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Analytical and Finite Element Approach for the In-plane Study of Frames of Non-conventional Civil Aircraft

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Abstract

In this paper, we study families of optimal frames of aircraft, as the fuselage cross-section vary, in the preliminary design. A complete closed-form solution of displacements and stresses for a circular arc, already introduced in a previous paper of the authors, is applied to study, in a wide generality, a fuselage cross-section made of tangent circular arcs, connected together in a \({\mathscr {C}}^1\)-class curve. The closed-form solution is used here for two optimization case studies involving such piece-wise tangent cross-sections. First, we obtain minimum weight configurations of frames under pressurization, and also the effect of a small eccentricity with respect to the perfect circular fuselage is investigated; then, the constraints due to the presence of two floor decks are introduced. Second, the analytic solutions are validated by means of a finite element simulation in Abaqus and, to show the generality of the closed-form solution, the case studies are dedicated to non-conventional aircraft. Finally, we investigate the effects of the ellipticity ratio and the presence of a vertical and horizontal truss by means of finite element beam models.

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Abbreviations

\(\alpha \) :

Arc amplitude, [rad]

\(\beta _y\) :

Shear factor

\(\varepsilon , \gamma , \chi \) :

Normal strain, shear strain, curvature

\(\theta \) :

Beam cross-section rotation, [rad]

\(\xi \) :

(pseudo) Ellipticity ratio

\(\mu \) :

Parameter for convexity of fuselage perimeter

\(\sigma \) :

Normal stress tension, \([{{\text{MPa}}}]\)

\(\tau \) :

Shear stress tension, \([{{\text {MPa}}}]\)

\(\varphi \) :

Generic angle, [rad]

\(\varPhi \) :

Optimization objective function

A :

Beam cross-section area, \([{{\text {mm}^{2}}}]\)

\(A_t\) :

Beam cross-section reactive area for shear stress, \([{{\text {mm}^{2}}}]\)

C :

Arc center

CG:

Center of gravity

E, G :

Young’s modulus, Shear modulus, \([{{\text {MPa}}}]\)

J :

Beam cross-section second order moment of inertia, \([{{\text {mm}}^4}]\)

M :

Internal bending moment, \([{\text {N mm}}]\)

N, T :

Normal, shear internal force, \([\text {N}]\)

P :

Edge point of arc

R :

Arc radius, \([{{\text {mm}}}]\)

a :

Frame reference vertical half-height, \([{{\text {mm}}}]\)

b :

Frame reference horizontal half-width, \([{{\text {mm}}}]\)

e :

Eccentricity, \([{{\text {mm}}}]\)

n :

Number of arcs

\(\mathbf{n}, \mathbf{t }\) :

Local normal and tangent unit vectors

p, q :

Tangential, radial load, per unit or arc length, \([{{\text {N mm}^{-1}}}]\)

s :

Curvilinear abscissa along an arc

\(\mathbf{u }\) :

Displacement vector

v, w :

Radial, tangential displacement, \([{{\text {mm}}}]\)

\(\mathbf{x }\) :

Optimization variables vector

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Acknowledgements

Authors thank Prof. G. Pannocchia, Prof. M. Gabiccini and Prof. A. Artoni, of the University of Pisa, for the “Fundamentals of Optimization” Ph.D. lectures and for the introduction to CasADi.

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Authors and Affiliations

  1. Department of Civil and Industrial Engineering, Aerospace Section, University of Pisa, Via G. Caruso 8, 56126, Pisa, Italy

    Marco Picchi Scardaoni & Aldo Frediani

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  1. Marco Picchi Scardaoni
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  2. Aldo Frediani
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Correspondence to Marco Picchi Scardaoni.

Additional information

The present paper presents part of the activities carried out within the research project PARSIFAL (“PrandtlPlane ARchitecture for the Sustainable Improvement of Future AirpLanes”), which has been funded by the European Union under the Horizon 2020 Research and Innovation Program (Grant Agreement n. 723149).

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Picchi Scardaoni, M., Frediani, A. Analytical and Finite Element Approach for the In-plane Study of Frames of Non-conventional Civil Aircraft. Aerotec. Missili Spaz. 98, 45–61 (2019). https://doi.org/10.1007/s42496-018-00004-z

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