Procesos gobernantes

In Chap. 2 we saw that the fuselage structure of a flying wing is quite different to a conven-tional aircraft. The former utilises a multi-bubble shell configuration, whilst the latter has a cylindrical pressure vessel. Both are subject to pressurisation and distributed loads; however, additional horizontal and vertical loads must be considered for tube-and-wing aircraft.

Multi-Bubble Pressure Vessel Geometry

Figure 3.11 shows a cross section of the multi-bubble pressure vessel perpendicular to the spanwise axis. For N cylindrical bubbles, there are N-1 vertical bulkheads of height 2hf,vb

and thickness t_f,vb. Each cylinder has a radius R_f and skin thickness t_f,s, and the overall section area is Af,s. The bubbles either terminate at wing spar locations, or are cylindrical.

The shell has the following geometric and section-area relations:

θ_f,vb = arcsin



 s

1− h_f,vb

Rf

2

 (3.31)

A_f,s = 2(θ_f,1+θ_f,2)R_ft_f,s+ (N−1)4θ_f,vbR_ft_f,s (3.32) A_f,vb = 2(N−1)hf,vbt_f,vb+ 2Rft_f,vb(cosθ_f,1+ cosθ_f,2) (3.33)

The total fuselage cross-sectional area and surface area of the cabin shell are denoted by Af

and S_f,mb, respectively:

A_f =

θ_f,1+θ_f,2+ 1

2(sin 2θf,1+ sin 2θf,2)

R²_f

+(N−1)(2θf,vb+ sin 2θf,vb)R²_f (3.34) S_f,mb = 2(θf,1+θ_f,2)Rfl_f,mb+ 2(N −1)lf,mbR_fθ_f,vb (3.35) wherel_f,mb is the spanwise length of the cabin (as shown in Fig. 3.12).

Figure 3.11: Multi-bubble geometric parameters (perpendicular to the spanwise axis).

Figure 3.12 shows a cross section of the fuselage perpendicular to the chordwise axis.

The multi-bubble cabin section is capped off from the unpressurised regions at the spanwise extremes by domed bulkheads of radius R_f and thickness t_f,db. Wing ribs (fuselage frames) extend through the lower surface to support the floor.

The dome-bulkhead surface areaS_f,db is an extension of the approximation for a spherical bulkhead given by Greitzer et al. (2010):

Sf,db '2 (θf,1+θf,2)R²_f+ (N −1)4θf,vbR²_f

. (3.36)

3.2. TECHNICAL APPROACH

Figure 3.12: Cabin cross section (perpendicular to the chordwise axis).

Shell

The skin resists internal pressurisation in hoop-wise tension provided vertical bulkheads are placed at the ‘bubble’ joins. The axial and hoop-wise skin stresses are:

σ_xx = N_s∆P A_f

A_f,s+A_f,vb (3.37)

σ_θθ = Ns∆PRf

t_f,s (3.38)

The ratio of these two stresses, as a function of the various geometric parameters that define the multi-bubble pressure vessel, is:

σxx

σ_θθ = 1 2

θf,1+θf,2+ (N −1)(2θf,vb+ sin 2θf,vb) θ_f,1+θ_f,2+ (N −1)(2θf,vb+ cosθ_f,vb)

. (3.39)

The ratio ^σ_σ^xx

θθ is plotted against θ_f,vb for extreme values ofN in Fig. 3.13 for the case where θ_f,1 =θ_f,2 =π/2. The figure shows that for the cylindrical case, i.e. one bubble, σxx/σ_θθ = 1/2, as expected; in contrast, for a large number of bubbles (N = 20), the maximum value of σ_xx/σ_θθ does not exceed around 0.6. Consequently, the skin thickness must be sized by the larger, hoop-wise, stress to meet the allowable stress σ_f,s:

t_f,s =Ns∆P Rf

σ_f,s. (3.40)

The hoop-wise membrane forces T make an angleφwith the vertical, and are reacted by a tensile force F in the vertical bulkhead at the multi-bubble joins, as illustrated in Fig. 3.14.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

θ_f,vb (rad) σ xx/σ θθ

N=20 N=1

Figure 3.13: Variation of the ratio of axial-to-hoopwise stress with θ_f,vb and N for θ_f,1 = θ_f,2 =π/2.

Assuming both components have the same material properties, the vertical bulkhead thickness

T T

F

ϕ ϕ

Figure 3.14: Vertical bulkhead force.

is given by

t_f,vb = 2tf,scosφ (3.41)

whereφ=π/2−θ_f,vb.

For a cylindrical pressure vessel with a semi-hemispherical end-cap, the stress in the cap is half the hoop-wise stress, and therefore the cap thickness is half the cylinder thickness. Such

3.2. TECHNICAL APPROACH

a discontinuity induces local bending moments, and a transition is required. Therefore, it is conservatively assumed that the domed-bulkhead thickness (tf,db) is equal to the fuselage-skin thickness (tf,s).

The volumes of material required for the domed bulkheads, multi-bubble cylinders and vertical bulkheads are:

Vf,db = S_f,dbt_f,s (3.42)

Vf,mb = A_f,sl_f,mb (3.43)

Vf,vb = A_f,vbl_f,mb (3.44)

A breakdown of the shell weight is therefore:

Wf,s = ρalg(Vf,mb+Vf,db) (3.45)

Wf,vb = ρalgVf,vb (3.46)

W_f,sh = W_f,s(1 +f_f,st) +W_f,vb (3.47) where ρal is the material density, and the weight fraction of the stringers is given by Howe (2004) to bef_f,st= 0.25.

Floor

Transverse floor beams are required to support the payload and additional weights (seats, galleys, etc), which are assumed uniformly distributed, and have a weight per unit area

Wf,f l =N_lN_s (Wpay +W_{f ix}) (lf,mb+ 2Rf)wf,f l

. (3.48)

In sizing the floor beams, the emergency landing case (Nl = 3) is taken as critical.

Figure 3.12 shows that the floor joints are modelled as pinned, with a separation lf,f l

equal to the rib/frame pitch. This arrangement leads to a more conservative weight estimate than a clamped model would. The load due to the floor self-weight is neglected, as it is typically much smaller than the payload. The maximum shear force and bending moment expressions are subsequently

Sf,f l = 1

2w_{f,f l}l_{f,f l}Wf,f l (3.49)

Mf,f l = 1

8w_{f,f l}l_{f,f l}² Wf,f l (3.50)

where the floor width

w_{f,f l} = 2(N −1)Rfsinθ_f,vb+R_fsinθ_f,1+R_fsinθ_f,2. (3.51) For a set of floor I-beams with height h_{f,f l}, the total average cross-sectional area and corresponding volume of beam material are given by Greitzer et al. (2010) as:

A_{f,f l} = 2.0Mf,f l

σ_{f,f l}h_{f,f l} + 1.5Sf,f l

τ_{f,f l} (3.52)

Vf,f l = w_{f,f l}A_{f,f l} (3.53)

(The coefficients of 2.0 and 1.5 assume no tapering of the cross section; the floor heighth_{f,f l} is assumed to be 10 cm.) The floor weight is then

W_{f,f l} =ρ_algVf,f l+w_{f,f l}(lf,mb+ 2Rf)Wf,f l−pl (3.54) whereWf,f l−pl is a specified floor-plank weight density.

Cylindrical Pressure Vessel

The reader is referred to the study of Greitzer et al. (2010), as this is followed directly.

However, a short summary of the sizing method follows.

The model consists of a cylindrical pressure vessel with an ellipsoidal nose endcap and a hemispherical tail endcap. The fuselage is subjected to internal pressurisation ∆P, vertical (Mv) and horizontal (Mh) bending, and torsion (Qv) loads. These are depicted in Fig. 3.15.

(In this study, the added weight W_padd and seat weight W_seat are not treated in the same way. Instead, a fixed equipment weight is includedWf ix.)

The skin thickness is sized based on the hoop-wise skin stress that results from cabin internal pressurisation. Reinforcements such as frames and stringers are assumed to con-tribute 25% each to the total shell weight (Howe (2004)). Additional vertical and horizontal bending material is added wherever the resulting bending moments exceed the capability of the pressure vessel’s thickness based on pressurisation considerations alone. The schematic diagram in Fig. 3.15 (a) shows that the moments acting over the front and rear of the fuselage match at the location xwing, which for simplicity is taken as the wing’s area centroid. The distributed loading, weight of the tailplane N W_tail and aerodynamic loading acting on the horizontal tailplane r_{M h}L_h contribute to Mh. Two scenarios are considered: in-flight at the maximum load factor, and an emergency landing impact. Meanwhile,Mv depends solely on the vertical fin loadingr_{M v}L_v. (The factorsr_{M v} and r_{M h} account for inertial relief.)

3.2. TECHNICAL APPROACH

(a) Conventional fuselage layout, loads, and bending moment and inertia distributions.

(b) Fuselage cross-section, and torsion load from ver-tical fin.

Figure 3.15: Conventional fuselage geometry and applied loads (from Greitzer et al. (2010)).

Secondary Structures

Secondary fuselage weights are given by empirical weight fractions:

1. Windows (conventional only): an area-distributed weight of 0.66 kg/m², estimated from the study of Greitzer et al. (2010).

2. Insulation: an area-distributed weight of 1.08 kg/m², estimated from the study of Greitzer et al. (2010).

3. An area-distributed floor-plank weight of 5 kg/m² (Torenbeek (1976)).