V-n Diagram and Load Factors: Method

V-n Diagram and Load Factors

What the diagram is

The V-n diagram plots the load factor n (lift divided by weight) against equivalent airspeed and marks the combinations an airplane's structure must carry. Two families of conditions bound it. The maneuvering envelope comes from the limit load factors the pilot may pull or push, capped by the maximum lift the wing can develop at each speed. The gust envelope comes from vertical gusts of prescribed strength met in level flight at the design speeds. 14 CFR 23.333 (and 25.333 for transports) requires compliance with the strength rules at every point on and within the envelope formed by both. The tool builds the envelope from the rule text, tabulates its corners at limit and ultimate level and draws it. Speeds are equivalent airspeeds in knots (KEAS) throughout: the rules write them so, and equivalent airspeed makes the stall curves independent of altitude.

Which rules the tool follows

For the normal, utility, acrobatic and commuter categories the tool follows 14 CFR Part 23 as it stood in the 2016 edition (Amendment 23-62). Amendment 23-64 (2017) replaced the prescriptive envelope rules by performance-based requirements with the formulas moved to accepted means of compliance (ASTM F3116), so the 2016 text is the last edition that carries them. For the transport category the tool uses the maneuvering factors of 14 CFR 25.337, unchanged since 1970, and the discrete-gust load-factor formula with the 66 / 50 / 25 ft/s gust velocities that § 25.341 prescribed until Amendment 25-86 (1996). The current § 25.341 requires a dynamic analysis of a one-minus-cosine gust with reference velocities of 56 ft/s at sea level reducing with altitude, plus a continuous-turbulence analysis; the simplified formula survives in § 25.335(d) only for the definition of V_B. The tool states this on the page: for a transport airplane its gust factors are the classical preliminary-design values, not a certification analysis.

Air density

The density at the flight altitude enters the mass ratio of the gust formula. The tool uses the 1976 U.S. Standard Atmosphere: geopotential altitude H = r₀Z/(r₀ + Z), a temperature lapse of 6.5 K/km to 11 km, an isothermal layer at 216.65 K to 20 km and the warmer layers above, pressure by the hydrostatic equation and density by the perfect-gas law with R = 287.053 J/(kg·K). Sea-level density is 1.2250 kg/m³ (0.0023769 slug/ft³). The implementation reproduces the printed Table IV rows to every digit (see the test script).

Stall speeds and the stall curves

The 1-g stall speed with flaps retracted is V_S1 = √(2W/(ρ₀ S C_Lmax)) with the sea-level density, so that it is an equivalent airspeed; the inverted stall speed uses the negative maximum normal-force coefficient. The maximum load factor the wing can develop at speed V is n = (V/V_S1)² above the axis and n = −(V/V_Sneg)² below it. These parabolas close the envelope at low speed: no maneuver or gust can load the structure beyond them because the wing stalls first. The stall speeds may be entered instead of the coefficients, in which case the coefficients they imply are printed.

Limit maneuvering load factors

§ 23.337(a): the positive limit maneuvering load factor may not be less than n = 2.1 + 24,000/(W + 10,000), W the design maximum takeoff weight in pounds, except that n need not exceed 3.8 for the normal and commuter categories; 4.4 for the utility category; 6.0 for the acrobatic category. The formula gives 3.8 or more up to W = 4,118 lb, so every light airplane in the normal category is designed to 3.8. § 23.337(b): the negative factor may not be less than 0.4 n (normal, utility, commuter) or 0.5 n (acrobatic). § 23.333(b)(3): the negative factor applies at V_C and varies linearly with speed to 0 at V_D for the normal and commuter categories, to −1.0 at V_D for the utility and acrobatic categories. § 25.337 uses the same formula bounded below by 2.5 and above by 3.8, with −1.0 up to V_C and zero at V_D. An airplane may be designed to higher factors; the tool accepts entered values and notes when they fall below the rule.

Design speeds

§ 23.335(a): V_C in knots may not be less than 33 √(W/S) for the normal, utility and commuter categories and 36 √(W/S) for the acrobatic category, with W/S the wing loading at the design maximum takeoff weight in lb/ft²; for W/S above 20 the multiplying factor may be decreased linearly to 28.6 at W/S = 100 (V_C need not exceed 0.9 V_H at sea level). § 23.335(b): V_D may not be less than 1.25 V_C, and with V_Cmin the required minimum cruising speed, V_D may not be less than 1.40 V_Cmin (normal, commuter), 1.50 V_Cmin (utility) or 1.55 V_Cmin (acrobatic), these factors decreasing linearly to 1.35 at W/S = 100. The tool takes the design speeds as these minima or as entered values checked against them. § 23.335(c): V_A may not be less than V_S √n and need not exceed V_C; the tool reports V_A = V_S1 √n⁺ as the positive corner of the stall curve and V_G = V_Sneg √|n⁻| as the negative one. Part 25 gives no formula for a minimum V_C or V_D; for a transport airplane both are entered.

Gust load factors

§ 23.341(c) (and § 25.341 before 1996, now § 25.335(d)) gives, in the absence of a more rational analysis,

n = 1 ± K_g U_de V a / (498 W/S),   K_g = 0.88 μ_g / (5.3 + μ_g),   μ_g = 2 (W/S) / (ρ c̄ a g)

with U_de the derived gust velocity in ft/s, V the equivalent airspeed in knots, a the slope of the airplane normal-force coefficient curve per radian (the wing lift-curve slope may be used when the horizontal-tail gust load is treated as a separate condition), W/S the wing loading in lb/ft², ρ the density at the flight altitude, c̄ the mean geometric chord and g the acceleration of gravity. The mass ratio μ_g measures the airplane's inertia against the air it displaces per chord of travel; a heavy, small-chord, high-altitude airplane has a large μ_g and feels more of the gust (K_g approaches 0.88), a light, large-chord one at low altitude rises with the gust and is alleviated. The expression for K_g is the fit of Pratt and Walker (NACA Report 1206, 1954) to the numerical rigid-airplane response to a one-minus-cosine gust of 25 chords total length with unsteady lift; the report states the fit's error at less than ±0.01. The constant 498 is 2/(ρ₀ × 1.688 ft/s per knot) with ρ₀ = 0.0023769 slug/ft³, which is 498.5; the tool uses 498 as the rule prints it (0.11 % more increment than the exact constant).

The derived gust velocities of § 23.333(c)(1) are 50 ft/s at V_C and 25 ft/s at V_D from sea level to 20,000 ft, reduced linearly to 25 and 12.5 ft/s at 50,000 ft; for the commuter category (and the pre-1996 transport rule) a rough-air gust of 66 ft/s at V_B, reduced to 38 ft/s at 50,000 ft, is added. Each gust line is the straight line n = 1 ± (Δn/V) V through the level-flight point (0, 1), drawn to its design speed; the gust envelope joins the end points, since § 23.333(c)(2) lets the gust load factor vary linearly between V_C and V_D. The design speed for maximum gust intensity V_B (§ 23.335(d)) is the lesser of the speed where the positive stall curve meets the rough-air gust line and V_S1 √n_gC, with n_gC the positive gust load factor at V_C, and need not exceed V_C.

The combined envelope

At every speed the structure must carry the more severe of the maneuver and gust load factors, within what the stall curves allow: n⁺(V) = min[(V/V_S1)², max(n_man, n_gust)] and the mirror image below the axis. The tool samples this envelope, lists its corners (A, C, D on the positive maneuver line; E, F, G on the negative; the primed points where the gust lines reach V_B, V_C and V_D) with the ultimate factors (1.5 × limit, § 23.303) and names the governing positive and negative load factors with the speed at which they occur. For light airplanes the negative gust at V_C usually exceeds the negative maneuver factor of 0.4 n, as it does in the default case (−1.66 against −1.52); for lightly loaded wings or high cruising speeds the positive gust can govern too. A gust point that lies beyond a stall curve is flagged as unreachable, and V_A greater than V_C is noted (the positive maneuver factor then cannot be developed below V_C).

Reading the diagram

The figure draws the stall curves in black (dotted where they lie outside the envelope), the maneuver envelope in green, the gust lines and the gust envelope in dashed red, and the combined envelope in blue with its corners marked; the two red points are the governing positive and negative load factors. The level-flight line n = 1 is the origin of the gust lines. The diagram is valid for one weight, one altitude and the clean configuration.

Default example

The default is a Part 23 normal-category light airplane of 2,400 lb with 160 ft² of wing (W/S = 15 lb/ft²), a mean chord of 4.5 ft, a lift-curve slope of 4.5 per radian and C_Lmax = 1.6 / −1.0 at sea level, with the design speeds at the rule minima. The formula gives n = 4.04, capped at 3.8, with −1.52 negative; V_S1 = 52.6 KEAS, V_A = 102.6, V_C = 33 √15 = 127.8, V_D = 1.40 V_C = 178.9 KEAS; μ_g = 19.4, K_g = 0.691, and the 50 ft/s gust at V_C gives Δn = 2.66, so n = 3.66 / −1.66: the positive envelope is governed by the 3.8 maneuver factor and the negative one by the gust. Every one of these values is checked in the tool's test script against the rule text, NACA Report 1206 and an independent numpy construction of the envelope.

Limits

Not covered: the flap-extended envelope of § 23.345 (25 ft/s gust with flaps down, n = 2.0), unsymmetrical, rolling and yawing conditions, the pitching-velocity terms of § 25.337(a), horizontal-tail and vertical-tail gust loads, the Mach-limited design speeds V_C/M_C and V_D/M_D and the 20-second upset of § 23.335(b)(4), weight and centre-of-gravity variations (one weight is analysed), and the continuous-turbulence and tuned discrete-gust analyses of the current Part 25. The gust formula assumes a rigid airplane that rises but does not pitch, with the gust uniform across the span; airplanes with unusual configurations (canards, tandem wings, very flexible wings) require a rational analysis (§ 23.341(b)).