Development of Analytical Methods for Fuselage Design

Development of analytical methods for fuselage design:validation by means of finite element analyses

L Boni and D Fanteria*

Dipartimento di Ingegneria Aerospaziale, Universita di Pisa, Pisa, Italy

Abstract: The paper presents the results of a set of finite element analyses (FEAs) carried out tosupport the development of an integrated design procedure that, based on semi-empirical andanalytical methods, is capable of defining generic fuselage sections of a transport aircraft. Theprocedure, which is implemented in a structural optimization code, defines a structure that, compliantwith durability and damage tolerance requirements, is characterized by a post-critical behaviour ofthe stiffened panels and by a design of the frames that takes the frame flexibility and the presence ofthe floor beams into account.FEAs, carried out on a reference configuration defined by the optimization code, are used to

acquire a deeper knowledge of the advantages and disadvantages of the analytical approach in thedesign of complex structures subjected to realistic load cases. In particular, the influence of the actualframe flexibility on the distribution of the skin shear flow induced by the frame is evaluated;moreover, the effects on the stress distribution in skin and frames, caused by the presence of thestringers, and of the stiffness concentration introduced by the floor beam are addressed.Finite element method results demonstrate the effectiveness of the analytical model of the flexible

frame in evaluating the shear flow that a single loaded frame transfers to the skin and highlight theeffects of the presence of adjacent loaded frames. By means of geometrically non-linear FEAs, theeffects of the stringers on the stress distribution of a pressurized cylinder are evaluated, as well as themagnitude and extension of the perturbation introduced by the floor beams.

Keywords: structural optimization, fuselage design, frame, finite element methods

NOTATION

D shell bending stiffness per unit lengthE Young’s modulus of the materialIx moment of inertia of the fuselage section

with respect to the X axisMx longitudinal bending momentMz axial torqueN load indexNz longitudinal membrane axial force per unit

lengthq shear flowR fuselage radiusSx static moment of the fuselage section with

respect to the X axist skin thicknessteq equivalent skin thickness

Ty shear load acting along the Y axisw skin radial displacementwf frame radial displacement

Dp pressure loadn Poisson’s ratio of the materialW anomaly along the fuselage perimeter

1 INTRODUCTION

The fuselage of a commercial aircraft has the funda-mental function of accommodating the payload, byguaranteeing a good level of comfort for the passengersand a high functionality in the cargo storage, in additionto containing the instrumentation and the flight equip-ment.

The fuselage structural layout is quite complex,because such a structure must withstand aerodynamicand mass loads (causing bending, torsion, and shear) aswell as internal pressure loads. As for every structural

The MS was received on 23 December 2003 and was accepted afterrevision for publication on 24 August 2004.* Corresponding author: Department of Aerospace Engineering, PisaUniversity, Via G Caruso, Pisa 156126, Italy. email: [email protected]

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G05603 # IMechE 2004 Proc. Instn Mech. Engrs Vol. 218 Part G: J. Aerospace Engineering

component of an aircraft, the fuselage weight must bekept to a minimum, while retaining adequate perfor-mances with respect to safety requirements and eco-nomical exigencies.

Conventional analysis methods have been developedsince the late 1940s, especially in the USA [1–3] andcontinuously refined through the years to obtain well-established simplified stress analysis methods; typicalexamples of such methods can be found in the literature[4–8].

Semi-empirical and analytical stress methods can beintegrated into design procedures, often implemented incomputer codes, which are suitable for profitable use inthe pre-development phase of new airframes [9].

By using such integrated design procedures a baselineconfiguration can be defined starting from a limitednumber of input parameters; this makes them the idealtool to carry out parametric and sensitivity studies in thepreliminary design phase. The results, which are rapidlyachieved through a process of constrained optimizationwith respect to given combinations of desirable features(mass, cost, etc.), constitute a good starting point formore sophisticated method of structural design andoptimization usually based on finite element analyses(FEAs).

The capability of a design procedure, based on semi-empirical and analytical methods, to generate significantresults relies upon the applicability of the methods, uponthe validity of the unavoidable simplifying assumptionsand on a correct management of the interfaces betweenthe number of procedures for the design of the structuralcomponents.

In this context, at the Dipartimento di IngegneriaAerospaziale, an integrated design procedure has beendeveloped and implemented in a computer code, whichpermits the minimum weight dimensioning of typicalcomponents of a transport aircraft fuselage, i.e. stiffenedpanels, frames, and passenger and cargo floor beams,supporting struts and tear straps [10].

During the development of the integrated designprocedure, specific FEAs have been carried out on areference configuration defined by the numerical code inorder to achieve a deeper insight into some aspects ofthe structural design of fuselage components.

In this paper some results of such FEAs arepresented and discussed, with the aim of developing anadequate sensitivity about the advantages and thelimitations of the analytical approach, when used todesign complex structures subjected to realistic loadcases and, possibly, to quantify the magnitude of suchlimitation with respect to more refined design andanalysis methods.

In particular, results obtained by two sets of FEAswill be discussed. The first group of analyses has theobjective of evaluating the influence of the frameflexibility and of the presence of mass load on multipleframes on the shear flow introduced into the skin. This

contributes to assessing the validity of the resultsobtained by using the ‘load coefficient method’ [1]which can cope with flexible frames, but it is able tomanage a design case characterized by a single loadedframe properly while loads are actually distributed on allthe frames.

The second set of analyses is aimed first at evaluatingthe effects on the stress distribution in skin and frames,caused by the presence of the stringers under the actionof the pressure loads, and then at estimating themagnitude and extension of the influence of the stiffnessconcentration introduced by the floor beams. The resultsof this group of FEAs will give a contribution to assessthe applicability of the theoretical design methods to ageometry representative of a wide-body fuselage section.The theoretical methods, which originate from thetheory of thin-walled cylinders stiffened by equallyspaced circular frames can be formulated, according toreference [4], in order to account for the presence of thelongitudinal stiffening elements (stringers) by adding anextra thickness to the skin, as the result of the spreadingof the stringers’ cross-sectional area over the skinperimeter. Such an approach takes into account theincrease in resistant area due to the stringer contribu-tion, but it is not able to predict the actual modificationin the stress field of the skin and frames induced by thestiffeners. Moreover, classical methods are based on thehypothesis of structural axial symmetry of the fuselagesection, so that it is not possible to take into account thepresence of cabin and lower-deck floor beams, whichintroduce major alterations in the stiffness of theframe.

The general structure of the integrated designprocedure is briefly reviewed in section 2 of the paper;then the attention is focused on a critical review of themethodologies and on some of the results obtained,which are used later to define the contributions to thedesign of the fuselage section of both the mass andpressure load cases. Section 4 presents finite elementmodels used to carry out the two sets of analyses,while the main results are presented and discussedin the following section. Finally a few concludingremarks and recommendations for future work aregiven.

2 DESCRIPTION OF THE INTEGRATED

DESIGN PROCEDURE

With reference to a conventional transport aircraftfuselage with circular cross-section, a flow diagram ofthe design procedure is shown in Fig. 1. The procedureis based on the simplifying hypothesis of symmetry ofthe geometry and of the design loads with respect to theaircraft longitudinal plane (YZ plane in Fig. 2). The

L BONI AND D FANTERIA316

Proc. Instn Mech. Engrs Vol. 218 Part G: J. Aerospace Engineering G05603 # IMechE 2004

reference section for design is located in the cylindricalpart of the aft fuselage; the latter is assumed to beclamped at the rear spar bulkhead and subjected to thefollowing design loads: longitudinal bending momentMx, shear load Ty, and torque Mz.

The design procedure is of iterative nature and startswith a preliminary evaluation of the load index N, given

by

N ¼ Mx

pR2ð1Þ

where R is the fuselage radius.Since the outer structure of the fuselage must be

designed according to specific requirements which vary

Fig. 2 Main components of the fuselage section

Fig. 1 Flow diagram of the design procedure relevant to a fuselage section

ANALYTICAL METHOD FOR FUSELAGE DESIGN BY MEANS OF FE ANALYSES 317


along the perimeter, three fundamental typologies ofstiffened panel are identified, according to the sketch inFig. 2, i.e. upper panel, lower panel, and lateral panels.At each iteration step the geometry of the panels isdefined, as the result of the design procedure and of thegeometrical features of the frame and of the passengerand cargo floor beams are evaluated. Once the geometryof the cross-section is known, load indices relevant toeach panel are re-evaluated and a new step of theiterative procedure is started. When convergence withrespect to the load indices is obtained, the final geometryof the fuselage cross section is completed by dimension-ing the tear strap*.

Analytical methods used in the procedure describedabove allow the design of a structure that is compliantwith durability and damage tolerance requirements[11, 12]. The design codes allow the lower and the upperpanels to be designed in such a way that instabilityphenomena under compression loads occur at anassigned value of the load factor. In particular, thepost-critical behaviour of these panels is accounted forin the case when the design instability load factor islower than the ultimate [3, 5, 6]. The lateral panels,according to the theory of the incomplete diagonal stressstate, are designed in post-critical field under combinedcompression and shear loads [2, 6, 10]. The design of theframes takes into account the mutual skin-frameflexibility and the presence of the floor beams [1, 10].

3 ANALYTICAL MODELS AND RESULTS

In this section, a critical review of the theoreticalmethods, and of the results achievable by using them,is reported; in particular, attention is focused on thedistribution of the skin shear flow, caused by the frame,along the section perimeter and to the effects of thepresence of a circular frame on the radial displacementdistribution of a non-stiffened cylindrical shell.

3.1 Models for the structural analysis of the frame

The roughest model that can be employed to calculatethe distribution of the shear flow in correspondence of aframe is based on the assumption that the frame isinfinitely stiff in its plane and has no stiffness for out-of-plane deformations; under such hypotheses, the shearflow is given by the Jourawsky equation

q ¼ TySx

Ixð2Þ

Ty being the shear load at the section, Sx the staticmoment, and Ix the moment of inertia of the sectionwith respect to the X axis (see Fig. 2).

By indicating the anomaly by the symbol W, equation(2) gives the expression

qðWÞ ¼ Ty

pRsinðWÞ ð3Þ

for the shear flow distribution along the sectionperimeter.

In the case of a transport aircraft fuselage, theframe is characterized by an intrinsic flexibility thatinvalidates the elementary approach just described, andtherefore more sophisticated analysis methods havebeen proposed in the past. Among these the followingthree are well established: the elastic centre method,the pressure circle method, and the load coefficientmethod [1].

Of the three approaches, the load coefficient methodis by far the most complete and thus has been used in thedesign code; it considers that the frame distorts underload and alters the shear flow distribution in the shell.Consequently the loads and stresses are resolved intotwo systems: the first is relevant to the equilibrium stressfield obtained from applied loads by means of theelementary theory, and the second consists of acorrective stress distribution deriving from the distor-tions of the frames and the skin. This second system isevaluated as a function of the relative-stiffness para-meter, which relates shell stiffness to frame stiffness;then, the solution is obtained by superposition of thetwo load–stress systems.

The load coefficient method has been used to evaluatethe shear flow distribution for the case of a single framewith constant mass load distribution along bothpassenger and cargo floor beams. The loads aretransferred to the frame, and then to the skin, atconnection areas with the floor beams and with thestruts supporting them (Fig. 2). At each of theseconnection areas between the frame and the floorbeams, a force and a bending moment are introducedinto the frame while, at each strut–frame connectionarea, only a force is exchanged. The final distribution ofthe skin shear flow along the section perimeter isobtained by superposition of four load systems, asshown in Fig. 3.

The load coefficient method is used in the optimiza-tion code to account for the mass loads in thedimensioning procedure of the frame. The latter isdesigned to withstand mass and pressure loads and incompliance of a number of static and fatigue require-ments.

In Fig. 4 the results obtained with the load coefficientmethod are compared with those given by the rigidframe model [equation (3)], for the case when sole massloads act on the frame.

* Connection elements (clips) between the skin and the frames (shearties) and between the frames and the stiffeners (stringer ties) are notincluded in the design procedure; however, their presence has beentaken into account in some of the FEAs.



From the results presented in Fig. 4, it is observedthat the presence of frame flexibility introduces thepossibility that the shear flow becomes negative incertain regions of the fuselage section; thus, for the sameresultant shear load, the maximum value of the shearflow exceeds (by more than four times in the exampleshown) the peak value of the sinusoidal distributionrelevant to the rigid frame case. Moreover, the locationof the peak value of the shear flow moves towards theregion between the two floor beams, while in the crownpart of the fuselage section the magnitude of the shearflow is quite small (about 4 per cent of the peak value inthe example shown).

3.2 Modelling of the effects of the pressure loads

The effects of the presence of circumferential stiffeningelements (frames) on the stress–strain field induced bypressure loads Dp in a thin-walled cylinder of radius Rand thickness t can be described by the classical theory[4], resulting in the differential equation in terms of theradial displacement w, namely

q4wqz4

�Nz

D

q2wqz2

þ tE

DR2w ¼ � Dp

Dþ nNz

DRð4Þ

where Nz is the longitudinal membrane axial force ðNz ¼

Fig. 3 Load systems on the frame

Fig. 4 Theoretical skin shear flow distribution for infinitely stiff and flexible frame



szt ¼ RDp=2Þ and D is the bending stiffness given by

D ¼ Et3

12ð1� n2Þ ð5Þ

E being Young’s modulus of the material and nPoisson’s ratio. The general form of the solution ofdifferential equation (4) is

wðzÞ ¼ wþX4i¼1

Ai expz

li

� �ð6Þ

where 1=li are the roots ð [CÞ of the characteristicequation associated with equation (4) and w is theparticular solution that can be derived by settingderivatives to zero in equation (4) according to

w ¼ � 1� n2

� �DpR2

Et¼ 1� n

2

� �wm ð7Þ

For the simple case of a single frame, solution (6)must satisfy the boundary conditions

wjz¼0 ¼0, ? rigid frame,

�wf , flexible frame,

�limz??

ðwÞ ¼ �ww

dw

dz

��z¼0

¼ 0, limz??

dw

dz

� �¼ 0

ð8Þ

In the case of four real roots*, the solution assumes theform

w

wm¼�1� n

2

��1� 1� wf=w

ðl2 � l1Þ

l2 exp � z

l2

� �� l1 exp � z

l1

� �� ð9Þ

where 1=l1 and 1=l2 are given by the relationships

1

l1¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR Dp4D

1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4E2t4

3ð1� n2Þ Dp2 R4

s !vuut1

l2¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR Dp4D

1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4E2t4

3ð1� n2Þ Dp2 R4

s !vuutð10Þ

For the case of multiple equally spaced frames,equation (4) yields similar results by adopting suitableboundary conditions; such results have been extended tothe more realistic case of a stiffened shell by using amodified equation proposed in reference [4], which isbased on the concept of spreading the stringer section

area along the fuselage perimeter, resulting in anunstiffened shell of equivalent thickness teq.

4 FINITE ELEMENT MODELS OF THE

FUSELAGE BARREL

The geometrical model of the fuselage barrel has beenbuilt by using the parametric modelling tool included inthe commercial software ANSYS 5.7; the same softwarehas been used for the meshing, solution, and post-processing phases.

Since the real structure is mainly constituted by thin-walled components, shell elements have been usedextensively in the meshing phase together with beamelements to model stiffening members (low-resolutionmeshes of frames and stringers). Shell element number63 has been selected [13], which is a four-node planeelement having six degrees of freedom per node andcapable of simulating bending and membrane beha-viour.

As far as meshing strategies are concerned, ananalysis region has been selected that is enclosedbetween two boundary zones where border effects dueto loads and constraints disappear. For such main zonesa structured mesh is adopted with a finer grid for theanalysis region and a coarser grid for the boundaryzones; between these two, transition regions exist forwhich it is convenient to use a free meshing approach(an example of such meshing strategy is given in Fig. 5).

4.1 Model for mass load analyses

For these kinds of analysis, the fuselage is assumed tohave a clamped section, corresponding to the wing rearspar, and to be subjected to constant mass load per unitlength on both the upper-deck and the lower-deck floorbeams [10, 14].

The analysis barrel has a length of seven bays and iscontained between two boundary zones: a first zone,composed of 12 bays, is on the constraint side, while thesecond zone extends for 19 bays towards the tail cone(Fig. 5). The presence of the stringers is accounted for bymeans of an equivalent shell thickness. As far as themodelling of the frames and of the floor beams isconcerned, in the boundary zones they are representedby means of beam elements, while in the analysis regionthey are modelled according to the example shown inFig. 6. Moreover, in the analysis region the frames andthe skin are connected by using node-to-node rigidelements (CERIG) [13].

The main data used for the analyses carried out tostudy mass loads effects are summarized in Table 1.They refer to the outputs of the optimization coderelevant to a one-g load index of 200N/mm, for thedesign of stiffened panels with hat stringers at a spacing

* The case of real roots is relevant to fuselages characterized by smallvalues of the ratio ðt=RÞ4 for typical pressurization levels (50–60 kPa).



of 200mm, and to mass loads equivalent to resultantsection shear loads of 10 000 and 5000N for upper-deckand lower-deck floor beams respectively. The dimen-sions refer to a Z geometry for the frame cross-sectionand an I geometry for the cross-section of both floorbeams.

All components are made of aluminium alloy 2024with Young’s modulus equal to 72 000MPa andPoisson’s ratio equal to 0.32.

4.2 Models for pressure load analyses

The effects of the pressure load have been studied bymeans of models of increasing geometrical complexity,in order to evaluate the contributions to the stress–strainfield of the different fuselage components [10, 15].

The models have been developed using the hypothesisof cylindrical symmetry, so that only a cylindrical sectorhas been modelled of finite extension along the long-itudinal axis of the fuselage (Fig. 7). To respect thesymmetry and equilibrium conditions, the cylindricalsector must be properly constrained and loaded at itsedges: those parallel to the fuselage axis (side edges) areconstrained against circumferential displacement whileon the edges orthogonal to the axis (normal edges)different conditions are applied; one is restrained againstaxial displacement and to the other a tensile load per

Fig. 5 Meshing strategies

Fig. 6 Frame meshing example

Table 1 Main dimensions used for the FEAs

Value (mm)

ComponentMass loadanalyses

Pressureloadanalyses

Fuselage diameter 5640Spacing of frames (bay length) 500Equivalent skin thickness 3.6

Skin thickness – 2

Spacing of stringers 200

Web height of hat stringers – 56Free flange width of hat stringers – 16.8Connection flange width of hat stringers – 56.6Thickness of hat stringers – 1.97Tear-strap width – 47Tear-strap thickness – 0.64

Frame web height 165Frame flange width 36.5Frame thickness 1.5Web height of upper-deck floor beam 240Flange width of upper-deck floor beam 156Thickness of upper-deck floor beam 2.5Web height of lower-deck floor beam 180Flange width of lower-deck floor beam 60Thickness of lower-deck floor beam 1.5

Fig. 7 Fuselage sector meshing example



unit length is applied in order to establish theequilibrium of the structure when subjected to a pressureload. The value of such an axial load corresponds to thatgiven by the analytical model for the case of acircumferentially non-stiffened shell (i.e. in absence offrames).

The sector extension in the circumferential directionhas been determined so that the effect of the constraintson the side edges is almost suppressed in the centralportion of the arc, where the analysis section is located.

As far as the extension in the longitudinal direction isconcerned, two cases have been analysed: for the modelscharacterized by a single central frame, the axial lengthof the sector has been selected in such a way as to allowfor the effect of the frame to be suppressed at the normaledges; for the models with multiple frames, a number ofbays in the longitudinal direction have been modelled sothat the stress–strain distribution in the central bay isnot affected by the boundary effects at normal edges.

In the model used to study the effects of thelongitudinal stringers, the real geometry of the framehas been accounted for, which includes the holesnecessary for the crossing of the stringers and thepresence of the tear strap between the skin and theframe.

Finally, a model has been developed that takes intoaccount the presence of a structural element thatinterrupts the axial symmetry of the fuselage section(Fig. 8). The upper-deck floor beam has been added,lying on the bisector to the fuselage cylindrical portion;symmetry constraints are applied to the floor beamat the longitudinal plane of symmetry of the fuselage(Fig. 9).

In all the models the cross-section of the frames has aZ shape and the tear strap is modelled as a simple strip;hat stringers have been used and the floor beam has an Icross-section. Main dimensions of the components aresummarized in Table 1 and correspond to the output ofthe optimization code relevant to a pressure load of68 000 Pa.

Also for this model all components are made ofaluminium alloy 2024, except for the tear strap, forwhich titanium alloy is used.

5 RESULTS OF THE FINITE ELEMENT

ANALYSES

In this section the results relevant to the models andload conditions described in the previous section arediscussed. In particular, as far as the mass load analysesare concerned, a comparison of analytical versusnumerical results of the shear flow transfer betweenthe skin and the frame is presented. Numerical resultshave been obtained by means of a linear FEA.

The results concerning the pressure load analyses aresubdivided into two parts: the first considers the effectsof the stringers while the second shows the modificationsof the stress field due to the presence of a structuralelement which causes the loss of axial symmetry. Theresults of the two subsets have been computed by using ageometrically non-linear finite element method (FEM)which had been previously tested against well-knowntheoretical results relevant to a constant thickness shellstiffened by means of equally spaced frames. For suchcases the agreement between the numerical results andthe theoretical data was found to be excellent.Fig. 8 Meshing example of a fuselage sector with floor beams

Fig. 9 Circumferential control stations



5.1 Skin-frame shear flow transfer: comparison of

analytical and numerical results

In Fig. 10, results relevant to the case of a single loadedframe are shown. It can be observed that the skin shearflow, induced by the frame, predicted by FEA is in fairlygood agreement with the theoretical distribution alongthe section perimeter. Both results highlight that theregion above the upper-deck floor beam is relativelyunloaded, while the major part of the overall shear loadis carried by the regions between the two floor beams.

Results relevant to the case of multiple loaded framesare compared with the results of a single-loaded frame inFig. 11. It is evident that the presence of loaded framesin the vicinity of the section under analysis redistributesthe skin shear flow, so that the peak value is shiftedtowards the upper-deck floor beam; as a consequence,the zone right above the floor beam, where the shearflow is close to zero in the previous case, results in asignificant amount of shear load.

As a final remark, it can be observed that the shearflow distribution in the case of multiple loaded frames is

%

Fig. 10 Single-loaded frame: comparison between theoretical and FEM results

Fig. 11 FEM results: comparison of a single loaded frame and multiple loaded frames



quite similar to the distribution predicted by theelementary theory.

5.2 Effects of the stringers on the stress–strain field due

to pressurization

In Fig. 12, results relevant to the case of a single andinfinitely stiff frame inserted in a shell stiffened by

longitudinal stringers are shown in terms of non-dimensional radial displacement versus longitudinalnon-dimensional coordinate*, while in Fig. 13 thevariation in the radial displacement along the section

Fig. 12 Comparison of FEM and theory for a single, infinitely stiff frame and stiffened shell

Fig. 13 FEM results for a single, infinitely stiff frame and stiffened shell

* The origin of the longitudinal coordinate is at the location of one ofthe frames; the reference length used to make the longitudinalcoordinate non-dimensional is l2&68:3 mm; the radial displacementis made non-dimensional by dividing it by wm ¼ Dp R2=Et&3:75 mm.



perimeter, for different sections distributed in long-itudinal direction, is presented.

About the radial displacement distribution along thelongitudinal axis, the followingobservations canbemade.

1. Far from the frame, the presence of stringersdiminishes the membrane axial stress in the skin,causing a larger radial expansion of the stiffened shellwith respect to the non-stiffened shell.

2. Close to the frame, halfway between two stringers,

the radial displacement approaches the asymptoticalvalue of the theoretical distribution relevant to thesolution with equivalent thickness teq; as the distancefrom the frame increases, the numerical solutiontends to the theoretical value relevant to the non-stiffened shell of thickness t.

3. Close to the frame, at the stringers location, theradial displacement is significantly lower than the onebetween the stringers; such difference disappears asthe distance from the frame increases.

Fig. 14 FEM results for multiple frames and stiffened shell: longitudinal variation in radial displacement

Fig. 15 FEM results for multiple frames and stiffened shell: circumferential variation in radial displacement



In Figs 14 and 15, results relevant to the case ofmultiple frames, of finite stiffness, in a longitudinallystiffened shell are shown in terms of variation in theradial displacement along the longitudinal axis andalong the section perimeter respectively.

Close to the frames, the radial displacement halfwaybetween two stringers is lower than that at the stringerslocation (Fig. 14); at mid-bay between the frames theopposite is true; i.e., the stiffened shell expands morehalfway between two stringers than in correspondenceto the stringers themselves. The behaviour mentionedabove can be observed also in Fig. 15 by comparing thecircumferential distribution of radial displacement atstations 1 and 4. An explanation of such behaviour canbe given by realizing that the stiffness of the frame is notuniform but decreases significantly at stringers locationsdue to the presence of the holes to allow for the crossingof the stringers themselves.

5.3 Perturbation of the stress–strain field due to

pressurization caused by the floor beams

Figure 16 shows the radial displacement along thelongitudinal axis, at several stations distributed alongthe circumferential direction (see Fig. 9), relevant to thecase of a stiffened shell with multiple frames and takinginto account the presence of the floor beam. It is evidentthat the high longitudinal stiffness of the floor beamstrongly modifies the distribution of the radial displace-ment in the vicinity of the floor beam itself (station 1);nevertheless, such effect decreases rapidly as the distanceof the control stations increases, so that at station 6 thevalue of the displacement at mid-bay between the frames

reaches the value calculated in absence of the floor beam(see also Fig. 13 for comparison).

6 CONCLUSIONS

The results of some specific FEAs, carried out tosupport the development of an integrated procedurebased on analytical and semi-empirical methods, for thedesign of aircraft fuselage components, have beenpresented and discussed with the aim of achieving adeeper understanding of the advantages and the limita-tions of the analytical approach.

FEM results demonstrate the effectiveness of the loadcoefficient method in estimating the shear flow that asingle loaded flexible frame introduces into the skin andhighlight the importance of the interactions betweenstress–strain fields induced by a set of equally spacedframes subjected to typical mass load distributions onthe floor beams. From this point of view, the elementarytheory provides anyway a solution that is a goodstarting point for further high-fidelity finite-element-based design activities.

Results of finite element models for pressure loadanalyses highlight that, as far as a first-approach designis sought, the theoretical methods can be regarded asfully satisfactory; nevertheless, more advanced tools areneeded when dealing with detailed design of the regionsin the vicinity of the stringers and of those structuralelements (such as the floor beams) that invalidate thehypothesis of axial symmetry on which theoreticalmethods are based.

Although the results presented cannot be regarded ascomplete and exhaustive, they are a first step towards thevalidation of the analytical design procedure; further

Fig. 16 FEM results for multiple frames and stiffened shell with floor beams



investigations are planned for which the use of morerefined FEAs are being considered. From this point ofview, the combined use of a three-dimensional parametricmodeller, which can manage assemblies of many compo-nents, and a FEM solver, with enhanced submodellingand interface capabilities, would be very advantageous.

ACKNOWLEDGMENTS

The authors would like to express their sincereappreciation to Professor Attilio Salvetti and ProfessorLuigi Lazzeri for the encouragement, the many helpfulsuggestions and fruitful discussions.

The help given by Massimiliano Cartolano andFrancesco Calvetti in carrying out the FEAs is gratefullyacknowledged.

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Canonsburg, USA).

14 Cartolano, M. Analisi numeriche di strutture di fusoliera

soggette a sollecitazioni di flessione e taglio. Final degree

thesis, University of Pisa, Pisa, Italy, 2002.

15 Calvetti, F. Analisi numeriche di strutture di fusoliera

soggette a sollecitazioni di pressurizzazione. Final degree

thesis, University of Pisa, Pisa, Italy, 2002.



Documents

Development of Analytical Methods for Fuselage Design