Design Oriented Analysis

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American Institute of Aeronautics and Astronautics

DESIGN-ORIENTED ANALYSISOF AIRCRAFT FUSELAGE STRUCTURES

Gary L. Giles*

NASA Langley Research Center, Hampton, Virginia

Abstract

A design-oriented analysis capability for aircraft fuselage structures that utilizes equivalent plate methodology isdescribed. This new capability is implemented as an addition to the existing wing analysis procedure in theEquivalent Laminated Plate Solution (ELAPS) computer code. The wing and fuselage analyses are combined tomodel entire airframes.

The paper focuses on the fuselage model definition, the associated analytical formulation and the approach used tocouple the wing and fuselage analyses. The modeling approach used to minimize the amount of preparation of inputdata by the user and to facilitate the making of design changes is described. The fuselage analysis is based on ringand shell equations but the procedure is formulated to be analogous to that used for plates in order to take advantageof the existing code in ELAPS. Connector springs are used to couple the wing and fuselage models.

Typical fuselage analysis results are presented for two analytical models. Results for a ring-stiffened cylindermodel are compared with results from conventional finite-element analyses to assess the accuracy of this newanalysis capability. The connection of plate and ring segments is demonstrated using a second model that isrepresentative of the wing structure for a channel-wing aircraft configuration.

Nomenclature

b, h, z = dimensions of rectangular portion of ringcross-section [see Fig. 3]

E = modulus of elasticity

k s = spring stiffnessQ = lamina stiffness matrixr = radius of shell reference surface

R 1 , R 2 = radii of curvature in meridional andcircumferential directions

s = meridional shell coordinatet = thickness of cover skin layeru,v,w = displacements in Cartesian coordinatesu v w, , = displacements in shell coordinates

x,y,z = Cartesian coordinates seg = deflection of segment at spring connection s = strain in meridional direction = strain in circumferential direction s = shear strain____________________

* Senior Aerospace Engineer, Systems Analysis Branch,Aeronautics Systems Analysis Division, Airframe Systems ProgramOffice. Associate Fellow AIAA.

Copyright 1998 by the American Institute of Aeronautics andAstronautics, Inc. No copyright is asserted in the United Statesunder Title 17, U.S. Code. The U.S. Government has a royalty-freelicense to exercise all rights under the copyright claimed herein forGovernmental Purposes. All other rights are reserved by thecopyright owner.

= angle between Cartesian and shell coordinates = Poissons ratio = strain energy = circumferential coordinate(,) = denotes differentiation, e.g., w, x = dw/dx ;

w, xx = d 2

w/dx2

Introduction

During the conceptual design of aircraft, manyalternative configurations must be evaluated inmultidisciplinary design trades to determine thecharacteristics of a candidate configuration which willbest meet specified measures of overall vehicleperformance and/or cost. Airframe weight is the keyparameter that is required from the structures discipline.The airframe should be lightweight but also have

sufficient strength and stiffness necessary to satisfy allthe requirements throughout the flight envelope.General-purpose finite-element structural-analysis codesare available to model and analyze the static anddynamic response of airframes in great detail. However,such analyses often require several months to generatethe finite-element model and repetitive analyses can becomputationally expensive. With the objective of


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reducing design-cycle time, equivalent platemethodology 1-4 has been developed and used for analysisof aircraft wings and empennage structures during earlyphases of design. In order to model entire airframes,similar capabilities are needed for modeling other

structural components.This paper describes a design-oriented analysis

capability for aircraft fuselage structures. This newcapability has been implemented as an addition to theexisting wing analysis procedure in the EquivalentLaminated Plate Solution (ELAPS) computer code 5.Thus, the wing and fuselage analyses are now combinedso that entire airframes can be modeled.

The paper focuses on the fuselage model definition,the associated analytical formulation and the approachused to couple the wing and fuselage analyses. Themodeling approach used to minimize the amount of preparation of input data by the user and to facilitate themaking of design changes is described. The fuselageanalysis is based on ring and shell equations but theprocedure is formulated to be analogous to that used forplates in order to take advantage of the existing code inELAPS. Connector springs are used to couple the wingand fuselage models.

Typical fuselage analysis results are presented for twoanalytical models. Results for a ring-stiffened cylindermodel are compared with results from conventionalfinite-element analyses to assess the accuracy of thisnew analysis capability. The connection of plate andring segments is demonstrated using a second modelthat is representative of the wing structure for a channel-wing aircraft configuration.

Analytical Modeling

Fuselage structures are modeled as ring-stiffened shellsegments, typical of transport aircraft construction, asshown in Fig. 1. Each shell segment is used torepresent large regions of a fuselage and only a smallnumber of segments are typically used to model anentire fuselage. These segments have circular crosssections with a radius that varies along the length to

represent area ruling and the necking down in the foreand aft regions of a typical vehicle. The skin of theshell segments are composed of layers of orthotropicmaterial. The properties of the layers can be defined torepresent composite laminates or a smearedrepresentation of longerons. Fuselage frames aremodeled with rings having cross sections that arecomposed of multiple rectangles.

The shell segments are defined over quadrilateralregions with boundary edges at constant values of xalong the length and at constant values of in thecircumferential direction as shown in Fig. 2a.Segments for half airplane models, that are symmetric

about the x-z plane, are often defined with extendingfrom - /2 to + /2. An appropriate set of boundaryconditions must be applied at = - /2 and at = + /2.Small segments with a negative thickness can besuperimposed on these larger segments to representcutouts such as doors or windows. The radius of thereference surface for the shell is defined by a polynomialfunction along the length of the segment.

r x r r x r x r x r xnn( ) = + + + + +0 1 2 2 3 3 L (1)

The skin of the shell segments consist of orthotropiclayers with the thickness of each layer being definedindependently by a two-dimensional polynomialfunction along the length and around the circumference.

t x t t x t x t t xk mnm n( , ) = + + + + +00 10 20 2 01 L (2)

Orientation of the stiffness properties and correspondingthickness are specified for each layer, and theorientations and thicknesses can be different in differentshell segments.Fuselage frames are modeled with rings having crosssections that are defined using up to three rectangles sothat a variety of shapes (e.g., zee, tee, cee) can be

represented. An example of a zee cross section isshown in figure 3 that indicates the width, b , height, h ,and the distance of the centroid from the shell referencesurface, z , must be defined for each rectangle in thecross section. The dimensions of cross sections at eachend of a shell segment must be defined along with thetotal number of frames, N , to be equally spaced alongthe length of the segment. At frames between the ends,the cross sectional dimensions are interpolated using alinear variation between dimensions of rectangles thatare defined at the ends of a shell segment. During theanalysis, the bending and extensional stiffnesses of these frames are smeared over the surface of the shell.This smeared approximation provides improvedcomputational efficiency. Individual, discrete framescan be modeled using additional narrow shell segmentswith a width equal to the width of the discrete frameshell. The behavior of discrete frames will be closelyapproximated when the frame stiffnesses are smearedover these narrow segments. The option of usingdiscrete or smeared modeling can be used to trade


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accuracy for computational speed, a desirable feature of design-oriented analysis methods.

Analysis Procedure

The analysis procedure is based on the Ritz method inwhich the deflection of the structure is described byassumed polynomial displacement functions. Axial,tangential and radial ( u, v, w ) deflections are defined interms of functions in the axial, x , and circumferential, , coordinates on the surface of a shell segment asshown in Fig. 2b. Note that the radial deflection isdefined to be normal to the x-axis rather than theconventional definition of being normal to the shellsurface. This definition of radial deflection facilitatesthe coupling of shell segments to plate segments. Theassumed displacement functions are substituted into theexpression for total energy. This equation isdifferentiated with respect to each of the displacementfunction coefficients to minimize the total energy. Aset of linear, simultaneous equations is produced thatcan be solved for the desired set of unknown polynomialcoefficients. These coefficients are used to calculatedeflections, strains and stresses at a user specified grid of points over the surface of the shell segments.

The total energy consists of the strain energy of thestructure and the virtual work associated with theapplied loads and assumed displacement functions. Thestrain energy of the structure has contributions from the

shell segments and the frames. Only the membraneenergy of the shell is used while energies associatedwith both extension and bending are included for thering frame. The strain equations are derived from ringand shell theory.

Strain Energy of Shell Segments The membrane strain energy of a shell segment is

given asshell T area Q dA= 1 2 { } [ ]{ } (3)

where{ } { }

T s s= (4)

Strain-displacement relations and the differential area interms of conventional shell coordinates are given inRef. 6 as

s su w R= +, 1 (5) = + +v r ur r w Rs, , 2 (6) s s su r v vr r = + , , , (7)

and dA rd ds= (8)

where is the circumferential coordinate and s is themeridional coordinate. The displacement w is normal

to the shell reference surface and the displacements uand v are in the meridional and circumferentialdirections respectively. In order to make the fuselageshell formulation be analogous to the plate formulationfor wings, quantities in these strain-displacementequations are transformed to the Cartesian coordinatesystem shown in Fig. 2b. Formulating energyexpression in this form, facilitates the coupling of theshell and plate segments. The quantities in theexpression for strain energy after transformation are

s x x x xu r w s= +( ) ( ), , , ,2

(9)

= +( )v w r , (10) s x x x xu r v rv r w rs= + +( ) ( ), , , , , , (11)

and dA rs d dx x= , (12)

where s r x x, ,= +1 2 (13)

The details of the transformation procedure are given inAppendix A.

Strain Energy of Ring Frames Only in-plane displacements of the ring frames are

included in this formulation; out-of-plane bending andtorsion are neglected. The general equations for thestrain energy of a ring are given in Ref. 6 in terms of the same Cartesian coordinate system that is used forthe shell in the preceding subsection. Therefore, the in-plane contributions are taken directly from theseequations and are written in the nomenclature of thispaper as

ring r A r r E rdAd = 1 2 { }[ ]{ } (14)

where r rv rw zv zw r = + + ( ), , , 2 (15)

and A refers to the cross section of the ring. The firstand second terms in Eq. (15) give the membrane strainin the ring and are seen to be the same as for the shellas given in Eq. (10). The third and fourth terms in Eq.(15) give the bending strain in the ring. In the presentformulation, it is assumed that the cross section of eachring is constant around the circumference. Therefore,the properties of the ring cross section given by the


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integrals z dA ii A =; , ,0 1 2 that occur in Eq. (14) arenot a function of and can be readily evaluated andsummed for the rectangles used to define a ring crosssection.

Implementation of MethodA goal in this development was to make the

formulation of the new, design-oriented fuselageanalysis method analogous to the formulation that wasused in the equivalent plate analysis method for wings.This approach facilitates the implementation of the twoanalyses in a single version of the ELAPS code. Inparticular, procedures that were developed for plateanalysis are also used in shell analysis. One example isthe use of the special library of subroutines that existsin ELAPS to perform the operations of addition,subtraction, multiplication, differentiation and

integration of the terms in the high-order, lengthypolynomial expressions that result when geometricalfunctions and assumed displacement functions aresubstituted into the energy equation. Use of thesesubroutines allow all operations to be performed in anexact, closed-form manner and result in astraightforward, simplified coding implementation of these lengthy, tedious polynomial expressions.However, in order to use these special subroutines, allexpressions must be in polynomial form.

Unlike the strains for plates, the strains for shells andrings contain terms that cannot be expressed in standardpolynomial form. Functions of the shell radius, r , that

is expressed as a polynomial in Eq. (1), are contained inthe denominator of expressions for the shell and ringstrains, Eqs. (9), (10), (11) and (15). In addition, thequantity s, x in Eq. (13) is the square root of apolynomial that cannot be manipulated by the specialset of subroutines. The terms 1/r and s, x are bothfunctions of the shell radius. In order to be able toutilize the advantages of the special subroutines,quantities containing these terms are approximated bystandard polynomial functions. Each quantitycontaining these terms is evaluated at a set of pointsalong the length of a fuselage segment and a least-squares fit is made through these points to obtain apolynomial that approximates the original geometricquantity. The resulting polynomials can be combinedwith other polynomials in the strain energy equationsusing operators in the special subroutines. Thisprocedure allows closed-form integration to be used toform sets of integral tables that are subsequently used toassemble the stiffness matrix for the structural model.This procedure has been demonstrated to be very

computationally efficient in applications of theequivalent-plate analysis. This approximation approachwas readily implemented in the ELAPS code and offeredefficiency advantages over the conventional approach of using Gaussian integration that was used to implement

similar ring-frame energy equations in Ref. 7.

Connecting Structural Segments Spring elements are used to connect the structural

segments that comprise an airframe model. Use of suchsprings to connect adjacent shell elements along theircommon circumferential boundaries is illustrated in Fig.4. A typical connection is shown by the springs thatare located along circumferential boundaries of the shellsegment between the second and fourth rings in theillustrative figure. In addition, the springs that areshown located along the longitudinal boundary are usedto impose appropriate boundary conditions at the planeof symmetry. Translational springs in the x , y , and zdirections and a rotational spring about the x-axis can bedefined at each location indicated by the symbols. Therotational springs are used to connect the circumferentialbending deformation that is governed by ring framestiffnesses of adjacent shell segments. The strainenergy of such connectors between segment A andsegment B is given by

spring s segA segBk = ( )1 22

(16)

where spring stiffnesses, k s, are defined for each of thethree translations and the rotation. In the displacementquantities, segA and segB , the radial and circumferentialdisplacements of a shell must be transformed todisplacements in the y and z directions beforecalculation of the springs contribution to the globalstiffness matrix. When the springs are used to imposedboundary conditions on a model, the displacements forone of the segments is taken to have the value of zero.These connector springs are also used to connectadjacent plate segments by locating translations springsat the upper and lower surfaces of their commonboundaries.

Finally, these connector springs are used to connectshell segments to plate segments. Typical shell-to-plate connections are shown in Fig. 4. The connectorsprings do not have to be located at any particularpoints along the intersection of two segments, sinceboth the model geometry and the assumed displacementsare defined as continuous functions over the segments.Individual springs can be defined as well as a set of evenly-spaced springs along a segment edge. This


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method of connecting segments provides significantversatility for constructing models and facilitatesmaking design changes such as shifting the longitudinallocation of the wing/body intersection. Suchdefinitions and modifications are time-consuming and

difficult to automate when using conventional finite-element methods.

Applications and Results

The fuselage structural analysis procedure will beapplied to examples that illustrate various features of the new method. Two examples are presented, (1) aring-stiffened cantilever cylinder that is used to assessthe accuracy of the fuselage formulation and (2) a plate-ring model that is used to demonstrate the connection of plate and ring segments.

Cantilever Cylinder ModelAn example application of a ring-stiffened cantilever

cylinder that has been studied in Refs. 8 and 9 is shownin Fig. 5. The cylinder has a length of 60.0 in., radiusof 15.0 in. and a wall thickness of 0.032 in. Fourstiffening rings are evenly spaced along the length of the cylinder. These rings have a rectangular cross-section with a height of 0.78 in. and a width of 1.00 in.The material in both the cylinder wall and rings isaluminum with a Youngs modulus of E = 10.6 x 10 6

psi and a Poissons ratio of = 0.33. The cylinder is

modeled with a single shell segment that spans theentire length. Only half of the cylinder is modeled sincesymmetry boundary conditions are applied byconstraining deflections in the y-direction and rotationsabout the x-axis along shell edges in the x-z plane, at = -90 o and at = +90 o. The four rings are modeled asdiscrete members by smearing their stiffness propertiesover 1.0 in. wide shell segments at each of the ringlocations.

Cantilever Cylinder Results A static analysis is performed with a radial point load

of 1000 lb. in the positive z-direction at the free end.

The deformed shapes of rings 1-3 from the ELAPSanalysis are shown in Fig. 6. The displacement valueswere multiplied by a factor of 40 for illustrativepurposes. The ring-stiffened cylinder was also analyzedusing a conventional finite-element method 10. In Table1, numerical values of radial displacements at = -90 o,0o, and +90 o on the free end that are calculated using theELAPS model are compared with results from the

finite-element model (FEM). A maximum power of 8was used on the x and terms in the assumeddisplacement functions for u , v and w in the ELAPSanalysis. Results are shown for three different levels of FEM modeling refinement. The entire circumference of

the cylinder was represented in the FEM, not asymmetric half-model. The cylinder wall was modeledusing a single row of finite elements between the rings.The least refined model, FEM 1, had 12 joints aroundthe circumference and four constant strain triangles wereused for each skin element as in Ref. 8. Linear strainquadrilateral elements were used to model the cylinderwall in FEM 2 and FEM 3 with the number of circumferential joints increased to 24 in FEM 3.Agreement between the displacements from the mostrefined FEM model , FEM 3, and displacements fromELAPS is within 5 percent.

The rings in the finite-element models are modeledwith straight beam elements between joints in thecircumferential direction. A comparison of themoments in ring 1 from the ELAPS and FEM 3analyses are shown in Fig. 7. Similar results for rings2 and 3 are shown in Fig. 8. The curves for momentsfrom ELAPS are in good agreement with the curvesfrom the FEM analysis. The largest differences occuron the ELAPS segment boundaries at = -90 o and =+90 o. The difference at = -90 o on ring 2 is the largest.It appears that the displacement functions that span theentire half model used by ELAPS give a goodrepresentation of the overall moments but provide lessaccuracy in localized areas. Comparisons of thedistribution of shear stresses in the cylinder wall fromELAPS and the FEM 3 analyses are shown in Figs. 9and 10. These shear stresses are evaluated mid-waybetween rings 1 and 2 for Fig. 9 and mid-way betweenrings 2 and 3 in Fig. 10. There is good overallagreement in the curves from both methods.

The comparisons of displacements, moments andshear stresses for the ring-stiffened cylinder indicate thatthe new design-oriented analysis method in ELAPSprovides adequate accuracy for use during conceptualdesign.

Plate-Ring ModelThe second example application is a plate-ring model

shown in Fig. 11. A half-cylinder is connected to twoplates to demonstrate the connection of plate and ringsegments. This relatively simple example isrepresentative of more general plate-shell connectionssince the bending of plate segments are reacted by thebending stiffness of the rings in a stiffened shell. The


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plates are connected to the ring by sets of connectingsprings shown as the large, solid circular symbols.Clamped boundary conditions are imposed at the inneredge of the inboard plate. The thicknesses of the upperand lower skins of the plates and the inner and outer

flanges of the ring have a constant value of 0.20 inches.The material properties of aluminum from the previousexample are used for analysis purposes. Thedimensions of this plate-ring structure are constant inthe direction normal to the cross-section shown in Fig.11. Therefore, only a strip of unit width is modeled.

Although the geometry of this model has beensimplified for illustrative purposes, it is still somewhatrepresentative of a semi-span wing model for a channel-wing aircraft configuration. Channel-wing aircraft havebeen investigated because of their potential benefits forshort take-off and landing operations 11. An engine ismounted inside of the semi-circle. The high velocity of the airflow in the semi-circular region that is producedby the engine generates increased lift over that portionof the wing at low velocities of the aircraft.

Plate-Ring Results A static analysis is performed with a 1.0 psi upward

pressure load applied to the outboard plate segment forillustrative purposes. The vertical deflection of theplate-ring model is shown as a function of semispanlocation in Fig. 12. The deflection of the plate-ringmodel shown by the dashed curve is compared to thedeflection of a cantilever plate model shown by the solid

curve. The cantilever plate has an overall length of 250.0 in. and a constant depth of 10.0 in. The ELAPSmodel used to generate the solid curve was created byreplacing the semi-circular ring segment with a 100.0inch long plate segment. The deflection of the plate-ring model is greater that the cantilever plate modelbecause the flexibility of the 100.0 in. diameter ringsegment is greater than that of a 100.0 in. long platesegment.

The tensile stress in the lower cover of the platesegments and outer flange of the ring segment is shownas a function of semispan location in Fig. 13. Again,the stresses for the plate-ring model are compared with

stresses for the cantilever plate model. The stresses forboth the inboard and the outboard plate segments are thesame for both models since the bending moments fromthe applied loads are the same. However, the stresses inthe outer flange of the ring differs from the stresses inthe lower skin of the cantilever plate model. Thisdifference is caused by the manner in which the pressureloading that is applied to the outer plate segment is

carried by the ring segment and the corresponding platesegment in the cantilever plate model. Tensile stressesin lower surface of the cantilever plate model is theresult of only the applied moment on the plate cross-section that is produced by the applied pressure loading.

Stresses in the outer flange of the ring segment is theresult of the applied moment and a vertical componentof force. This vertical component of force increases theouter flange stress in the outer 90 o portion of the ringand decreases the stress in the inner 90 o portion.

Concluding Remarks

A design-oriented analysis capability for aircraftfuselage structures is described. The analytical model isdefined using polynomial functions that minimize theamount of input data preparation and also facilitate anysubsequent modifications that are made to the modelduring design. The analysis is based on the Ritzmethod and uses strain-energy equations from ring andshell theory. Example results are presented to indicatethat the accuracy of this new structural analysis methodis sufficient for use in conceptual design. The use of connector springs to couple plate segments and ringsegments is demonstrated. The capability to couplering-stiffened shell segments with plate segmentswithin the ELAPS code provides a new design-orientedtool for modeling and analyzing entire airframes.


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Appendix A Shell Strain Coordinate Transformation

The detailed steps of transforming the shell strains inshell coordinates of Eqs. (5)-(7) to strains in Cartesiancoordinates are presented in this Appendix. First , thedisplacements u , v and w in shell coordinates and thedisplacements u, v, and w in Cartesian coordinates areconsidered. The circumferential displacements are thesame in both coordinate systems, i.e., v = v. Theremaining two sets of displacements are shown in thefollowing sketch.

ww

u

u

x

r

The transformation equations between the two sets of displacements are

u

w

u

w=

cos sin

sin cos

(A.1)

The trigonometric terms in Eq. (A.1) can be convertedto differential form using the quantities shown in thenext sketch

dx

dr

ds

u

u

with cos = d x /ds = 1/s, x and sin = d r /ds = (d r /d x)/(d s /d x) = r, x /s, x.

Substituting Eqs. (A.1) in differential form into Eqs.(5)-(7) gives

s sss

s s sss

s s xr

Ru x u r

x

Rw r w=

+ + +

+, , , , , , , ,1 1

(A.2)

= + + + x r

r r R

u vr

r r

x R

ws s s s s, , , , , ,2

2

2(A.3)

s s s s s x u r v rv r w r = + +( ), , , , , , (A.4)

Now, the derivatives of quantities, [ ], with respect to sare expressed as derivatives with respect to x.

[ ] = [ ]= [ ] =[ ], , ,s x xd

ds

d

dx

dx

dss (A.5)

and

[ ] =[ ]

=[ ] [ ] ,

,, , , , ,

ss

s

xx x x x xx

d

dx

dx

dss s s2 3 (A.6)

Next, the differential quantities in shell coordinates, s, xand s, xx, are converted to differentials quantities in theradial coordinate, r, x and r, xx, using the differentialrelation from consideration of the preceding sketch

ds dx dr 2 2 2= + (A.7)

or s r x x, ,2 21= + (A.8)

then s r x x, ,= +1 2 (A.9)

and s r r r xx x xx x, , , ,= +1 2 (A.10)

The shell curvatures, 1/ R1 and 1/ R2, in shell coordinatescan be expressed in Cartesian coordinates as

1

11 23

2 R

r

r

xx

x

= +( )

,

,(A.11)

and1 1

2 R r rs x= =cos

,

(A.12)

Substitution of Eqs. (A.5)-(A.12) into Eqs. (A.2)-(A.4)gives the strain equations in Cartesian coordinates asshown in Eqs. (9)-(11). Note that values of zeros areobtained when the coefficients for the u and wdisplacements in Eq. (A.2) are evaluated. Similarly, thecoefficient of the u displacement in Eq. (A.3) is foundto be equal to zero.


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References

1 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.

2 Giles, G.L.: Further Generalization of anEquivalent Plate Representation for Aircraft StructuralAnalysis. J. of Aircraft, Vol. 26, No. 1, January 1989,pp. 67-74.

3 Livne, E.; Sels, R.A.; and Bhatia, K.G.: Lessonsfrom Application of Equivalent Plate StructuralModeling to an HSCT Wing. J. of Aircraft, Vol. 31,No. 4, July-Aug. 1994, pp. 953-960.

4 Livne, E.: Equivalent Plate Structural Modeling forWing Shape Optimization Including Transverse Shear.AIAA Journal, Vol. 32, No. 6, June 1994, pp.1278-1288.

5 Giles, G.L.: Equivalent Plate Modeling forConceptual Design of Aircraft Wing Structures. AIAAPaper No. 95-3945, 1st AIAA Aircraft Engineering,Technology and Operations Congress, Los Angeles,CA, Sept. 19-21, 1995.

6 Bushnell, D.: Computerized Analysis of Shells -Governing Equations. Computers & Structures,Pergamon Press Ltd., Vol. 18, No. 3, 1984, pp. 471-536.

7 Bowman, L.M.: Analysis of Fuselage Frames withGeneral Circumferential Shape using the Ritz Method.George Washington University Masters Thesis, July1991.

8 Turner, M.J.; Martin, H.C.; and Weikel, R.C.:Further Development and Applications of the StiffnessMethod. In Matrix Methods of Structural Analysis,Pergamon Press, 1964, pp.204-266.

9 Langelaan, J.W.; and Livne, E.: AnalyticalSensitivities and Design Oriented Structural Analysisfor Airplane Fuselage Shape Synthesis. Computers &Structures, Pergamon Press Ltd., Vol. 62, No. 3, 1997,pp. 505-519.10 Whetstone, W.D.: EISI-EAL Engineering Analysis

Language Reference Manual--EISI-EAL System Level2091, Engineering Information Systems, Inc., July1983.11 Blick, E.F.; and Homer, V.: Power-on ChannelWing Aerodynamics. J. of Aircraft, Vol. 8, No. 4,April 1971, pp. 234-238.

Table 1. Comparison of displacements

Model Radial displacement of free end

= -90 o = 0 o = 90 o

ELAPS a -0.2056 0.1177 -0.0671FEM 1 b -0.1903 0.1030 -0.0601FEM 2 c -0.1982 0.1086 -0.0644FEM 3 d -0.2037 0.1098 -0.0660

a The ELAPS model uses maximum power of 8 on x and in the displacement functions.b FEM 1 has 12 joints per ring and four constant strain triangles are used for each skin element.c FEM 2 has 12 joints per ring and linear strain quadrilateral elements are used for the skin.d FEM 3 has 24 joints per ring and linear strain quadrilateral elements are used for the skin.


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Membrane shell with

layered composite skin.

Frame cross-section definedusing three rectangles.

Fig. 1 Analytical modeling of fuselage structure.

(a) Geometry definition

X

Y

Z

r(x, )

t(x, )

u(x, ) v(x, )

w(x, )

(b) Displacement functions

Fig. 2 Shell segment definition.

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Carry thru Floor

Side body

Connector springs

Fig. 4 Coupling of segments using connector springs.

Fig. 3 Fuselage frame definition.

Frame 1 Frame N

b

hz

N equally spaced frames. Dimensions vary linearly.

Between ends of segment.

Shellreferencesurface


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15

4 @15F=1000 lbs

Ring 1 2 3 4

Fig. 5 Cantilevered ring-stiffened cylinder.

Fig. 6 Ring displacements.

y

z

RING 1

RING 2

RING 3

ORIGINAL


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-1000

-500

0

500

1000

1500

2000

2500

-90 -60 -30 0 30 60 90

, degrees

Moment,in-lb FEM

ELAPS

-600

-400

-200

0

200

400

600

800

1000

1200

-90 -60 -30 0 30 60 90

,degrees

Moment,in-lb

ELAPS, RING2FEM, RING2ELAPS, RING3FEM, RING3

Fig. 8 Moments in ring 2 and ring 3.

Fig. 7 Moments in ring 1.


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-400

0

400

800

1200

-90 -60 -30 0 30 60 90 , degrees

Stress,psi

Series1

Series2

FEM

ELAPS

-400

0

400

800

1200

1600

-90 -60 -30 0 30 60 90 , degrees

Stress,psi

FEMELAPS

Fig. 10 Shear stress between ring 2 and ring 3.

Fig. 9 Shear stress between ring 1 and ring 2.


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0.0

1.0

2.0

3.0

4.0

5.0

0 50 100 150 200 250Semispan location, in

Deflection,in CANTILEVER PLATE

PLATE-RING

50.0 100.0 100.0

10.0

10.0

Fig. 11 Plate-ring model of channel wing structure.

Fig. 12 Vertical deflection of wing structures.

Fig. 13 Stress in lower cover skin.

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CANTILEVER PLATEPLATE-RING

Design Oriented Analysis

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