Wing Span Loads: Method

Wing Span Loads: Schrenk's Approximation, Inertia Relief, and the Shear, Moment and Torque Along the Span

The problem

The first structural question about a wing is how much shear, bending moment and torque each station of the spar or wing box has to carry in the critical load case of the V-n diagram. The answer needs the spanwise distribution of the lift, which the exact theories (Prandtl's lifting line as computed by Glauert, Anderson or Multhopp, or a lifting-surface or panel method) give at the cost of a Fourier or matrix solution for every planform, twist and angle of attack. Schrenk (1940) observed that for the purposes of a load estimate the distribution of plain wings lies very close to the arithmetic mean of two curves that need no computation at all: the planform chord itself and the semi-ellipse of the same area. His paper compares the approximation with Multhopp's exact method for rectangular, tapered, triangular, cut-out and rounded planforms, with twist and with flaps, and finds agreement "in general satisfactory, and to some extent even surprising". The approximation has been the standard preliminary-design and loads-course method ever since (Peery Chapter 5, Bruhn A5, Niu Chapter 3). This tool applies it, adds the inertia relief of the wing's own mass, and integrates the result into the loads a structural analysis needs.

Conventions and units

One half wing is analysed, from the plane of symmetry (the root, y = 0) to the tip (y = s = b/2); the other half is its mirror. Lift, net load and shear are positive upward; the bending moment is positive when it bends the tip up (upper surface in compression); the torque is positive nose up (leading edge up). Chordwise positions are fractions of the local chord from the leading edge: the elastic axis x_ea (the shear centre of the wing box, about which the torque is taken), the aerodynamic centre x_ac (where the section lift acts with the pitching moment c_m0), and the centres of gravity of the structure, the fuel and the masses. Lengths, forces and the airspeed go through the unit system (ft, lb, kts or m, N, m/s); the load factor, the angles, the chord fractions, c_m0 and the section slope are plain numbers. Weights are entered per side. The load factor n multiplies the weight for the lift (L = n W, both sides) and every inertia load; a negative n gives a negative-g case.

Schrenk's additional lift distribution

Schrenk's formula for the untwisted wing (TM 948, p. 3) is, in his notation with t the chord, t̄ the mean chord F/b and dc_a/dα the wing's lift-curve slope,

dA/dx = ½ q (dc_a/dα) α { t(x) + (4/π) t̄ √(1 − (x/(b/2))²) },

the mean of the planform-proportional distribution q (dc_a/dα) α t and the elliptical distribution of the same area. Since the wing's lift coefficient is C_L = (dc_a/dα) α and L = q S C_L, the running lift is l_a(y) = ½ [c(y) + c_ell(y)] L/S with c_ell(y) = (4S/(πb)) √(1 − (2y/b)²), and the loading parameter is c c_l = ½ (c + c_ell) C_L. Both curves enclose the area S/2 over the semi-span, so the approximation carries exactly L/2 per side whatever the planform, and it does not depend on the airspeed or the lift-curve slope: only on the planform and the total lift. The elliptical curve is the distribution of minimum induced drag and constant downwash, "independent of the wing shape"; the planform curve is the lift the sections would carry at one angle of attack with no induced effects, "determined in a simple manner by the wing shape"; the real distribution lies between them (TM 948, p. 2). For a straight-tapered wing of taper ratio λ the root ordinate is ½ (c_r/b + 4S/(πb²)) and the centroid of the additional lift lies at ½ [(1 + 2λ)/(3(1 + λ)) + 4/(3π)] of the semi-span; the fully elliptical load has its centroid at 4/(3π) = 0.4244 s and a root bending moment of L b/(3π) per side, which the tool prints beside Schrenk's value as a reference.

Accuracy. Schrenk's Figures 2 to 6 compare the approximation with Multhopp's method at an aspect ratio of 6.67 for taper ratios of 1, 0.75, 0.5, 0.25 and 0: the curves lie within a few percent of each other inboard of about 90 % of the semi-span; at a square tip the approximation keeps a finite ordinate where the exact curve falls to zero, so it is some 10 % high in the last few percent of the span, which is conservative for the loads and of no consequence for the root moment. The test script filed with the tool compares the approximation with Anderson's lifting-line tables for tapered wings with rounded tips (NACA Report 572, Table II, aspect ratio 6, taper 0.5): the additional distribution agrees within 3.5 % at every station inboard of 80 % semi-span (within 2 % inboard of 60 %), and the root bending moment of Anderson's illustrative example (Table IV) within 2 %. Schrenk himself judged the method "suitable for all problems where too great accuracy is not required", which in loads work it is, and "not directly applicable" to the induced drag or the downwash, which it is not.

The basic lift of a twisted wing

A twist (geometric or aerodynamic: a change of section, or a flap or aileron, is a change of the zero-lift direction) adds a distribution that carries no net lift when the angles are reckoned from the zero-lift direction of the whole wing. Schrenk takes that direction as the chord-weighted mean of the local twist, δ̄ = ∫ δ(x) t(x) dx / ∫ t(x) dx (TM 948, p. 2), and writes the twist term of his formula as ½ q (dc_a/dα_∞) δ(x) t(x) with the section slope dc_a/dα_∞ and δ measured from δ̄: the mean of the chord-proportional twist distribution and the zero line, "which has the zero line instead of the ellipse as the ideal distribution, and for which the twist angle must always be considered". The tool writes it l_b = ½ q a₀ [ε(y) − ε̄] c(y) with a linear aerodynamic twist ε = ε_t y/s, for which ε̄ = ε_t y_MAC/s = ε_t (1 + 2λ)/(3(1 + λ)) on a trapezoid; the basic lift is positive inboard of y_MAC and negative outboard for washout, integrates to zero, and scales with the dynamic pressure rather than with the load factor, so it matters more at high speed and low n. Schrenk's Figures 9 to 11 show the twist term against Multhopp for a 3° linear and a parabolic washout: the zero crossing (at 4/9 of the semi-span for the linear case on a 0.5 taper, as the formula gives) and the shape agree, the magnitude less closely than for the additional lift, and Anderson's Table IV example (a 0.5 taper wing with −7.1° of aerodynamic twist) gives a basic root load about 17 % below Schrenk's. For a twist that is not linear, a flap or an aileron, Schrenk's prescription is to apply the formula to the actual δ(x), rounding the corners of a step by hand with his Figures 12 and 13 as a guide; the tool offers the linear twist only.

Lift-curve slope. Schrenk refines the usual correction of the section slope to the wing, dc_a/dα = (dc_a/dα_∞) / [1 + (dc_a/dα_∞) (1 + κ) F/(πb²)], with a factor κ "determined by trial in such a way that the lift determined with our approximation method is given as correctly as possible"; his Figure 1 plots κ against the taper ratio t_a:t_i, from 0.19 for the pointed wing through zero near a taper ratio of 0.3 to 0.175 for the rectangular wing, where "the value thus determined agrees with the theoretically determined value of Glauert". For planforms that are not trapezoids "the nearest value of the taper ratio is used"; the tool fits a straight line to the inboard 80 % of a typed chord table for that purpose. The slope and the resulting angle of attack C_L/a (measured from the mean zero-lift direction) are reported for information; the load distribution does not use them.

Inertia relief

Everything attached to the wing is accelerated with the airplane: in a pull-up at load factor n a mass W/g on the wing exerts a downward inertia force n W on the structure, which relieves the lift and is one of the reasons wing-mounted engines, tip tanks and fuel in the outer wing reduce the root bending moment. The tool takes the structural weight of one side distributed over the semi-span with a shape proportional to the chord (the usual first estimate), to the chord squared (the depth of a box of constant thickness ratio times its width) or uniform; the fuel of one side between two span stations with the same choice of shape; and any number of concentrated masses (engine, landing gear, tank, store) at a station and a chordwise position. The distributed weights integrate exactly to the values entered; a concentrated mass is a downward step n W_j in the shear and a kink in the moment. The net running load is p = l_a + l_b − n (w_w + w_f). The critical bending case is usually the one with the least relief (tanks empty, no stores) at the highest load factor, and the critical torque case can be a different one (full tanks aft of the elastic axis, or a high-speed case where the pitching moment dominates); run them separately.

Shear, bending moment and torque

From the free tip inward,

V(y) = ∫_y^s p(ξ) dξ − Σ_{yj ≥ y} n W_j, M(y) = ∫_y^s p(ξ) (ξ − y) dξ − Σ_{yj > y} n W_j (y_j − y),

so that the shear at a mass station is the value just inboard of the mass (the table gives the value just outboard in parentheses) and the moment is continuous. The torque is taken about the elastic axis with the running torque t = (l_a + l_b)(x_ea − x_ac) c + q c² c_m0 − n w (x_ea − x_cg) c, nose up positive: lift ahead of the elastic axis twists the nose up, a negative c_m0 (cambered sections) twists it down, and a weight aft of the axis twists it up; a mass adds −n W_j (x_ea − x_j) c(y_j). Each integral is evaluated panel by panel with six-point Gauss–Legendre quadrature on a grid of more than 200 panels that contains every planform corner, mass station, fuel limit and output station as a node; the integrands are polynomials except for the square root of the elliptical chord at the tip, for which the last panel is refined geometrically, and the result is converged to about one part in 10⁹ (the test script checks the trapezoid, rectangular and triangular root moments against their closed forms and the whole station table against an adaptive scipy integration). The elliptical reference L b/(3π) and the ratio of Schrenk's additional-lift moment to it are printed with the root values.

Swept elastic axis. The loads act at streamwise sections while the structure runs along a straight elastic axis swept Λ aft. Resolving the moment of a load p at (x_ξ, ξ) about the elastic-axis point at station y into components along the axis and perpendicular to it, the terms in (ξ − y) tan Λ cancel in the axial component, and the structural pair is T_ea = T₀ cos Λ for the torque about the elastic axis and M_n = M / cos Λ − T₀ sin Λ for the bending moment about the axis perpendicular to it, where M and T₀ are the unswept moment and torque computed with the streamwise arms. For Λ = 0 the two pairs coincide. The sweep enters nowhere else: Schrenk's method is a straight-wing method, and the lift distribution of a swept wing (which moves outboard with sweepback) is outside its scope.

Limits

The approximation is for plain monoplane wings of moderate to high aspect ratio without sweep: Schrenk's comparisons run from an aspect ratio of 5 to 10. It gives the spanwise distribution only, not the chordwise pressure distribution, the induced drag or the downwash; part-span flaps and ailerons are handled by his hand-rounding of the twist step and are not offered here; the lift distribution of a swept wing, a wing with a fuselage carry-over or nacelles, dihedral and chordwise (drag) loads, and the aeroelastic redistribution of a flexible wing are not represented. The basic lift of the twist is the least accurate part of the method and should be checked by a lifting-line or panel calculation when the twist is large. The resulting V, M and T are limit loads for the load factor entered; the wing-box stresses and margins follow from the Beam Stress and Box Beam Shear Flow tools with the section properties of the station.

Reading the results

The planform section prints the area, aspect ratio, taper, mean chord and mean aerodynamic chord with the planform figure. Schrenk's section gives the total lift, the dynamic pressure and the lift coefficient, the elliptical root chord, the c c_l and the running lift at the root, the centroid of the additional lift, the mean twist and the root basic lift when a twist is entered, and κ with the corrected lift-curve slope and the angle of attack; Figure 2 draws the chord curves and the c c_l distributions. The inertia section lists what was entered and Figure 3 the running lift, inertia and net loads. The loads section gives the root shear, the parts of the root bending moment (additional lift, twist, structure, fuel, masses) and the root torque (lift arm, pitching moment, inertia), the swept-axis pair when a sweep is entered, the elliptical reference and the extremes along the span, then the station table and the shear, moment and torque diagrams.