Effective Sheet Width

Effective Sheet Width

Effective Sheet Width

What the effective width is

A sheet riveted to stringers and loaded in compression along the stringers buckles between them at a stress far below the strength of the stringers: a 0.040 in aluminium sheet on stringers 4 in apart buckles at about 4 ksi, while the stringers may carry 35 ksi or more. After buckling the sheet does not fail. The bowed middle of each bay sheds load, but the strips along the stringers are held straight by the stringers and keep following the stringer stress. The compressive stress across a bay is therefore highest at the stringers and lowest at mid-bay, and the sheet still carries a load that grows, slowly, with the stringer stress.

Von Kármán, Sechler and Donnell (1932) replaced the real stress distribution by two strips of width w, one on each side of the stringer, carrying the stringer stress uniformly and the same total load as the real sheet. The total width 2w is the effective width b_e. Its use is simple: the area 2w t is added to the stringer section, and the stringer with its effective sheet is then checked as a column or for crippling at the stringer stress. The effective width is not a property of the sheet alone; it falls as the stringer stress rises, so it has to be evaluated at the stress of the check.

Sheet buckling stress

The bay between stringers is a long plate of width b (the stringer pitch) and thickness t, loaded along the stringers. Its elastic buckling stress is

σ_cr = k π² E / [12 (1 − ν²)] (t/b)²,

with k = 4.0 when the stringer lines act as simple supports (the usual design assumption for a riveted flange) and k = 6.98 when they clamp the sheet. The tool also accepts an entered coefficient for an elastic restraint between the two. Below σ_cr the sheet is flat and the whole pitch is effective.

Von Kármán's formula

Von Kármán argued that the two effective strips, taken together as a simply supported plate of width b_e, are exactly at their own buckling stress when that stress equals the stringer stress σ: setting σ_cr(b_e) = σ gives

b_e = 2w = C t √(E/σ), C = √(k π² / [12 (1 − ν²)]) = 1.901 for k = 4 and ν = 0.3, printed as 1.9.

The constant is often quoted as 1.7 for riveted aluminium panels (Sechler's tests gave somewhat less than the theory) or 1.9 for the theoretical simply supported strips; the tool uses 1.9, the value of Bruhn's and Niu's design formulas, and shows the exact constant for the entered k and ν beside it. At σ = σ_cr of the bay the formula returns the whole pitch, which is the consistency check of the test script. The formula ignores the stress still carried by the middle of the bay, so it is conservative at moderate stress ratios and close to the large-deflection solutions at high ones.

Marguerre's formulas

Marguerre (1937, translated as NACA TM 833) solved the large-deflection equations of a simply supported plate with an assumed buckle shape and gave the "apparent width" (his term) as two fits to his solution. The first, eq. (7.8) of the memorandum, is

b_e/b = 0.19 + 0.81 √(σ_cr/σ),

which Gerard's Handbook of Structural Stability Part IV reproduces as eq. (5) for a square simply supported plate with straight unloaded edges free to move laterally. The second, eq. (7.9), is the cube-root formula

b_e/b = (σ_cr/σ)^1/3,

which Marguerre recommends for its simplicity and its behaviour at the buckling stress; it runs a little low for σ above about 20 σ_cr. Both give b_e = b at σ = σ_cr. Marguerre's figure 3 compares them with the Lahde–Wagner test points.

Koiter and Argyris–Dunne (Gerard Part IV)

Gerard's Part IV (NACA TN 3784) gives two more results of large-deflection theory. For a long plate with rotationally restrained unloaded edges held straight and free to move laterally (Koiter's analysis), eq. (6):

b_e/b = 1.2 r^2/5 − 0.65 r^4/5 + 0.45 r^6/5, r = σ_cr/σ.

For stress ratios up to 3, Argyris and Dunne's near-buckling relation, eq. (10): b_e/b = r [1 − β + β/r], with β from TN 3784 Table 1; β = 0.500 for a long simply supported plate whose unloaded edges are held straight by stringers of negligible area (A_r/at = 0), rising to 0.746 for very heavy edge members and 0.583 for clamped edges. The tool uses β = 0.500 and shows this relation only inside its range. Gerard also notes that small initial imperfections lower the effective width near the buckling stress (up to about σ/σ_cr = 2) and that plasticity, by Mayers and Budiansky's analysis of 2024-T3 plates, reduces it further for edge strains up to about three times the critical strain.

Winter's formula and the AISI form

Winter's Cornell tests (1947) on cold-formed steel flanges showed von Kármán's formula to be unconservative for real, imperfect plates, and he added a correction term:

b_e = 1.9 t √(E/σ) [1 − 0.475 (t/b) √(E/σ)].

The 1968 AISI specification reduced the coefficient to 0.415, and the specification now writes the rule as ρ = (1 − 0.22/λ)/λ for λ > 0.673, with λ = √(σ/σ_cr) and b_e = ρ b; for k = 4 and ν = 0.3 this is the 1.9 / 0.418 form, and it takes any buckling coefficient. The form gives b_e < b already at σ = 0.45 σ_cr, because it stands for an imperfect plate. The tool prints Winter's 1947 formula with its own constants and the AISI form with the entered k.

Plasticity correction

Aircraft stringers work at stresses near the compressive yield strength of the sheet, where the modulus that governs buckling is no longer E. The tool follows Gerard's handbook: every formula is written with ηE in place of E, η being Stowell's plasticity-reduction factor (NACA Report 898) for a long plate with the chosen edge condition,

η = (E_s/E) [1/2 + 1/2 √(1/4 + 3 E_t/(4 E_s))] (simply supported), η = (E_s/E) [0.352 + 0.648 √(1/4 + 3 E_t/(4 E_s))] (clamped),

with the secant and tangent moduli of the Ramberg-Osgood curve ε = σ/E + 0.002 (σ/F_cy)ⁿ evaluated at the stringer stress. The factor enters the von Kármán width as √η; the ratio formulas see it through σ_cr, which becomes η σ_cr. The buckling stress of the full bay itself is also reported, solved as σ_cr = η(σ_cr) σ_cr,e; for a thin sheet it buckles elastically and the two agree. With the correction off the formulas are elastic.

Inter-rivet buckling

The sheet between fasteners along the attachment line can buckle as a wide column before the stringer stress is reached, especially with widely spaced or countersunk rivets. Gerard's Part V (NACA TN 3785) gives Howland's form, eq. (10),

σ_ir = c π² η E / [12 (1 − ν²)] (t/p)²,

with the end-fixity coefficient c of Table 5: flathead rivet 4, spot weld 3.5, brazier-head rivet 3, countersunk rivet 1 (also suggested for dimpled joints). The plasticity factor of a plate column is E_t/E, so σ_ir cannot rise much above the compressive yield strength whatever the pitch. When σ_ir is below the stringer stress, the skin keeps carrying σ_ir on its width effective at σ_ir but no more (TN 3785 eq. 11); the tool reports that load and the equivalent width at the stringer stress, which is what should be added to the stringer.

Reading the results

The table lists every method's width, its ratio to the pitch and the sheet area it adds. The design value is the von Kármán width, the rule the stiffened-panel methods were calibrated with; the spread of the other values shows the uncertainty of the quantity. The chart plots the effective-width ratio against the stress ratio for all methods with the case marked, and the sketch shows the stress across two bays with the equivalent rectangles. The default case, a 0.040 in 2024-T3 sheet (E_c = 10.7 Msi, F_cy = 39 ksi, n_c = 20 from MIL-HDBK-5J Table 3.2.3.0(b)) on stringers 4 in apart at a stringer stress of 35 ksi, buckles at 3.87 ksi and has an elastic von Kármán width of 1.329 in, reduced to 1.206 in by the plasticity factor η = 0.824.

Limits

The formulas are for flat sheet in uniform compression along the stringers with the bay long compared with its width. Curved sheet has a larger effective width that is semi-empirical (TN 3784 Appendix A); shear or combined loading, bending of the stringer with its sheet, the stiffness-type reduced effective width (dσ̄/dσ_e, TN 3784 eq. 11) and the strength of the stringer itself are outside the tool. Riveted panel tests (Sechler) gave constants somewhat below 1.9; a constant of 1.7 is sometimes used for them.