Shear Lag in Box Beams

Shear Lag Method

Shear Lag in Box Beams: Kuhn's Substitute Single-Stringer Method

What shear lag is

The engineering theory of bending assumes that plane sections remain plane, so that every stringer of a flat box-beam cover at the same distance from the neutral axis carries the same stress M y / I. In a real stringer-sheet cover the bending load enters at the corner flanges (the spar caps), where the shear webs deliver it, and reaches the stringers only through the shear deformation of the cover sheet. Near a built-in root, or near any station where the shear flow changes abruptly, the sheet has not had the length to transfer the load: the flanges carry more than their M y / I share and the stringers near the centre line carry less. This departure from the elementary theory is the shear-lag problem. Paul Kuhn's work at NACA between 1937 and 1948 (Reports 608, 636 and 739, Technical Note 739, ARR L-401, TN 1728) produced the practical method reproduced by this tool; the definitive statement is NACA Report 739 (Kuhn and Chiarito, 1942), which also reports the strain-gage tests that verified it.

Idealization

The structure is symmetric about its longitudinal centre line, so one half of one cover is analysed. Stringers are concentrated at their centroids and carry only longitudinal stress; the sheet carries only shear, its own longitudinal capacity being added to the stringers as an effective width. The shear web's share of the bending is added to the corner flange as an area h t_w / 6, which makes the idealized section modulus equal to the actual one. Infinitely many ribs of infinite chordwise stiffness are assumed; the report's tests showed the finite rib stiffness of real beams to have a small influence. The compression and tension covers are analysed independently, each with its own effective areas.

The half cover is then reduced to a single-stringer structure: the corner flange A_F, a substitute single stringer of area A_L = Σ (stringer areas) placed at the force centroid of the stringers, a distance b_S from the flange, and the sheet of thickness t between them. The skin strip between the flange rivet line and the first stringer is treated as a small stringer sitting at the flange, a rule that covers the limiting case of a cover without stiffeners and halves the error of the finite stringer count.

Fundamental relations and the shear-lag parameter

Two statements of equilibrium and one of elasticity govern the single-stringer beam of Report 739 (equations 1a to 1c): the flange force changes by the web shear S_W/h minus the sheet shear flow, the stringer force changes by the sheet shear flow, and the sheet shear stress changes along the span in proportion to the stress difference between flange and stringer, dτ = −(G / (E b)) (σ_F − σ_L) dx. Combined, they give a second-order equation whose characteristic length is set by the shear-lag parameter

K² = (G t / (E b)) (1 / A_F + 1 / A_L)     (eq. 4)

A large K (stiff sheet, slender flanges) means the stringers pick up their share within a short distance and the elementary theory holds almost everywhere; a small K (thin sheet, heavy flanges, wide cover) spreads the transfer over a long length. G is the effective shear modulus of the sheet, reduced when the sheet has buckled.

The recurrence formula

For a beam whose cross section and web shear vary along the span, Kuhn divides it into bays within which both are taken constant and writes the forces in two groups: the P-forces of the elementary theory, which are in equilibrium by themselves, and the X-forces that express the shear-lag correction. At any station the flange gains X and the stringers lose X, so the X-forces are self-equilibrated. Equating the shear deformation of the sheet at the common end of two adjoining bays gives a three-moment-type recurrence formula (equation 2),

X_n−1 q_n − X_n (p_n + p_n+1) + X_n+1 q_n+1 = −γ_n + γ_n+1

with p = K / (G t tanh KL), q = K / (G t sinh KL) and γ = S_W A_L / (h A_T G t) for each bay (L the bay length). The common factor G t may be dropped. At a free tip X₀ = 0. At the root three cases arise: (1) a rigid foundation, the plane of symmetry of a wing continuous from tip to tip, where no shear deformation occurs inboard of the root (p_r+1 = γ_r+1 = 0); (2) carry-through members passing through the fuselage, whose extensional flexibility over the length L_c adds p_r+1 = (1 / A_F + 1 / A_L) L_c / (b E); (3) stringers not connected at the root, where X_r = (M / h) A_L / A_T is known and the last equation is dropped. The stresses follow from equations (9a) and (9b): σ_F = σ^P + X / A_F and σ_L = σ^P − X / A_L, with σ^P = M / (h A_T) the elementary stress of a flat cover. The running shear in the sheet next to the flange at the two ends of a bay follows from equations (11a) and (11b), and its bay average is the statics value (F_L,n − F_L,n−1) / L.

Substitute width and successive approximations

The substitute stringer sits at the force centroid of the stringers, which is unknown until the chordwise distribution of stress is known. In the first approximation the My/I stress is uniform across the stringers and the centroid is the area centroid b_S1. Because the stresses actually fall from the flange toward the centre line, the force centroid moves toward the flange: b_S = 2 b_S1 (1 − y_L/b) (equation 17a), where y_L is the distance of the centroid of a cosh(Yy) distribution from the centre line. The report reads 1 − y_L/b from its figure 15; the closed form, used here, is tanh(Yb/2) / (Yb), which starts at ½ for Yb = 0 and tends to 1/(Yb). Yb for each station comes from the ratio σ_L/σ_F of the previous approximation (equation 15), and its spanwise average fixes the new substitute width. Rather than recomputing b_S, the report divides every K by √(2 (1 − y_L/b)); the tool reports this K factor after each approximation. Report 739 states that the second approximation may be taken as final when the shear-lag parameter KL over the whole beam exceeds about 4 in the first approximation, and the first when KL exceeds about 7; its own example stops at the second approximation when the K factor has moved by 1 %.

Chordwise distribution of the stringer stresses

The analysis of the substitute beam gives the flange stress exactly (within the method) but only the average stress of the stringers. Solutions for the limiting case of infinitely many stringers show that the stringer stress across the cover follows a hyperbolic-cosine law, σ = σ_CL cosh(Yy), y measured from the centre line (equation 14). Two conditions fix the unknowns: the average over the half cover equals σ_L, and the stress at y = b equals σ_F, which gives tanh(Yb) / (Yb) = σ_L/σ_F (equation 15, the report's figure 13) and σ_CL = σ_F / cosh(Yb) (equation 16). The forms apply for 0 < σ_L/σ_F < 1, the range of every design-critical region; outside it the report proposes σ = σ_CL (2 − cosh Yy) with the corresponding equations (15a) and (16a). With a finite number of stringers the summed stringer forces do not balance the external force; the report's practice (figure 14, Table VI) is to leave the skin strip at the flange at σ_F and to multiply the other stringer stresses by the ratio of the force required to the sum of the forces computed, so that statics is restored. NACA ARR L-401 (Kuhn 1943) offers a simpler cubic form of the correction, Δσ = X/A_F − D [1 − (y/b)³] with D = (4/3) X (1/A_F + 1/A_L), whose average over the width is −X/A_L exactly; the tool prints it beside the cosh values. It breaks down near stations where the stringers stop, where the stresses are small anyway.

Constant-section closed forms

For a cantilever of constant cross section the recurrence formula has a closed solution, Report 608 Appendix B case 3(a): with x measured from the tip, under a tip load P the sheet shear is τ = P (1 − cosh Kx / cosh KL) / (t h (1 + A_F/A_L)) and the stringer stress σ_L = P (x − sinh Kx / (K cosh KL)) / (h (A_L + A_F)), which is the X-force X = (γ/K) sinh(Kx) / cosh(KL) with γ = P A_L / (h A_T); at the root X = (γ/K) tanh KL. Under a uniform running load w the same equation X″ − K² X = (A_L/(h A_T)) M″ with τ = 0 at the root and X = 0 at the tip gives X_root = (c/K²) [(1 + KL sinh KL)/cosh KL − 1], c = w A_L / (h A_T). The tool evaluates both with the root-bay properties and prints them against the recurrence result: they agree closely for a slowly varying section because tanh KL is near 1 and the root value is set by the local properties. As K → ∞ the X-forces vanish and M y / I is recovered; as K → 0 the flange carries the whole M / h. Report 739's Appendix B shows that the earlier constant-section solutions of Younger (1930), Kuhn (1937) and Reissner (1938) all reduce to a constant flange efficiency η = tanh F / F; the recurrence formula was preferred because the measured flange stress of a real beam is not a straight line along the span.

How to read the results

Table 1 lists the station data and the elementary stress; Table 2 follows the substitute-width approximations; Table 3 gives the recurrence coefficients of the final approximation; Table 4 the X-forces, the flange stress, the average stringer stress, their ratio and Yb at every station, with the flange efficiency η = σ^P/σ_F. The root lines give the shear-lag increase of the flange stress and the deficiency of the centre stringer, the two numbers a stress analyst takes from a shear-lag analysis. Table 5 works the chordwise distribution at the chosen station in the report's Table VI form, Table 6 lists the corrected stringer stresses at every station (Table VII), and Table 7 the running shear in the cover sheet with the panel shear flows by statics. Figure 1 draws the idealized half cover, Figure 2 the sheet shear along the span, Figure 3 the stresses along the span and Figure 4 the chordwise distribution. The margins of safety close the page.

Margins of safety and the load-level convention

Three places are checked: the corner flange at the station of largest shear-lag stress (the root for a cantilever), the most-loaded stringer with its corrected chordwise stress at any station, and the most-loaded sheet panel, the largest of the running shear next to the flange at either end of a bay (equations 11a and 11b) and of the panel flows by statics, as a shear stress q / t. The Cover select says which cover the entered areas describe. For the tension cover the flange and stringer stresses are compared with F_tu (ultimate) and F_ty (yield); for the compression cover with the crippling stress F_cc of the section (ultimate, from the Crippling tool; left blank, F_cy stands in and the page says so, which is right only for a stable section) and with F_cy (yield). The sheet shear is compared with F_su (ultimate) and with a shear yield allowable F_sy = (F_ty / F_tu) F_su, derived in the ratio of the tensile yield to the tensile ultimate strength because the handbooks list no shear yield strength.

The Load Level radio says what the entered loads P and w are, ultimate or limit. Strength margins (F_tu, F_cc, F_su) are written at ultimate load and labelled ULT: with limit loads entered the stress is multiplied by the factor of safety FS (default 1.5), with ultimate loads entered it is taken as entered and never multiplied again. Yield margins (F_ty, F_cy, F_sy) are written at limit load times the yield factor FS_yld (default 1.15, the factor several certification agencies require on limit load for no detrimental permanent set) and labelled LIM: with limit loads entered the stress is multiplied by FS_yld, with ultimate loads entered it is divided by FS to recover the limit stress and then multiplied by FS_yld. FS therefore stays active whatever the radio says. Each box names the level of its allowable, not of the entered loads. The running shear next to the flange, which moves the shear-lag load between flange and stringers (and rises above the elementary value where the stringers are not connected at the root or the flange load changes quickly), must also be carried by the fastener row between sheet and flange at the station of largest flow (fastener shear, sheet bearing, net shear of the sheet, Fastened Joint Strength), and a thin sheet that buckles in shear carries the load in diagonal tension, which the shear-resistant analysis here does not represent (Diagonal Tension, Shear-Resistant Web).

The Report 739 example

The default inputs are the beam of Report 739 Figure 50, an idealization of the report's test beam 1: 600 lb at the tip of a 48 in cantilever in six 8 in bays, half-width 10.875 in with three stringer pitches, G/E = 0.40, sheet 0.015 in, depth 6.18 in, flange area 0.229 to 0.392 in², stringers 0.194 to 0.318 in², skin strip 0.018 to 0.030 in². The report's Tables I to VII are reproduced: first approximation X at the root 1054 lb and σ_F 6525 psi, second approximation 987 lb and 6355 psi, station 5 stringer stresses 5000 / 3320 / 2203 / 1873 psi. The tool's values differ from the printed ones by up to 2 to 3 % because the report worked with a 10-inch slide rule, read Yb from its figure 13, and solved its second approximation with a miscopied coefficient (p₄ = 0.1336 for 0.1386 in Table IV, which the test script confirms by reproducing the printed X-forces from the printed coefficients).

Limits

Not covered: taper in depth (part of the shear is carried by the inclined flanges, equation 12 of the report), taper in width (the report replaces the beam by an untapered one with the sheet thickness increased in the ratio b_r/b), cambered covers (equation 1c′), cut-outs (the liquidating-force method of part I), chordwise variation of stringer size or sheet thickness (the method of successive shear-fault reduction), a concentrated load away from the tip (treat as two problems as Report 608 figure 23 shows, or divide the bays so that the load falls on a station and enter the shear by bay in a later version), and the effective width of buckled compression sheet, which must be entered through the stringer areas and an effective shear modulus. The method assumes a flat cover with a line of symmetry, uniform stringer spacing across the half cover and areas varying linearly along the span.