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V Tail Box v0.3.257 - 2026-09-22 (Created from TOOLtemplate.html v0.1.6)

V Tail Box

Explore tail geometry, maneuver loads, and ultimate structural design limits from a JET11 aircraft definition.

Choose or drop a JET11 fileJET11 .xlsm or .xlsx workbook, or JET JSON export
No file loaded.

Structural load requirements

-- NOTE: changing limits here will not be saved to your aircraft project. They are for exploration and trade study only.

The V–n diagram combines aerodynamic stall boundaries with positive and negative structural limits. Ultimate loads multiply the structural limits by the selected Factor of Safety.

Positive corner—
Negative corner—
Positive ultimate load—
Negative ultimate load—

Gust load factor

LaTeX: n_{\mathrm{gust}}=1\pm\Delta n=1\pm\frac{K_g C_{L_\alpha}\rho V_{\mathrm{gust}}V_\infty}{2(W/S)}ngust= 1 ± Δn= 1 ±KgCLαρVgustV∞2(W/S)

Corner velocity

LaTeX: V_{corner}=\sqrt{\frac{2n(W/S)}{\rho C_{L,\max}}}Vcorner=2n(W/S)ρCL,maxLaTeX: n_{ultimate}=FoS,n_{limit}nultimate=FoS×nlimit

Tail design

-- NOTE: changing limits here will not be saved to your aircraft project. They are for exploration and trade study only.

The root leading-edge station is fixed. Drag the root trailing edge or either tip corner, or use the sliders and manual-entry boxes. Both tail halves remain symmetric.

Editable JET11 tail planform
Front view — drag either tail tip to set dihedralAirfoil profile at the fuselage attachment point

Original Tail

Span—
Root chord—
Tip chord—
MAC—
Quarter-chord sweep—
Trailing-edge sweep—
NACA
4-digit analytic profile
Aileron geometry

Exposed Tail Geometry (one Tail)

Sw · exposed area—
bw · exposed span—
ARw · exposed AR—
MACw · exposed MAC—

Tail-box Design

A representative closed tail box is cut from the current NACA section at the fuselage attachment point. Adjust its dimensions and wall thicknesses to affect strength and rigidity.

Representative tail-box section at the fuselage attachment point
Right-tail spar planform

Tail Bending

Ixx formula

LaTeX: I_{xx}=\frac{w\cdot h^3-(w-2t_w)(h-2t_s)^3}{12}+4c_w\cdot t_w\left[\frac{t_w^2}{12}+\left(\frac{h}{2}-t_s-\frac{t_w}{2}\right)^2\right] I x x = w · h 3 − ( w − 2 t w ) ( h − 2 t s ) 3 1 2 + 4 c w · t w t 2 w 1 2 + ( h 2 − t s − t w 2 ) 2

Tail fuel-tank volume

Internal volume between the front and rear spars, less spars, caps, ribs, and stiffeners. Packing factor accounts for unusable volume.

Gross internal volume · one tail—
Net geometric volume · one tail—
Usable tank volume—
Assumed JP-8 density50.0 lbm/ft³
Maximum fuel · both tails—
Selected fuel · both tails0.0 lbm
Click a smaller panel to move it into the main display.
Tail Bending three-dimensional structural view. Left-drag rotates, right-drag pans, and the mouse wheel zooms.
Maximum bending stress versus aircraft y station
Section Ixx versus aircraft y station
Maximum bending stress top-down tail heat map
Tail Buckling
UNDER CONSTRUCTIONPreliminary calculations and visualizations are still being developed and validated.

Preliminary stiffened-skin stability analysis. Panel coefficients use each local rib bay and stringer/spar gap; stringer columns use the selected Johnson/Euler treatment. Local-element buckling is reported separately from empirical crippling.

Stringer mass—
Root Ixx contribution—
Minimum margin—
Governing mode—
Stringer slenderness, λ / Cc—
Column equation used—
Root governing panel—
Spar-cap assessment—
Simply supported skin-panel coefficients
LaTeX: k_c(m)=(\frac{mb}{a}+\frac{a}{mb})^2 k c ( m ) = ( m b a + a m b ) 2

kc is the minimum over integer m. For shear: ks = 5.34 + 4(b/a)² when a/b ≥ 1; otherwise ks = 4 + 5.34(b/a)².

Stringer transition-column selection
LaTeX: r=\sqrt{\frac{I_{min}}{A}},\lambda=\frac{K_eL}{r},C_c=\sqrt{\frac{2\pi^2E}{F_y}} r = I m i n A , λ = K e L r , C c = 2 π 2 E F y Johnson for λ < Cc LaTeX: F_{cr,J}=F_y[1-\frac{F_y}{4\pi^2E}(\frac{K_eL}{r})^2] F c r , J = F y [ 1 − F y 4 π 2 E ( K e L r ) 2 ] Euler for λ ≥ Cc LaTeX: F_{cr,E}=\frac{\pi^2E}{(K_eL/r)^2} F c r , E = π 2 E ( K e L / r ) 2
Stringer and spar-cap local-element buckling
LaTeX: F_{cr,plate}=\frac{k\pi^2E}{12(1-ν^2)}(\frac{t}{b})^2 F c r , p l a t e = k π 2 E 1 2 ( 1 − ν 2 ) ( t b ) 2

Stringer web/flange elements use their own unsupported widths, with the selected attachment reduction applied. Spar caps are checked separately as outstanding plate elements. This is not an empirical crippling allowable.

Stringer contribution to Ixx
LaTeX: I_{xx,st}=\sum\left[I_{xx,st,c}+A_{st}(z_{st}-\bar z)^2\right]Ixx,st = Σ [ Ixx,st,c + Ast(zst − z̄)² ]
Margins of safety versus aircraft y station
Governing failure mode versus aircraft y station

Skin stiffeners (stringers)

Skin stiffeners—usually called stringers—are spanwise structural members attached to the tail skin. Together, the skin, stringers, spars, and ribs form a semi-monocoque tail structure.

What they do

  • Divide the skin into narrower panels, substantially increasing its resistance to compression and shear buckling.
  • Carry part of the tail’s spanwise axial load.
  • Help the skin retain the aerodynamic contour.
  • Increase the section’s bending stiffness Ixx.
  • Allow a thinner skin than an otherwise unstiffened tail.
  • Transfer loads through their attachments into ribs, spars, and neighboring skin.

For positive tail bending, the upper skin and its stringers are generally compression-critical, while the lower structure is predominantly in tension. NASA describes spanwise stringers as the conventional means of making thin tail skin effective under compression. NASA structural-design review

Why spacing matters

A first-order skin-buckling relation is:

LaTeX: \sigma_{\mathrm{cr}}=\frac{k\pi^2E}{12(1-\nu^2)}\left(\frac{t_s}{b_s}\right)^2σcr =kπ²E12(1−ν²)(tsbs)²
  • σcr = critical elastic skin-buckling stress
  • k = buckling coefficient determined by loading and edge restraint
  • E = elastic modulus
  • ν = Poisson’s ratio
  • ts = skin thickness
  • bs = unsupported width between stiffeners

Reducing stiffener spacing bs raises buckling stress approximately with 1/bs². This is their principal advantage.

Common configurations

Metal tails commonly use angles, Z-stringers, hat sections, J-sections, blades, and integrally machined stiffeners. Composite tails commonly use blades, hats, or T-shaped stiffeners that are bonded, cocured, stitched, or mechanically fastened.

Important failure modes

  • Skin compression and shear buckling between stringers
  • Stringer local buckling and crippling
  • Column buckling between ribs
  • Combined compression and shear
  • Fastener bearing, pull-through, and fatigue
  • Skin–stringer separation
  • Composite delamination and impact damage
  • Overall stiffened-panel buckling

NASA testing shows that postbuckling deformation can redistribute skin, stringer, and attachment loads, while composite panels can fail through skin–stringer separation after local buckling. NASA buckled-panel study, NASA composite separation study

Method used by this tool

  • Skin panels: a is the local distance between adjacent ribs, b is each individual gap between spars and stringers, and the compression coefficient is minimized over positive integer half-wave counts m. The shear coefficient varies with a/b.
  • Stringer columns: the tool calculates the full principal section inertias (including the product of inertia for unsymmetric sections), radius of gyration, slenderness λ = KeL/r, and Cc. The course-default control forces Johnson; Automatic selects Johnson below Cc and Euler at or above Cc.
  • Local elements: web and flange plate buckling is shown as “stringer local-element buckling,” not as empirical crippling.
  • Spar caps: upper and lower caps receive separate outstanding-plate local-buckling checks. A true empirical crippling allowable still requires section-specific test constants or approved material data and is not claimed here.

Preliminary-design caution: the live calculations are classical screening equations, not certification substantiation. Attachment, cutouts, imperfections, postbuckling, fatigue, and local load introduction require detailed analysis and test.

Tail Twisting

The torsion properties are evaluated at the selected station on the exposed tail.

Tail twisting evaluation section
Tail twisting evaluation station on the tail planform
Torsion constant, J (Resists Twisting)—
Kθ (V Tail Box Torsional Spring Constant)—
Chord at Evaluation
(exposed yMAC)—

J formula

For a rectangular box,

LaTeX: J=\frac{2(hw)^2}{\frac{h}{t_w}+\frac{w}{t_s}} J = 2 ( h w ) 2 h t w + w t s Rectangular tail-box dimensions used by the specialized torsion-constant equation Am h w tw ts

Kθ formula

LaTeX: K_\theta=\frac{G\,J}{b_w} K θ = G J b w

CLδa — aileron lift contribution

Current NACA airfoil with a deflected trailing-edge aileron
Original max camber—
Deflected max camber—
Deflected ΔαL=0—
ΔCL =—
CLδa—
Aileron span fraction, ba/bw0.500
LaTeX: \Delta h=c_a\sin(\delta_a) Δ h = c a sin ( δ a ) LaTeX: \Delta_{\mathrm{max\ camber}}=\frac{\Delta h}{c} Δ max camber = Δ h c
LaTeX: C_{L_{\delta_a}}=\frac{b_a}{b_w}C_{L_\alpha}\frac{\Delta\alpha_{L=0}}{\delta_a}\mathrm{deg}^{-1} C L δ a = b a bw C L α Δ α L = 0 δ a deg − 1

CMδa — aileron pitching-moment contribution

The selected airfoil family uses deflected maximum camber for 4-digit, 5-digit, and modern high-lift sections, or the designation design-lift coefficient for NACA 6-series sections.

Current NACA airfoil with pitching-moment camber construction
Original max camber—
Deflected max camber—
Deflected CMac—
ΔCM—
CMδa—
Aileron span fraction, ba/bw0.500
LaTeX: C_{M_{ac}}=-2.5\frac{\mathrm{Camber}_{\mathrm{deflected}}}{c} C M a c = − 2 . 5 Camber deflected c
LaTeX: C_{M_{\delta_a}}=\frac{b_a}{b_w}\frac{\Delta C_M}{\delta_a}\mathrm{deg}^{-1} C M δ a = b a bw Δ C M δ a deg − 1

Teaching approximations from the supplied guide: use CMac = −2.5(Camberdeflected/c) for 4-digit, 5-digit, and modern high-lift sections; use CMac = −0.15cl,design for NACA 6-series sections, with cl,design expressed as 0.2, 0.3, and so on.

Divergence airspeed

Torsional stiffness resists the aerodynamic moment caused by the offset between the aerodynamic center and elastic axis.

Divergence geometry at the representative MAC sectionLaTeX: \theta=\frac{qS_weC_{L_\alpha}(-\alpha_{L=0})}{K_\theta-qS_weC_{L_\alpha}} θ = q S w e C L α −αL=0 K θ − q S w e C L α
Vdiv——
qdiv—
Kθ—
Exposed tail Sw—CLα
(at selected Airspeed/M)—
LaTeX: q_{div}=\frac{K_\theta}{S_w e C_{L_\alpha}},\quad V_{div}=\sqrt{\frac{2K_\theta}{\rho S_w e C_{L_\alpha}}}qdiv=KθSweCLαVdiv=2KθρSweCLα LaTeX: C_{L_\alpha}=\frac{a_0}{\sqrt{1-M^2}+\frac{a_0}{\pi e AR}},\quad M\leq 0.85 C L α = a 0 1 − M 2 + a 0 π e A R , M ≤ 0.85
Prandtl–Glauert
LaTeX: C_{L_\alpha}=\frac{4}{\sqrt{M^2-1}+\frac{4}{\pi e AR}},\quad M\geq 1.20 C L α = 4 M 2 − 1 + 4 π e A R , M ≥ 1.20
K theta and q S w e C L alpha versus airspeed

control surface reversal speed

Aileron lift twists the exposed tail until the elastic rolling effectiveness crosses zero.

Aileron-reversal geometry at the exposed-tail evaluation section
Vrev—
qrev—
Kθ—
Rigid CLδa—
Elastic CLδa
(at selected V)—
Air density, ρ—
CLα
(at selected V)—
Exposed tail Sw—
Assessed chord, c—
CMδa—
Elastic aileron effectiveness and its zero crossing versus airspeed
LaTeX: q_{\mathrm{rev}}=\frac{-K_\theta C_{L_{\delta_a}}}{S_wcC_{L_\alpha}C_{M_{\delta_a}}} q rev = − K θ C L δ a S w c C L α C M δ a
LaTeX: V_{\mathrm{rev}}=\sqrt{\frac{-2K_\theta C_{L_{\delta_a}}}{\rho S_wC_{L_\alpha}cC_{M_{\delta_a}}}} V rev = − 2 K θ C L δ a ρ S w C L α c C M δ a

Flutter speed

The supplied thin-wall teaching relation combines the current tail-box torsional stiffness, exposed-tail aspect ratio, NACA thickness ratio, and atmospheric pressure.

Tail box used for flutter-speed estimate
Flutter speed, Vf—
Speed of sound, a—
Effective shear modulus, GE—
Exposed-tail AR—
Thickness ratio, t/c—
Pressure, P—
Representative taper, λ—
Shear modulus, G—
Tail-box J—
Chord, c—
Normalized flutter-speed margin and its zero crossing
V_f=a\sqrt{\frac{G_E(AR+2)(t/c)^3}{1.337(AR)^3P(\lambda+1)}}Vf=aGE(AR+2)(t/c)³1.337AR³P(λ+1)
LaTeX: G_E=\frac{6GJ}{c^4(t/c)^3} G E = 6 G J c 4 ( t / c ) 3

Moment of inertia formulas

Moment of inertia reference chart showing rectangular, hollow rectangular, circular, hollow circular, and I-section formulas, plus the bending-strength relation.

CMac source relations

Reference chart showing pitching-moment coefficient relations for NACA 4-digit, NACA 6-series, and modern aft-loaded high-lift airfoils

Lift-curve slope versus Mach number

Piecewise lift-curve-slope model versus Mach number, including subsonic, transonic interpolation, and supersonic relations

Subsonic compressibility and finite-Tail correction

finite-tail lift-curve-slope equation that incorporates Prandtl–Glauert compressibility:

LaTeX: C_{L_\alpha}=\frac{a_0}{\sqrt{1-M^2}+\frac{a_0}{\pi e AR}},\quad M\leq 0.85 C L α = a 0 1 − M 2 + a 0 π e A R , M ≤ 0.85

where:

  • a0: incompressible 2-D airfoil lift-curve slope
  • √(1 − M2) = β: Prandtl–Glauert compressibility term
  • AR: tail aspect ratio
  • e: span-efficiency factor
  • M: Mach number
  • CLα: resulting finite-tail lift-curve slope

The pure Prandtl–Glauert correction is only:

LaTeX: a_{\mathrm{2-D,comp}}=\frac{a_0}{\sqrt{1-M^2}} a 2−D,comp = a 0 1 − M 2

The additional term

LaTeX: \frac{a_0}{\pi e AR} a 0 π e A R

accounts for finite-tail induced downwash. Therefore, the displayed equation should be described as:

Subsonic finite-tail lift-curve slope with Prandtl–Glauert compressibility correction

Rib Lightening

Properly designed lightening holes are common in aircraft ribs and can remove substantial mass without significantly reducing the rib’s intended strength.

Ribs credited—
Both-tail mass saved—
Minimum shear MS—
Minimum buckling MS—

Preliminary screen: one or more automatically positioned and packed circular, flanged openings in each eligible region forward of the front spar, within every active spar bay, and aft of the rear spar. Each rib is checked at its ultimate tributary lift using its remaining shear ligaments and an elastic local-buckling estimate. Manufacturer data, detailed finite-element analysis, joints, fatigue, damage tolerance, and testing still govern a real design.

Aircraft wing structure under construction showing multiple ribs with circular lightening holes
Example tail ribs with circular lightening holes.

A Tail rib primarily:

  • Maintains the airfoil shape.
  • Transfers skin pressure and shear into the spars.
  • Stabilizes the skins and spar webs against buckling.
  • Distributes concentrated loads from hinges, fittings, landing gear, engines, and other attachments.

Industry practice generally follows these principles:

  • Use circular, oval, or smoothly rounded openings—never sharp corners.
  • Preserve continuous material around the rib perimeter, skin flanges, spar attachments, and concentrated-load fittings.
  • Place holes in relatively low-stress regions identified through analysis.
  • Form a flange or bead around sheet-metal holes. The turned edge greatly improves local stiffness and buckling resistance.
  • Use several appropriately spaced holes instead of removing one uncontrolled large region.
  • Add doublers or thicker/flanged edges where an opening interrupts an important load path.
  • Check web shear, net-section stress, local buckling, crippling, fatigue, vibration, and damage tolerance—not merely the remaining area.

FAA structural-repair guidance illustrates both circular and rounded triangular lightening holes in metal ribs, including formed or reinforced edges. It also emphasizes following manufacturer-approved structural data where available. FAA AC 43.13-1B

Modern practice increasingly uses finite-element and topology optimization to locate material along the actual load paths. NASA research specifically evaluates lightening holes and rib/spar topology against static, buckling, aeroelastic, and flutter requirements. NASA aircraft rib and spar topology study NASA testing has also evaluated intermediate rib webs containing flanged lightening holes under representative shear loading. NASA structural test report

Thin-airfoil flap-effectiveness model

LaTeX: C_{L_{\delta_a}}=(2\pi\frac{\pi}{180})\eta f_{\mathrm{span}} C L δ a = ( 2 π π 1 8 0 ) η f span

Variables

CLδa
The tail lift-coefficient derivative with respect to aileron deflection: the change in CL produced by one degree of δa. Its units are per degree, and its sign follows the selected aileron-deflection convention.
δa
Aileron deflection in degrees. It appears in the derivative subscript; it is not multiplied directly into this equation. Multiplying CLδa by δa gives the estimated ΔCL.
2π
The ideal two-dimensional thin-airfoil lift-curve slope in units of per radian.
π/180
The radians-per-degree conversion, which changes the 2π lift-curve slope from per radian to per degree.
η
The dimensionless plain-flap effectiveness factor. The tool calculates it from the aileron hinge position xh/c; it does not depend on the selected aileron-deflection angle.
fspan
The dimensionless fraction of the exposed tail span affected by the aileron, evaluated as ba/bw.
π
The mathematical constant pi, approximately 3.14159.

The thin-airfoil result is idealized and can be high because it assumes inviscid, attached, two-dimensional flow. Real effectiveness may be reduced by finite-tail effects, hinge gaps, boundary layers, separation, and control-surface geometry.

The thin-airfoil flap-effectiveness model may yield a 40–60% higher CLδa, but it does not prove that the higher value is more accurate for the actual aircraft. You may consider taking the worst-case scenario for your design.

control surface reversal speed

This panel uses the lift-contribution form of the aileron-reversal model. Let L denote lift force, lowercase l rolling moment, and M twisting moment. Corresponding coefficient subscripts are capital L, lowercase l, and capital M.

1. Relate control surface lift to rolling moment

If the aileron lift contribution acts at a fixed, nonzero spanwise moment arm ya, its rolling moment is:

LaTeX: l_a=y_aL_a l a = y a L a

Differentiating with respect to aileron deflection gives:

LaTeX: \frac{\partial l_a}{\partial\delta_a}=y_a\frac{\partial L_a}{\partial\delta_a} ∂ l a ∂ δ a = y a ∂ L a ∂ δ a

Because ya is fixed and nonzero, rolling-moment reversal and aileron lift-contribution reversal occur at the same condition:

LaTeX: \frac{\partial L_a}{\partial\delta_a}=0 ∂ L a ∂ δ a = 0

2. Direct control surface lift and twisting moment

The rigid-tail aileron lift contribution is:

LaTeX: L_a=qS_wC_{L_{\delta_a}}\delta_a L a = q S w C L δ a δ a

with dynamic pressure:

LaTeX: q=\frac{1}{2}\rho V^2 q = 1 2 ρ V 2

The aileron also produces a twisting moment about the elastic axis:

LaTeX: M_{\mathrm{aero}}=qS_wcC_{M_{\delta_a}}\delta_a M aero = q S w c C M δ a δ a

Equating that moment to the tail-box restoring moment gives:

LaTeX: K_\theta\theta=M_{\mathrm{aero}} K θ θ = M aero

Therefore, the elastic twist per degree of aileron deflection is:

LaTeX: \frac{\theta}{\delta_a}=\frac{qS_wcC_{M_{\delta_a}}}{K_\theta} θ δ a = q S w c C M δ a K θ

This teaching form intentionally neglects the separate aerodynamic twisting-stiffness term CMα; it retains the aileron-induced twisting moment CMδa.

3. Twist changes the lift contribution

The lift caused by elastic twist is:

LaTeX: L_\theta=qS_wC_{L_\alpha}\theta L θ = q S w C L α θ

Adding direct aileron lift and twist-induced lift:

LaTeX: L_{\mathrm{total}}=qS_w(C_{L_{\delta_a}}\delta_a+C_{L_\alpha}\theta) L total = q S w ( C L δ a δ a + C L α θ )

So the effective lift derivative is:

LaTeX: C_{L_{\delta_a}}^{\mathrm{elastic}}=C_{L_{\delta_a}}+C_{L_\alpha}\frac{\theta}{\delta_a} C elastic L δ a = C L δ a + C L α θ δ a

4. Solve the reversal condition

Set the effective derivative to zero at qrev and substitute the twist relation:

LaTeX: C_{L_{\delta_a}}+C_{L_\alpha}\frac{q_{\mathrm{rev}}S_wcC_{M_{\delta_a}}}{K_\theta}=0 C L δ a + C L α q rev S w c C M δ a K θ = 0

Solving for the physically meaningful positive dynamic pressure gives your equation:

LaTeX: q_{\mathrm{rev}}=\frac{-K_\theta C_{L_{\delta_a}}}{S_wcC_{L_\alpha}C_{M_{\delta_a}}} q rev = − K θ C L δ a S w c C L α C M δ a

Using q = ½ρV² gives the reversal true airspeed:

LaTeX: V_{\mathrm{rev}}=\sqrt{\frac{2q_{\mathrm{rev}}}{\rho}}=\sqrt{\frac{-2K_\theta C_{L_{\delta_a}}}{\rho S_wC_{L_\alpha}cC_{M_{\delta_a}}}} V rev = 2 q rev ρ = − 2 K θ C L δ a ρ S w C L α c C M δ a

Units used by this tool: Kθ is in ft·lbf/deg, CLα, CLδa, and CMδa are per degree, Sw is exposed-tail area in ft², c is the evaluated-section chord in ft, and ρ is in slug/ft³.

The sign of CMδa depends on the chosen aileron convention. The tool searches the Mach-dependent CLα model and selects the lowest positive zero crossing.

Flutter-speed teaching estimate

Flutter is a coupled dynamic instability: aerodynamic forces feed energy into structural bending and twisting until damping can no longer suppress the motion. This panel uses the supplied empirical thin-tail estimate:

LaTeX: V_f=a\sqrt{\frac{G_E(AR_w+2)(t/c)^3}{1.337AR_w^3P(\lambda+1)}} V f = a G E ( A R w + 2 ) ( t / c ) 3 1 . 3 3 7 A R 3 w P ( λ + 1 )

Unit-consistent torsional rigidity

Tail Twisting reports Kθ per degree. The empirical relation requires GJ, so the calculation converts back to a per-radian rigidity:

LaTeX: GJ=K_\theta b_w\frac{180}{\pi} G J = K θ b w 1 8 0 π LaTeX: G_E=\frac{6GJ}{c^4(t/c)^3} G E = 6 G J c 4 ( t / c ) 3

The atmosphere at the selected altitude supplies the local speed of sound a and static pressure P. The displayed Vf is true airspeed.

Zero-crossing chart

To show the predicted boundary clearly, the chart uses this normalized flutter-speed margin:

LaTeX: F(V)=1-\left(\frac{V}{V_f}\right)^2 F ( V ) = 1 − ( V V f ) 2

It is positive below Vf, zero at Vf, and negative above Vf.

Teaching limitation: this margin is not modal damping. A rigorous flutter prediction requires structural mass distribution, bending and torsional natural frequencies, damping, mode shapes, and unsteady aerodynamic derivatives.

Classical Tail-divergence derivation

Classically, divergence airspeed is found by equating torsional restoring stiffness to the destabilizing aerodynamic pitching moment caused by twist.

For a typical section:

LaTeX: K_\theta\theta=M_{\mathrm{aero}} K θ θ = M aero

Kθ is torsional stiffness and θ is elastic twist. If the aerodynamic center is a distance e behind the elastic axis, twist produces additional lift:

LaTeX: \Delta L=qSC_{L_\alpha}\theta Δ L = q S C L α θ

and aerodynamic twisting moment:

LaTeX: M_{\mathrm{aero}}=qSC_{L_\alpha}e\theta M aero = q S C L α e θ

with

LaTeX: q=\frac{1}{2}\rho V^2 q = 1 2 ρ V 2

At divergence the moments balance:

LaTeX: K_\theta\theta=q_DSC_{L_\alpha}e\theta K θ θ = q D S C L α e θ

Cancel θ:

LaTeX: q_D=\frac{K_\theta}{SC_{L_\alpha}e} q D = K θ S C L α e

Since

LaTeX: q_D=\frac{1}{2}\rho V_D^2 q D = 1 2 ρ V 2 D

the classical divergence speed is

LaTeX: V_D=\sqrt{\frac{2K_\theta}{\rho S C_{L_\alpha}e}}VD=2KθρSCLαe

Here, e is the dimensional distance between the elastic axis and aerodynamic center.

Multi-cell thin-wall torsion method

One closed cell

LaTeX: J=\frac{4A_m^2}{\displaystyle\oint\frac{ds}{t}}J=4Am2dst

While there is more stiffness to be gained by adding spars, it is marginal and not worth the computational complexity. Multiple spars primarily aid in bending strength. This tool assumes a single closed cell for torsion. That said, here is the math...

Multiple closed cells

Each cell has an unknown constant shear flow qi. All cells must have the same twist rate:

LaTeX: Cq=2G\theta'A C q = 2 G θ ' A
  • Ai is the median-line area of cell i.
  • Cii is the sum of ds/t around cell i.
  • Cij is negative ds/t along the spar shared by cells i and j.
  • θ′ is twist per unit span.

The total torque is

LaTeX: T=2\sum_iA_iq_i T = 2 ∑ i A i q i

and the equivalent torsion constant is

LaTeX: J=4A^TC^{-1}A J = 4 A T C − 1 A

This reduces to the single-cell relation when only one closed cell is present.

Do the spars contribute?

Yes. A spar joining the upper and lower skins is a shared wall between closed cells. Its thickness and length alter the shear-flow distribution and therefore J. Additional closed-cell spars generally increase torsional stiffness.

Spar caps

  • Their primary contribution is bending stiffness.
  • They contribute to torsion by adding material at the skin–spar junction.
  • The thin-wall model represents each cap as a localized increase in wall thickness over its actual width.
  • A concentrated cap area alone is insufficient for ordinary Saint-Venant torsion.

V Tail Box calculates multi-cell J from every active skin segment and spar web at the selected Tail Twisting section. Cap-covered skin segments use the combined local thickness.