I

NASA-TM-111263

AIAA 95-3945

Equivalent Plate Modelingfor Conceptual Designof Aircraft Wing Structures

Gary L. GilesNASA Langley Research CenterHampton, VA

1st AIAA Aircraft Engineering,Technology and Operations Congress

Los Angeles, CASeptember 19-21, 1995

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics370 L'Enfant Promenade, S.W., Washington, D.C. 20024

https://ntrs.nasa.gov/search.jsp?R=19960016182 2020-04-11T21:48:17+00:00Z

EQUIVALENT PLATE MODELING FOR CONCEPTUAL

DESIGN OF AIRCRAFT WING STRUCTURES

Gary L. Giles*

NASA Langley Research Center, Hampton, Virginia

Abstract

This paper describes an analysis method that generatesconceptual-level design data for aircraft wing structures.A key requirement is that this data must be produced in atimely manner so that it can be used effectively bymultidisciplinary synthesis codes for performing systemsstudies. Such a capability is being developed byenhancing an equivalent plate structural analysiscomputer code to provide a more comprehensive, robustand user-friendly analysis tool.

The paper focuses on recent enhancements to theEquivalent Laminated Plate Solution (F_LAPS) analysiscode that significantly expands the modeling capabilityand improves the accuracy of results. Modeling additionsinclude use of out-of-plane plate segments forrepresenting winglets and advanced wing concepts suchas C-wings along with a new capability for modeling theinternal rib and spar structure. The accuracy of calculatedresults is improved by including Wansverse shear effects

in the formulation and by using multiple sets of assumeddisplacement functions in the analysis.

Typical results are presented to demonstrate these newfeatures. Example configurations include a C-wingtransport aircraft, a representative fighter wing and ablended-wing-body transport. These applications aaeintended to demonstrate and quantify the benefits of usingequivalent plate modeling of wing structures duringconceptual design.

Introduction

The design process for aerospace vehicles progressesthrough conceptual, preliminary and detailed designphases in which the sophistication of the analysismethods and the quality of the design data increases.During conceptual design, many alternativeconfigurations are evaluated in multidisciplinary designtrades to determine the values of system-level variables(e.g., type and number of engines), the size (grossweight) and shape (external geometry parameters) of acandidate configuration which will best meet a specified

Copyright © 1995 by the American Institute of Aeronautics andAstronautics, Inc. No copyrightis assertedin the United States underTitle 17, U.S. Code. The U.S. Government has a royalty-free licenseto exercise all rights under the copyright claimed herein forgovernment purposes. All other rights are reserved by the copyrightowner.

*SeniorResearch Engineer,Systems Analysis Branch,AeronauticsSystems Analysis Division, Associate Fellow AIAA.

measure of overall vehicle performance. A representativeconfiguration sizing code for use in conceptual design isthe Flight Optimization System (FLOPS) 1 whichoperates on such system-level design variables.

Typically in such codes, use is made of relativelysimple, experience-based equations or elementaryanalytical models to relate these system design variablesto the important vehicle characteristics from eachengineering discipline such as aerodynamic lift and dragand structural weight. This implicit, experience-baseddesign data should not be used beyond its limits ofvalidity. However, characteristics of advanced aircraftconcepts often exceed these limits and user judgment isrequired to adjust and extrapolate the available data inorder to perform system studies. Hence, there is acontinuing need to improve the quality of conceptualdesign data through use of explicit, physics-basedanalysis methods early in the design process

Airframe weight is the key structural parameter used inaircraft system studies. In conceptual and earlypreliminary design, empirical weight equations are oftenused to calculate this data. These equations meprincipally functions of the vehicle class and externalgeometry and require very little information about theinternal structural details. In later design phases, thelocations and sizes of an assemblage of structural

members making up the airframe are determined. Theairframe should be lightweight but also have sufficientstrength and stiffness necessary to satisfy all therequirements throughout its flight envelope. Generalpurpose finite dement structural analysis codes gateavailable to model and analyze the static and dynamicresponse of airframes in great detail. However, suchanalyses usually require considerable calendar time togenerate the finite dement model and repetitive analysescan be computationally expensive.

This paper will describe an analysis method that is

intended to generate conceptual-level design data foraircraft wing structures. It is a design-oriented structuralanalysis method 2 that is intended to bridge the gap,between weight equations and detailed finite elementanalyses. This capability is being developed byenhancing the Equivalent Laminated Plate Solution(ELAPS) computer code 3"5 to provide a more

comprehensive, robust and user-friendiy analysis tool.A key requirement is that the design data must beproduced in a timely manner so that it can be usedeffectively by multidisciplinary synthesis codes for

performing systems studies.The present paper will focus on recent enhancements

to the ELAPS code that significantly expand the

modelingcapability and improves the accuracy of results.Modeling additions include use of out-of-plane platesegments for representing winglets and advanced wingconcepts such as C-wings along with a new capability

for modeling the internal rib and spar structure. Theaccuracy of calculated results is improved by includingtransverse shear effects in the formulation and by usingmultiple sets of assumed displacement functions in theanalysis.

Typical results are presented to demonstrate these newfeatures. Example configurations include a C-wingtransport aircraft, a representative fighter wing and ablended-wing-body transport. These applications meintended to demonstrate and quantify the benefits of usingequivalent plate modeling of wing structures during

conceptual design.

Equivalent Plate Modeling

A wing box structure is represented as an equivalentplate in this formulation. Planform geometry of thisequivalent plate is defined by multiple trapezoidalsegments as illustrated by the two-segment box in Fig.1. Each plate segment has upper and lower cover skinswhich may contain multiple layers of compositematerial. The cross-sectional view of a typical segment

illustrates the generality to define out-of-plane shapessuch as the twist and camber characteristics of an aircraft

wing. The cross-sectional dimensions of wing depth,camber definition and cover skin thicknesses are defined

by polynomials which vary over the planform of eachsegment. For static analysis, loading is applied to thewing box as concenWated forces or distributed loads.Mass properties for dynamic analysis are defined byconcentrated or distributed quantities.

The specification of model Characteristics ascontinuous distributions in polynomial form on only a

few members requires only a small fraction of thevolume of input data for a corresponding l'mite elementstructural model where geometry and stiffness propertiesare specified at discrete locations. The resultingreduction in model preparation time is important duringearly design phases when many candidate configurationsare being assessed. Also, the geometric locations of themass quantities and the applied loadings can beindependently defined, i.e., they are not referenced to a setof joint locations as in a finite element model. The easeof relocating these quantities without disrupting otheraspects of the model is important during early designwhen such changes often occur. Finally, the polynomialdescription of model characteristics lends itself to usewith optimization algorithms since the polynomialcoefficients can be used directly as design variables.Therefore, the analytical formulation in ELAPS requiresminimal time for model preparation and modification.

The analysis procedure, outlined in Fig. 2, is based onthe Ritz method in which the deflection of the structure

is described by assumed polynomial displacementfunctions. In order to achieve a high level ofcomputational efficiency, the displacement functions U,V and W are defined by combinations of terms from apower series in the chordwise coordinate x and a power

series in the spanwise coordinate y. as indicated in thefigure. These x and y coordinates are non-dimensionalized by dividing by a reference length(usually the wing semispan) so that the coefficients allhave the same units of length. Substituting thesefunctions into the expression for total energy aiddifferentiating to minimize the energy produces a set oflinear, simultaneous equations which can solved for thedesired set of unknown polynomial coefficients. Thesecoefficients are used to calculate deflections, strains and

stresses over the planform of the plate segments.Typically, the number of terms in the polynomialdisplacement functions is relatively small (around ahundred) and results in efficient computation. Thenumber of terms is selected by the user. In this manner,the user is given the capability to trade accuracy forspeed. However, there are upper limits on the degree ofthe polynomials that can be specified for thedisplacement functions. This limit results from usingpower series terms which are non-orthogonal. High-degree terms produce nearly linearly dependent equationsand cause the set of governing equations to become ill-conditioned. Typical, practical upper limits on thepower series terms are fourth degree in x and seventhdegree in y.

2

Modeling Enhancements

A design oriented structural analysis code needs to becapable of modeling a wide variety of advanced conceptsand needs to include all the primary structural members.Recent enhancements to ELAPS include modeling ofwings composed of segments that are not all in the sameplane and adding capability to include rib and spar shearwebs in the structural model. In order to calculate the

deformation and stresses of the rib and spar webs,transverse shear considerations must be included in the

analytical formulation. These additions are discussed in

this section. The details of the approach used to includetransverse shear effects are described in Appendix A.

Out-of-plane modelingThe addition of out-of-plane modeling provides the

capability to perform structural analysis ofunconventional, advanced wing concepts 6 such as the C-wing 7 shown in Fig. 3. The various portions of such awing are modeled as shown in Fig. 4a by usingequivalent plate segments whose positions are defined bynewly added local analysis reference planes. These

reference planes are connected by sets of very stiffsprings with the goal of providing approximate,displacement continuity at the juncture of adjacent platesegments. This approach provides a good approximationto having equal displacements in each plate, ifsufficiently large values of spring stiffnesses are used.The motions of the upper and lower wing cover skinsgoverned by motions of the reference planes via rigid,straight lines that remain normal to these deformedsurfaces as illustrated in Fig. 4b for a single point that isrepresentative of all points over the surface. Since onlythe reference planes are connected in this formulation, iteliminates the time-consuming complexity of matching

all the joint locations along boundaries of adjacentcomponents (at both the upper and lower surfaces) of theactual wing structure that is required for conventionalf'mite element modeling.Each of the localanalysisreferenceplanesare defined

withrespecttotheglobalreferenceframe. In the currentimplementation,the inboardand outboardedges of the

local reference frames ate defined to be parallel to theglobal x-axis that is oriented in the chordwise directionof the wing, usually along the centerline of the vehicle.Therefore, the position of each reference plane is definedby an angular rotation about the x-axis relative to the x-yplane. These reference planes provide the capability tomodel wings with dihedral and/or tip fins as well as more

complex geometries such as C-wings. The sets of stiffsprings are not restricted to lie in the analysis referem:eplane. Hence, they can be defined near the mid-camberlocation of adjacent segments of the wing SmlCtural boxor located along the hinge line of control surfaces or atlocations of boundary condition constraints.

The coordinate systems for the stiff springs are definedwith respect to the global coordinate system in order toresolve the different spatial orientations of the platesegments. The strain energy of the springs must beexpressed in terms of the displacement functioncoefficients so that contributions to the overall stiffness

matrix can be evaluated. This relationship is expressedin terms of a transformation matrix [T#] as

{_ 6_ c$_0j 02 0j}r = [T4] {Cu Cv Cw C_, c_}r (1)

The terms _j,t$2,83 defme springtranslationsand 01,02,

03definespringrotations.The terms C u, C v,C w _c

coefficientsfor displacementfunctions in the x,y,z

directionsrespectivelyand C_ and C_y arecoefficientsfortransversesheardeformationto be discussedin the

next section.

The overall transformation matrix, [T4], is formed asthe product of three individual transformation matrices as

It41= [rl][r21[r l (2)

Taken individually, the elements of [T_], [/'2], [T3]are straightforward to evaluate. Therefore, for brevityonly a brief narrative description of each matrix is givento avoid a lengthy and detailed geometric definition anddescription of the various coordinate systems that meinvolved.

[T1] is used to transform deformations from the globalcoordinate system to the same deformations in a springcoordinate system.[1"2] is used to transform deformations from the

coordinate system of a local analysis reference plane tothe same deformations in the global coordinate system.[Ts] relates the deformations of a local analysis

reference plane that correspond to deflections that areevaluated at the locations of the springs to thecoefficients of the polynomial displacement functions.The reference plane deformations are calculated by

multiplying the polynomial coefficients by [1"3].

Elements of the [T 11 and [T 21 matrices are composedof direction cosines of a spring coordinate system and acoordinate system of a local analysis reference planesince both of these coordinate systems are defined withrespect to the global coordinate system. Calculation of

elements of [T3] involves evaluation of the expressionsfor translations and rotations given in Eqs. (A.1)-(A.5),where z is the perpendicular distance from the analysisreference plane to a spring of interest and x and y are its

coordinates in the analysis reference plane.The stiffness matrix of a spring, [Ks], is composed of

six diagonal elements corresponding to values for springconstants in the three translational and three rotational

directions. The contribution of an individual spring,

[Ks], to the overall or system stiffness matrix, [K_y,] isgiven by

[K,_,I= [T,], r [K,] [T,], - [T,], r [K,I [T,], + [T,]t,r [r,] [T,]_ (3)

where the subscripts a and b refer to two different localanalysis reference planes that are connected by the spring.

For static analysis, rigid-body motion of the equivalentplate must be conslrained. These constraints are oftenrefened to as boundary conditions. The stiff springs canbe used to constrain the translations and/or rotations at

selected locations. For this use, the displacement systemdenoted by the subscript b is taken to be zero, thus onlythe fast triple product in Eq. (3) is evaluated. Althoughdisplacements cannot be specified to be exactly zero at aselected location, use of sufficiently stiff springs willprovide a good approximation to the desired condition.An additional method of specifying boundary conditionsis provided for use in vibration analysis. A eigenvalueshift parameter has been included that allows a vibrationanalysis to be performed on a model with unconstrained(rigid-body) motions.

Modeling of ribs and sparsThe capability to model ribs and spars has been added

through the use of caps located at the upper and lowercover skins and corresponding shear webs through thedepth of the wing. Sets of ribs or sets of spars amdefined with respect to the corresponding plate segmentplanform definitions. Examples of sets of ribs and sparsare illustrated by the planform layouts shown in Fig. 5.A minimal number of input quantities are used to definethe geometric layout of these ribs and spars.

A set of ribs is defined by specifying the total numberof ribs that are to be evenly spaced between the inboardand outboard edges of a plate segment. This numberincludes the ribs along the inboard and outboard edges inaddition to the intermediate ribs. The rib definition is

very straightforward since all ribs are parallel and amlocated in the chordwise direction.

Three options are available for defining sets of spars.In the first option the spars are all parallel to the trailingedge of the plate segment and the specified number ofspars are spaced equidistant along the inboard edge asshown in Figs. 5a and 5b. The second option is similarto the first option except that the spars are all parallel tothe leading edge of the segment as shown in Fig. 5c. In

thethirdoption,the spars are "fanned out" between theleading and wailing edges with the ends evenly spaced onboth the inboard and outboard edges as shown in Fig. 5d.

The cross sectional areas of the caps and thethicknesses of the shear webs are specified by

polynomials that are defined over the planform of a platesegment. The values for areas and thicknesses at anylocation on an individual fib or spar are obtained byevaluating the appropriate polynomial at the location ofinterest. Therefore, the member sizes within a set can

change from fib to fib or spar to spar and also vary alongtheir lengths. This polynomial description of rib andspar member properties is useful in design studies in thatit links all member sizes using a small number ofpolynomial coefficients that can be used as designvariables. Also, the cross-sectional dimensions can be

evaluated along the length of each individual memberduring structural weight calculations.

In the analysis formulation, the discrete stiffnessproperties of these members are "smeared" to form anapproximate, continuous representation over theplanform of a segment. This approach permits tables ofintegrals that are calculated over the segment planformsfor generating the stiffness matrices for the cover skinsto also be used for the fibs and spars. Hence, thesmeared representation provides a significant reduction incomputational time compared to forming the requiredintegrals for each individual fib and spar.

Terms for the shear web contribution to the overall

stiffness matrix am obtained from the equations that meproduced by differentiating the energy expression of thewebs with respect to the coefficients of polynomials that

describe the transverse shear. The energy of a set ofsmeared webs is given by

E = (G/2) Sarea SZ [ (h) (t ,M,,a) (0,) 2 ] dzdA (4)

where G is the shear modulus of the material,h is the depth of the wing,t _,_,_,d is the equivalent thickness of the web,_ is the transverse shear strain in the web.

Integration is performed between the upper and lowerskin surfaces and over the planform of a plate segment

For fibs:

t .... d = t _,c,,,, * (no. of fibs) / (span of segmen0and y_ = (_

For spars:t _=_,,d = t d_,,u * (no. of spars) / (local chord * dy/dl)

and Ytz= _?x* dx/dl + (py* dy/dl

where t ,_,c,,,, is the thickness of a discrete web,

Cxand _y are transverse shear displacementfunctions described in the next section,

I is along the length of a web.

These calculations for shear webs involve considerationof transverse shear effects that is discussed in the next

section.

Transverse shear consideration

During early studies of a High Speed Civil Transport(HSCT) wing at the Boeing Company s, it was concludedthat for some applications it was important to includethe effects of transverse shear in the equivalent plateformulation in order to improve the accuracy of theanalysis results. The option to include transverse sheareffects has been added to ELAPS in conjunction with themodeling of rib and spar webs.

The analysis formulation of Ref. 4 is developed in

terms of expressions for the bending and stretching of areference plane. This formulation has three sets ofassumed displacement functions; U and V am in-planeand W is normal to the reference plane. The motions ofthe upper and lower wing cover skins am governed bymotions of the reference plane via rigid, straight linesthat remain normal to the deformed reference plane. Inorder to include transverse shear in the formulation

additional degrees of freedom am added which allow thestraight lines robe oriented at angles other than 90degrees (not normal) to the reference surface. Thesedegrees of freedom am formulated as additional rotations

in the x and y directions with respect to the referencesurface normal and am defined using two new sets of

assumed displacement functions _ and ¢y. The detailsof the transverse shear formulation used herein me

presented in Appendix A. This formulation results intransverse shear strain and stresses that am constant

throughout the depth of the wing but vary over theplanform. Thus, the shear strain is constant through thedepth of a fib or spar web but can vary along its length.

In the present analysis, the assumed displacementfunctions am formulated in a manner that allows the

shear deformation terms to be either included or neglected(set to zero) through specification of a few user inputparameters. Therefore, the importance of transverse sheareffects for a particular application can be assessed quicklyand may be neglected if the effects am small in order tominimize computational time. This ability to neglecttransverse shear during early design cycles and to includeit in later cycles provides a hierarchical analysisprocedure with the option to trade accuracy forcomputational speed; a desirable feature for use duringconceptual design. In contrast, the displacementformulation for conventional fn-st order shear deformation

plate theory, as used in Ref. 9, does not allow the sheardeformation terms to be simply set to zero. Acomparison of the two formulations is given inAppendix A.

Mass modeling enhancementsDuring conceptual design studies, one of the key

considerations is the modeling of overall vehicle weightor mass. Enhancements that have been made to ELAPS

which facilitate modeling of masses that am importantfor wing design am discussed in this section. Majorconsiderations include items such as the main structural

box, the leading and trailing edges including any controlsurfaces, fuel carded in the wing, and wing mountedengines and pylons. The mass of the idealized structuralbox is calculated as the product of the density and

volume of material in the equivalent plate structural

4

model. Currently, this idealized structural mass ismultiplied by a correlation factor to approximate a valuefor actual structural mass. This correlation factor is

usually obtained by performing an equivalent plateanalysis of an existing, baseline vehicle with knownmeasured masses.

User input of concentrated masses was available inearlier versions of ELAPS for representing a variety ofsources of non-strncmral mass. The capability to readilymodel mass that is distributed over the wing planform

has been added to the present version. This mass can beuniformly distributed over the planform area and/orthroughout the internal volume between the upper andlower surfaces of each plate segment. The user is giventhe option to input either total or unit massescorresponding to the area or volume of each platesegment. This capability allows leading and trailingedge structural mass to be easily distributed over theappropriate areas of the wing planform and fuel mass tobe readily distributed within designated volumetricregions of the wing.

This user defm_ mass model is used to calculate a

variety of mass related quantities. The integral tables thatwere formed to generate the stiffness matrix of thestructure are used in these calculations. Total mass of allstructural members and all non-structural items that ate

contained in the model is calculated along with the firstand second moments of inertia with respect to the globalaxes. The center of gravity location is then calculated bydividing the first moments of inertia by the total mass.This total mass and inertia data is available for use in

stability and control calculations during early conceptual

design.Inertia relief loads are calculated based on input of a

value of maneuver load factor. These inertia relief loads

are combined with the aerodynamic loads to give loadsthat are used for structural design. In addition, a massmatrix is generated for the entire model for use invibration analysis to calculate natural modes andfrequencies.

Solid Modeling for Aeroelastic CalculationsThe integral tables for the planfolm of the equivalent

plate segments were extended to include terms forintegration through the entire depth of the wing inaddition to the previous terms for integration through thecover skins. These new tables can be used to form the

stiffness matrix for wings that have solid materialbetween the upper and lower surfaces. Wind tunnelmodels are often constructed in this manner in an attemptto minimize model deflections to a level that resultingeffects on measured aerodynamic data can be neglected.

The ELAPS solid modeling capability was used toanalyze a solid wind tunnel model of a high aspect ratiotransport wing that was recently tested in the NationalTransonic Facility. t° This equivalent plate model wascoupled with an unstrnctured-grid Euler flow solver withan interacting boundary layer method to perform a seriesof aeroelastic analyses as described in Ref. 11.Including the aeroelastic effects in the calculated wingpressures improved the comparison with measured windtunnel data. Up to 19,917 surface nodes were used in the

unstructured aerodynamic grid. The effort required tocombine the loads and the deflections from the

unslluctured aerodynamic analysis and the equivalentplate model was minimal because of the continuousdefinition of the wing deflection in ELAPS. Also, whenthe number and location of the surface nodes in the

aerodynamic grid was changed, the corresponding forceswere calculated in a consistent manner without anymodifications to the codes.

The relatively small size of the ELAPS code offers thepracticality of embedding it directly into the iterativeprocedure of any CFD code so that such a combined codewould converge to an aerodynamic solution on adeformed shape with a minimal increase incomputational time over that required for the analysis ofa rigid shape. In addition to the application outlinedherein, equivalent plate models can be used effectivelyvery early in the design process to assess the importanceof aeroelastic effects on advanced concepts before detailedfinite element models are available.

Levels of Analysis

Input parameters are available in ELAPS to provideuser control of the level of detail to be included during ananalysis. In this manner, the user is given the capabilityto trade accuracy for computational speed. Thiscapability is beneficial during conceptual design sinceinitial design cycles can be performed at a low level ofdetail that can be easily increased during later cycles.This approach can be particularly advantageous whenELAPS is used in a structural optimization procedurethat requires a large number of analyses during eachiterative design cycle. The level of analysis in ELAPScan be controlled without changing the geometricdefinition of the model. Rather, the total number ofunknown quantities in the resulting set of governingequations is readily changed by specifying a few inputparameters. Three methods are available for selecting thelevel of detail to be included in an analysis.

Method 1: Select which deformations (U, V, W, ¢_, Cy)are to be included in a set of displacement functions.

Method 2: Specify the number of terms used to defineeach deformation quantity.

Method 3: Specify multiple sets of displacementfunctions to be used in an analysis.

With Method 1, the most elementary level of analysisis selected by using only the out-of-plane bendingdeformation, W. This modeling is valid for pure bending(no transverse sheafing) of symmetric equivalent platemodels of thin wing structures representative of fighteraircraft and for modeling of aerodynamic control surfaces.The next level of analysis includes the inplanedeformations U and V in addition to W. This level c_

be used to represent unsymmetric plate structures thatresult from wing camber and twist and/or cover skinsthat are not the same thicknesses on the upper and lowersurfaces. Finally, transverse effects can be included with

theaddition of Cx and Cy to the set of displacementfunctions. Transverse shear deformation can be

significant for thick wing configurations and in designswith low shear stiffness resulting from a small numberof ribs and spars and/or from small web thicknesses.The effects of neglecting some of these deformations canbe quickly assessed for a given configuration to establishthe appropriate level of analysis needed for a particularaspect of conceptual design.

With Method 2, the degree of the polynomials that _esummed to form each displacement function atespecified. Combinations of terms from a power series inthe chordwise direction x and a power series in thespanwise direction y were used in the previous versionsof ELAPS. There are upper limits on the degree of theterms in the power series. These upper limits meproblem dependent but typical exponents are 6 or 7.Increasing the degree produces an ill-conditioned set ofgoverning equations and the upper limit is reached whenthe solution subroutine terminates with a message toindicate excessive numerical error. The previousversions of ELAPS used an equation solver based onCholesky decomposition. In the present version, anoption is provided to use a solver that incorporatesGaussian elimination with a full pivoting strategy. Itwas found that when the Gauss solver is used the

exponents could be increased by 1 or 2 over the upperlimit used by the Cholesky solver.

Two approaches were studied in an attempt toovercome the ill-conditioning problem. In the firstapproach, direct generation of the governing equationsusing orthogonal polynomials was evaluated. Thisapproach was found to require significantly morecomputational time than the very efficient power seriesimplementation. The second approach was investigatedin which the set of governing equations were generatedusing powers series and then transformed to a set ofequations with basis functions from a family oforthogonal polynomials. This approach was

implemented by appropriate pre- and/or post-multiplyingboth the stiffness and mass matrices and load vectors bya transformation matrix that related the orthogonal

polynomials to the polynomials composed of powerseries. Options to select Legendre, Chebyshev orHermite orthogonal polynomials were implemented andtested. This approach proved to be unsuccessful in that

attempts to use exponents for terms in the oahogonalpolynomials that were greater than the upper limit usedfor power series either caused the solution subroutines toterminate or the analysis results to diverge. Considerableeffort was devoted to the second approach because of thepotential simplicity that would be gained through use ofa single set of very high-order, orthogonal displacementfunctions to represent the behavior of all plate segmentsin a wing structure.

Method 3 was tested by using multiple sets ofdisplacement functions with each set governing thebehavior of a selected grouping (subset) of the platesegments in the model. These sets of displacementfunctions were connected using the stiff springs discussed

in the out-of-plane modeling section. Method 3 wasfound to give increased accuracy while not producing

excessively large matrices and provides a reomunendedalternative to the implementation of Method 2.

6

Applications and Results

Results are presented in this section from selectedexample applications that demonstrate the new equivalentplate modeling features of the ELAPS code. Theseexamples are intended to illustrate the variety of

stt'uctural analyses that can be performed and the kinds ofcalculated data and the levels of accuracy that can beproduced for use in conceptual design studies.

C-wing ExampleIn the fn'st example, the wing structure for a C-wing

transport configuration shown in Fig. 3 is analyzed todemonstrate the use of the new out-of-plane modelingcapability. A schematic of a semispan equivalent plateof this wing is shown in Fig. 6. Four plate segments,numbered 1-4, are used to model the C-wing structure.Segments 1 and 2 represent the main wing box m_dsegments 3 and 4 represent the vertical and horizontal finportions of the C-wing respectively. Segment 3 has aconstant chord and has the same sweep as the leadingedge of the main box. One set of stiff springs are usedto provide cantilever boundary conditions at thecenterline of the wing and two other sets are used toconnect segment 2 to segment 3 and to connect segment3 to segment 4. A conventional planar wing that hasextended tip added to the main box (outlined in dashedlines) is also shown superimposed to the same figure.The planar wing is composed of segments 1, 2 and 5.Segment 5 of the planar wing is the same size and shapeas segment 4 of the C-wing. The depths of thesegments, h, have a linear spanwise variation and have aconstant value in the chordwise direction

Analyses ate performed to compare results for the C-wing and the planar wing under comparable appliedloads. A uniform pressure load of 1.0 psi acting upwardis applied to the planar wing. The C-wing is analyzedfor two separate loading cases. In both cases, uniformupward pressure is applied on segments 1 and 2 and noload is applied to segment 3. However, the pressure loadon segment 4 acts upward in the first load case and actsdownward in the second load case. A pressure load of 1.0psi is used for the first load case which gives the sametotal lift as for the planar wing. The pressure load isincreased to 1.306 psi in the second load case to provideto same total lift because of the downward pressure onthe horizontal fin.

The vertical deflections of the leading and trailingedges along the semispan of the main box (segments 1and 2) are shown in Fig. 7. The planar wing has thelargest tip deflection. The C-wing with an upward loadedhorizontal fin has less vertical tip deflection but hasmore tip twist as indicated by the larger difference invertical deflections of the leading and trailing edges. TheC-wing with a downward loaded fin has significantly lessdeflection and the tip is twisted to have a positive angle-of-attack.

Spanwise stresses in the upper and lower cover skinsat the trailing edge of the main box are shown in Fig. 8.

Theplanarwing has compressive stresses in the uppercover skin and tensile stresses in the lower cover skin.These stresses increase from tip to root which is expectedfor a constant cover skin thicknesses used for this

illustrative example. The C-wing has some unusualstress distributions at the tip of the main wing box.These unusual distributions are caused by the moments

that are produced by loading on the horizontal fin. In thecase of the upward loading, the cover skin stressesactually change sign at approximately 85 percentsemispan. In the case of downward loading, the coverstress is approximately zero in the 75 tmr,,ent semispanregion but remains in compression in the upper skin andtension in the lower skin of the outboard region.

Additional studies am required to determine if control ofthe loading on the horizontal fin can be used to reducethe structural weight of the main wing box as a result ofthe lower stresses. This potential structuralimprovement must be traded against the effects onaerodynamic drag that would occur. The ELAPSanalysis code can provide the structural data needed toinvestigate such questions during early conceptualdesign.

In addition, the first ten vibration frequencies for theC-wing and the planar wing are compared in Fig. 9. Thefrequencies of the C-wing are significantly lower that forthe planar wing indicating that aeroelastic phenomenonsuch as flutter should be analyzed in subsequent studies.

Clipped-delta ExampleA representative example of a fighter wing structure

that was studied in Ref. 4 is used herein to assess the

accuracy of the newly implemented smeared webmodeling capability and the new transverse shearformulation in ELAPS. The planforms of the equivalentplate model and a finite element model used forcomparison are shown in Fig. 10. The planforms arecomposed of a clipped-delta outer segment with a 45

degree leading edge sweep and an inner segment that isused to represent a carry-through structure.

In this study, rib and spar shear webs are incorporatedin the equivalent plate model. These rib and spar websare located vertically through the depth of the wing boxalong the lines that define the mesh shown in the finiteelement model. A smeared representation of these websis used in the equivalent plate formulation while discreteshear elements are used in the finite element model.

Static analyses were performed with three differentthicknesses in all rib and spar webs; tw,b = 0.01, 0.10and 5.0 inches. The web thickness of 5.0 is

unrealistically large but is used in the finite elementmodel to approximate the condition of infinite shearstiffness (no shear flexibility). For this case ELAPSwas run with the transverse shear displacement functions

set equal to zero. A uniform pressure load of 1.0 psi isapplied to the outer segment of the wing. Values ofvertical displacement at the trailing edge of the wing tipas indicated in Fig. 10 are compared. These results meshown in Fig. 11. The tip deflection increases only

slightly when the web thickness is reduced from 5.0 to0.1. The finite element and equivalent plate results _'e

in close agreement for twe b ---- 5.0 and 0.1. Transverse

shear effects am more pronounced when the webthickness is reduced to 0.01. In this case ELAPS gives a

7 percent larger deflection than the finite elementanalysis.

The shear stress distributions are compared along the

semispan of the selected spar that is indicated in Fig. 10.These results am shown in Fig. 12 for a web thicknessof 0.1. The shear stresses for ELAPS are greater than forthe finite element analysis. Also, a discontinuity inshear stress occurs between the inner and outer platesegments. The stiff springs used to connect the twosegments enforce approximate continuity ofdisplacements but the strains and stresses may bediscontinuous as shown in Fig. 12. The stresses fromthe two analyses exhibit the same trends and theequivalent plate results am shown to be of sufficientaccuracy for use in conceptual design studies.

Blended-Wing-Body ExampleThe use of ELAPS for structural sizing is illustrated

by this example. Also, use is made of the newcapabilities for mass modeling. The aircraftconfiguration used in this study is an advanced blended-wing-body t2 large subsonic transport that offers potentialperformance benefits. A schematic of a representativeconfiguration is shown in Fig. 13. The inboard regionof this vehicle has sufficient depth to accommodate adouble deck passenger cabin with a theater-like seatingarrangement. Candidate structural concepts for carryingthe cabin pressure loads include (a) a self-containedpressure vessel that is suspended inside of the wing boxstructure and (b) use of sandwich skins with sufficient

depth to carry the local bending moments produced bythe cabin pressure. The present study is representative ofconcept (a) in that cover skins of the main box structureare sized to carry only the aerodynamic loads.

A planform view of a semispan equivalent plate modelof this vehicle is shown in Fig. 14. A total of 18 plate

segments are contained in the model. Segments 1-6represent the main structural box; segments 7-12 are usedto represent the leading edge; and segments 13-18 areused to represents the trailing edge region of the wing.The spanwise stiffness of the material in the leading mltrailing edge is reduced so that no spanwise wing bendingloads are carried by these regions. However, any loadsthat are applied to these regions are properly transmittedfore and aft to the main box structure. The depth andcamber of each plate segment is specified as products ofcubic polynomials in the spanwise and chordwisedirections. A single set of displacement functions isused to govern the displacements in all 18 platesegments.

Representative non-structural masses that are includedin the model are also indicated in Fig. 14. Thecapability to distribute mass uniformly over theplanform area of a segment is used to represent the massof passengers in segments 1 and 2. The capability todistribute mass within the volume between the upper m-dlower surfaces of a segment is used to represent the fuelmass in segments 3 and 4. The contributions to theovemU mass matrix of the model am calculated in

ELAPS by integrating the products of the masses in

theseareasand volumes with the displacement functionsover the planforms of these segments. Total massesassociated with each segment is all the user input that isrequired. Concentrated masses are used to represent

localized components such as landing gear and engines asshown in Fig. 14. A single mass is used for eachlanding gear and the engine weight is distributed amongfive masses along the length of the engine. The x, y, zcoordinates and magnitude of each mass is all the requireduser input. This straightforward, simple method ofdefining a mass model is very useful during conceptualdesign. The time-consuming operation of distributingthese various masses to grid points in a finite elementmodel is avoided in the equivalent plate approach. Thestructural and non-structural masses are used to calculatethe first and second moments of inertia of the vehicle

along with the total mass and center of gravity location.Inertia relief loads are also calculated corresponding tomaneuver load factors that can be input by the user. A2.5g symmetric pull-up maneuver is used as arepresentative design condition in this study. Anelliptical spanwise distribution of aerodynamic pressureis assumed. This pressure is distributed over all 18planform segments.

The thickness distribution of the cover skins in the

main wing box is sized using a manual iterative

procedure. In each iteration cycle, Von Mises stresses inthe cover skins are calculated using ELAPS. Then, theskin thickness values over the planform are increased ordecreased by the ratio of the calculated Von Mises stressto the allowable stress of the material. The thicknesses

of the upper and lower skins are assumed to be the same.This so-called fully-stressed design procedure convergedin three cycles. The resulting skin thickness distributionalong the semispan of the wing is shown in Fig. 15.The large thickness that occurs in the vicinity of thetrailing edge break, y/semispan = 0.45, could be reducedby increasing the wing chord and corresponding wingdepth at that location. Such a design change to improvethe structure would have to be traded against the effect onaerodynamic performance that would occur. Thisexample indicates the necessity to be able to use ELAPSin a multidisciplinary design system to perform overallsystem level design Wade studies. The recentenhancements to ELAPS have been demonstrated to

provide an effective design tool for such purposes in thatthe major considerations in the design of aircraft wingstructures can be analyzed in a timely manner.

Concluding Remarks

Recent developments in equivalent plate modeling aredescribed. These modeling methods are for use in designand analysis of aircraft wing structures during conceptualdesign studies before the external shape of the vehicle isfixed. The capability to define out-of-plane platesegments has been added for representing winglets ardadvanced concepts such as C-wing configurations. Amore complete representation of a wing structure isprovided by the addition of rib and spar shear webs. Theanalysis formulation is expanded to include the effects oftransverse shear deformations. This formulation provides

shear stresses in the rib and spar webs and also improvesthe aconacy of the wing deflections and cover skinstresses over previous versions of ELAPS that neglectedtransverse shear effects. These enhancements gae

implemented such that the user can easily control thelevel of detail to be included during an analysis. Thiscontrol is beneficial during conceptual design sinceaccuracy of results can be waded for computational speedin order to maximize the effectiveness of overall

structural design procedures. An additional capabilityallows leading and wailing edge structural mass to beeasily distributed over appropriate areas of the wingplanform and fuel mass to be readily distributedthroughout designated regions of a wing. Overall massand inertia properties are calculated along with inertiarelief loads. These additions provide a versatile

equivalent plate analysis tool that can model the physicalbehavior of the major components that must beconsidered in wing design and can effectively producestructural-related data for use in performing systemsstudies during conceptual design.

Typical results are presented to demonstrate these newfeatures. A C-wing transport aircraft is used to illustrateout-of-plane modeling capability. A representativefighter wing is use to illustrate the modeling of rib andspar webs along with the use of the transverse shearformulation. A blended-wing-body transport is used toillustrate the use of new mass modeling capabilities andto demonstrate a manual structural sizing procedure.These applications demonstrate and quantify the benefitsof using equivalent plate modeling of wing structuresduring conceptual design.

Appendix ATransverse Shear Formulation

The analysis formulation of Ref. 4 is developed interms of expressions for the bending and stretching of areference plane. The motions of the upper and lowerwing cover skins are governed by motions of thereference plane via rigid, straight lines that remainnormal to the deformed reference plane. In order toinclude transverse shear in the formulation additional

degrees of freedom are added which allow the straightlines to be oriented at angles other than 90 degrees withthe reference surface. These degrees of f_edom are

defined as additional rotations in the x and y directionswith respect to the reference surface normal.

Displacements and rotations throughout the wing are

approximated by

u = Uo - zWo,x + zCx (A.1)

V = Vo - z Wo,y + z Cy (A.2)

w = Wo (A.3)

Ov= Wo,y- _y (A.4)

Ov= Wo,x- (_x (A.5)

where¢xand_yarethetwonew rotational displacement

fields. In the Ritz analysis procedure, these displacementfields are specified as products of terms from a powerseries in the x direction with terms from a power series

in the y direction as is done for the U, V, and Wcomponents of the deformation.

Corresponding strains in the skins in the x and y

directions are given by

ex = Uo,x - zWo,xx + ZgPx,x (A.6)

ey = Vo,y - zWo,yy + z¢y,y (A.7)

exy = Uo, y + Vo, x - 2zWo,xy +

zCx,y + zCy,x (A.8)

and shear strains in the rib and spar webs are given by $x = 7ix + W, x (A.19)

)'xz = t_x (A.9) q'x -- Cx- W, x (A.20)

Tyz = 4/y (A.10)

Thus, in this formulation these shear strains in thewebs are constant through the depth of the wing but vary

over the planform of the wing. These strains are used inthe Ritz procedure, as described in Ref. 4 to produce asystem of simultaneous equations which can be solvedfor the unknown coefficients to minimize the total

energy expression. These equations have the same formas illustrated in Fig. 2. The addition of the tworotational deformation fields results in a 5 x 5 set ofsubmatrices in the stiffness matrix.

The kinematic assumptions of fn'st order sheardeformation plate theory is used in the equivalent plateformulation of Ref. 9.

U = Uo + z 7"x (A.11)

V = Vo + z Wy (A.12)

W = Wo (A.13)

With corresponding strains in the skins in the x and ydirections are given by

ex = Uo, x + ZWx, x (A.14)

ey = Vo,y + z_"y,y '(A.15)

exy = Uo,y + Vo,x + Z_Px, y + ZT_y,x (A.16)

As shown, the formulation in this paper and theformulation in Ref. 9 are equivalent. However, theformulation in this paper offers the option to neglect thetransverse shear effects in a direct manner by simplysetting this rotation equal to zero through use of an inputparameter to the code. The present formulation isdirectly suited for use in a hierarchical optimizationstrategy where transverse shear is neglected during earlyoptimization cycles and is included in only the finaloptimization cycles where the additional accuracy isneeded. In addition, for symmetric segments such assome control surfaces, W only can be used withsignificant saving of time. The formulation in Ref. 9does not lend itself to setting the shear terms to zero andappears to require a separate analysis code to includetransverse shear considerations.

References

1 McCullers, L. A.: Aircraft Configuration

Optimization Including Optimized Flight Profiles.NASA CP-2327, Proceedings of Symposium on Recent

Experiences in Multidisciplinary Analysis andOptimization, April 1984, pp. 395-412.

2 Sobieszczanski-Sobieski, J.: Multidisciplinary

Design Optimization: An Emerging New EngineeringDiscipline. Presented at The World Congress onOptimal Design of Structural Systems, Rio de Janeiro,.Brazil, August 2-6, 1993.

and shear

_txz =

_'yz =

strains in the rib and spar webs are given by

_gx + Wo,x (A.17)

_Py + Wo,y (A.18)

A comparison of the transverse shear rotations is givenin the following sketch.

3 Giles, G. L.: Equivalent Plate Analysis of AircraftWing Box Structures with General Planform Geometry.J. of Aircraft, Vol. 23, No. 11, November 1986, pp.859-864.

4 Giles, G. L.: Further Generalization of an

Equivalent Hate Representation for Aircraft StructuralAnalysis. J. of Aircraft, Vol. 26, No. 1, January 1989,

pp. 67-74.

5 Giles,G.L.: DesignOrientedStructuralAnalysis.In Advancesin StructuralEngineeringComputing;Proceedingsof 2nd InternationalConferenceonComputationalStructuresTechnology,Athens,Greece,Aug.30-Sept.1,1994,pp.l-10. Also published asNASA TM-109124, June 1994.

6 Bushnell, D. M.: Potential Impacts of AdvancedAerodynamic Technology on Air Transportation System

Productivity. NASA TM 109154. September 1994.

7 McMasters, J. H.; Kroo, I. M.; Bofah, K.; Sullivan,

J. P.; Drela, M.; and Gallman, J." AdvancedConfigurations for Very Large Subsonic TransportAirplanes. NASA Contract: NAS1-20269, NASA CR inpreparation.

8 Livne, E.; Sels, R. A.; and Bhatia, K. G.: Lessons

from Application of Equivalent Plate StructuralModeling to an HSCT Wing. AIAA Paper No. 93-1413-CP, 34th AIAAIASMEIASCEIAHSIASCStructures, Structural Dynamics, and MaterialsConference, La Jolla, CA, April 1993, pp. 959-969.

9 Livne, E.: Recent Developments in Equivalent PlateModeling for Wing Shape Optimization. AIAA PaperNo. 93-1647-CP, 34th AIAAJASME/ASCE/AHS/ASCStructures, Slructural Dynamics, and MaterialsConference, La Jolla, CA, April 1993, pp. 2998-3011.

10 Igoe, W. B.: Characteristics and Status of the U.S.National Transonic Facility. Cryogenic Wind Tunnels,AGARD-LS-111, July 1980, pp. 17-1-17-11.

11 Cavallo, P. A.: Coupling Static Aeroelastic

Predictions with an Unstructured-Grid Euler/InteractingBoundary Layer Method. Master's Thesis, GeorgeWashington University Joint Institute for Advancementof Flight Sciences, NASA Langley Research Center,Hampton, VA, July 1995.

12 Liebeck, R. H.; Page, M. A.; Rawdon, B. K.;Scott, P. W.; and Wright, R. A.: Concepts forAdvanced Subsonic Transports. NASA CR-4624,September 1994.

conoosblayeredcovel

Multiple trapezoidal plates

skin thickness

Applied loads- Point forces

- Pressures

- Temperatures

depthMid-camber

surface

Dimensions are defined

by polynomials

distance

out-of-plane

Reference

plane

Fig. 1 Analytical modeling of wing box structure.

10

• Ritzsolution technique used

• Polynomial displacement functions

U = _._.A U xiYJzj

V - _Bkl Xkyik ]

W= _, Cmnxmy nIll n

[] Minimization of energy

I in-plane stretching

I bending

/,-

= 1/2 Jvol ( ET Q E - 2 A T (xT Q E) dV

[] Solve equations for polynomial coefficients; A, B and C

Kenm

Kuv Kuw

Kw Kvw

Kvw K_

Bkl{ = ' Pv

CmnJ

Fig. 2 Outline of equivalent plate analysis procedure.

Fig. 3 C-wing aircraft configuration.

11

Spring _ of analysisreference surfaces. Kinem_lc relationships

Fig. 4 Out-of-plane equivalent plate modeling.

(a) Parallel to T.E. (c) Parallel to L.E.

\

(b) Parallel to T.E. (d) Equal chord fraction.

Fig. 5 Modeling of rib and spar structure.

12

FRONTVlEW

Fig. 6 C-wing and planar wing equivalent plate models.

Vertical

deflection,

In.

too

$o

eo

4o

zo

.2c :o.o

• Planar wing: TE: up

/ Planar wing: LE: upLE = Wing leading edge , F.TE = Wing trailing edge ,w ._ C-wing: TE: up

down = Downloaded wing tip *** j"up = Uploeded wing ,p

:..y/__ C-wing: LE: down

C-wing: TE: down

• . • . , . • . •

o.2 o.4 o.e o.a t.o

y I wing semispan

Fig. 7 Vertical deflection of wing leading and trailing edges.

13

_. "'le.,,o_i.o,,_n /_ C-wing: lower: down

_6_ -- " _" _b. _ "'" ''-. aL ,'_ G-.wing: upp.r: up

I ." O'_ "" W / .o._:ole Planar wing: upper: up

_ ..._..._ /I ...o'""_ _.-'* -_. c.,g:°pp.r:down•.21; .,, _'" _ C-wing: lower: up

.40 *°

_ _°.-" down . downloaded wing Up

7S "'_°'" up- uploaded wing tip

O.O 0.2 0.4 0.6 0.8 1.0

y / wing eemispan

Fig. 8 Spanwise stress in cover skin along wing trailing edge.

C-win9

SO

2O

10

0

Frequency,Hz.

1 2 3 4 ti ti 7 ti 9 10

Mode number

Fig. 9 Vibration frequency comparison for C-wing and planar wing.

14

I

Stressescorn

Deflections

compared

here 7

Finite element model

984 elements to define1565 equations to solve

edge

f Trailing edge I

Equivalent plate model

2 plates to define80 equations to solve

Fig. 10 Clipped-delta wing structural models.

Wing tip

deflection,

in.

• Finite element analysis

[] Equivalent plate analysis

O.Ol 0.1 $

Shear web thickness

Fig. 11 Effect of transverse shear on wing tip deflection.

15

2O0O

1000

Shear stress,

psi.

---o - - gimps plate 1

r _ "ram"m<I"m FllnaP": P'llsld:le2nl mnmlysls

: i

o •

am"al

%:..;

-'%.° 0'2 o', o,' o'. ,.o

y I wing somlspan

Fig. 12 Shear stress distribution along a wing spar.

" t i

Fig. 13 Blended-wing-body aircraft configuration.

16

7

8 14

Landing gear mass

Engine mass

Passengers

Fuel

10 16

11 17

12 18

Fig. 14 Equivalent plate model of blended-wing-body.

Cover. skin

thickness,

In.

0.6

o.t5

0.4

0.3

O.2

0.1 ¸

°_:o oi_ o:, o:, oi, ,o

y I wing semlspen

Fig. 15 Spanwise distribution of cover skin thickness.

17