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

  • Marco Picchi Scardaoni
  • Aldo Frediani
Original article


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.


Aircraft structure Closed-form solution Optimization Non-conventional aircraft PrandtlPlane Parsifal project 

List of Symbols

\(\alpha \)

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


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


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


Arc center


Center of gravity


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


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


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


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


Edge point of arc


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


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


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


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


Number of arcs

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

Local normal and tangent unit vectors


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


Curvilinear abscissa along an arc

\(\mathbf{u }\)

Displacement vector


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

\(\mathbf{x }\)

Optimization variables vector



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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Copyright information

© AIDAA Associazione Italiana di Aeronautica e Astronautica 2019

Authors and Affiliations

  1. 1.Department of Civil and Industrial Engineering, Aerospace SectionUniversity of PisaPisaItaly

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