ANAI,YrI'IC SENSITIVITY AND APPROXIMATION OF SKIN DUCKLING CONSTRAINTS IN WING SHAPE SYNTHESIS

Eli ~ i v n e * and Radomir ~ i l o s a v l j e v i c * *

Dcparlment o f Aeronautics and Astronautics, FS-I0

University o f Washington, Seattle, WA 98195

ABSTRACT Explicit expressions for terms of the stiffness and

I geometric stiffness matrices are derived for the buckling ~ analysis of trapezoidal fiber composite wing skin panels.

Thc formulat~on is based on Ritz analysis using simple polynomials, and leads to explicit expressions for the

1 analytic sensitivities of thc stirfness and geoniclric stiffness

I m:llrlcc< \\ 11t1 I L > \ ~ C C I I L ~ 1:1>1.~r ~ti~ckri:~\. l . i h x r ( i ~ r cc~~ou< ~ I I N ~

pancl shapc. Inicgr:rt~on \\ i t11 \\~rlg box analys~s usng either the equivalent plate approach or [he finite elcrnent method, makes it possible to oblain sensilivi~ies of panel buckling constraints with respect 10 wing planforrn shape or locations of internal ribs and spars. The analytic sensitivities derived are used to consmuct approximations of pancl buckling constraints for inregratcd using / panel design synthesis.

NOMENCLATURE

m,n - powers of x and y in a polynomial term.

mz,n; - powers of x and y in polynomial series for thickness of layer i.

m,", n," - powers of x and y in R i a functions (Eq. 17)

m,, n, - powers of x and y in the wing depth series.

m:.n,y.n: - powers of x and y in polynomial terms of

mf 2y,nf 2: - powers of x and y in elements of [F,] (Eq. 51). - - m ,,.,,, n,,,,, - powers of x and y for elements of [a] . M I , M y , M , - internal moments per unit length.

N , - number of terms in the thickness polynomial for layer 1.

[ A ] - in-plane local stiffness matrix for the plate (3 x 3) N , - number of layers. [D l - out of plane local stiffness malrix for the plate (3 x 3) [N] - 2 x 2 matrix of in plane loads (Eq. 5). f, ( x , y) - admissible functions. N x , N,, N , - in plane loads per unit length. [F , ] , [F , ] , [F , ] - matrices containing admissible functions - and heir derivatives (Eqs. 2-4). [c] - 3 x 3 constitutive matrix for a layer (Eq. 22).

F , ( x , y ) - weight function ensuring zero displacement on [a 1 - material pro~erties matrix (Eqs. 20,21)

panel boundary.

I F y - coefficients ~n the polynomial expression for [F,] I 0%. 30)

h - total thickness of panel

. H ( x , y ) - depth of wing. - H,, - coefficients of wmg depth polynom~al.

I i,i, ,i, - indices of terms in layer thickness polynomials.

I , , - integral of a simple polynomial tcrm over ~rapezoidal panel area. [K ] , [K, ] - stiffness and geometric stiffncss matrix,

I respectively.

* Assistant Profcssor, senior member AIAA. ** Graduate student. Copyright3 1995 by Eli Livne and Radomir Milosavljevic. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

q,,{q) - generalized displacement and the vector of generalized displacements for panel. R, S - coefficients of Front line or aft line of panel. li ( x , y) - thickness of layer i.

Ti - coefficient j in the polynomial series for layer i

U,, V , - shape dependent coefficients of FB (Eqs 15, 16)

~ ' ( x , ~ ) - panel elastic out of plane displacement.

W ~ ~ , ~ w - coefficients of polynomial elements of [a]. [ a ] - matrix of function derivatives defined in Eq. 41.

1 - buckling eigenvalue. 6 - fiber orienlation angle. subscripts A - aft (rear) F - front. L - left R - right.

l N ~ l ' U O l J ~ ~ ~ ' I ' l 0 N Bucklmg ol skm pancls In t1111i wallcd struc~urcs 1s

cwc o f thc ~iiost Iml,orl;lnt fn~lrrrc motlc.; to hc cor~sidcwd iri

m y tlcs~gri s y r ~ t l ~ c s ~ . ~ A rrch body ol I~tcr;~turc cxr\Ls today o n thc buckling of isotropic and aniso~ropic pancls (Rcfs. 1 - 5 , for cx:~~i~plc). I t rcflccts the cxtcrisivc cxpcricncc in this ,rrl.;r. h \ c d on tl~cor~.r~cal t lcvclop~~~cnt . ~ iu~i~<. r~ i .a l ~nvcstrgilt ~ons and nunlcrtm tcsLs.

Whilc thc study of rsolatcd pancls of rccungular shapc has rcached certain maturity, rescarch addressing the buckling of pancls functioning as components in a largcr strucrural systcm is strll evolving. In thc cornplcte structure. such as an airplane wing or fuselage, buckling can occur locally (when a particular panel buckles) or globally (when a segment of the structure, containing several panels. buckles). The need to assess global bchavior (for internal force disuibotions or global buckling) and local behavior (pancl buckling) simultaneously, leads to computationally intensive analysis. The complex global-local interactions can also makc buckling constraints highly nonlinear in terms of thc structural dcsrgn variables. Thus, in the conlcx I of airframe struct~rral optimizarion, proper represen~ation of bucklin~ constrants is a major challcngc. md 11 i \ \ 1 1 1 l ; I I ~ ;irca O ~ ; I < . [ I \ C r c \ ~ * . l r ~ . h (RL,~ \ 6 - I i \

\\'hen u ~ n g pldnlornl hcionlc~ \uhjc';t lo dcslgn optimimtion in addition to the s~zing of structural mcnibcrs, or when variation of the internal structure is allowed, the panel buckling problem becomes more complex. In the more general case, panels (as defined by the array of ribs and spars that supports them) are rarely rectangular. They are usually trapezoidal in shapc. Xlorcover, whrle in high aspect ratio wings i t may be acceptable to assume unidirectional compression on the skin panels, in low aspect ratio wings panels are usually loaded by in plane loads in an inherently combined manner.

It is important, then, lo develop an efficient buckling analysis methodology, applicable to trapezoidal panels in combined in-plane loading (Refs. 16-18). Such a methodology should be design oriented, that is, issues of sensitivity analysis and approximation coilcepts should be addressed from the outset. Integration of the wing box global analysis with the local panel buckling analysis should be fast computationally. The result should be an efficient structural analysis, sensitivity and approximation tool for wing shape optimization, including stress, deformation, dynamic behavior and panel buckling consuainu.

In the present article, a method for buckling analysis of trapezoidal wing composite skin panels is presented. Analytic sensitivities of buckling consmints w ih respect to sizing, fiber directions and panel shape are derived. Integradon of the resulting buckling analysis with an approximate wing box structural analysis based on the equivalent plate approach (Refs. 19-21) or finite element analysis is discussed. Alternative approximation techniques are assessed. Lessons learned and the resulting recommendations will help guide further developments on the way to effective wing shape optimization in the preliminary design srage.

The article opens by deriving thc general equations for panel buckling analysis for simply supported uapezoidal panels. The Ritz method, using polynomial admissible

i~ rnc~~ons , is cmploycd. Dctailcd dcrival~oris, taking advantage of the simplicity of manipulation of polynomials, dcrnons~ratc how stiffncss and gcomctric stiffness rcrms can bc cxprcsscd In tcrrns of sr~.ing, ribcr dircclion, and sh;ipc design variables cxplicilly. Thc integration of wing box structural analysis with pancl buckling analysis is discussed ncxt, followcd by a rnctliod for thc con~pul;~tion of analytic scnsitivitics. Rc.sulls of numerical evaluations and lcssons lamed concludc the prcsenlation.

RlTZ PANEL RUCKLING ANALYSIS A N D CONSTRAINT FORMULATION

Stability equations Based on the principle of virtual work, a

variational equation for a symmetrically layered composite plate under h e action of internal bending moments and transverse shear forces in the abscnce of transverse load or

(1) initial deformation is

whcre the vertical deflection due to toad is w l ( ~ , ~ ) . The matrices [Dl (and a corresponding matrix [A] ) are the 3 x 3 local transverse stiffness (and in-plane stiffness) matrices. respectively (Refs. 1,2). In the case of a symmetrically layered panel, the in-plane loads Nx, Ny, Nxy are related to

the in-plane strains (&: & Y:~) through h e matrix

[A] The bending moments Mx, My, and Mxy are related to the bending curva tures due t o load

( - w ' . ~ - w ' , ~ - 2 ~ ' , ~ ~ ) . through the matrix [Dl (Refs. 1.2).

The unknown elastic panel deflection Wlcx.,, is approximated by a series of admissible functions:

The fust derivatives, then, can be expressed as:

ar~d tlic sccond tlcr~varlvcs arc g1vcv1 by:

In the above expressions the matrices [F,]. [F*]

and [ F ~ ] are all functions of x ;~nd y. Using Eqs. 2-4 to

express virtual displ;~ccmcnts and thcir dcriva~ivcs in tcrms

of virtual gcncralircd dis~,l;iccmcnts. m1 = [F,]{&} . Eq. 1 Icad, to:

where

The matrix equa~ion for the linear buckling analysis of the panel is thus:

[[KI + l]M= I01 (6) where the stiffness matrix is:

and the geometric stiffness matrix is:

The scalar A is used as a scaling parameter increasing or decreasing the given in-plane loads Nij simullaneously, to determine whether the panel is stable or unstable. Equation 6 is a generalized linear eigenvalue problem. Since the stiffness and geometric stiffness matrices are real and symme~ic, the eigenvalues are real. The buckling constraint for he panel is in the form:

1. -A,, 5 0. (9) assuring that the given in-plane loads has to be increased to reach instability.

Simple polynomials for modeling and Approximation Admissible functions based on simple

polyno~n~als. ils 1s wcll kllown (Ref. 28). lead to i l l - condlr~oncd sys i c~ i~ nlelrices. As the order or the approximalion polynomials is increased, high ordcr tcrms ;lnd lou ortkr rcrrw ;ypXar si~nul~~ncously in I I IC s~ill'ncss and gmmetric stiffness matrices. Large differences in ordcr of magnitude among thosc terms will lead to ill-conditioned lincirr quation and cigc~lvalue solutions on a finite word- lcngth computer. Unless convergence of the solution is obtaincd with low order polynomials, i t may bccome impossible to achieve convergence at all, if the order of plynomial functions is increased lo the point where the equations bccoming i l l conditioned.

Still, simple polynomials offer significant advantages in structural analysis, especially in the case of beam and plate like structures. Simple polynomials are easily manipulaled, and therefore the formulation of the problem for solution becomes quite simple. They can also lead to integrals which may be evaluated analytically. Numerical quadrature is not required, and h e solution process becomes significantly faster. Simple polynomials have been used as admissible functions in plate vibrations problems (Refs. 29-30) and in shells (Ref. 31). In the context of wing optimization, simple polynomials were found 10 k:td ro acccplablc dcfornia~ion, slrcss and mode >ll'lpc> I11 v. l l l g h 01 ~ [ L l l l C L ~ \ ~ I I ~ ~ I C A pl~lllf~l-lll~ (UcI.5. 2 1-27), The resulting suucrural analysis capabilities were found to be very efficient computationally.

When planform shape variations are allowed (in addition to sizing changes, during wing optimization) using simple polynomials for structural analysis makes it possible to obtain structural shape sensitivities analytically. As Ref. 25 shows, it is possible to obtain closed form expressions for the shape derivatives of wing box stiffness and mass terms avoiding numerical quadrature. In wing design- oriented structural analysis involving shape design variables, there is a need to evaluate buckling constraints for panels whose shape is changing. Buckling constraint sensitivities with respect to the shape of a panel, linked to variations in the shape of the wing, are also required. Based on existing experience with wing box modeling and shape sensitivity analysis, simple polynomials (multiplied by appropriate weight polynomial functions) are used in the present work for the panel buckling analysis. The simplicity of formulation, compuiational efficiency and ease with which analytical shape sensitivities are obtained provide h e motivation. Convergence studies show fast convergence with a small number of polynomial terms and no ill conditioning problems. Straightforward integration of local level panel and global level wing box solutions is an additional advantage.

Modeling In the polynomial based wing box analysis, the

thickness of layers of fibers in different directions is given by simple polynomials. For layer i (out of NL layers)

The powers mf ;1nd 11' are x ;lnd y powers of k k thc k - t h term ol' ~ h r thlrknc\\ wrlcs Tor thc I-th Iaycr Thc

serve s r r ~ r l ~ i g typc design variables. cocrl'lcrcnts 7" IJnlikc. many stutl~cs, i n wt11c.h wing [r;~pcr.oltlal scglncnls ;lrc lrarlsforjn~(j Into ;I i ~ r l r l \qir;rrc lor nulrlcrlc;ll qu:\dr;itur~. hcrr thc thickness p a l y ~ ~ ~ ~ n i ; ~ l I S g~vcn i n tcrrns ol' ~ h c physical x , y coordinalcs (Fig. 1 ). The ovcr;rll [h~ckncss of sk~n panels is givcn by (Eq. 10)

( 1 1 )

In a slmilar manner, cap arcas for the spars and ribs of [he wing box modcl ;\rc also expressed as simple polynomials of ei~her x (for ribs) or y (for spars) (Refs. 21- 23, 25-27). Depth of the wing box is also givcn by a simple polynomial in x and y, to be dclined later.

Admissible functions Figure I shows a tr;~puoidnl panel defined by

coordinate\ of its vcrticcs in thc *,y asc<. Thc rul-rscripts L and R d c n ~ ~ c Icl-I ; ~ n d r r g h ~ srdc's, r i ' \ p ~ i l ~ \ ~ I > . . Thc subscripts F and A dcnotc front and aft lines, respectively, and XF and XA are the x coordinates of the forward and rear points on a line parallel to the sides of the panel. Based on Fig. I , we can write thc follo\t ing equation for points on the front line:

( 1 2)

In a similar way, on thc r w l i n t , expressing x~ in terms of y leads to:

(1 3)

The function F (x, y ) = B

satisfies the zero displacement boundary conditions on the circumference of the panel. Using Eqs. 12 and 13. and expanding in terms of x and y yields

(14)

and, using index notation,

whcrc thc constants U enti V arc yivctl in tcrms of pancl vcrtcx crwrdinarcs by: U , = yL.VR .U2 = -(?I + 1' ).U3 = 1 L . R (16)

v 4 = I . \ ' 5 = - ( s A + s F ) , v 6 = s F s A Powers of x and y corresponding to constants U

and V (Eq. 15) arc given in Tables 1 and 2. Multiplying by a gcncral polynomial series, adn~issible functions for the simply supported trapezoidal panel are obtained:

where the cocfficicnts q p are the generalized

displaccn~cnls. Substituting the expression for FB, from Eq. 15, u~ can write:

(18)

The admissible functions are thus expressed in terms of simple polynomials, where the p'th admissible function (Eq. 2) is given by:

(19)

STIFFNESS MATRIX Polynomial description of skin layer thicknesses in

terms of the global x-y coordinates was given in Eqs. 10.11. This thickness distribution is for an entire wing box, or segments of the wing box. The panels analysed for buckling are those trapezoidal skin segments defined by the supporting internal spar/rib array (Fig.1). For each such panel containing NL layers of fibers, the in-plane stiffness matrix [A] can be expressed in terms of individual layer thicknesses and fiber orientation angles as.:

(20) N,

[ A ] = .$ [[Q,] + [ Q , ] C O S ~ O ~ + [Q&s4ei + f =1

Lct a matcrial and Sibcr orientation dcpcndcnt

matrix. [g(9, ) ] , a 3x3 rn;,six. bc defined :is:

The ~n-planc sullncs lnalrix [ A ] u n now b~ cxprcmi'd 111

terms of the sizing design variables, T1j , and fiber directions as a polynomial:

For unidirectional, orthotropic or quasi homogeneous laminates (Refs. 7,3? and 35) the in-plane and bending stiffness nia~rices are related ~hrough ,.

Using Eq. 1 1 to express h2 in terms of sizing type design variables, double summation is needed. The indices 11 and 12 are used for summation of polynomial terms associated with each layer, as follows:

! 2 3

The bending stiffness matrix, [Dl, can now be written in polynomial form as

(26)

Thc dcpcndcncc of [D l on lhc si/.ing design \,ariablcs (thickness cocfficicnrs) and fiber angles is now c\prcsscd in cxplicit form. With thc polynomial Rirz funcrions prcscntcd In Eqs. 2,19, and thc stiffncss matrix h:i.wd on Eq. 7 , cxprcssions for tcrms of the s~iffncss mauix ~n pdyno~nial form can now bc dcrivcd. Elcnicnu of the nutrix IF31 (Eq. 4) i n polynomial form are gcncratcd firsl, rw follows: Sccond dcriviitivcs of the f p (Eq. 19) functions ;ire cxprcsscd In polynomial I'orm.

(27)

\;:i~ii(i ti<ri\.;tfi\,c\ of /cro ordcr or fir51 ordcr tcrms) S ~ n ~ i l i ~ r l ~

(28)

u v w where fpYyy = 0, if n. + n . + np 5 1. Finally 1 J

(29)

In the last expression the term fpVxy is set to zero in two cases:

The elements of the [F3] matrix are. thus, all polynomials, and the q,p element of [F3] can be written as:

where the coefficients P~~ and corresponding pwers of i r /

and y, mf? and nf qP respectively, are shown in the 11 11

Table 2. Equalion 30 together with Eq. 26 are substituted into h e equation for the stiffness malrix (Eq. 7). The rs'th term of ~e stiffness matrix is:

and thc final c\prcs\lon f o r thc K r s clcmcnl in poIyntu~ii;~l for111 IS: ( 3 1 )

whcrc the powcrs or the x and y Lcrms in the arca intcgral are

All clcmcnts of h e stiffness n u t s i x are, thus, linear comhirlarions of inlegrals o\.cr the p;~ncl's arca of the form

Note the explicit tlcpcndcnce on thickness coefficients and fiber dircctions. Dcpcndencc on pancl shape is more conlplex. Thc cocfficicnts U ; and Vj (Eqs. 16) depend on the x,y poslions of rhc lcrtlces oC ihc pancl. These coefficients, in turn, determine the F coefficients in Table 2. In addition, he area iniegrals (Eq. 33) depend on the shape of the panel. since the polynomial terms are defined in global (physical) coordinates, and the limits of integration change when [he planform shape is changing. The integrations (Eq. 33) can be carried out analytically, as shown in detail in Refs. 23, 25 and 27.

THE GEOMETRIC STIFFNESS MATRIX In-plane loads from wing box stress analysis The geometric stiffness matrix in the Rayleigh-Ritz buckling analysis formulation for skin panels is given in Eq. 8. and it depends on the matrix [F2] (Eq. 3), containing derivatives of the admissible functions, and the mamx containing the in-plane loads. In classical linear buckling analysis of the panel, these in-plane loads are assumed given, and they are determined by a separate suess analysis of the wing box, before the buckling analysis for the panel

I is carried out. In the case of wing structures, wing box stress analysi: can be based on standard finite element techniques. In this case. in-plane forces along the sides of

I h e are obtained by some functional approximation based on either nodal forces acting on nodes surrounding the panel, or in-plane stresses e~aluatcd for the finite elements surrounding the panel. Alternatively, based on stress distribulion in the panel, in-plane loads for buckling analysis can be evaluated - throughout the panel, to be inlegrated over the area of the panel to obtain the matrix [KG]. (Refs. 8-10,12,15-17,33,34).

For preliminary design purposes, if the skin panels are small relative to the wing, buckling evaluation may be

;I(.~w;tlc c l~wgh 11 .Iwri1gc N x . N y and N x y arc uwl I'or the I X I I I C I Irirc'kllng analysis. Thcsc in-planc strcsscs arc ;~zsu~ncd c m s m t throughout thc pancl in this casc. This s i l~l l ) l~l~c\ ihc r i~ i~ .gr ;~~~cns i n Eq. H, anti makcs I I po\s~blc to usc interaction formulas for fast approximau: buckling analysis (Rcfs. 10.16 and 33).

Wl~cn an cquivalcnl plale nodc cling approach is used for rhc wing box analys~s (Rcfs. 21-23), thc in-pli~nc skin sircsscs arc ohlaincd from thc wing yencralizcd displaccmcn~s calculated in the wing box slrcss analysis stage. In the formulation used in Refs. 21-23 admissible Ritz functions for the wing box analysis arc givcn as polynon~ials in x and y (the global coordinates uscd LO dcfinc thc planform of the wing and the shape of all panels, Fig. 1). In this casc, the transverse displaccmcnl of the wing is -

where a bar will associate variables with the wing box - - analysis. I n Eq. 34, ihe powers mi , ni , and the number of

- tcrlns Nu arc known from the Ritz series used for the - \\ Ing Ixj\ d~ i l ' l a i ; n l~ .n l i c ~ l t 1 1 l O r l . The cocfficrcnl~ (1 , . arc thc gcncral~/.cd n lrig box d~qlacenicnts. Mult~ple load cases can be accoimred for in the wing analysis, lading to - differen1 generalized displacement vectors (4) .

I n cquivalcnt plate wing suuctural analysis based on classical plate theory (Refs. 21-23), the vertical - displaccrnent under loading 1s W(x,y), and using Kirchoffs kinematics for a wing with a symmetric cross section, the engineering strains in the x and y directions are given by (Ref. 2, p. 19) - -

- - yxy = -2zW.xy

Let the wing depth be given by R(I.y,. Then, when skins are thin compared to the deplh, hey can be - assumed located at z = k ('.~d. Focusing on h e upper

skin, in-plane strains are (Eqs. 35)

Now, in a polynomial based formulation for the wing box, the depth of the wing is given in polynomial form

win): box disilaccments by integration through the

- Eq. 4 1 is, thus, 3 by N w . Each clcmcnt of his matrix is of

111c cr~l'fic~cnt W(l\,,,pw, and powers of x and y.

f i ( i ~ ~ , p w and fig,,.,pw, rcq~c~ively. arc dcscribed in Teblc 3. Whcn lhc indcx qw dcnotcs the row of the matrix (its valucs varying from 1 to 3) and the indcx pw specifies icrms 01' the polynomial displnccmcnt series for hc wing

box (varying from 1 to Nw), then (Eqs. 40), the in-plane load N x for the panel can bc expressed in polynomial form 3s

leading lo

leading to (Eqs. 20-23)

We now turn attention to the derivatives of the wing displacemenl, in order to express Eq. 40 in terms of

che generalized displacements (q) calculated for the wing box solution. We define a new vector (a) of wing box curvatures as follows: (41)

- - z p w - I npw-I mpwnpwx Y

where pw is an indcx for the p'th clcmen~ in thc wing Riu - series. Thcrc arc Nw Lcrms in the Ri t z series for wing

Siniilarly the expressions for Ny and Nxy can be derived:

Now the polynomial expression for wing depth (Eq. 37) can be substituted into Eqs. 42-44. The general expression for terms of h e [N] mauix ,

I f i )p = I and qq = 1, then Appp ,(p,, = Al.qw

Ij'pp = 2 and qq = 2, rhcn = A 2.qw (40)

The polynomial expression for ~lic [ A ] nialrix (Eq. 23) is now used: (4 7)

where the index ppp means !he I'ollow~ng:

- - - ff pp = 2 and qq = 2 , then QpppSv = Q 2 p (48)

The matrix [N], then, can be expressed as a polynomial in x and y according to Eq. 48. This equation shows how [N] depends on the wing box solution . the depth of the wing, the thickness coefficients for layers in the panel, material properties and fiber directions. Of

course. the wing solution (GI , also depends on depth. thickness of layers, fiber directions and material properties. These affect the stiffness matrix of the wing as described in Refs. 21-23 and 25-27. I t should be emphasized again, that polynomial in-plane loads for the buckling analysis, can be obtained in a similar manner from wing box analysis based on first order shear deformation plate theory (Ref. 27) or from finite element results, when skin stresses are approximated by polynomials uhing least square fitting (Ref. 35)

The geornelric sliffnrss matrix Rascd on thc Ritz polynomial functions (Eq. 19).

the malrix [F2], nceded for the cvnluation of the geometric stiffness ma~rix (Eq. 8), can bc written in polynomial form as follows: Derivatives with respect to x and y of he Ritz functions are

(rrrw+mv-1) ( n w + n u + n \ i ) .t I-' J P i j

, Y

Thus, the clcmcnt q,p of the matrix [F2] is of the form: (51)

The coefficients k.,?'!) and powers of x and y, - 1 . 1

r/ I' n l j 2 ' . and ,I~.L!"' , defining the clements of the IF21 1-j 1,j

matrix, are described in the table 4. Elements of the matrix [KG] can now be expressed in polynomial form as follows:

The indices r and s identify terms in the panel Ritz displacement series for thc panel. Using polynomial expressions for (F2] and [N], the r,s element of [KG] is written as: (53)

N. N t .

where (54)

The index ppp used in qppp.qv (*, is defined in

Eq. 48. As in the case of the stiffness matrix [K], the geometric stiffness mauix is represented as summation of surface integrals of polynomial terms calculated over h e area of the panel. Inregrals of h e same family ITR(m.n) are

b'tirthvr disct1skio11 of intcgr; i t io~~ M it11 v ing ~trt1~11iri11 analysis

Rcc.311 that 1Itc rcsul~x ol the wing analysts al'fcct buckl~nl; analysis of the panel through the in-plane load

i matrix IN]. This matrix is obtained by intcgrating skin s t r c s s~~s (due to \r.lrig dcl'or~li;~tion) through thc lhlckncss of the sktn, and thc rcsultlng Nx, Ny and Nxy are functions of x and y ns evidcnt in Eqs. 45-48. Now, whilc equivalent platc wing structural analysis Icads to good skin stress predictions in stiff, low aspect ralio wings (Refs. 21-23), still cquilibriu~n IS not g u a r a n ~ ~ c d for isola~cd skin panels. Thus, thc in-planc loads (as given by Eq. 47) arc gcncrally not in equilibrium (This problcm is not cncoun~crcd whcn wing analysis is based on thc finilc c,lcrnent tnelhod, whcn in-plane loading on rhc circtr~nfcrcncc of an isolated pancl is detcm~incd from nodal forces on t t i ; i~ circumference).

One way to avoid this difficulty and simplify I

bucklins ari:~I>.sis of p n c . 1 ~ 1.: ro u.;c a n avcr-arc ~n-pl;~ric

i . . I0;idln~ 1 ~ ) tY ; t . \ . \ l l l l l ~ d ~ ~ ~ l l \ ~ ~ l l l l \ l \ < , r l I lC pdilcl (kc1 >!. I 1 point (xo, yo) inside the panel is used to evaluale these average in-plane loads, then thc elen~ents of [N] become (Eq. 47)

( 5 5 )

01 111c p;111cI) a\allahlc in thc forni of Eq. 32, 11 I S

stra~glitloru.ard to obtain sensitivities analylically. Thrcc typcs of scnsi~ivitics can hc cvalualcd: a) scnsi!ivitics with

rcspccl to tti~ckncss dcsign variables, Ti ; b) scnsitivilics

with rcspcct ro Inycr oricntarion angle, 6 , . und c) scns~tivitics \\ i th rcspcct lo planform shapc variables.

StilTnrss stwsiti~.ities with Respect to Thickness Dcsign

Variables Tk Plunform shapc variablcs and orientation anglcs

are fixed in this case. The thickness coefficients Ti appears in thc exprcssion for thc stiffness matrix (Eq. 32) explicitly in a triple summation over thc indices k. I1 and 12. Scnsiliviry IS then obtained by direct diffcrcnriation, noting

that i f the design variable involved is T~ then r '

Sensitivities with Respect to Fiber Direction 6) , Now the planform is fixed and thickness is fixed.

The angle 0 rcpresenls the direction of fibers in h e i-th

layer, and the stiffness matrix Krs depcnds on 6) i through

elcrnenls of rhe rnauix (Eq. 22). The derivative of

each term in the stiffness matrix will be calculated in the (m. +m ) . (nih+irqW,pW) . ipw H . lh 4W7PW yo following manner. In the summation (Eq. 32) over i = l t

I lh 0 NL, 311 matrices Q(6,) are set 10 zero, except the malrix [= I The expression for elements of the geometric

stiffness matrix (Eq. 53) has now lo be modified, since [Q(oi)] corresponding to [he 8 , variable considered.

terms including xu and yo become constant for the area integration, and can be laken out of the, integral. is replaced (in Eq. 32) by the

(56) following expression (Eq. 22): 1 2 2 3 6 3 6 N h

KG r,s =-5 X X .E .E -2 -1 . Z , a=l b=l~=l j=liZl j=lih=l a 8,0,, = - 2 [ ~ ] s i n 2 6 , - 4 [ ~ ] s i n 4 6 , +

26, (57) i 3 N w N ~ h ' t i - ~ , . - b S

Z 1 .x ~ h i , ' p 2 + + 2 [ ~ , ] cos 2 8 + 4 [ ~ ] cos 4 ei I qw=l pw=lit=l k = l I Y J Equation 32 is , thus. used for the sensitivity of the stiffness

term, w i ~ h the derivative Eq. 57 replacing Q(0;) . [- I xo

(mih +fiqw,pw) . Jo ( n i h + f i q ~ , ~ w ) . Jj[ Stiffness sensitivities with respect to panel planform variables y ~ , y ~ , XFL, XFR, XAL and XAR (Fig. 1)

( , n f ~ ~ ~ ~ + n ~ 2 b ? + r n r i t ) (n f2~+n/2 ! . i+nr i ' ) Thickness coefficienls and orientation angles are X 1,) i ,J k Y L,J i j drdykeld tired. ll-ie t e n s Kn of lhe stiffness rnauix depend on

h e plarrform through matrix [F3] @q. 30 and Table 2) and h e integrah I T R ( ~ , ~ ) (q. 33). Coefficienls Ui and Vj are

ANALYTIC SENSITIVIITES defined through expressions 16 and 18. If x is any StiITncss Matrix Sensitivities planform design variable, then:

With the explicit exprcssion of stiffness matrix elcmcnis (in tcrms of thickncsh, fibcr directions and shape

4P . The dcrivattves of he terms F.. . can now be prepared 1 1

d ITR is (Eq. 30, Table 2). The tables for the derivatives --- d x

prepared using Refs. 23 and 25. I t should be noted that in evaluating ~ h c shape derivatives oi thcsc tnrcgrals ~herc is no need for more integrations over the panel's area. The shape sensitivities of the area integrals are linear combinations of other members of the same table of integrals - a table already available from the analysis step.

After collecting all the information necessary, one can calculate the derivative of the term Krs as follows:

(59)

Gcon~ctric stiffness sensitivities with Respect lo

'l'hickncss Iksign Vilriablrs 7' it k ' The ticsign variablc in this casc IS the k - t h

ctxffic~cnt in rhe polynomial thickness scrics for thc i-lh laycr. Examination of Eqs. 5 3 5 4 rcvcals that Ihe gcomcuic stirfncss ~natrix is explicitly linear i n thc thickness

cwfficiena T; . 11 is, of course, also dependent on those

cocfficicnts via thc wing box solution {Q) , unless i t is assumed that in-plane loads [N] do not change. Differentiation of Eq. 53, using Eq. 33 leads to

(0)

where rif 'R is equal to I only when it=R an6 k=S. k , S

Othcrwse, it is zero.

Geometric stiffness sensitivities with Respect to Fiber Direction ei,,

Layer orientations affect the geometric stiffness - -

matrix through the material matrices QpppVqw (8. ) It

and the wing box generalized displacements (i) . The analytic sensitivity with respect to fiber direction in a given

Geometric stiffness sensilivities with respect to planform variables y ~ , g ~ , XFL, XFR, XAL and XAR

Thickness coefficients and orientation angles are held fixed. The geometric stiffness matrix KG depends on the planform variables through Ui and Vj terms (Eqs.

whcrc s represents any of thc planform v;rriables, and tllc p ~ \ \ L * I \ , ) I ' in!~.gr:ind\ In 17-k ;1 r~ , I I I = I ~ I C ; ~ , \ :111ci

n=nC;r,S (E+. 3). 111 Eq. 07 11 1s ~ S U I I I I C ~ 111;11 o\crall u.111g planform is fixed, and pancls are changing shape and location due to moving of control surfaces, ribs and spars. If overall planform shape of the wing is changing, [hen derivatives of the wing depth coelficicnts with rcspcct 10

the shape design variables. d ~ ~ ~ ~ / d ' , must be addcd.

since the wing depth is defined in global x,y coordinares (Eq. 37).

It is clear, examining the analytic sensitivities of stiffness and geometric stiffness derived in the previous sections, that all involve area integrals of polynonlial terms over the panel's area. Since members of the family of integrals 1 ~ ~ ( m , n ) are generated for the analysis slage, there is no need to generate [hem again for the sensitivity calculation stage. No numerical integration is therefore needed for either analysis or sensitivity calculations. This is similar to the process by which analytic sensitivities are obtained for the wing box analysis (Ref. 25). A table of area integrals over the wing surface has to be generated once (using the same analytic formulas used here for the

I

i panel). The integrals are subsequently used for wing box analysis and sensitivities.

Of course, when the wing changes shape. panels change shape too. There is linking, therefore, between overall wing planform design variables and panel vertex locations. This linking has to be accounted for in the sensitivity compulations when overall wing / panel shapes are changing.

When constant in-plane load matrix [N] is assumed for the panel buckling analysis, and when i t is based on evaluation of in-plane loads at a point (xo, yo) on the panel (Eqs. 55,56), then when shape sensitivities are required, the motion of point (xo, yo) when the panel changes shape has to be ken into account.

Eigenvalue Sensitivities Now that the sensitivities of the [K] and [KG]

hcncc. lhc scnslilvliy ol huckl~ng conslralnls IS glvcn by. (63)

This analytic sensitivity can be used to consvucl direct, reciprocal or hybrid approximations of thc conslraint (Ref. 37). Al~ernativcly, the sensitivities of [K] and [KG] can bc uscd in a Raylcigh Quoticnt approximation (Refs. 38 and 39) , whcrc, bascd on direct or reciprocal

approximations, approximale stiffness , [ K ~ ~ ~ ~ ] , and

geometric stiffncss, [ K ~ ~ P " ] . matrices are constructed.

TEST CASES A N D RESULTS

To assess the new capability, test cases were chosen to address the following issues: a) convergence rate with increased polynomial order (Fast convergence using low order polynomial R i~z functions is important not only to avoid i l l conditioning, but also to reduce computational resources needed for design optimization); b) accuracy of analysis results for isotropic and anisotropic panels of general shape (rectangular, skewed and triangular shapes were studied); c) reliability of analytic sensitivities and accuracy of finite difference derivatives; d) quality of conventional approximation techniques based on first order sensitivity analysis in the presence of shape variations; and e) integration of wing box structural analysis with panel buckling analysis for fast structural analysis of airplane wings.

Analysis results from Ref. 3 are used to test accuracy and convergence rate of the present technique in the case of rectangular anisotropic panels. Simply supported, angle ply laminates made of 20 plies of DLP material (Ref. 3). with a=Y, b=10", and combined in plane loading consisting of Nx=l, Ny=l and Nxy=l, are considered. Fig. 2 shows convergence of the buckling load (the critical eigenvalue) as a function of the order of polynomial Ritz series Nw (Eq. 17). Figure 3 shows critical buckling eigenvalues, calculated with Nw=4 (15 term complete polynomial), for cases in which the fiber orientation angle varies. Good accuracy and fast convergelice are demonstrated. With only 6 Lerms (order 2) the present results are only within 3% error of Ref. 3 results. For automated synthesis, where the need to have very fast analysis and sensitivity results is critical, this fast convergence makes i t possible to use low order models

angles. Thc a/l) I- ;IIIO for t l~c pl;ltcs I S I , ;~nd lllc t l ~ ~ c k ~ ~ c s s ovcr h r;~tio ib O . ( K ) I . All A ~ I I I I I I I I U I I ~ ~ I ; I I C S ;Ire analy/.ctl w~th a=h= lm. Clni;ni;il co~nprc~sion of N x = l is applictl.

In Fig. 5 . ~ h c conib~nctl clYccts of skcw an& and anisotropy arc cxamincd, and thc prcscnt rcsulrs iirc cornp;~rcd to RI I I x ~ d hnitc c lcn~cn~ rcwlts I'rom Rcl'. 20. and to results I'rom Ref. 40. Orthotrop~c panels w~th a=b=lm, thickness of O.001 m, a ratio of Dl 1/D22=5 and D66/D22=0.5, arc cxan~incd. Using 111c lay-up, thickness and material distributions allowed i n the present analysis, equivalent panels arc crcritcd to malt h those i n Refs. 20 atxi 40. Compressive in plane loads of Nx=Ny=l and a shear load of Nxy=0.5 were applied. Accuracy of thc present resulls, obtained with a 15 term Riv polynomial (Eq. 17). is good, and the effects of anisouopy and skcw angle prcscnt no difficulty.

Buckling rcsults for a simply supported Aluminum syrn~nctric trapc/oidal pancl. in which one of [hc sidc.; \ h i ~ r i h i In <I/<.. :rr; ~ I i o \ \ ~ i 111 1-1s 6 I rirlor~ii 111 11lan~- cornprcsswn (Nx=Ny=l) I S applrcd. Starting w ~ t h a rectangular geometry, the figure shows how the critical buckling cigenvalue varies u n t i l the trapezoid become a triangle. With analytic shape sensitivity with respect to the trapcmid's uppcr sdc (ob~lined as described i n this article), direct Taylor scries approximations are constructed (Ref. 37) and shown in the figure. The reference design used 1s the triangular design (upper b=O). The present capability is fasr (10 Rilz terms are used in Eq. 17) and reliable i n both analysis and sensitivity computations.

Accuracy. behavior sc'nsiti\.ity and Taylor scrics approximations for ~riangular panels of variable shape are shown i n Figs. 2 and 7. Simply supported c,quilatcral aluminum panels are analyzed for buckling, and the effec~ of varying ratio of altitude to base (ah) is examined (Ref. 33). A typical convergence study for the case of an equilateral panel is shown in Fig. 2. Fast convergence is achieved to the result of Ref. 41, allowing the use of low roder models in subsequent studies. In Fig. 7, the variation of critical buckling eigenvalue with changes i n a h is shown. "Exact" results (obrained with 10 Ritz terms) are shown, as well as direct and reciprocal Taylor series approximations based on analysis and analytical sensitivity at the reference design (a/b=l). Rayleigh Quotient Approximations (RQA) are also shown. They perform quite well for values of a/b greater than the refcrcnie value. However, when a/b is smaller than h c reference value, both RQA approximations fail beyond a 10% move limit. This approximation failure is similar to the problems encountered when aeroservoelastic poles are approximated in wing planform shape synthesis based on the equivalent platc approach (Ref. 42). Resolution of this problem and improved approximations for the case of planlorrn shape variations, is beyond the scopc of the present article, and will bc reported separately.

Figure 8 shows comparisons of first order finite difference sensirivities to analytic sensitivities obtained with the present formulatron. Thc rcct;~ngular panel used

I'or thc L . O I I \ ~ I . ~ ~ I I L I . s~udich ol' Fig. 2 IS c~scd hcrc. Dcs~gn v;~riiitdcs ;\IT ~hickncss, fihcr angle and two shapc variables. 'I'R ; I I I ~ X I % (WC, F I ~ . 1 ) . Rcs~~lls wcrc o lwi~wl with I0 R I ~ Z ~crtns. As can bc sccn, rcliable finitc diffcrcncc s cns i t i v~~~cs can be obtained with step sixes bctwecn 0.001 0; ~ n t i 0. 1 %. 11 sho~rld he rccogni/cd. that whcn sl~nplc p ~ I y l o n ~ i ; ~ l s are U S C ~ lor R i t s wing hox analysis (Ref. 25). i l l coutlitioning can l ~ t l to situalions in which no rc1i;lblc first order finitc diffcrcnce scnsitiv~tics can be ohiaincd duc to scvcrc roundoff crrors. This is not the case hcrc, as Fis. H shows. This, and the fast convcrgcncc dcmonstrstcd, ~ndicatc that there arc pracrically no i l l conditioning problems using simple polynomials in the present pancl buckling analysis

We ncxr move to examine the inlegration of pancl buckling analysis and sensitivity computalions as presented here, with a wing box structural analysis based on the equivalent plate approach. As has already bccn discussed, the main appcal of thc equivalent plate approach in wing structural analysis is in the simplicity of rnodcling and speed of cornputntion - both extremely imporlant for cfficicnt n~i~ltidisciplinary wing optiniization. A fighter typc all nluminurn u,ing is shown in Fig. 9, togcthcr with I I ~ C ;III. ,I! ,\! \ ~ ; I I \ ;1ri(1 r i b 0 1 I I I [ C ' I - I I ; I ~ \ I ~ I I ( . I I I I C . A 1.5C: par;~bol~c ~\lrf'o~I IS used. An alrplane weight 01'25,000 Ibs is assumed, and the wing is loaded in a uniform manner to support a 9g pull-up maneuver (multiplied by 1.5 for safety factor). Skin thickness over the wing varies linearly from root to t ~ p . r(x,y)=0.015 - 0.0026 y (meters). Shape variation of the internal structure are due to changes in the position of the rear spar, supporting a flaperon. As the flaperon chord to wing chord ratio varies, the design variable XCL (Fig. 9) changes. With the spars linked to each other to be equally spaced between X F L and XCL, all thc spars move as a result of changes in XCL. We focus on one skln panel, shown in Fig. 9. I t s change in shape and position, as the flaperon grows from 5 % ~ to 3 5 % ~ is shown in Fig. 10.

The wing box analysis in h i s case is based on a 10 term Ritz polynomial satisfying cantilever boundary conditions at y=O (Ref. 23). Ten terms are also used in h e Ritz series for the panel buckling analysis (Eq. 17). Analytic sensitivities of wing box response are discussed in Ref. 25. Analytic sensitivities of panel critical buckling eigenvalue were obtained by the technique described here. In Fig. 10, critical buckling eigenvalues for the panel are shown as a function of global changes in the wing structure (flaperon chord size). The in-plane loads are obtained from stresses in the skin predicted by the wing analysis. When constant NxjVySxy are used (based on their values at the center of Lhe panel xO,yO), buckling eigenvalues are smaller compared wilh the case, in which Nx,Ny and Nxy vary in a polynomial manner throughout the panel (as obtained from Eq. 45). The difference is smaller when the flaperon's chord is smaller. In hat case, h e panel is bigger and moves from h e front to the center chord of the wing, thus carrying a larger in plane load. An approximation based on direct Taylor series is shown in Fig. 10. No difficulties are encountered in the integration of wing analysis and sensitivity with panel buckling analysis and sensitivity computations, thus demonstrating the feasibility of combining lwo cfficicn~ modeling techniques for wing

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1992. 19. Smith, J.P., "BUCKY - A p F~r i~ tc Elcrncnt Progranl I'or Pl>atc Analysic", .41AA 03-i?OI. ProcciTd~ngs of thC 15111 AIAA/ASME/ASCE/AHS/ASC Structures, Structur;~l Dynamics and hlatcrials Confcrcncc, H~lton Kcad, SC', April 18-20, 1994, pp. 694-701. 20. Jounky, N., Knight J r . , N.F., and Amhur, D.R., "Buckling of Arbitrary Quntir~latcral Aniwtropic Plates", AIAA 9 4 - 1 3 6 9 , P roceed ings of the 35th AIAA/ASME/ASCE/AHS/ASC Strucrures, Structural Dynamics and h,laterials Conference. Hilton Kcad, SC, April 18-20, 1994, pp. 493-503. 21. Giles, G.L., "Equivalent Plate Analysis of Aircraft Wing Box S~ructures with Gcncral Planform Gcomctry", Journal of Aircraft, Vol. 23, No. 11, November 1986, pp. 859-864. 22. G ~ l e s , G.L., "Further Gcncralization of an Equi\ralcnt Platc Representation for Aircraft Structural Analysis", Journal of Aircraft, Vol. 26, No. I , January 1989, pp. 67-74. 23. Livne, E., "Integrated Multidisciplinary Optin~ization of Actively Controlled Fiber Composite Wings", Ph.D. dissertation, Mechanical, Acrospacc and Nuclear Engrncering Dcparlrncnl, Unr\ cr.;i~!. 01' Califor-nia. 1 ~ i Angclcs, 1900.. 24. S ~ r i ~ l i \ ~ . S.. :ind K : I ~ I I I I ; I . U,K . ' . - \ I I ; I I > ~ I I C ; I I Sh:tpC Scnsit iv~t~cx and Approx~~ii; l l~or, \ ol hlod:~l Kcspon\c (11. Gencrally Lamina~cd Tapered Skcu. Plalcs", AIAA 92- 2 3 9 1 , Proceedings o l t h e 3 3 t h AIAA/ASME/ASCE/At(S/ASC Strucrurcs, Structural Dynamics and Materials Conlcrcncc, Dallah. Tcxas, April l9W. 25. Livne, E., "Analy~ical S c n s ~ t r v r t ~ c ~ for Shapc Optimization i n Equivalent Plate Structural Wing hlotfcls", Journal of Aircraft, to be published 1994. 26. Livne, E., Sels, R , and Bhatia, K., "Lessons from Application of Equivalent Platc Structural Modeling to the HSCT Wing". Journal o f A~rcraft. LO be publ~shcd 1994. 27. Livne, E., "Equivalent Platc Structural Modcling for Wing Shape Optimization including Transverse Shcirr", AIAA Journal, to be published 1093. 28. Fletcher, C.A.J., "Compu~alional Calerkin Methods", Springer Verlag, New York 1984. 29. Leissa, A,, and Jacob, K.I., "Three Dimensional Vibrations of Twisted Cantilevered Parallelepipeds", Journal of Applied Mechanics. Seprernber 1986, Vol. 53, pp. 614-618. 30. McGee, O.G., and Lc.issa, A.W., "Three Dimensional Free Vibrations of Thick Skewed Can~ilevcred Plates", Journal of Sound and Vibrat~on (1991) 143(2), 305-3112. 31. Koch, R.M. , "Thrcc Dimensional Vibralions of Doubly- Skewed, Hclicordal. Ci~ntilcvcrcd Thick Shclls". AIAA 93- 1 6 4 I , P r o c c c ' d i n g . 3 0 1 ' t h e 3 51h AIAA/ASh4E/ASCE/AHS/ASC' S~ructurcs. Structural Dynarn~cs and M a ~ c r ~ a l s Conicrcncc, H~lton Kcad, SC, April 18-20, 1994, pp. 818-818. 32. Tsai. S.W., and Hahn, H.T., "lnuoduction to Composite Materials", Technomic Puhli\li~ng Conipany, 1980 (page 204). 33. Bruhn, E.F., Analysis and Design of Flight Vehicle Structures, Tri State Offset Company, 1973, Chapter C5. 34. Antona, E. , and Romeo, G. . "Analytical arid Experimental Investigation on Advanced Composite Wing Box Strucrures in Bending Ini.luding Effects of Initial

Impcrl'cctronx and Crushing Prcssurc", ICAS-86-4.2.3, 15th Congress of the International Council of the Aeronautical Scicnccs. I.ondon, England. Scptcnibcr 1986, pp. 1155-261. 35. Harvey, M., "Automatcd Finite Element Modeling of Wing Structures for Shapc Optimization", Masters Thesis, Dcparrmcnt of Aeronautics and Astronautics, University of Washingmn, Scattle, Washingmn, 1993. 36. Milosavljcvic, R., "Buckling Analysis and Analytic Shape Scnsitivitics for Trapezoidal Panels in Wing Preliminary Design", Masters Thesis, Deprtment of Aeronautics and Astronautics, University of Washington, Seaulc, Washington, July 1994. 37. Haftka, R.T., and Gurdal, Z., "Elements of Structural Optimization", Third Edition, Kluwer Acadcmic Publishers, Dordrccht, Boston, London 199 1. (Chaptcr 6) 38. Canfield, R.A., "High Quality Approximation of Eigcnvalues in Srructural Optimization", AIAA Journal, Vol. 28,No. 6., June 1990, pp. 1116-1122. 39. Canficld, R.A., "Design of Frames Against Buckling Using a Rayleigh Quotient Approximation", AIAA Journal, Vol. 31, No. 6, June 1993, pp. 1143-1 149. 30. Kennedy, J.B., and Prihhakara, M.K., "Combined-Load Buckling of Orthorropic Skew Plates", ASCE Journal of the E n g i n ~ > ~ r ~ n _ c hlcc~hanic\ D~vision. Vol. 105, So. EMI, February 1979, pp. 71 -7'9. 41. Timoshcnko, S.P., and Gere, J.M., "Theory of Elastic Stability", McGraw-Hill, New York, 1961, p. 393. 32. Livnc. E., and Li, W-L., "Acroservoelastic Aspects of Wing / Control Surracc Planform Shape Opt~mization", AIAA Paper No. 93-1483, presented at the AIAA/ASME/ASCE/AHS/ASC 35th Structures, Structural Dynamics and hlaterials Conference, Hilton Head, SC, April 18-20, 1994.

TABLE 1: Constants Ui and Vj, and their corresponding powers of x and y, associated with the admissible displacement series function FB.

4 (Row of fi]

Mauix)

1

2

3

mf?

(Power of x)

q' (Power of y )

n r + n j + n ;

n r + n j + n;-2

n;+n;+n;-l

TABLE 2: - - .

Coeeficients and powers of polynomial terms in the [F3] matrix

(if any of the powers of x and y in columns three and four is less than zero, the corresponding coefficient is set to zero)

+ Figure 1: Wing planfornl, internal sfrucfure, a 1 2 skin panels geometry: panel shape design variables.

TABLE 3: Elements of the matrix [a] : coefficients and powers of x and y.

q=1 (Row I of [Fz] Matrix

q=2 (Row 2 of IF2] Matrix

TABLE 4: Coefficients and powers of terms in the [Fz] matrix

\

1 0 rectangular, anisotropic (Ref. 3) triangular, isotropic (Ref. 41)

a l o

b U

0 1 2 3 order of pol{nomhl

6 7 V

Figure 2:Convergence of critical buckling cigenvalue for a 30 degrees angle ply rectangular panel in combined in-plane loading, and an isotropic triangular panel

545 under uniform in-plane pressure.

Figure 3: Critical buckling cigcn\~alues for recfangular angle-ply panels (Ref. 3).

STAGS (Ref. 20)

present

1 5 2 0 2 5 4 0 4 5 skew an y e (&)

Figure 5: Cr i t ical buckling loads for anisotropic skewed panels.

o 1 skew 25 angle 2 2 5 (degf 33. 5

4 5

Figure 4: Critical buckling loads for isotropic s k e ~ e d panels.

I computed , direct I

Figure 6: Critical buckling load for trapezoidal isotropic panels of varying shape.

Figure 7: Taylor series and Ragleigh Quotient approximations for crit ical buckling loads o f triangular panels of variable shape.

Figure 9: Fighter type wing, its spar and r i b arrangement, and panel analyzed for buckling.

2 - 0.001 0.1 1 0 normalized step size (%)

Figure 8: Accuracy o f f ini te difference derivatives o f panel buckling eigenvalue with respect to sizing, fiber orientation and shape design variables.

variable Nx,Ny,Nxy (from wing (4) fixed Nx,Ny,Nxy at (x0,yO)

Q) direct Taylor for variable Nx,Ny,Nxy

3 1.5

Figure 10: Variation o f critical panel buckling eigenvalue due to changes i n wing spar locations.