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  • American Institute of Aeronautics and Astronautics

    1

    Local optimisation of long anisotropic laminated fibre composite

    panels with T shape stiffeners

    J. Enrique Herencia

    * and Paul M. Weaver

    University of Bristol,

    Bristol BS8 1TR, UK

    Mike I. Friswell

    University of Bristol,

    Bristol BS8 1TR, UK

    A method to locally optimise long anisotropic laminated fibre composite panels with T

    shape stiffeners is presented. The technique splits the optimisation problem into two levels.

    At the first level, composite optimisation is performed using mathematical programming

    (MP), and the skin and stiffeners are modelled using lamination parameters that account for

    their membrane and flexural anisotropy. Skin and stiffener laminates are assumed to be

    symmetric, or mid-plane symmetric laminates, with 0, 90, 45, or -45 degree, ply angles. The

    skin-stiffener configuration is further idealised as a group of flat plate laminates that are

    rigidly connected. The panel is subjected to a combined case of loading under strength,

    buckling and manufacturing constraints. At the second level, the actual skin and stiffener

    lay-ups are obtained using a genetic algorithm (GA) and considering the ease of

    manufacture. This approach offers the advantage of introducing accurate analysis methods

    such as finite elements at the first level, without significant increases in processing time.

    Furthermore modelling the laminate anisotropy enables the designer to explore and

    potentially use elastic tailoring in a beneficial manner.

    Nomenclature

    a = panel length

    A = membrane stiffness matrix, membrane

    Aij = terms of the membrane stiffness matrix, i=j=1,2,6

    Asf = area of the stiffener flange

    Askin = area of the skin

    Astg = area of the stiffener

    Asw = area of the stiffener web

    b = panel width

    bsf = stiffener flange width

    B = membrane-bending coupling stiffness matrix

    c = continuous

    C = compression

    d = discrete

    D = bending stiffness matrix, bending

    Dc = bending stiffness of a skin-stiffener element

    Dij = terms of the bending stiffness matrix, i=j=1,2,6

    Eij = Youngs modulus in ij direction, i=j=1,2

    EIc = longitudinal stiffness of a skin-stiffener element

    Fc = longitudinal force applied at skin-stiffener centriod

    * Marie Curie Research Assistant, Department of Aerospace Engineering, Queens Building,

    AIAA Student Member. Reader, Department of Aerospace Engineering, Queens Building, AIAA Member. Sir George White Professor of Aerospace Engineering, Department of Aerospace Engineering, Queens Building,

    AIAA Member.

    47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Confere1 - 4 May 2006, Newport, Rhode Island

    AIAA 2006-2171

    Copyright 2006 by J.E. Herencia, P.M. Weaver and M.I. Friswell. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

  • American Institute of Aeronautics and Astronautics

    2

    Gij = shear modulus i=1, j=2

    Gi = ith design constraint swxyG = shear modules of the stiffener web

    h = laminate thickness

    hsw = stiffener web height

    Ki = buckling coefficient in i direction, i=x,xy

    M = mass of a skin-stiffener element, running moment vector

    MS = middle-plane symmetric

    N = running load vector

    Ni = load per unit of width in i direction, i=x,y,xy

    Nicr = critical load per unit of width, i= x,xy,wx,sh

    Pcr = critical buckling load of a column accounting for transverse shear load

    Pe = Euler buckling load

    Qij = terms of the reduced stiffness matrix, i=j=1,2,6

    RFij = reserve factor, i=px,pxy,pb,wb,cb,sb,cs,T,C; j=x,y,xy

    S = symmetric

    Skn = skin

    Stg = stiffener flange and web

    Stf = stiffener flange

    Stw = stiffener web

    t = thickness of the skin

    ta = thickness of the stiffener flange

    tsf = total thickness of the stiffener flange

    tsw = total thickness of the stiffener web

    tw = thickness of the stiffener web

    T = tension

    Ui = material invariants, i=1...5

    Wi = weight, i=c,d

    wfij = weighting factor for lamination parameters, i=1,2,3; j=A,D

    xr = vector of design variables, abscise

    yr = gene, ordinate

    ,,, = non-dimensional parameters

    = middle-surface curvatures i

    j = applied strain, i=T,C; j=x,y,xy

    aij = allowable strain, i=T,C; j=x,y,xy

    i = eigenvalue, i=b,s

    = tolerance value

    i = Poisons ration in i direction, i=12,21

    = density

    ij = lamination parameters, i=1,2,3; j=A,D

    = fibre orientation angle , , = encoded ply angle for the skin, the stiffener flange and web, respectively

    I. Introduction

    he use of composite materials as primary structures in the commercial aviation industry, has been gradually

    increasing over the last decade. This has culminated in programmes such as the Airbus A3501 and the Boeing

    7872, where composite materials will play a major role. Primary flight composite structures such as wings or

    fuselages are mainly designed using stiffened panels. In general, composite materials present high specific strength

    and stiffness ratios3. Furthermore, structures made of composite materials can be stiffness tailored potentially

    offering a significant advantage over their metallic counterparts. This latter feature is intimately related to their

    design and manufacture. Due to practical, yet often limiting, manufacturing considerations, laminated fibre

    composite panels have been restricted to symmetric, or middle plane symmetric laminates, with 0, 90, 45, and -45

    degree, ply angles. The manufacture of the T shape stiffeners adds an additional degree of complexity since it allows

    the modification of the stiffener web and flange by adding extra and capping plies, respectively. This paper is

    T

  • American Institute of Aeronautics and Astronautics

    3

    inspired by the desire to design elastically tailored stiffened composite panels that consider manufacturing

    requirements, commonly used in the aerospace industry.

    Over the years, optimisation techniques have been developed to assist engineers with composite design4-33. The

    nature of composite optimisation is non-linear. In the seventies, early attempts on optimisation of laminated fibre

    composites were performed by Schmit and Farshi4,5. They optimised symmetric laminated fibre composite materials

    having homogeneous and orthotropic properties, considering the ply thicknesses as continuous variables. They

    transformed the non-linear problem into a sequence of linear problems. In the same light, Stroud and Agranoff6

    optimised composite hat-stiffened and corrugated panels using non-linear mathematical techniques with a simplified

    set of buckling equations as constraints. The width and thickness of the elements of the dimensioned cross section

    were the design variables. They assumed that the laminates were orthotropic. However, although designed carefully

    composites might exhibit some degree of flexural anisotropy. Ashton7 initially showed the effect of the flexural

    anisotropy on the stability of composite plates. Chamis8 concluded that neglecting flexural anisotropy of the

    composite in the evaluation of buckling behaviour could lead to non-conservative results. Later, Nemeth9

    characterised the importance of flexural anisotropy and provided bounds within which its effect would be

    significant. Recently, Weaver10 developed closed form solutions to quantify the effect of flexural anisotropy on

    compression loads. Flexural anisotropy is intrinsically related to laminate stacking sequence. The addressing of the

    laminate stacking sequence and hence the identification of the number of plies of each fibre orientation, converts the

    lay-up optimisation problem into a non-linear problem with discrete variables which a non-convex design space.

    Tsai and Pagano11 and Tsai and Hann

    12 gave an alternative representation of the stiffness properties of a

    laminated fibre composite panel by the use of lamination parameters. Miki and Sugiyama13 proposed the use of

    lamination parameters to deal with the discrete laminate stacking sequence problem. They assumed symmetric and

    orthotropic laminates. Optimum designs for the required in-plane stiffness, buckling strength, and so on, were

    obtained, from geometry relations between the lamination parameters feasible region and objective function.

    Fukunaga and Vanderplats14 used lamination parameters and MP techniques to perform stiffness optimisation of

    orthotropic laminated composites. Cylindrical shells under combined loading were used as a practical application.

    Haftka and Walsh15 used integer programming techniques to carry out laminate stacking sequence optimisation

    under buckling constraints on symmetric and balanced laminated plates. They used zero-one integers as design

    variables that were related to stiffness properties via lamination parameters and showed that the problem was linear.

    Flexural anisotropy was limited to manually modifying the optimum design and they used the branch and bound

    method to solve the problem. Nagendra, Haftka and Grdal16 extended the previous work and optimised the stacking

    sequence of symmetric and balanced composite laminates with stability and strain constraints. Unfortunately, integer

    programming techniques require large computational resources especially when structure complexity increases.

    Fukunaga and Sekine17 presented an approach to maximise buckling loads under combined loading of symmetrically

    laminated plates including the bending-twisting couplings or flexural anisotropy. They employed MP techniques and

    the lamination parameters as design variables. They confirmed the detrimental effect of the flexural anisotropy on

    the buckling load of panels under normal loading and highlighted that under shear and shear-normal loading flexural

    anisotropy could increase or decrease the critical buckling load. Although an optimal laminate stacking sequence

    with optimal fibre orientation was presented, neither discrete nor practical laminates were shown.

    A different strategy was adopted by Le Riche and Haftka18 and later by Nagendra, Haftka and Grdal

    19,20. They

    employed GAs to solve the integer stacking sequence problem. GAs are search algorithms based on the mechanics

    of natural selection and natural genetics21, which do not require gradient information to perform the search. GAs are

    widely used for their ability to tackle search spaces with many local optima22 and therefore a non convex design

    space. Nagendra also investigated the application of a GA to the design of blade stiffened composite panels.

    VIPASA23 was used as the analysis tool and results were compared with PASCO

    24, which uses VIPASA as the

    analysis tool and CONMIN25 as optimiser. It was concluded that the designs obtained by the GA offered higher

    performance than the continuous designs. However, it was recognised that great computational cost was associated

    with the GA.

    Yamazaki26 initially proposed a two level optimisation strategy combining lamination parameters-MP and GAs.

    The optimisation was split into two parts. Firstly, a gradient based optimisation was performed using the in-plane

    and out-of-plane lamination parameters as design variables. Secondly, the lamination parameters from the first level

    were targeted using a GA. In this paper, the volume, buckling load, deflection and natural frequencies of a

    composite panel were optimised without accounting for either membrane or flexural anisotropy. Autio27 following a

  • American Institute of Aeronautics and Astronautics

    4

    similar approach, investigated actual lay-ups. The approach adopted was similar to Yamazaki, with the difference

    being that commercial uni/multiaxil plies were considered and certain lay-up design rules were introduced as

    penalties in the fitness function of the GA code. Earlier, Todoroki and Haftka28 proposed a more sophisticated

    approach. They divided the optimisation into two stages. First, the lamination parameters were used in a continuous

    optimisation to identify the neighbourhood of the optimum design. Subsequently a response surface approximation

    was created in that neighbourhood and the GA was applied to that approximation. They applied this procedure to

    buckling load maximisation of a composite plate. However, with the exception of Fukunaga and Vanderplats14, none

    of the previous authors considered the feasible region in the lamination parameter space that relates in-plane,

    coupling and out-of-plane lamination parameters.

    Liu, Haftka and Trompette29 employed lamination parameters and defined the feasible region between two of the

    four membrane and bending lamination parameters to maximise the buckling load of unstiffened composite panels

    with restricted ply angles. They compared their approach against one using a GA and concluded that the use of

    lamination parameters in a continuous optimisation produced similar results to those obtained by the GA except in

    cases where laminates were thin or had low aspect ratios. Furthermore, Liu and Haftka30 proposed a single level

    weight minimisation of composite wing structures using flexural lamination parameters. They assumed continuous

    thicknesses for a set of restricted ply angles and two flexural lamination parameters, as design variables. Their

    constraints were strength, buckling and the flexural lamination parameters domain. The wing consisted of several

    unstiffened composite panels with orthotropic properties. They compared their work against a two level wing

    optimisation strategy using GAs and concluded that both approaches produced similar results and that the single

    level approach provided a lower bound to the true optima.

    Diaconu and Sekine31 performed lay-up optimisation of laminated composite shells for maximisation of the

    buckling load, using the lamination parameters as design variables and including their feasible region. They fully

    defined, for the first time, the relations between the membrane, coupling and bending lamination parameters for ply

    angles restricted to 0, 90, 45, and -45 degrees, in order to identify their feasible region within the design space. Note

    that although developed independently from each other, their definition of the feasible region for the lamination

    parameters was consistent with the one provided by Liu and Haftka30 (only defined for two membrane and bending

    lamination parameters with ply angles restricted to 0, 90, 45, and -45 degrees).

    The aim of the present paper is to provide an approach to locally optimise long anisotropic laminated fibre

    composite panels with T shape stiffeners. The technique splits the optimisation problem into two levels. At the first

    level, composite optimisation is performed using MP, the skin and stiffeners are modelled using lamination

    parameters accounting for their membrane and flexural anisotropy. Skin and stiffener laminates are assumed to be

    symmetric, or mid-plane symmetric laminates, with 0, 90, 45, or -45 degree, ply angles. The skin-stiffener

    configuration is further idealised as a group of the flat laminated plates rigidly connected. The panel is subjected to a

    combined case of loading under strength, buckling and manufacturing constraints. At the second level, the actual

    skin and stiffener lay-ups are obtained using a GA and considering the ease of manufacture. This approach offers the

    advantage of introducing accurate analysis methods such as finite elements at the first level, without significant

    increases in processing time. Furthermore, modelling the laminate anisotropy enables the designer to explore and

    potentially use elastic tailoring in a beneficial manner. The novelty of the current approach is based upon: the

    inclusion of membrane and flexural anisotropy for elastic tailoring purposes; manufacturing constraints; practical

    design constraints on lay-ups and the interaction between membrane and flexural lamination parameters for stiffened

    panels.

    II. Panel geometry and loading

    The composite stiffened panel is assumed to be wide and composed of a series of skin-stiffener configurations or

    panels. Each local skin-stiffener panel consists of three flat plates that will be considered to be rigidly connected (all

    degrees of freedom match at the interface), corresponding to the skin, stiffener flange and web, respectively. The

    behaviour of the panel can be modelled by a skin-stiffener repeating element. Figure 1 defines the skin-stiffener

    geometry, the material axis, and the positive sign convention for the loading.

    The geometry of the stiffener is affected by its design and manufacturing process. For this study four different

    stiffener configurations were considered. The stiffener is manufactured as a back to back angle (Fig. 2a), considering

  • American Institute of Aeronautics and Astronautics

    5

    capping plies in the flange (Fig. 2b), adding extra plies at the stiffener web (Fig. 2c), and finally the combination of

    the previous configurations (Fig. 2d).

    Figure 1. Skin-stiffener configuration.

    Figure 2. T-shape stiffener types.

    b

    a

    Fc

    Ny

    Nxy

    x

    y

    0o

    45o90

    o

    A

    A'

    Stiffener

    Skin

    bsf

    hsw

    tsw

    t

    tsf

    Skin

    Stiffener web

    Stiffener flange

    Section AA'

    twa

    b

    c

    d

    ta

    ta

  • American Institute of Aeronautics and Astronautics

    6

    III. Laminate constitutive equations

    Laminate constitutive equations for the skin, stiffener flange and web, respectively, are obtained by applying the

    classical laminate theory (CLT)3 to each of them. Thus,

    =

    o

    DB

    BA

    M

    N (1)

    The above properties can be expressed in terms of material stiffness invariants (U) and twelve lamination

    parameters ()11,12

    . As laminates are considered to be symmetric or mid-plane symmetric, the membrane-bending

    coupling matrix (B) will vanish. This also reduces the number of the lamination parameters to eight. In addition, as

    plies are assumed to be orthotropic, with only 0, 90, 45, and -45 degree, fibre angles, the lamination parameters are

    further decreased to six. The expressions for the membrane and bending stiffness terms are,

    =

    5

    4

    3

    2

    1

    3

    3

    2

    21

    2

    21

    26

    16

    66

    22

    12

    11

    0002

    0

    0002

    0

    1000

    001

    0100

    001

    U

    U

    U

    U

    U

    h

    A

    A

    A

    A

    A

    A

    A

    A

    A

    AA

    A

    AA

    (2)

    =

    5

    4

    3

    2

    1

    3

    3

    2

    21

    2

    21

    3

    26

    16

    66

    22

    12

    11

    0002

    0

    0002

    0

    1000

    001

    0100

    001

    12

    U

    U

    U

    U

    U

    h

    D

    D

    D

    D

    D

    D

    D

    D

    D

    DD

    D

    DD

    (3)

    The material stiffness invariants (U) are given as follows,

    =

    66

    22

    12

    11

    5

    4

    3

    2

    1

    4121

    4161

    4121

    0404

    4323

    8

    1

    Q

    Q

    Q

    Q

    U

    U

    U

    U

    U

    (4)

    The lamina stiffness properties (Q) are related to the ply Youngs modulus and Poisson ratio by the following

    equations,

    2112

    1111

    1 =

    EQ (5)

  • American Institute of Aeronautics and Astronautics

    7

    2112

    221212

    1

    =E

    Q (6)

    2112

    2222

    1 =

    EQ (7)

    1221 QQ = (8)

    1266 GQ = (9)

    11

    221221

    E

    E= (10)

    The membrane and bending lamination parameters are calculated, respectively, by the following integrals,

    [ ] [ ] dzh

    h

    h

    A = 2

    2

    321 2sin4cos2cos1

    (11)

    [ ] [ ] dzzh

    h

    h

    D = 22

    23321

    2sin4cos2cos12

    (12)

    IV. Local optimisation strategy

    The local optimisation strategy is shown in Fig. 3. It is divided into two stages. At the first stage, the skin-

    stiffener panel is optimised using lamination parameters and gradient based techniques. Values of the lamination

    parameters for an optimum skin-stiffener design are obtained. At the second stage, a GA is used to target the

    optimum lamination parameters to obtain the actual and manufacturable stacking sequence for both skin and

    stiffener.

    A. First level-Gradient based optimisation At this level a non-linear constrained local optimisation is performed. The basic mathematical optimisation

    problem can be expressed as follows,

    Minimise )(xMr Objective function (13)

    Subject to 0)( xG jr

    Gnj ,....,1= Inequality constraints (14)

    uii

    li xxx ni ,....,1= Side constraints (15)

    where { }nxxxx ,....,, 21=r

    Design variables (16)

    In this case the objective function is the mass of the stiffened panel per unit of width (or skin-stiffener panel), the

    inequality constraints are strength, local and global buckling, as well as practical design requirements. The design

    variables are the thicknesses of the skin, stiffener flange and web as well as their related membrane and bending

    lamination parameters, depending on the stiffener type. The side constraints are the bounds of those design

    variables. MATLAB32 is employed to conduct the gradient based optimisation.

  • American Institute of Aeronautics and Astronautics

    8

    Figure 3. Optimisation flow chart.

    1. Objective function

    The objective function is the mass of the skin-stiffener panel. The mass as a function of the design variables,

    materials properties and geometry is given by,

    ( ))()()( xAxAaxM stgstgskinskin rrr += (17)

    where the skin and the stiffener area respectively, are defined as follows,

    btAskin = (18)

    swswsfsfswsfstg htbtAAA +=+= (19)

    2. Design variables

    The design variables for a skin-stiffener panel, depending on the stiffener type are listed in Table 1.

    For stiffener types a, c and d, sub-laminates are employed to calculate the membrane and bending stiffness

    properties for the stiffeners web (stiffener types a, c and d) and flange (stiffener type d), respectively.

    1 Gradient based

    optimisation

    Loads GeometryMaterial

    properties

    2 GA

    optimisation

    1 level

    Constraints

    check

    NO

    YES

    Final local

    design

    1 level Constraints

    2 level Constraints

  • American Institute of Aeronautics and Astronautics

    9

    Table 1. Table of design variables.

    Design variables

    xr

    Stiffener type

    (ST)

    Stiffener

    configuration

    Skin Stiffener

    flange

    Stiffener

    web

    h

    [ ]DA,

    321

    ta bsf

    [ ]DA,

    321

    hsw

    a

    ht =

    asf tt =

    asw tt = 2

    b

    As stiffener type a, knowing that

    asf tt =

    asw tt =

    h

    [ ]DA,

    321

    ta bsf

    [ ]DA,

    321

    tw

    hsw

    [ ]DA,

    321

    c

    ht =

    asf tt =

    wasw ttt += 2

    d

    As stiffener type c, knowing that

    asf tt = 2

    3. Design constraints

    Four sets of design constraints are considered in the local optimisation of the skin-stiffener panel. The following

    sections describe those constraints in detail.

    1) Lamination parameter constraints It is shown (e.g. Ref. 13) that the lamination parameters either in membrane, coupling or bending must be bound.

    For a symmetric or mid-plane symmetric laminate with ply angles limited to 0, 90, 45 and -45 degrees, the

    expressions for the membrane and bending lamination parameter constraints are given by13,31

    ,

    012 ,2,

    1 DADA (20)

    012 ,2,

    3 +DADA (21)

    Additional constraints between the membrane, coupling and bending lamination parameters were introduced by

    Diaconu and Sekine31. Their aim was to define the feasible region in the lamination parameter design space with

    special consideration given to the compatibility between the membrane, coupling and bending lamination

  • American Institute of Aeronautics and Astronautics

    10

    parameters. Since there is a dependency between the membrane, coupling and bending properties, it is this

    compatibility which enables the production of laminate designs that possess consistent properties. Accounting for

    symmetric laminates and rearranging terms those expressions are,

    ( ) ( ) ( ) 01141 4 AiDiAi 3,2,1=i (22) ( ) ( ) ( ) 01141 4 +++ AiDiAi 3,2,1=i (23) ( ) ( ) ( ) 012121612 2121421 AADDAA (24) ( ) ( ) ( ) 012121612 2121421 ++++++ AADDAA (25) ( ) ( ) ( ) 032321632 2121421 +++ AADDAA (26) ( ) ( ) ( ) 032321632 2121421 +++ AADDAA (27) ( ) ( ) ( ) 012121612 2323423 +++ AADDAA (28) ( ) ( ) ( ) 012121612 2323423 +++ AADDAA (29) ( ) ( ) ( ) 032321632 2323423 AADDAA (30) ( ) ( ) ( ) 032321632 2323423 ++++++ AADDAA (31) ( ) ( ) ( ) 01141 3131431 AADDAA (32) ( ) ( ) ( ) 01141 3131431 ++++++ AADDAA (33) ( ) ( ) ( ) 01141 3131431 +++ AADDAA (34) ( ) ( ) ( ) 01141 3131431 +++ AADDAA (35)

    The above constraints are imposed on the skin, stiffener flange and web laminates, respectively.

    2) Strength constraints Strength constraints are introduced to limit the magnitude of strains in tension, compression and shear taken by

    the laminate. This is conducted in terms of allowable strains. Strains in x, y and xy directions are restrained. The

    strains, under the applied in-plane loads, are calculated using CLT3. Hence,

    =

    xy

    y

    x

    oxy

    oy

    ox

    N

    N

    N

    AAA

    AAA

    AAA1

    662616

    262212

    161211

    (36)

    A reserve factor or ratio between the allowable and applied strain, is defined as

    ji

    jaij

    iRF

    = CTi ,= ; xyyxj ,,= (37)

  • American Institute of Aeronautics and Astronautics

    11

    where T and C denote tension and compression, respectively. The strain constraints for both tension and

    compression take the following expressions.

    01 jiRF CTi ,= ; xyyxj ,,= (38)

    These constraints are applied to the skin, stiffener flange and web laminates, respectively.

    3) Buckling constraints Buckling constraints are assessed in terms of local buckling (failure of the skin, the stiffener web or the local

    skin-stiffener interaction) and global buckling (failure of the stiffened panel) criteria. Local and global buckling

    constraints on anisotropic composite stiffened panels are considered using analytical (closed form solutions) and

    numerical (finite elements) methods.

    Closed Form (CF) solutions

    This section describes closed form solutions used to assess the local and global buckling behaviour of an

    anisotropic composite stiffened panel.

    Local buckling

    Local buckling of the skin and the stiffener, are considered separately assuming no interaction between them.

    Buckling of the skin

    The skin is assumed to be a long flat plate simply supported along the edges under normal and shear load.

    Weaver10,33

    has recently provided a comprehensive set of closed form solutions for long flexural anisotropic plates

    under compression and shear loading. Note that in addition to presenting a closed form solution for the uniaxial

    compression case, Weaver10 details a procedure to identify exactly the critical uniaxial compression load. Non-

    dimensional parameters were used to calculate buckling coefficients, following Nemeth9, in order to obtain the

    critical buckling load. The non-dimensional parameters (e.g. Ref. 9) are defined in terms of the flexural stiffness as

    follows,

    4

    11

    22

    D

    D= ,

    2211

    6612 2

    DD

    DD += ,

    422

    311

    16

    DD

    D= , and

    4 32211

    26

    DD

    D= (39)

    Normal buckling

    Weaver10 approximated the critical buckling load of a long anisotropic plate with simply supported conditions

    along the edges and under normal loading as follows,

    22112

    2

    DDb

    KN xcrx

    = (40)

    where Kx is a non-dimensional buckling coefficient given by,

    ( ) ( )( )( )

    ( )( )( )3

    33

    2

    22

    3

    324

    3

    323212

    +

    ++

    +

    ++++=

    xK (41)

    When and < 0.4, Kx is expected to give sufficient accuracy. For laminates with and > 0.4 an iteration scheme to calculate Kx is applied. For further details see Ref. 10.

    The reserve factor for the uniaxial compression loading is given by,

  • American Institute of Aeronautics and Astronautics

    12

    x

    crx

    pxN

    NRF = (42)

    Shear buckling

    Weaver33 defined the shear buckling coefficient in terms of the non-dimensional parameters as,

    ( ) ( )22 225.0236.089.679.113.005.242.3 ++++=xyK (43)

    The critical shear buckling load has the following expression,

    4 322112

    2

    DDKb

    N xycrxy

    = (44)

    The reserve factor for the shear loading is thus,

    xy

    crxy

    pxyN

    NRF = (45)

    In the case of negative shear the shear buckling coefficient is calculated assuming that each ply angle is reversed

    in sign. This is the same as changing the sign of the non-dimensional parameters and .

    Normal-shear bucking interaction

    The following formula34 is used to address the interaction,

    2)(

    111

    pxypxpb RFRFRF+= (46)

    The local buckling of the skin in terms of constraints is given by

    01 pbRF (47)

    Buckling of the stiffener web

    The stiffener web is assumed to be a long flat plate simply supported along three edges (two short edges and one

    long edge) and one edge free (long edge) under normal load. Herencia and Weaver35 have recently developed a

    closed form solution for this loading case that includes the effects of flexural anisotropy. In this case, the critical

    buckling load is given by,

    =

    22

    226

    662

    12

    D

    DD

    bN

    crwx (48)

    Note that when there is no flexural anisotropy the above formula reduces to the orthotropic expression36.

    The reserve factor for the stiffener web instability is given by,

  • American Institute of Aeronautics and Astronautics

    13

    wx

    crwx

    wbN

    NRF = (49)

    Global buckling

    Global panel buckling is considered by evaluating column buckling accounting for the shearing force induced at

    the stiffener web, panel shear buckling and the interaction column-shear buckling.

    Column buckling

    The panel is assumed to behave as a wide column with pinned ends. The critical buckling load for this case

    accounting for the shearing force induced at the stiffener web during buckling6,37 is given by,

    swxysw

    e

    ecr

    GA

    P

    PP

    +

    =1

    (50)

    where

    2

    2

    a

    EIP ce

    =

    (51)

    The reserve factor between the critical column load and applied load is defined as,

    c

    crcb

    F

    PRF = (52)

    Shear buckling

    The stiffened panel is assumed to be infinitely long with simply supported conditions along the long edges6. The

    critical shear load is derived from Ref. 33 considering that the width of the plate (b) coincides with the length of the

    stiffened panel (a) and the longitudinal and transversal bending stiffness of the stiffened panel are given,

    respectively by,

    2211 DD = (53)

    b

    EIDD cc ==22 (54)

    Substituting the above expressions into Eq. (44), the non-dimensional parameters can be calculated, and thus a

    new shear buckling coefficient shK is found. Hence, the critical shear load is as follows,

    422

    3

    2

    2

    DDKa

    N cshcrsh

    = (55)

    The reserve factor between the critical and applied shear load is defined as,

  • American Institute of Aeronautics and Astronautics

    14

    xy

    crsh

    sbN

    NRF = (56)

    In the case of negative shear, the shear buckling coefficient is calculated assuming that each ply angle is reversed

    in sign, which is equivalent to changing the sign of the non-dimensional parameters and .

    Column-shear buckling interaction

    An interaction formula6 is used to address the overall buckling constraints. Hence,

    2)(

    111

    shcbcs RFRFRF+= (57)

    The global buckling of the stiffened panel in terms of constraints is given by

    01 csRF (58)

    Finite Element (FE) Analysis

    MSC/NASTRAN is used to perform linear buckling analysis (SOL 105)38. The skin-stiffener section is modelled

    using quadrilateral elements of 4 nodes (CQUAD4). A minimum of five nodes are used per half wave length38.

    PSHELL and MAT2 cards are utilised to idealise the skin and stiffener flange and web respectively, in terms of their

    membrane and bending properties. Rigid body elements (RBE2s) are employed to simulate rigid connections and to

    account for the offsets between the skin and the stiffener flange as well as the stiffener flange and web, respectively.

    The skin-stiffener section is assumed to be simply supported along the short edges and restrained in rotations along

    the long edges. This rotation provides symmetry conditions and simulates that the panel consists of several skin-

    stiffener repetitive elements. Normal loading is introduced via RBE2 elements, whereas transverse and shear loading

    are applied by nodal forces. This FE modelling technique captures both local and global buckling behaviour of an

    anisotropic stiffened panel. Figures 4-5 show in detail the features of the FE modelling set up.

    Figure 4. FEM boundary constraints.

    Rotation in x is

    constrained along the long edges

    (Symmetry condition)

    Displacement in z is

    constrained along the short edges

    (Simply supported condition)

    Rotation in x is

    constrained along the long edges

    (Symmetry condition)

    Displacement in z is

    constrained along the short edges

    (Simply supported condition)

  • American Institute of Aeronautics and Astronautics

    15

    Figure 5. FEM normal and shear loading.

    4) Practical design constraints Niu

    39 provided a comprehensive summary of design practices for composites. A reduced set of those rules are

    used as design constraints. For a composite panel configuration the design considerations are addressed by limiting

    the percentages of 0, 90, 45, and -45 degree, ply angles, the skin-stiffener flange Poissons ratio mismatch and skin

    gauge. The design constraints are described in the following sections.

    Percentages of ply angles

    At least 10% of each ply orientation should be provided39. Maximum and minimum percentages of the ply

    angles for the skin, stiffener flange and web are limited. The percentages of the 0, 90, 45, and -45 degree, ply angles

    for each of those elements are,

    1002

    =h

    tp ii 45,45,90,0 =i swsf ttth ,,= (59)

    The maximum and minimum allowable ratios are given, respectively, by,

    i

    ipi

    p

    pRF

    maxmax = (60)

    and min

    min

    i

    ipi

    p

    pRF = (61)

    The maximum and minimum percentages of 0, 90, 45, and -45 degree, ply angles are implemented in terms of

    design constraints as follows,

    Shear loading

    (Nodal forces)

    Normal loading

    (RBE2s Force @

    skin-stiffener centriod)

    Offsets

    (RBE2s Skin-stiffener flange

    & Stiffener flange-web)

    Shear loading

    (Nodal forces)

    Normal loading

    (RBE2s Force @

    skin-stiffener centriod)

    Offsets

    (RBE2s Skin-stiffener flange

    & Stiffener flange-web)

  • American Institute of Aeronautics and Astronautics

    16

    01max piRF (62)

    011min

    piRF

    (63)

    Skin-stiffener flange Poissons ratio mismatch

    The reduction of the Poissons ratio mismatch is critical in composite bonded structures39. The difference

    between the skin and stiffener flange Poissons ratio is limited by a small number to reduce the mismatch. An

    acceptable value of is assumed to be 0.05.

    The Poissons ratio mismatch design constraint between the skin and stiffener flange is given by,

    0 stgfxyskinxy (64)

    Skin gauge

    The minimum skin gauge is determined by the danger of a puncture due to lightning strike. Niu39 suggests a

    minimum skin thicknesses of 3.81 mm. Skin gauge is addressed in terms of maximum and minimum skin thickness.

    The maximum and minimum skin thickness ratios are given by,

    t

    tRFt

    maxmax = (65)

    and min

    min

    t

    tRFt = (66)

    Thus, the design constraints for the skin gauge are given by,

    01 max tRF (67)

    011min

    tRF

    (68)

    4. Sensitivities

    When finite elements are used to provide the buckling constraints, sensitivities40,41

    are supplied to MATLAB32 to

    decrease the number of FE runs and to speed up the optimisation process. Buckling sensitivities are computed in

    MSC/NASTRAN using the design sensitivity and optimisation solution (SOL 200)40. Strength, lamination parameter

    and practical design constraints sensitivities are calculated by the forward finite difference approximation given by

    ( ) ( )j

    iji

    j

    i

    x

    xGxxG

    x

    xG

    +=

    rrr)(

    (69)

    where jx is a small perturbation applied to the jth design variable. After a trial error exercise, a suitable step size

    for the perturbation was determined as 0.0001.

    B. Second level-GA based optimisation A standard GA

    22,42 is employed at this level to solve the lay-up discrete optimisation problem. The lamination

    parameters from the first optimisation level are targeted to obtain the laminate stacking sequence for the skin, the

  • American Institute of Aeronautics and Astronautics

    17

    stiffener flange and web, respectively. The GA is applied separately to the skin, stiffener flange and web. The

    structure of a standard GA is well reported in the literature18-22,42

    .

    5. Fitness function

    The fitness function is expressed in terms of the square difference between the optimum and the targeted

    lamination parameters31. Extra penalty terms are added to account for design rules such as maximum plies of the

    same orientation15-16,18-20,42

    . Thus,

    ( ) ( ) kki

    Diopt

    Di

    Di

    i

    Aiopt

    Ai

    Ai wfwfyf ++=

    ===

    4

    1

    3

    1

    23

    1

    2)( r

    (70)

    where yris the design variable vector or gene representing the laminate stacking sequence, DAiwf

    , are weighting

    factors and k are the penalty functions terms to limit the number of plies of the same orientation stacked together.

    When more than 4 plies of the same orientation are stacked together (e.g. Ref. 14) the value of k is one otherwise is

    zero.

    6. Genes

    The design variables are the thickness and the 0, 90, 45, and -45 degree, ply angles that constitute the laminate

    stacking sequences for the skin, stiffener flange and web. Those variables are encoded and modelled as

    chromosomes in genes within the GA. The corresponding encoded chromosomes to ply angles are: 1, 2, 3, 4, 5, 6

    and 7 for 45, 902, 02, 45, -45, 90 and 0 degrees, respectively. Figure 6 shows the modelling of the gene for the

    skin. The total skin thickness is given by h, the encoded ply angle is and n corresponds to half or half plus one

    plies if the skin laminate is symmetric or middle plane symmetric.

    h 1 2 n-1 n

    7,6,5,4,3,2,1=i

    Figure 6. Gene with chromosomes for skin.

    The modelling of the genes for the stiffener flange and web, respectively, depending on the stiffener type are

    given in Fig. 7. The variables at and wt are defined in table 1, and are the encoded ply angle for the stiffener flange and web respectively, m and p are half or half plus one plies if the stiffener flange and web laminate is

    symmetric or middle plane symmetric, respectively.

    Stiffener types a-b

    at 1 2 1m m

    7,6,5,4,3,2,1=i

    Stiffener types c-d

    at 1 2 1m m

    wt 1 2 1p p

    7,6,5,4,3,2,1== ii

    Figure 7. Genes with chromosomes for stiffener types.

    V. Numerical examples

    Ref. 20 provides a set of optimum composite panels with blade stiffeners under strength, buckling and continuity

    constraints, where a GA is employed to carry out the optimisation. This work has been used, initially, as a

    benchmark to compare this two stage optimisation approach. Note that the set of constraints found in Ref. 20 differ

  • American Institute of Aeronautics and Astronautics

    18

    slightly from the design constraints described in this paper especially strength constraints. Ref. 20 assesses strength

    failure at ply level whereas in this paper strength failure is considered at the laminate level. In order to carry out a

    direct comparison, an optimum composite stiffened panel under buckling and continuity ply constraints only was

    taken from Ref. 20. The properties of the skin-stiffener section selected corresponding to the minimum weight

    design are listed in Table 2. Material properties are described in Table 3. The composite stiffened panel is under

    normal and shear loading. The smear normal and shear loads are -3502.54 N/mm and -875.63 N/mm, respectively.

    Table 2. Skin-stiffener properties from Ref. 20. Dimensions in mm.

    W [kg] a b bsf hsw Lay-up

    2.58 762 203.2 60.96 82.55

    Skin (32 plies)

    [45/904/(45)5]S

    Stiffener (68 plies)

    [(45)4/02/(45)2/(04/45)3/02]S

    Table 3. AS4/3502 material properties as in Ref. 20.

    Material AS4/3502

    E11[N/mm2] 127553.8

    E22[N/mm2] 11307.47

    G12[N/mm2] 5998.48

    12 0.3

    [kg/mm3] 1.578 10

    -6

    tp[mm] 0.132

    First of all, the selected design from Ref. 20 was assessed employing the buckling methods described in section

    A.3.3. Results are collected in Table 4. Good agreement in results was found employing finite elements. However,

    closed form solutions showed significant discrepancies in results relative to those in Ref. 20. The main reason for

    these differences lies in the fact that the closed form solutions used in this paper do not account for the interaction

    between the skin and stiffener. The stiffener flange and web will have an impact on the local and global buckling

    capabilities of the skin-stiffener section. When a local buckling mode occurs, the skin and stiffener will usually

    share the same number of longitudinal half wave lengths. This phenomenon is normally associated with a lower

    energy state than the one resulting from the buckling of the skin or stiffener web in isolation. In addition, the

    stiffener flange will act as a reinforcement that will locally increase the stiffness of the skin, therefore improving its

    resistance to buckling. Buckling of the skin or the stiffener web in isolation implies that either the skin or the

    stiffener web presents high stiffness and therefore has no contribution to the local buckling. The closed form

    solutions used, addressed the buckling of the skin and stiffener web in isolation without relating buckling patterns or

    accounting for the stiffening effect of the stiffener flange in the skin. As a result, closed form solutions predicted

    lower buckling loads.

    Table 4. Buckling methods results.

    Ref. 20 FE CF

    b b RFpx RFpxy RFpb RFwb RFcb RFsh RFcs 1.0090 1.0299 0.4869 0.6434 0.2238 0.7019 2.0933 5.7728 1.9698

    Subsequently, the two level optimisation approach was applied. The first level optimisation was set up using

    stiffener type b since this is equivalent to the blade stiffener used in Ref. 20. The stiffener flange width was also

    Shared by the skin and stiffeners

  • American Institute of Aeronautics and Astronautics

    19

    fixed since it is not a design variable in Ref 20. At the second level, a GA code was used with a population of 40,

    200 generations, a 0.7 probability of crossover, a 0.05 probability of mutation and assuming that all weighting

    factors for the lamination parameters were equal to 1.

    Table 5 details the optimum designs obtained by this two level optimisation using finite elements (FE) and

    closed form (CF) solutions to assess buckling constraints. The weights of the continuous (Wc) and discrete (Wd)

    optimisation are provided. The first and fourth optimum designs do not include ply continuity constraints. The

    second, third and fifth optimum designs include ply continuity constraints. Additionally, the third design is further

    constrained to locate at least one set of 45 degree plies at the outer surface of the skin laminate.

    Table 5. Skin-stiffener type b optimum designs considering buckling/continuity constraints.

    Method

    (Mtd)

    Wc/Wd

    [kg]

    b/RFb hsw Lay-up

    FE 2.45/2.49 1.0004 74.77

    Skin (33 plies)

    [904/453/45/90/(45)3/45]MS Stiffener (66 plies)

    [(45)2/-45/45/(453)/08/45/012]S

    FE 2.45/2.51 0.9906 74.77

    Skin (33 plies)

    [(902/45)2/90/45/(45)4/45]MS Stiffener (67 plies)

    [(45)2/-45/45/(04/45)3/02/(02/90)2/0/0]MS

    FE 2.45/2.51 0.9940 74.77

    Skin (33 plies)

    [45/904/45/902/452/(45)2/45/-45]MS Stiffener (67 plies)

    [(45)3/02/45/(02/45)2/04/45/(04/-45)2/0/0]MS

    CF 2.94/2.96 0.9972 75

    Skin (45 plies)

    [(45)2/453/(902/45)2/904/45/904/45]MS Stiffener (70 plies)

    [(45)7/02/(45)2/015]S

    CF 2.94/2.96 0.9862 75

    Skin (45 plies)

    [(45)2/453/(902/45)2/904/45]MS

    Stiffener (71 plies)

    [(45)6/02/45/04/45/02/90/04/-45/04/45/-45]MS

    Results show that when employing finite elements to assess buckling constraints, a lighter design than the one in

    Ref. 20 is obtained even at the expense of adding ply continuity constraints. Without ply continuity constraints a

    3.5% weight saving is achieved. When ply continuity constraints are applied at the second level the weight saving is

    2.7%. It is interesting to see in this case that although ply continuity constraints were not included at the first

    optimisation level, they can still be met at the second level with a small weight penalty. From the best optimum

    solution it is observed that the skin laminate presents flexural anisotropy and no 0 degree plies. For this specific

    case, the laminate anisotropy is used in our advantage to improve the buckling load carrying capability of the panel.

    In contrast, the stiffener shows high percentage of 0 degree plies and no 90 degree plies. As one might expect, the

    skin loses stiffness in the longitudinal direction whilst simultaneously improving its buckling resistance. This is

    compensated with an increase of stiffness in the longitudinal direction in the stiffener to prevent global buckling

    failure. Closed form solutions, as expected, offered heavier solutions than the one in Ref. 20.

    Table 6 shows the lamination parameters obtained at the first and second level optimisation, respectively, for the

    skin and stiffener. It is clearly observed that a good correlation often exist between the lamination parameters at both

    levels. When ply continuity constraints are added small discrepancies are observed. However, in this case, the

    optimums still satisfy the design requirements. Note that ply continuity constraints will limit the bending lamination

    parameter 2D.

  • American Institute of Aeronautics and Astronautics

    20

    Table 6. Optimum and actual lamination parameters considering buckling/continuity constraints.

    Membrane lamination

    parameters

    Bending lamination

    parameters

    h 1

    A 2

    A 3

    A 1

    D 2

    D 3

    D

    1st Level 4.2574 -0.2710 -0.4581 0.1922 -0.6126 0.2251 0.3042

    Skn 2nd Level 4.3560 -0.3030 -0.3939 0.2121 -0.5980 0.1960 0.2693

    1st Level 8.6626 0.6248 0.2555 0.0179 0.2413 -0.5000 0.0268

    Stg

    FE

    2nd Level 8.7120 0.6061 0.2121 0.0303 0.2447 -0.5107 0.0307

    1st Level 4.2574 -0.2710 -0.4581 0.1922 -0.6126 0.2251 0.3042

    Skn 2nd Level 4.3560 -0.3030 -0.3939 0.2121 -0.5973 0.1946 0.2900

    1st Level 8.6626 0.6248 0.2555 0.0179 0.2413 -0.5000 0.0268

    Stg

    FE

    2nd Level 8.8440 0.5224 0.2836 0.0000 0.4065 -0.1776 0.0134

    1st Level 4.2574 -0.2710 -0.4581 0.1922 -0.6126 0.2251 0.3042

    Skn 2nd Level 4.3560 -0.3636 -0.2727 0.2121 -0.5179 0.0358 0.1545

    1st Level 8.6626 0.6248 0.2555 0.0179 0.2413 -0.5000 0.0268

    Stg

    FE

    2nd Level 8.8440 0.5821 0.1642 0.0000 0.3648 -0.2704 0.0780

    1st Level 5.9234 -0.5681 0.1363 0.2838 -0.1834 -0.6332 0.4046

    Skn 2nd Level 5.9400 -0.5333 0.0667 0.2889 -0.2494 -0.5012 0.3280

    1st Level 9.1306 0.4652 -0.0695 0.0000 0.1007 -0.7986 0.0000

    Stg

    CF

    2nd Level 9.2400 0.4857 -0.0286 0.0000 0.1347 -0.7305 0.0322

    1st Level 5.9234 -0.5681 0.1363 0.2838 -0.1834 -0.6332 0.4046

    Skn 2nd Level 5.9400 -0.5333 0.0667 0.2889 -0.2347 -0.5307 0.3428

    1st Level 9.1306 0.4652 -0.0695 0.0000 0.1007 -0.7986 0.0000

    Stg

    CF

    2nd Level 9.3720 0.4225 -0.0423 -0.0141 0.1867 -0.5941 0.0261

    Furthermore, the effect of practical design (PD) constraints on the optimum design, was assessed. Table 7 gives

    the real optimum designs obtained for stiffener type b when practical design rules such as at least 10% of each ply

    orientation (10%) or skin-stiffener flange Poissons ratio mismatch (NU) as well as ply continuity and at least one

    set of 45 degree plies at the outer surface of the skin laminate, are included in the optimisation.

    Table 7. Skin-stiffener type b optimum designs considering practical design rules.

    PD Mtd Wc/Wd

    [kg]

    b/RFb hsw Lay-up

    FE 2.51/2.54 0.9967 74.97

    Skin (38 plies)

    [45/904/452/90/452/0/45/45/02/45]S

    Stiffener (61 plies)

    [45/(45)4/-45/04/45/04/902/03/90/04/90]MS 10%

    CF 2.97/3.01 0.9837 75

    Skin (62 plies)

    [(45)6/02/45/(04/902)2/90/02]S Stiffener (47 plies)

    [(45)4/-45/(45)4/902/03/90/0]MS

    FE 2.50/2.54 1.0011 74.67

    Skin (38 plies)

    [45/902/454/90/45/02/(45)2/0/0]S Stiffener (61 plies)

    [(45)3/-45/45/04/45/03/90/04/-45/04/90/0/0]MS NU

    CF 2.94/2.98 0.9907 75

    Skin (47 plies)

    [45/45/(45)3/-452/(45/-454)2/45/-45]MS Stiffener (68 plies)

    [(45)8/(04/45)2/04/45/0]S

  • American Institute of Aeronautics and Astronautics

    21

    Results show a weight penalty (2% when FE is used in comparison with the first design in Table 5) when

    practical rules are applied. Those practical rules reduce the size of the membrane and bending lamination parameter

    design space. For example, when the 10% rule is applied the new feasible membrane lamination design space pass

    from Eq. (20) to the following expressions,

    06.02 21 AA (71)

    08.02 A (72)

    Table 8. Optimum and actual lamination parameters considering practical design rules.

    Membrane lamination

    parameters

    Bending lamination

    parameters

    h 1

    A 2

    A 3

    A 1

    D 2

    D 3

    D

    1st Level 4.9039 -0.0252 -0.2293 0.2103 -0.4878 0.0165 0.3393

    Skn 2nd Level 5.0160 -0.1053 -0.1579 0.2632 -0.4114 -0.0459 0.2282

    1st Level 8.0145 0.4258 0.2516 0.0154 0.0540 -0.5000 0.0120

    Stg

    FE

    2nd Level 8.0520 0.3770 0.2131 0.0000 0.2091 -0.4945 0.0734

    1st Level 8.1473 0.3120 0.0239 0.0001 0.0056 -0.7316 0.0781

    Skn 2nd Level 8.1840 0.2258 0.0968 0.0000 0.1303 -0.6428 0.0334

    1st Level 5.9670 0.0734 -0.4532 -0.0001 0.0184 -0.9591 0.0000

    Stg

    CF

    2nd Level 6.2040 0.0213 -0.4468 -0.0426 -0.0076 -0.9577 0.0034

    1st Level 4.9546 0.1316 -0.2708 0.1559 -0.2951 -0.3487 0.4727

    Skn 2nd Level 5.0160 0.0526 -0.2632 0.2105 -0.2282 -0.3664 0.3286

    1st Level 7.9078 0.5621 0.2599 0.0217 0.2466 -0.5000 -0.0003

    Stg

    FE

    2nd Level 8.0520 0.4754 0.2131 -0.0656 0.2561 -0.4251 -0.0404

    1st Level 6.0678 -0.0002 -0.9996 -0.3976 -0.0006 -0.9989 0.3172

    Skn 2nd Level 6.2040 0.0000 -1.0000 -0.3191 0.0000 -1.0000 -0.0071

    1st Level 8.9045 0.3999 -0.2001 0.0000 0.0640 -0.8721 0.0000

    Stg

    CF

    2nd Level 8.9760 0.3824 -0.2353 0.0294 0.1148 -0.7704 0.0350

    As previously stated, differences between finite elements and closed form solutions were found. The practical

    design rule of minimum skin gauge was not considered since in this case it had no effect on the optimum design.

    Table 8 gives the lamination parameters obtained at the first and second optimisation levels.

    Finally, the effect of the stiffener type on the optimum design under strain, buckling and practical design rules

    was evaluated. For this case, at the first level, the stiffener flange width was freed and considered as a design

    variable. Common aerospace design strain levels of 3600 in both tension and compression and 7200 in shear were imposed. Stacking sequence constraints such as ply continuity and at least one set of 45 degree plies at the

    outer surface of the skin laminate, were added at the second level. Table 9 provides the optimum designs obtained

    under those constraints.

    Under these circumstances the optimum design constraints obtained by finite elements and closed form solutions

    did not differ so significantly. This is because the strain limits are the driven constraint. The stiffener type does not

    seem to have a major impact on the design. Note that in the cases of stiffener types c and d, the stiffener flange

    minimum thickness was considered to be at least four plies. It was observed that for these two stiffener types the

    thickness of flanges tended to a minimum. That suggests that, in this case, no flanges might be needed. However, if

    T shape stiffeners are used the flanges have to provide a certain degree of integrity to the joint with the skin. The

    first and second level lamination parameters associated with the results shown in Table 9, are listed in Table 10.

    Adequate to good agreement is found in all cases.

  • American Institute of Aeronautics and Astronautics

    22

    Table 9. Skin-stiffener optimum designs for different stiffener types considering buckling, strain and

    practical design constraints.

    ST Mtd Wc/Wd

    [kg]

    b/RFb s/RFs bsf hsw Lay-up

    FE 2.74

    /2.89 1.0741 0.9952 60 69.96

    Skin (59 plies)

    [45/45/45/902/45/04/45/02/45/04/45/04/

    90/0/0]MS Stiffener (31 plies)

    [45/90/-45/04/45/04/90/0/0]MS a

    CF 2.90

    /3.05 1.0111 0.9824 60 70

    Skin (65 plies)

    [(45)2/0/(45)2/04/45/02/90/02/45/(04/90)2/

    90/0/0]MS

    Stiffener (30 plies)

    [(45)3/-45/03/90/02/90/0/0]S

    FE 2.74

    /2.84 1.0639 1.0034 60 69.96

    Skin (58 plies)

    [45/45/902/45/452/04/-45/03/45/04/45/

    04/90/0]MS Stiffener (47 plies)

    [(45)2/02/902/04/-45/04/90/03/45/0]MS b

    CF 2.99

    /3.10 1.0439 10024 60 70

    Skin (66 plies)

    [(45)2/0/45/45/02/-45/02/90/04/45/90/

    (90/04)2/45/02]S

    Stiffener (47 plies)

    [(45)5/02/45/902/02/90/04/90]MS

    FE 2.73

    /2.89 1.0641 0.9905 59.99 69.67

    Skin (61 plies)

    [45/453/45/02/-45/0/902/04/90/

    (04/45)2/45/0/0]MS

    Stiffener flange (18 plies)

    [45/02/45/02/90]S

    Stiffener web (32 plies)

    [02/90/02/-45/02/45/04/90/0]S c

    CF 2.79

    /2.87 1.0109 0.9925 60 70

    Skin (65 plies)

    [(45)3/-45/45/03/90/04/45/04/902/

    04/45/04/90]MS

    Stiffener flange (9 plies)

    [45/45/90/0]MS

    Stiffener web (44 plies)

    [(45/02)2/02/-45/03/90/04/90/02]S

    FE 2.72

    /3.21 1.3438 1.0876 57.85 68.13

    Skin (68 plies)

    [(45)2/45/0/902/45/04/45/04/45/04/902/

    04/45/02]S

    Stiffener flange (10 plies)

    [45/0/90/-45]S

    Stiffener web (57 plies)

    [04/45/0/45/02/45/04/-45/04/45/04/ 902/0/0]MS d

    CF 2.79

    /3.02 0.994 1.0154 60 70

    Skin (65 plies)

    [(45)2/90/(02/45)2/04/45/0/903/04/-45/

    04/45/0/0]MS

    Stiffener flange (8 plies)

    [45/0/90]S

    Stiffener web (53 plies)

    [45/0/(45)2/(04/-45)2/02/-45/902/04/90]MS

  • American Institute of Aeronautics and Astronautics

    23

    Table 10. Optimum and actual lamination parameters for different stiffener types considering buckling,

    strain and practical design constraints.

    Membrane lamination

    parameters

    Bending lamination

    parameters

    ST

    h 1

    A 2

    A 3

    A 1

    D 2

    D 3

    D

    1st Level 7.6119 0.459 0.3181 0.1199 0.0095 -0.3202 0.3528

    Skn 2nd Level 7.788 0.4237 0.2542 0.1695 0.144 -0.2366 0.2553

    1st Level 3.6771 0.5839 0.5678 -0.0129 0.158 -0.0366 0.004

    a

    Stg

    FE

    2nd Level 4.092 0.4839 0.4839 0.0000 0.2261 0.0102 -0.0532

    1st Level 7.5502 0.4550 0.3101 0.1450 0.0431 -0.1357 0.2125

    Skn 2nd Level 7.6560 0.4483 0.3103 0.1379 0.1226 -0.1384 0.1994

    1st Level 5.7506 0.5936 0.5872 0.0070 0.1676 -0.0003 0.0036

    b

    Stg

    FE

    2nd Level 6.2040 0.4894 0.4894 0.0000 0.2737 0.0848 -0.0072

    1st Level 8.0192 0.4853 0.3706 0.0865 -0.0176 -0.1629 0.2785

    Skn 2nd Level 8.0520 0.4426 0.2787 0.0984 0.2220 -0.1948 0.2036

    1st Level 1.8398 0.3928 0.1856 -0.0198 0.3239 -0.2747 0.1032

    Stf 2nd Level 2.3760 0.3333 0.1111 0.0000 0.3333 -0.3278 0.0988

    1st Level 3.9344 0.7541 0.9082 -0.0188 0.6046 0.7369 -0.0521

    c

    Stw

    FE

    2nd Level 4.2240 0.6250 0.7500 0.0000 0.6016 0.7559 -0.0396

    1st Level 8.0737 0.4887 0.3775 0.0844 -0.0243 -0.1351 0.2668

    Skn 2nd Level 8.9760 0.4412 0.3529 0.0882 0.2132 -0.0890 0.0973

    1st Level 0.5280 0.0467 -0.3838 -0.3654 0.2814 -0.2760 0.1284

    Stf 2nd Level 1.3200 0.0000 -0.2000 -0.2000 0.0960 -0.5840 0.1840

    1st Level 7.2105 0.6810 0.7441 0.0535 0.5989 0.3263 0.1439

    d

    Stw

    FE

    2nd Level 7.5240 0.6491 0.5789 0.0702 0.7402 0.4872 0.1286

    1st Level 8.3976 0.5111 0.4221 0.0889 0.0968 -0.2810 0.0985

    Skn 2nd Level 8.5800 0.4462 0.3846 0.0615 0.3420 -0.1995 0.0659

    1st Level 3.6517 0.2543 0.0441 0.0000 0.0949 -0.7154 0.0000

    a

    Stg

    CF

    2nd Level 3.9600 0.2667 0.0667 -0.0667 0.1043 -0.6966 -0.0003

    1st Level 8.3976 0.5111 0.4221 0.0889 0.0968 -0.2810 0.0985

    Skn 2nd Level 8.7120 0.4545 0.3939 0.0606 0.3128 -0.1155 0.0536

    1st Level 5.9974 0.2035 -0.0172 0.0000 -0.0348 -0.7627 -0.0001

    b

    Stg

    CF

    2nd Level 6.2040 0.1915 -0.0213 0.0000 0.0597 -0.7231 0.0379

    1st Level 8.3976 0.5111 0.4221 0.0889 0.0968 -0.2810 0.0985

    Skn 2nd Level 8.5800 0.4769 0.3846 0.0615 0.2881 -0.1923 0.0356

    1st Level 1.0560 -0.0584 -0.3900 0.3777 0.0463 -0.8612 0.0255

    Stf 2nd Level 1.1880 -0.1111 -0.4832 0.2222 -0.0343 -0.9259 0.0960

    1st Level 5.7957 0.6562 0.5592 0.0000 0.4563 -0.0523 0.0000

    c

    Stw

    CF

    2nd Level 5.8080 0.5909 0.5455 -0.0455 0.5157 0.1020 -0.0159

    1st Level 8.3976 0.5111 0.4221 0.0889 0.0968 -0.2810 0.0985

    Skn 2nd Level 8.5800 0.4462 0.3846 0.0615 0.2991 0.0503 0.1065

    1st Level 0.5280 0.0177 -0.2417 0.0000 0.0547 -0.6526 0.0000

    Stf 2nd Level 1.0560 0.0000 0.0000 0.0000 0.0938 -0.7500 -0.2813

    1st Level 6.7535 0.5195 0.3884 0.0000 0.2879 -0.3309 0.0000

    d

    Stw

    CF

    2nd Level 6.9960 0.4717 0.3208 -0.1132 0.4112 -0.1380 -0.0382

    VI. Conclusions

    A method to locally optimise long anisotropic laminated fibre composite panels with T shape stiffeners has been

    developed. The optimisation problem is divided into two levels. At the first level, the skin-stiffener panel is

    optimised using lamination parameters accounting for their membrane and flexural anisotropy and MP techniques.

    The panel is subjected to a combined case of loading under strength, buckling, stiffener manufacturability and

  • American Institute of Aeronautics and Astronautics

    24

    practical design rules. Skin and stiffener laminates are assumed to be symmetric, or mid-plane symmetric laminates

    with 0, 90, 45, or -45 degree, ply angles. Values of the lamination parameters for an optimum skin-stiffener design

    are obtained. At the second level, a GA code is used to target the optimum lamination parameters to find the actual

    stacking sequence for both skin and stiffener considering ply continuity and stiffener manufacture.

    This two level approach has shown good performance when compared with other work (Ref. 20). Optimised

    panels obtained under buckling and ply continuity constraints here are approximately 2.7% lighter than those

    optimised and reported in Ref. 20. The inclusion of membrane and flexural anisotropy in the optimisation procedure

    herein has enabled more designs to be explored. As such, elastic tailoring has been used to an advantage. Employing

    finite elements to assess buckling behaviour has highlighted the importance of considering skin-stiffener interaction.

    Closed form solutions did not consider this interaction and the resulting structures were heavier. However, closed

    form solutions provide the designer with a valuable understanding of the buckling constraints.

    When considering practical design rules, slight weight penalties have been observed. If the design is driven by

    strain constraints, the use of finite elements or closed form solutions to evaluate buckling response has shown

    similar results. Stiffener manufacture does not seem to have a major impact on weight when the design is strain

    driven. However, it might affect the design when buckling considerations are the driving constraint.

    In general, good agreement has been found between the lamination parameters obtained at the first level and

    those determined from the second level (where the actual stacking sequence is identified). Note that although

    sometimes the lamination parameters at both levels do not completely match, good designs have still been produced.

    It is also clear that the designs at the first level will always be lighter than the second level designs since at the latter

    level a rounding process occurs.

    The computational cost associated with both gradient and GA optimisation was acceptable in this study.

    Furthermore, it is hoped that this two level optimisation approach will be a module within a global optimisation

    procedure that could perform elastic tailoring in more complex structures.

    Acknowledgments Authors thank the EC for the Marie-Curie Excellence Grant. J.E. Herencia thanks Tim Edwards and Yoann

    Bonnefon, for their interest, help and valuable comments especially on finite elements as well as Mike Coleman and

    Andrew Main for their assistance on the creation of finite elements models. J.E. Herencia and P.M. Weaver thank

    Cezar Diaconu for his contribution and support, particularly on lamination parameters. J.E. Herencia thanks Mark

    Lillico for his comments on the modelling of stiffened panels. J.E. Herencia thanks Imran Khawaja, Alastair Tucker,

    Stuart King, Pedro Raposo and Paul Harper for their general remarks.

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