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Light-airplane tapered wing at a 3.8 g maneuver: Schrenk's lift distribution with inertia relief, and the root shear, bending moment and torque

Tool	Wing Span Loads
Source	Schrenk, O. (1940). A Simple Approximation Method for Obtaining the Spanwise Lift Distribution. NACA Technical Memorandum 948, National Advisory Committee for Aeronautics, Washington, DC. Pages 2–3 (the additional and basic distributions, the lift-slope correction with κ from Figure 1); Anderson, R. F. (1936). Determination of the Characteristics of Tapered Wings. NACA Report 572, Tables II–IV (the lifting-line values the approximation is compared with).
Level	Undergraduate

Problem

A straight-tapered wing of 36 ft span, 5.5 ft root chord and 3.5 ft tip chord (S = 162 ft², A = 8) carries an airplane of 2,400 lb at a limit load factor of 3.8 and 120 knots equivalent airspeed; the sections have a lift-curve slope of 5.73 per radian, c_m0 = −0.05, and the tip is washed out 3°; the elastic axis is at 35 % chord, the aerodynamic centre at 25 %. Each half carries 150 lb of structure (proportional to the chord, c.g. at 40 % chord), 150 lb of fuel between 2 and 9 ft (proportional to the chord, 35 % chord) and a 300 lb mass at 6 ft on the elastic axis. Find Schrenk's lift distribution, the net running load, and the shear, bending moment and torque at the root.

Inputs to Enter

Planform trapezoid b = 36 ft, c_r = 5.5 ft, c_t = 3.5 ft; elastic axis 0.35 c, aerodynamic centre 0.25 c, sweep 0
W = 2,400 lb, n = 3.8, V_e = 120 kts, c_m0 = −0.05, twist included: ε_t = −3°, a_0 = 5.73 /rad
Per side: W_w = 150 lb ∝ chord at 0.40 c; W_f = 150 lb from y = 2 to 9 ft ∝ chord at 0.35 c; mass 300 lb at y = 6 ft, 0.35 c; 11 output stations

Open the Wing Span Loads tool (loaded with this example)

Answers

S = 162 ft², A = 8, λ = 0.636, MAC 4.574 ft at y = 8.333 ft; L = n W = 9,120 lb (4,560 lb per side); q = 48.75 lb/ft², C_L = 1.155
Elliptical root chord 4S/(πb) = 5.730 ft; Schrenk's c c_l at the root ½ (5.5 + 5.730) × 1.155 = 6.484 ft, running lift 316.1 lb/ft; centroid of the additional lift 0.4437 s (elliptical 4/(3π) = 0.4244)
Basic lift: mean twist ε̄ = −3° × y_MAC/s = −1.389°, root l_b = ½ q a_0 (0 − ε̄) c_r = 18.62 lb/ft, zero at y_MAC; κ(0.636) = 0.054 from TM 948 Figure 1, wing slope a = 4.620 /rad
Root shear V = 4,560 − 3.8 × 600 = 2,280 lb; root bending moment M = 36,418 (additional) − 874 (twist) − 4,750 (structure) − 3,082 (fuel) − 6,840 (mass) = 20,872 lb-ft; elliptical reference L b/(3π) = 34,836 lb-ft
Root torque about the elastic axis 2,113 (lift arm) − 903 (c_m0) + 130 (inertia) = 1,340 lb-ft nose up; SI: 10,142 N, 28,298 N·m, 1,817 N·m

Notes

The tool's default inputs are this wing; press Submit. The case is the tool's own worked example: every station of the table is checked against an adaptive scipy integration (References/Wing-Span-Loads/checks/spanloads_check.py) and the root moments against the closed forms (L/2) s [(1 + 2λ)/(6(1 + λ)) + 2/(3π)] and L b/(3π) in test_wing_span_loads.php, which also reproduces the root ordinates of Schrenk's comparison Figures 2–6 and 10 and compares the approximation with Anderson's lifting-line Tables II and IV (within 3.5 % inboard of 80 % semi-span for the additional lift; Anderson's example wing can be entered as a chord table).