Fuselage Model

The purpose of a conventional commercial aircraft fuselage can be decomposed into two primary functions: integrating and connecting all of the subsystems (e.g. wing, tail, landing gear), and carrying the payload, which typically consists of passengers, luggage, and sometimes cargo. The design of the fuselage is therefore coupled with virtually every aircraft subsystem.

A detailed but still approximate analysis of fuselage structure and weight is done in TASOPT [Drela, 2011] , considering pressure loads, torsion loads, bending loads, buoyancy weight, window weight, payload-proportional weights, the floor, and the tail cone. The majority of the constraints in this model are adapted directly from these equations.

Model Assumptions

This model assumes a single circular-cross-section fuselage. This is an approximation, since narrow-body aircraft like the Boeing 737 and Airbus A320 do not have perfectly circular cross sections.

The floor structural model and the horizontal bending model assume uniform floor loading. The model leverages the analytical bending models from [Drela, 2011], which makes assumptions about symmetry in bending loads. Shell buckling is not explicitly modeled while designing bending structure, but is accounted for by the implementation of a lower yield stress for bending reinforcement material relative to the nominal yield stress of the material.

Model Description

Variable tables are available for download below:

• Free variables
• Fixed variables

Cross-sectional geometry constraints

The fuselage must be wide enough to accommodate the width of the seats in a row and the width of the aisle.

\[{2w_{fuse}} \geq (\mathit{SPR}) {w_{seat}} + {w_{aisle}} + 2{w_{sys}}\]

The cross sectional area of the fuselage skin is lower bounded using a thin walled cylinder assumption.

\[{A_{skin}} \geq 2 \pi {R_{fuse}} {t_{skin}}\]

The cross sectional area of the fuselage is lower bounded using the radius of the fuselage.

\[{A_{fuse}} \geq \pi {R_{fuse}}^{2}\]

Pressure loading constraints

The axial and hoop stresses in the fuselage skin are constrained by the pressurization load due to the difference between cabin pressure and ambient pressure at cruise altitude. The thickness of the skin is therefore sized by the maximum allowable stress of the chosen material.

\[\begin{split}\begin{aligned} {\sigma_x} &= \frac{{\Delta P_{over}}}{2}\frac{{R_{fuse}}}{{t_{shell}}}\\ {\sigma_{\theta}} &= {\Delta P_{over}} \frac{{R_{fuse}} }{{t_{skin}}} \\ {\sigma_{skin}} &\geq {\sigma_x} \\ {\sigma_{skin}} &\geq {\sigma_{\theta}}\end{aligned}\end{split}\]

Floor loading constraints

The floor must be designed to withstand at least the weight of the payload and seats multiplied by a safety factor for an emergency landing.

\[{P_{floor}} \geq {N_{land}} ({W_{payload}} + {W_{seat}})\]

The maximum moment and shear in the floor are determined based on this design load and the width of the floor, assuming that the floor/wall joints are pinned and there are no center supports.

\[\begin{split}\begin{aligned} {S_{floor}} &= \frac{P_{floor}}{2}\\ {M_{floor}} &= \frac{{P_{floor}} {w_{floor}} }{8}\end{aligned}\end{split}\]

The floor beam cross sectional area is constrained by the maximum allowable cap stress and shear web stress for the beams.

\[{A_{floor}} \geq 1.5\frac{{S_{floor}}}{{\tau_{floor}}} + 2\frac{{M_{floor}}}{{\sigma_{floor}} {h_{floor}}}\]

3-dimensional geometry constraints

The nose must be long enough to have an aerodynamic profile and to accommodate the cockpit. A reasonable, but arbitrary, lower bound is employed for this work [Drela, 2011].

\[{l_{nose}} \geq 5.2 \hspace{0.2cm} \rm{m}\]

The cylindrical shell of the fuselage sits between the nosecone and tailcone. The variables \(x_{shell1}\) and \(x_{shell2}\) define the beginning and end of the cylindrical section of the fuselage, respectively, in the aircraft x-axis.

\[\begin{split}\begin{aligned} {x_{shell1}} &= {l_{nose}} \\ {x_{shell2}} &\geq {l_{nose}} + {l_{shell}}\end{aligned}\end{split}\]

The number of seats is equal to the product of the seats per row and the number of rows. Note that non-integer numbers of rows are allowed and necessary for GP compatibility. It is assumed that the load factor is one, so that the number of passengers is equal to the number of seats.

\[\begin{split}\begin{aligned} {n_{seat}} &= {(\mathit{SPR})} {n_{rows}} \\ {n_{pass}} &= n_{seat}\end{aligned}\end{split}\]

The seat pitch and the number of rows of seats constrain the length of the shell. The passenger floor length is lower bounded by the shell length and twice the fuselage radius, to account for the space provided by pressure bulkheads.

\[\begin{split}\begin{aligned} {l_{shell}} &\geq {n_{rows}} {p_s} \\ {l_{floor}} &\geq 2{R_{fuse}} + {l_{shell}} \end{aligned}\end{split}\]

The length of the fuselage is constrained by the sum of the nose, shell and tail cone lengths. A signomial equality is needed, because increased \(l_{fuse}\) is not coupled directly to increased structural weight although it results in improved tail control authority.

\[l_{fuse} = l_{nose} + l_{shell} + l_{cone}\]

Other locations to constrain are the wing mid-chord and the wingbox fore and aft bulkheads, which serve as integration limits when calculating bending loads.

\[\begin{split}\begin{aligned} x_f \leq x_{wing} + 0.5 c_0 r_{w/c}\\ x_b + 0.5 c_0 r_{w/c} \geq x_{wing}\end{aligned}\end{split}\]

The skin surface area, and, in turn, skin volume for the nose, main cabin, and rear bulkhead are constrained. The surface area of the nose, which is approximated as an ellipse, is lower bounded using Cantrell’s approximation [Drela, 2011].

\[\begin{split}\begin{aligned} {S_{nose}}^{\frac85} &\geq \left(2 \pi {R_{fuse}^2}\right)^{\frac85} \left(\frac13 + \frac23 \left(\frac{l_{nose}}{R_{fuse}}\right)^{\frac85} \right) \\ {S_{bulk}} &= 2 \pi {R_{fuse}}^{2} \\ {V_{cyl}} &= {A_{skin}} {l_{shell}} \\ {V_{nose}} &= {S_{nose}} {t_{skin}} \\ {V_{bulk}} &= {S_{bulk}} {t_{skin}} \end{aligned}\end{split}\]

The cabin volume is constrained assuming a cylinder with hemispherical end caps. This is necessary for capturing buoyancy weight.

\[{V_{cabin}}\geq{A_{fuse}}\left(\frac23{l_{nose}} + {l_{shell}} + \frac23{R_{fuse}} \right)\]

Tail cone constraints

The tail cone needs to be able to transfer the loads exerted on the vertical tail to the rest of the fuselage. The maximum torsion moment imparted by the vertical tail depends on the maximum force exerted on the tail as well as its span and taper ratio. This torsion moment, along with the cone cross sectional area and the maximum shear stress of the cone material, bounds the necessary cone skin thickness. The cone cross sectional area, which varies along the cone, is coarsely approximated to be the fuselage cross sectional area (i.e. the cross sectional area of the cone base).

\[\begin{split}\begin{aligned} \label{eq:Qv1} {Q_v} &= \frac{{L_{vt_{max}}} {b_{vt}}}{3} \frac{{1 + 2{\lambda_v}}} {{1 + {\lambda_v}}} \\ \label{eq:Qv2} {t_{cone}}&= \frac{Q_v}{2{A_{fuse}} {\tau_{cone}} }\end{aligned}\end{split}\]

The volume of the cone is a definite integral from the base to the tip of the cone. This integral is evaluated [Drela, 2011] and combined with Equations and to give a single signomial constraint on the cone skin volume.

\[R_{fuse}\tau_{cone}(1+p_{\lambda_v})V_{cone} \frac{1+\lambda_{cone}}{4 l_{cone}} \geq L_{vt_{max}} b_{vt} \frac{p_{\lambda_v}}{3}\]

A change of variables is used for compatibility with the tail model, which uses \(p_{\lambda_v} = 1 + 2\lambda_v\) to make a structural constraint -compatible. The same taper lower bound is introduced as in the tail model.

\[{p_{\lambda_v}} \geq 1.6\]

The cone skin shear stress is constrained to equal the maximum allowable stress in the skin material.

\[{\tau_{cone}} = {\sigma_{skin}}\]

The tail cone taper ratio constrains the length of the cone relative to the radius of the fuselage.

\[{l_{cone}} = \frac{{R_{fuse}}}{{\lambda_{cone}}}\]

Fuselage area moment of inertia constraints

The fuselage shell consists of the skin and stringers. Its area moment of inertia determines how effectively the fuselage is able to resist bending loads. A shell with uniform skin thickness and stringer density has a constant area moment of inertia in both of its bending axes, shown by the dark red line in the lower plot of Figure [fig:fuse_bending_loads].

To be consistent with [Drela, 2011], the horizontal bending moments are defined as the moments around the aircraft’s y-axis, caused by horizontal tail loads and fuselage inertial loads, and vertical bending moments as the moments around the aircraft’s z-axis, caused by vertical tail loads.

The effective modulus-weight shell thickness is lower bounded by assuming that only the skin and stringers contribute to bending. This constraint also uses an assumed fractional weight of stringers that scales with the thickness of the skin.

\[{t_{shell}} \geq {t_{skin}}\left(1 + {f_{string}} {r_E} \frac{{\rho_{skin}} }{{\rho_{bend}}} \right)\]

It is important to consider the effects of pressurization on the yield strength of the bending material. Since pressurization stresses the airframe, the actual yield strength of the fuselage bending material is lower than its nominal yield strength, an effect captured using posynomial constraints.

\[\begin{split}\begin{aligned} \sigma_{M_h} + r_E \frac{\Delta P_{over} R_{fuse}}{2 t_{shell}}&\leq \sigma_{bend} \\ \sigma_{M_v} + r_E \frac{\Delta P_{over} R_{fuse}}{2 t_{shell}}&\leq \sigma_{bend}\end{aligned}\end{split}\]

The aircraft shell, which is composed of the pressurized skin and stringers, must satisfy the following horizontal and vertical area moment of inertia constraints.

\[\begin{split}\begin{aligned} I_{hshell} &\leq \pi R_{fuse}^3 t_{shell} \\ I_{vshell} &\leq \pi R_{fuse}^3 t_{shell}\end{aligned}\end{split}\]

Horizontal bending model

There are two load cases that determine the required : maximum load factor (MLF) at \(V_{ne}\), where

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} N &= N_{lift} \\ L_{ht} &= L_{ht_{max}} \end{aligned}\end{split}\\and emergency landing impact, where\end{aligned}\end{align} \]

\[\begin{split}\begin{aligned} N &= N_{land} \\ L_{ht} &= 0. \end{aligned}\end{split}\]

Both load cases are considered at the aircraft’s maximum takeoff weight (MTOW). The constraints for each case are distinguished by the subscripts \(MLF\) and \(Land\). Assuming the fuselage weight is uniformly distributed throughout the shell, the bending loads due to fuselage inertial loads increase quadratically from the ends of the fuselage shell to the aircraft , as shown by the blue line representing \(M_h(x)\) in Figure [fig:fuse_bending_loads]. The tail loads are point loads at \(x_{tail}\), and so the horizontal tail moment increases linearly from \(x_{tail}\) to the aircraft’s . In the maximum load factor case, the maximum moment exerted by the horizontal tail is superimposed on the maximum fuselage inertial moment at load factor \(N_{lift}\) to size the required. For the emergency landing impact case, only the fuselage inertial loads are considered at \(N_{land}\), assuming an unloaded horizontal tail.

Several intermediate variables are introduced and used in constraints that capture relationships. \(A_{0h}\) represents the area that is contributed by the aircraft shell.

\[A_{0h} = \frac{I_{hshell}} {r_{E} h_{fuse}^2}\]

Variables \(A_{1h_{Land}}\) and \(A_{1h_{MLF}}\) are the lengths that are required to sustain bending loads from the tail. Note that as the distance from the tail increases, the moment exerted from the tail increases linearly.

\[\begin{split}\begin{aligned} A_{1h_{Land}} &\geq N_{land} \frac{W_{tail} + W_{apu}}{h_{fuse} \sigma_{M_h}}\\ A_{1h_{MLF}} &\geq N_{lift} \frac{W_{tail} + W_{apu} + r_{M_h} L_{ht_{max}}}{h_{fuse} \sigma_{M_h}}\end{aligned}\end{split}\]

Variables \(A_{2h_{Land}}\) and \(A_{2h_{MLF}}\) represent the required to sustain the distributed loads in the fuselage. As the distance from the nose or the tail increases, the moment exerted due to the distributed load grows with the square of length.

\[\begin{split}\begin{aligned} A_{2h_{Land}} &\geq N_{land} \frac{W_{payload} + W_{padd} + W_{shell} + W_{window} + W_{insul} + W_{floor} + W_{seat}} {2 l_{shell} h_{fuse} \sigma_{bend}} \\ A_{2h_{MLF}} &\geq N_{lift} \frac{W_{payload} + W_{padd} + W_{shell} + W_{window} + W_{insul} + W_{floor }+ W_{seat}} {2 l_{shell} h_{fuse} \sigma_{M_h}}\end{aligned}\end{split}\]

Bending reinforcement material in the aircraft exists where the shell inertia is insufficient to sustain the local bending moment. Constraints are used to determine the location over the rear fuselage \(x_{hbend_\zeta}\) forward of which additional is required. Some simple constraints on geometry are added to ensure a meaningful solution. Constraints through occur for both aforementioned load cases in the model (with subscript \(\zeta\) replaced by \(MLF\) or \(Land\)) for worst-case fuselage sizing, but have been included once in the paper to reduce redundancy.

\[\begin{split}\begin{aligned} \label{eq:dupBend_1} A_{0h} &= A_{2h_\zeta} (x_{shell2} - x_{hbend_\zeta}) ^ 2 + A_{1h_\zeta} (x_{tail} - x_{hbend_\zeta}) \\ x_{hbend_\zeta} &\geq x_{wing}\\ x_{hbend_\zeta} &\leq l_{fuse} \end{aligned}\end{split}\]

To be able to constrain the volume of required, the area of required must be constrained and integrated over the length of the fuselage. As shown by [Drela, 2011], with some conservative approximation, the volume of may be determined through the integration of the forward and rear wingbox areas over the rear fuselage.

\[\begin{split}\begin{aligned} A_{hbendf_\zeta} &\geq A_{2h_\zeta} (x_{shell2} - x_{f})^2 + A_{1h_\zeta} (x_{tail} - x_{f}) - A_{0h} \\ A_{hbendb_\zeta} &\geq A_{2h_\zeta} (x_{shell2} - x_{b})^2 + A_{1h_\zeta} (x_{tail} - x_{b}) - A_{0h}\end{aligned}\end{split}\]

volumes forward, over and behind the wingbox are lower bounded by the integration of the areas over the three fuselage sections.

\[\begin{split}\begin{aligned} V_{hbend_{f}} &\geq \frac{A_{2h_\zeta}} {3} ((x_{shell2} - x_{f})^3 - (x_{shell2} - x_{hbend_\zeta})^3) \\ &+ \frac{A_{1h_\zeta}} {2} ((x_{tail} - x_{f})^2 - (x_{tail} - x_{hbend_\zeta})^2) - A_{0h} (x_{hbend_\zeta} - x_{f})\nonumber\\ V_{hbend_{b}} &\geq \frac{A_{2h_\zeta}}{3} ((x_{shell2} - x_{b})^3 - (x_{shell2} - x_{hbend_\zeta})^3) \\ &+ \frac{A_{1h_\zeta}}{2} ((x_{tail} - x_{b})^2 - (x_{tail} - x_{hbend_\zeta})^2) - A_{0h} (x_{hbend_\zeta} - x_{b}) \nonumber\\ V_{hbend_{c}} &\geq 0.5 (A_{hbendf_\zeta} + A_{hbendb_\zeta}) c_{0} r_{w/c} \label{eq:dupBend_2}\end{aligned}\end{split}\]

The total volume is lower bounded by the sum of the volumes of required in each fuselage section.

\[V_{hbend} \geq V_{hbend_{c}} + V_{hbend_{f}} + V_{hbend_{b}}\]

Vertical bending model

The is constrained by considering the maximum tail loads that a fuselage must sustain. The vertical bending moment, shown in red as \(M_v(x)\) in Figure [fig:fuse_bending_loads], increases linearly from the tail to the aircraft , since the tail lift is assumed to be a point force.

As with horizontal bending, several intermediate variables are introduced and used in constraints that capture relationships. \(B_{1v}\) is the length required to sustain the maximum vertical tail load \(L_{vt_{max}}\). When multiplied by the moment arm of the tail relative to the fuselage cross-sectional location, it gives the local area required to sustain the loads.

\[B_{1v} = \frac{r_{M_v} L_{vt_{max}}} {w_{fuse} \sigma_{M_{v}}}\]

\(B_{0v}\) is the equivalent area provided by the fuselage shell.

\[{B_{0v}} = \frac{{I_{vshell}}}{{r_E} {w_{fuse}}^{2}}\]

Since tail loads are the only vertical loads to consider, the location forward of which additional bending material is required can be determined. \(x_{vbend}\) is the location where the vertical bending moment of the inertia of the fuselage is exactly enough to sustain the maximum vertical bending loads from the tail, expressed by a signomial equality.

\[\begin{split}\begin{aligned} B_{0v} &= B_{1v} (x_{tail} - x_{vbend}) \\ x_{vbend} &\geq x_{wing} \\ x_{vbend} &\leq l_{fuse} \end{aligned}\end{split}\]

The area required at the rear of the wingbox is lower bounded by the tail bending moment area minus the shell vertical bending moment area.

\[A_{vbend_{b}} \geq B_{1v} (x_{tail} - x_{b}) - B_{0v}\]

The vertical bending volume rear of the wingbox is then constrained by integrating \(A_{vbend}\) over the rear fuselage, which yields the following constraint.

\[V_{vbend_{b}} \geq 0.5 B_{1v} ((x_{tail}-x_{b})^2 - (x_{tail} - x_{vbend})^2) - B_{0v} (x_{vbend} - x_{b})\]

The vertical bending volume over the wingbox is the average of the bending area required in the front and back of the wingbox. Since no vertical bending reinforcement is required in the forward fuselage, the resulting constraint is simply:

\[V_{vbend_{c}} \geq 0.5 A_{vbend_{b}} c_{0} r_{w/c}\]

The total vertical bending reinforcement volume is the sum of the volumes over the wingbox and the rear fuselage.

\[V_{vbend} \geq V_{vbend_{b}} + V_{vbend_{c}}\]

Weight build-up constraints

The weight of the fuselage skin is the product of the skin volumes (bulkhead, cylindrical shell, and nosecone) and the skin density.

\[{W_{skin}} \geq {\rho_{skin}} {g} \left({V_{bulk}} + {V_{cyl}} + {V_{nose}} \right)\]

The weight of the fuselage shell is then constrained by accounting for the weights of the frame, stringers, and other structural components, all of which are assumed to scale with the weight of the skin.

\[{W_{shell}} \geq {W_{skin}}\left(1 + {f_{fadd}} + {f_{frame}} + {f_{string}} \right)\]

The weight of the floor is lower bounded by the density of the floor beams multiplied by the floor beam volume, in addition to an assumed weight/area density for planking.

\[\begin{split}\begin{aligned} {V_{floor}} &\geq {A_{floor}} {w_{floor}} \\ {W_{floor}}&\geq{V_{floor}}{\rho_{floor}}{g}+{W''_{floor}}{l_{floor}} {w_{floor}}\end{aligned}\end{split}\]

As with the shell, the tail cone weight is bounded using assumed proportional weights for additional structural elements, stringers, and frames.

\[{W_{cone}}\geq{\rho_{cone}}{g}{V_{cone}}\left(1+{f_{fadd}}+{f_{frame}} + f_{string}\right)\]

The weight of the horizontal and vertical bending material is the product of the bending material density and the and volumes required respectively.

\[\begin{split}\begin{aligned} W_{hbend} &\geq \rho_{bend} g V_{hbend} \\ W_{vbend} &\geq \rho_{bend} g V_{vbend}\end{aligned}\end{split}\]

The weight of luggage is lower bounded by a buildup of 2-checked-bag customers, 1-checked-bag customers, and average carry-on weight.

\[{W_{lugg}} \geq 2{W_{checked}} {f_{lugg,2}} {n_{pass}} + {W_{checked}} {f_{lugg,1}} {n_{pass}} + {W_{carry on}}\]

The window and insulation weight are lower bounded using assumed weight/length and weight/area densities respectively. It is assumed that only the passenger compartment of the the cabin is insulated and that the passenger compartment cross sectional area is approximately 55% of the fuselage cross sectional area.

\[\begin{split}\begin{aligned} {W_{window}} &= {W'_{window}} {l_{shell}} \\ {W_{insul}} &\geq {W''_{insul}} \left( 0.55\left({S_{bulk}} + {S_{nose}} \right) + 1.1\pi{R_{fuse}} {l_{shell}} \right) \end{aligned}\end{split}\]

The APU and other payload proportional weights are accounted for using weight fractions. \(W_{padd}\) includes flight attendants, food, galleys, toilets, furnishing, doors, lighting, air conditioning, and in-flight entertainment systems. The total seat weight is a product of the weight per seat and the number of seats.

\[\begin{split}\begin{aligned} {W_{apu}} &= {W_{payload}} {f_{apu}} \\ {W_{padd}} &= {W_{payload}} {f_{padd}} \\ {W_{seat}} &= {W'_{seat}} {n_{seat}} \end{aligned}\end{split}\]

The effective buoyancy weight of the aircraft is constrained using a specified cabin pressure \(p_{cabin}\), the ideal gas law and the approximated cabin volume. A conservative approximation for the buoyancy weight that does not subtract the ambient air density from the cabin air density is used.

\[\begin{split}\begin{aligned} \rho_{cabin}&= \frac{p_{cabin}}{{R} {T_{cabin}}} \\ {W_{buoy}} &= \rho_{cabin} {g} {V_{cabin}}\end{aligned}\end{split}\]

There are two methods in the model that can be used to lower bound the payload weight. The first is the sum of the cargo, luggage, and passenger weights (Constraint ). The second is through the definition of variable \(W_{avg. pass_{total}}\), which is an average payload weight per passenger metric (Constraint ). For the purposes of this paper, the second method is used, and as a result Constraint  is inactive.

\[\begin{split}\begin{aligned} W_{pass} &= W_{avg. pass} n_{pass} \\ {W_{payload}} &\geq {W_{cargo}} + {W_{lugg}} + {W_{pass}}\label{eq:payload1st} \\ {W_{payload}} &\geq {W_{avg. pass_{total}}} {{n_{pass}}} \label{eq:payload2nd}\end{aligned}\end{split}\]

The total weight of the fuselage is lower bounded by the sum of all of the constituent weights. The fixed weight \(W_{fix}\) incorporates pilots, cockpit windows, cockpit seats, flight instrumentation, navigation and communication equipment, which are expected to be roughly the same for all aircraft [Drela, 2011].

\[\begin{split}\begin{aligned} {W_{fuse}} &\geq {W_{apu}} + {W_{buoy}} + {W_{cone}} + {W_{floor}} + W_{hbend} + W_{vbend} + {W_{insul}} \\ &+ {W_{padd}} + {W_{seat}} + {W_{shell}} + {W_{window}} + {W_{fix}} \nonumber\end{aligned}\end{split}\]

Aerodynamic constraints

The drag of the fuselage is constrained using \(C_{D_{fuse}}\) from TASOPT, which calculates the drag using a pseudo-axisymmetric viscous/inviscid calculation, and scaling appropriately by fuselage dimensions and Mach number.

\[D_{fuse} = \frac{1}{2} \rho_{\infty} V_{\infty}^2 C_{D_{fuse}} \left( l_{fuse} R_{fuse} \frac{M^2}{M_{fuseD}^2} \right)\]