is assembled from element contributions

g

e

?B T

e

áA eás e:e18TThe derivative of the inplane displacement vector is ob-tained by di?erentiation of(6)as

K má

d u

d b

?à

d K m

d b

áu:e19TDe?ning the adjoint displacement vector v as the solution of the problem

K máv?àg:e20TThe strain term can be simpli?ed to

S b22?àv Tá

d K m

d b

áu e;e21Twhich can be also evaluated locally.

Thus,all the calculations required to evaluate the sensi-tivity of the buckling load with respect to local change of sti?ness of the i th element can be calculated using informa-tion at the element level.The global redistribution of loads is accounted for totally through the evaluation of the adjoint displacement vector v.Substituting the above sensi-tivity equations back to(10),d k

d b

can be written as

d k

d b

?k a T

i

á

d K b

i

d b

áa i

tk2s T

i

á

d A i

d b

áe itv T

i

á

d K m

i

d b

áu i

:e22T

Recalling Eqs.(8)and(9),the element sensitivities is decomposed into two separate bending and membrane parts as follows(see the Appendix for more details on the element matrices):

S.Setoodeh et al./Composite Structures87(2009)109–117111

/ebTab e

à

o k

o S e

ab

?k D br D qa a T

e

á

d K b

e

d D qr

áa e

e23T

and

/emTab e

à

o k

o R e

ab

?k2A br A qa s T

e

á

d A e

d A qr

áe etv T

e

á

d K m

e

d A qr

áu e

:

e24T

5.Reciprocal interpolation

An important requirement in the design optimization of variable-sti?ness panels is to ensure the continuity of the distribution of the?ber orientation angles.Traditionally, design variables are linked to element properties.This is the intuitive approach suggested by the use of the?nite ele-ment method for analyzing the response of a structure. When independent design variables are linked to element properties to simulate the variable-sti?ness concept,there is no guarantee that the distribution of the design variables is going to be smooth.This issue is well known and espe-cially severe in topology optimization,where it results in checkerboard patterns.

To ensure the smoothness of the optimal?ber orienta-tion angles distribution,following the methodology intro-duced in[1,20,22]is proposed here,where the design variables are associated with nodes rather than elements. For the construction of element matrices,the variation of the sti?ness properties over the element is required.The simplest rule is to use an average value over the element. The use of an average value simpli?es the construction of element matrices.It also simpli?es the calculation of sensi-tivities with respect to nodal variables.

The key-point in introducing average element properties is that instead of de?ning the element sti?ness tensor as the average of the sti?ness tensor at the nodes,the element compliance tensor is calculated as the average of the com-pliance tensors at the nodes.This approach,?rst intro-duced in[1,22]was shown to be e?ective in producing smooth distribution of design variables for topology opti-mization problems.In this paper,the work of[1,22]is expanded to the case of composite sti?ness properties design rather than density measure.

The average element compliances are given by

eR e;S eT?

X

c2I e

w e;ceR c;S cT;e25T

where superscript c denotes node numbers and I e is the set of nodes connected to element e.The sum is weighted by integration weighing coe?cients w c such that for a smooth function f,

Z X e f d X%

Z

X e

d X

X

c2I e

w e;c f c:e26T

The e?ective element sti?ness tensor is calculated as the in-verse of the average compliance tensor given by(25).6.Design update rule

The reciprocal interpolation scheme introduced in Sec-tion5?ts nicely with the generalized reciprocal approxima-tion introduced in Section3.In this section,an approximation of the eigenvalue of the panel is obtained in terms the nodal values of the compliance tensors R and S,and derive the corresponding optimality conditions.

It is presumed here that an initial design R c and S c are known together with the corresponding mode shape a. The eigenvalue of the system is expanded in a?rst order Taylor expansion

k% kt

o k

e

ab

eR e

ab

à R e

ab

Tt

o k

e

ab

eS e

ab

à S e

ab

T;e27TSubstituting the expressions for the eigenvalue sensitivity (23)and(24),and the interpolation formulas(25)into (27),the following compact form of the generalized reci-procal approximation in terms of nodal values is obtained:

k%e ktUemTc

ab

R c

ab

tUebTc

ab

S c

ab

TàUemTc

ab

R c

ab

àUebTc

ab

S c

ab

;e28Twhere

e/emTc;/ebTcT?

X

e2I c

w e;ce/emTe;/ebTeT;e29T

in which I c is the set of elements connected to node c.

The generalized reciprocal approximation(28)has the important property of being a separable approximation [11].Each term in the summation depends only on the com-pliance tensor at one node,and hence only on the?ber ori-entation angles at this node.In order to update the design for the next iteration,the reciprocal approximation of the eigenvalue is maximized.Because of separability,the max-imization can be carried out at each node separate from the others.This makes the algorithm particularly suited to par-allel computations.

The design update rule for the nodal values of the?ber orientation angles can thus be expressed as

min

h c

eUemTc

ab

R c

ab

eh cTtUebTc

ab

S c

ab

eh cTT...no sum on c;e30TIn numerical implementation of the update rule a move limit of2.5°is used to stabilize the iterative design process. Note that the design update involves a minimization prob-lem because of the minus sign in(28).

If the optimization is carried out for a constant-sti?ness panel,the compliance tensors of all nodes are identical,and the update rule is no longer dependent on the node num-ber.The?ber orientation angles are thus updated as

min

h c

X

c

eUemTc

ab

R abeh cTtUebTc

ab

S abeh cTT:e31T7.Multimodal design

The multimodal case can be handled in a similar fashion to the method employed by Olho?[19]where the problem is regularized by introducing an independent parameter b

112S.Setoodeh et al./Composite Structures87(2009)109–117

max b

s :t :b 6a i k i ;

e32T

where a 1?1,and a i 61;i ?2;3;....

The purpose of the a i coe?cients is to be able to impose mode spacing constraints.This can be particularly useful when modal interaction leads to unstable post buckling behavior.

The problem described by (32)can be solved using the dual method by Fleury [6].The Lagrangian is written as,

L ?X l i à1 b àX a i l i k i ;e33Twhere l i are Lagrange multipliers.The complementary Lagrangian is de?ned as the minimum over the design vari-ables of the Lagrangian.This leads to two conditions,

L C ?min àX

a i l i k i e34Tand X

l i ?1:

e35T

It is noted that the auxiliary variable b is now eliminated from the formulation.

The Lagrange multipliers are determined through the dual problem,max L C ;

e36T

subject to (35),and non-negativity of the Lagrange multi-pliers l i .The gradient of the complementary Lagrangian with respect to the Lagrange multipliers is readily available as

o L C

o l i

?àa i k i :e37T

The Lagrange multipliers can now be solved for by using any gradient based optimization routine to solve the dual problem.1This can be e?ciently accomplished since the number of the Lagrange multipliers is small (the number of considered modes).The evaluation of the complemen-tary Lagrangian requires the update of all design variables through the minimization problem (34).When the proce-dure is applied to the reciprocal approximation of the eigenvalues rather than the actual objective,the evaluation of the complementary Lagrangian is reduced to the solu-tion of a separable set of optimization problems similar to the uni-modal case.8.Numerical results

Based on the design formulation presented,a Fortran code was implemented.The design and analysis cycle is repeated until the relative change in the buckling load is smaller than a given tolerance as mentioned earlier.In order to stabilize this iterative process,a move limit of 2.5°is used in design update rule.The mode spacing coef-?cients in Eq.(32)are set to a i ?1:0;i ?1;2;3;...with three modes considered in all of the examples.The buckling load P of a rectangular plate of dimensions a ?b is non-dimensionalized using the following relation N ?12a 2

p 2Q 22h P ;

e38T

where h is the total laminate thickness and Q 22is the re-duced sti?ness transverse to the ?ber.In the following examples,symmetric rectangular plates are considered with di?erent boundary conditions and di?erent loadings (see Fig.1).

First design of single layer uni-directional constant-sti?-ness panels is considered based on the design update rule (31).Numerical results for design of panels with di?erent aspect ratios against shear buckling eN xx ?N yy ?0Tare given in Table 1.As this table shows,there is a good agr-ement between the present results and those reported by Grenestedt [9].The buckling loads for the Grenestedt results are computed using the present ?nite element code resulting in di?erences less than %0.15with the present optimal buckling loads.The di?erence in the optimal orien-tation angle is attributed to the fact that the buckling load

1

DNCONG of IMSL version 3.0is used in the present study.

Table 1

Design of constant-sti?ness single layer plates with di?erent aspect ratios for shear buckling eN xx ?N yy ?0Ta =b Present Grenestedt [9]%Di?erence

h °N xy h °N xy 1.044.7104.6545.0104.6à0.011.249.089.4849.489.480.001.553.679.1354.479.130.001.756.075.9956.675.980.012.057.373.4358.073.420.012.557.169.8357.869.820.023.058.968.2159.868.180.054.0

59.5

66.57

60.7

66.48

0.14S.Setoodeh et al./Composite Structures 87(2009)109–117113

is rather?at close to the optimal design[9]and di?erences in the analysis tools used in the two studies.

To validated the formulation for multilayer symmetric composite panels for uni-axial and bi-axial loading the results are compared to those found by Narita and Turvey [18],who implemented an optimization routine using a lay-erwise approach.As can be seen from Table2,the results correlate exceptionally well,di?ering no more that0.5%.

The results of the proposed approach are further com-pared to those of Erdal and Sonmez[4]who optimized a 64ply balanced symmetrical rectangular plate with simple

supports using simulated annealing.The optimal results found by the present formulation are compared to the best three designs found by Erdal and Sonmez in Table3.The di?erence in optimal buckling load of approximately3% can be attributed to the continuous nature of the presented formulation as opposed to the discrete set of ply angels used in Ref.[4].The last row shows the optimal buckling load when designing the plate using the closed form solu-tion of simply supported plates and lamination parameters [14].The results are obtained using the reciprocal approx-imation following the formulation presentated in[14].It is to be seen that the values of the optimum lamination parameters are quite di?erent from the corresponding val-ues at the continuous and discrete optima.On the other hand,the value of the optimum buckling load is fairly close.This can be explained by the relative insensitivity of the buckling load to changes in lamination parameters in certain directions[9].

Now to demonstrate the performance of the proposed design methodology for design of variable-sti?ness panels, design of a rectangular composite panel as depicted in Fig.1is considered with the following orthotropic material properties:

E11

E22

?25;

G12

E22

?0:5;m12?0:25:

Optimal buckling loads for single layer constant-sti?ness and variable-sti?ness designs are listed in Table4for sim-ply supported and clamped plates.The corresponding var-iable-sti?ness designs are depicted in Fig.2a and b. Optimization process for these examples was started from a design with all?bers in the90°direction(direction of the loading).Similar designs were also reported in Lund et al.[16].For both boundary conditions considered sub-stantial improvement in the buckling load of the variable-sti?ness panels with respect to their constant-sti?ness coun-terparts were observed.Also note that in the case of simply

Table2

Comparison of the critical buckling factor of an8-layer,symmetric constant-sti?ness square panel(a/b=1)

k h1h2h3h4

Uni-axial,simply supported

Present320.9744.96à44.93à44.93à44.93 Narita[18]321.0045.00à45.00à45.00à45.00 Uni-axial,clamped

Present687.56à0.01à0.01à0.01à0.01 Narita690.900.000.000.000.00 Bi-axial,simply supported

Present160.4944.92à44.96à44.96à44.96 Narita160.5045.00à45.00à45.00à45.00 Bi-axial,clamped

Present347.29à0.0990.0090.00à90.00 Narita348.000.0090.0090.0090.00 Table3

Comparison of critical buckling factor of a64-layer,simply supported symmetric constant-sti?ness rectangular panel(a/b=2)

k W1W3 Present4097.3à0.06690.0243 Erdal and Sonmez[4]

(1)?9010=?452=902=?453=902=?454 s3973.0à0.06190.0405

(2)?908=?45=902=?45=902=?45=902=?456 s3973.0à0.06190.0405

(3)?9010=?45=902=?457=902=?45

s 3973.0à0.06190.0405

Optimal lamination parameters(W1;W3)[14]4102.3à0.79480.2660Table4

Design of single layer square panels for uni-axial buckling;cs:constant-sti?ness;vs:variable-sti?ness(a=b?1,N xx?N xy?0:,30?30nodes) Design N1N2N3%Improvements Simply supported

cs?h 28.5050.98102.75–

vs?h 75.6876.6492.07165.58 Clamped

cs?h 106.87140.32208.52–

vs?h 136.46160.22203.76

27.70

114S.Setoodeh et al./Composite Structures87(2009)109–117

supported boundary conditions optimal buckling load shows coalescence of the?rst and second eigenloads.The iteration histories for single layer variable-sti?ness design of simply supported and clamped plates are depicted in Fig.3a and b respectively.Convergence is typically fast requiring less than50analysis iterations.In the case of modal coalescence of simply supported plates the optimiza-tion requires and increased number of iterations.

Next design of balanced symmetric??h

s panels with

only one design variable per node is considered.The con-

stant-sti?ness designs in this case are??45

s and?90

s

lay-

ups for simply supported and clamped plates respectively. The corresponding variable-sti?ness design are shown in Fig.2c and d for which the buckling loads are listed in Table5.

In all the cases presented in Fig.2the?ber orientation towards the center of the panel is perpendicular to the applied loading.This reduces the local extensional sti?ness at the center,resulting in the the majority of the load being carried by the region near the edges of the plate,where the boundary conditions suppress the out of plane displace-ments while reducing the compressive load at the center.

It is important to mention here that both the buckling maximization problem and the local minimization problem of Eq.(30)are not convex(as most design optimization problems when parameterized using?ber orientation angles)and hence the result of the optimization process is highly dependent on the starting?ber orientation angles. Such problems are often remedied by parametrizing the design using lamination parameters[7].It is interesting to note that the lamination parameters for all the optimal laminates found by Erdal et al.[4](see Table3)are identi-cal to each other.It is also clear from Table3that the buckling performance can be improved even further when optimizing using lamination parameters.The authors believe that using the approximate feasible lamination parameters obtained by Setoodeh et al.[23]it is possible to design panels for maximum buckling based on the pres-ent work and similar to the previous studies by the authors for compliance[24]and frequency[2]design of variable-sti?ness composite panels.

9.Conclusions

A reciprocal approximation is proposed to design vari-able-sti?ness panels for maximum buckling load.Fiber ori-entation angles at?nite element nodes are treated as design variables and exact sensitivity analysis was performed based on an adjoint formulation.The adjoint sensitivity analysis requires only one back substitution using an already factored left hand side(the inplane sti?ness used in the buckling analysis)with a di?erent right hand side to compute the sensitivities for all design variables.The approximation assumes a separable form and therefore

Table5

Design of balanced symmetric??h

s square panels for uni-axial buckling;

cs:constant-sti?ness;vs:variable-sti?ness(a=b?1;N xx?N xy?0:; 30?30nodes)

Design N1N2N3%Improvements Simply supported

cs??h

s 38.4347.6671.72–

vs??h

s

64.0673.65128.8166.71 Clamped

cs??h

s 106.87140.32208.52–

vs??h

s 139.28144.98190.7530.33

S.Setoodeh et al./Composite Structures87(2009)109–117115

can be easily solved at each node.The case of multimodal design is handled using a dual formulation of a bound formulation.

Numerical results are compared with existing constant-and variable-sti?ness results in the literature and show good agreement between the two.In general,signi?cant improvements in the buckling loads of variable-sti?ness plates are obtained as compared to constant-sti?ness designs.Moreover,the proposed generalized reciprocal approximation exhibits good convergence behavior.How-ever,due to the non-convexity of the design problem when parametrized using?ber orientation angles,results depend on the starting points.This problem can be potentially remedied by using lamination parameters as design vari-ables instead of the?ber orientation angles.Additionally by introducing lamination parameters the in-plane and out of plane sti?ness properties can be prescribed indepen-dently,allowing the in?uence of extensional sti?ness and ?exural sti?ness distribution to be investigated individu-ally.This will allow for a better understanding of the load redistribution mechanism responsible for an increased buckling load.

Appendix A

The bending sti?ness matrix according to the classical laminate theory is computed as follows[21]:

K b

e

?D11T xxxxtD12eT xxyytT yyxxTt2D16eT xxxytT xyxxT

t2D26eT xyyytT yyxyTt4D66T xyxytD22T yyyy;e39Twhere the elements are of T xxxx;T xxyy,etc.depend on the ele-ment geometry and bending shape functions u i

T ngfl ij ?

Z

X e

o2u i

o n o g

o2u j

o f o l

d x d yei;j?1;2;...;16T:e40T

Notice that d K b e

d D qr terms as appear in Eq.(23)can now b

e eas-

ily obtained,for example

d K b

e

d D11

?T xxxx;

d K b

e

d D12

?eT xxyytT yyxxT=2:

The inplane sti?ness matrix on the other hand,can be writ-ten in the following form:

K m

e ?

k11k12

sym:k22 "#

:

The above4?4sub-matrices are given in terms of the lam-inate extensional sti?ness A as follows[21]:

k11?A11S xxtA16eS xytS yxTtA66S yy;

k12?A12S xytA16S xxtA26S yytA66S yx; k22?A66S xxtA26eS xytS yxTtA22S yy;e41T

where the elements of the S xx;S xy;and S yy matrices depend

on the inplane shape functions w

i

and are de?ned as

S ng

ij

?

Z

X e

o w i

o n

o w j

o g

d x d yei;j?1;...;4T:

In this paper,only square bilinear elements are used with

?xed side length s.Therefore matrices S xx;S xy;T xxxx,etc.

are the same for all elements and can be computed and

stored.The d K m e

d A qr

terms appearing in Eq.(24)can be obtained similar to their bending counterparts.

Finally,the average strain displacement matrix as used in Eq.(5)for an square element with side s is computed

from

B e?

1

2s

à111à10000

0000à1à111

à1à111à111à1

2

64

3

75:e42T

References

[1]Abdalla MM,Gu¨rdal Z.Structural design using optimality based

cellular automata.Proceedings of the43rd AIAA/ASME/ASCE/

AHS structures,structural mechanics and materials conference.Den-

ver,CO:AIAA;2002.

[2]Abdalla MM,Setoodeh S,Gu¨rdal Z.Design of variable sti?ness

composite panels for maximum fundamental frequency using lami-

nation parameters.J Compos Struct2007;81(2):283–91.

[3]Banichuk NV,Saurin V,Barsuk AA.Optimal orientation of

orthotropic materials for plates designed against buckling.Struct

Opt1995;10:191–6.

[4]Erdal O,Sonmez FO.Optimum design of composite laminates for

maximum buckling load capacity using simulated https://www.360docs.net/doc/c34344001.html,pos

Struct2005;71:42–52.

[5]Evans DO,Vaniglia MM,Hopkins PC.Fiber placement process

study.In:34th international SAMPE symposium and exhibition,

May1989.p.1822–33.

[6]Fleury C,Schmit LAJ.Dual methods and approximation concepts in

structural synthesis.Technical Report CR-3226,NASA,December

1980.

[7]Foldager J,Hansen JS,Olho?N.A general approach forcing

convexity of ply angle optimization in composite laminates.Struct

Opt1998;16:201–11.

[8]Fukunaga H,Sekine H,Sato M,Iino A.Buckling design of

symmetrically laminated plates using lamination https://www.360docs.net/doc/c34344001.html,put

Struct1995;57(4):643–9.

[9]Grenestedt https://www.360docs.net/doc/c34344001.html,yup optimization against buckling of shear panels.

Struct Opt1991(3):115–20.

[10]Gu¨rdal Z,Olmedo R.In-plane response of laminates with spatially

varying?bre orientations:Variable sti?ness concept.AIAA J

1993;31(4):751–8.

[11]Haftka RT,Gu¨rdal Z.Elements of structural optimization(solid

mechanics and its applications).Kluwer Academic Publishers;

1992.

[12]Hyer MW,Charette https://www.360docs.net/doc/c34344001.html,e of curvilinear?ber format in

composite structure design.In:30th structures,structural dynam-

ics,and material conference.Mobile,AL:AIAA;April1989.p.

1011–15.

[13]Hyer MW,Lee HH.The use of curvilinear-?ber format to improve

buckling resistance of composite plates with central https://www.360docs.net/doc/c34344001.html,pos

Struct1991;18(3):239–61.

[14]IJsselmuiden S,Seresta O,Abdalla M,Gu¨rdal Z.Multi-step blended

stacking sequence design of panel assemblies with buckling con-

https://www.360docs.net/doc/c34344001.html,pos Part B,submitted for publication.

116S.Setoodeh et al./Composite Structures87(2009)109–117

[15]Liu B,Haftka RT,Trompette P.Maximization of buckling loads of

composite panels using?exural lamination parameters.Struct Mul-tidiscip Opt2004;26:28–36.

[16]Lund E,Ku¨hlmeier L,Stegmann J.Buckling optimization of

laminated hybrid composite shell structures using discrete material optimization.In:6th World Congress on Structural and Multidisci-plinary Optimization,Rio de Janeiro,June2005.

[17]Moruzzi M,Oldani T,Tatting BF,Gu¨rdal Z,Blom AW.Tailoring of

composite layup through tow-placement manufacturing techniques.

In:Proceedings of the SAMPE2006conference,Long Beach,CA, May2006.

[18]Narita Y,Turvey GJ.Maximizing the buckling loads of symmetri-

cally laminated composite rectangular plates using a layerwise optimization approach.Proc Ins Mech Eng Part C J Mech Eng Sci 2004;218:681–91.

[19]Olho?N.Multicriterion structural optimization via bound formula-

tion and mathematical programming.Struct Opt1989(1):11–7. [20]Rahmatalla S,Swan CC.Continuum topology optimization of

buckling-sensitive structures.AIAA J2003;41:1180–9.

[21]Reddy JN.Mechanics of laminate composite plates,theory and

analysis.CRC Press;1997.[22]Setoodeh S,Abdalla MM,Gu¨rdal https://www.360docs.net/doc/c34344001.html,bined topology and?ber

path design of composite layers using cellular automata.Struct Multidiscip Opt2005;30(6):413–21.

[23]Setoodeh S,Abdalla MM,Gu¨rdal Z.Approximate feasible regions

for lamination parameters.In:11th AIAA/ISSMO multidisciplinary analysis and optimization conference,Portsmouth,VA,September6–8,2006.

[24]Setoodeh S,Abdalla MM,Gu¨rdal Z.Design of variable-sti?ness

laminates using lamination https://www.360docs.net/doc/c34344001.html,pos Part B Eng 2006;195(9–12):836–51.

[25]Tatting BF,Gu¨rdal Z.Analysis and design of tow-steered variable

sti?ness composite laminates.In:AHS Meeting,Williamsburg,VA, October30–November1,2001.

[26]Tatting BF,Gu¨rdal Z.Design and manufacture of elastically tailored

tow placed plates.Technical Report CR-211919,NASA,August 2002.

[27]Walker M,Adali S,Verijenko V.Optimization of symmetric

laminates for maximum buckling load including the e?ects of bending-twisting https://www.360docs.net/doc/c34344001.html,put Struct1996;58(2):313–9.

S.Setoodeh et al./Composite Structures87(2009)109–117117