Stiffened Panel Compression: Method

Stiffened Panel Compression: Crippling, Effective Width, Johnson–Euler, Inter-rivet Buckling and Wrinkling

What the tool does

A compression cover of an aircraft wing or fuselage is a thin sheet stiffened by riveted stringers and supported by frames or ribs. Loaded along the stringers, such a panel fails in one of several modes, and the design allowable is the lowest of them. The tool evaluates, for one stringer pitch b_s of a flat panel between frames, (1) the crippling stress of the stringer alone, (2) the stringer with its effective sheet as a column between frames, (3) buckling of the sheet between rivets, (4) wrinkling (forced crippling) of the sheet, which drags the stringer attachment flange along and cripples the stringer, and (5) the crippling strength of the sheet-stringer element as if it were monolithic. The loads of the modes are compared per stringer pitch and the lowest gives the allowable running load and the margin of safety.

Stringer crippling (Gerard)

Crippling is the failure of a thin-walled compression member after local buckling of its flanges. Gerard correlated the crippling tests of hundreds of sections in the NACA Handbook of Structural Stability (Part IV, TN 3784; Part V, TN 3785) with one form of equation. For formed Z- and channel sections (TN 3784 eq. 61) the correlation is F_cc/F_cy = β [(t²/A)(E/F_cy)^1/3]^0.75 with β = 4.00 for 2017-T3 and 2024-T3 formed to a bend radius of about 3t, 3.78 for FS-1h magnesium and 3.42 for alclad 7075-T6, whose corners strain-harden less. Extruded Z- and channel sections follow the same equation with β = 2.14 α (b_w/t)^0.25 and α = 0.80 (eq. 63). For angles (eq. 54) and for extruded sections made of T-type elements Gerard uses the generalized form F_cc/F_cy = β_g [(g t²/A)(E/F_cy)^1/2]^m, where g is the number of flanges plus the number of cuts needed to divide the section into angles: an angle counts 2, a tee 3, a cruciform 4, an H 7, a square tube 12, a formed multicorner section 3c − 1 with c corners. The constants are β_g = 0.558 and m = 0.85 for angle-type elements (0.665 for formed angles, 0.630 for 7075-type alloys, 0.55 for formed multicorner sections) and β_g = 0.670 and m = 0.40 for T-type elements (TN 3785 Table 4), valid to F_cc/F_cy = 3/4. The tool caps every crippling stress at F_cy and notes when the handbook's own validity limit is exceeded. An entered crippling stress (from a test, as NACA Report 1255 did in its example) replaces Gerard's value in the rest of the analysis.

Effective sheet and the Johnson–Euler column

After the sheet between stringers buckles, the strip next to each stringer still carries the stringer stress. Von Kármán, Sechler and Donnell (1932) wrote this strip as 2w = 1.9 t_s √(E/σ), half on each side of the stringer; the tool evaluates it at the stringer crippling stress and caps it at the stringer pitch. The stringer plus its effective sheet is the column: area, centroid, moment of inertia and radius of gyration ρ follow, and the effective length is L′ = L/√c with the end-fixity coefficient c of the frame attachment. The column allowable is the Johnson parabola F_c = F_cc − F_cc² (L′/ρ)²/(4π²E), which meets the Euler hyperbola π²E/(L′/ρ)² tangentially at L′/ρ = π √(2E/F_cc) where the stress is F_cc/2; beyond that point the Euler stress governs. TN 3785 writes the equivalent transition parabola from the local buckling stress (its eq. 28 and 31). Because the effective width is held at F_cc, the load of a long panel, whose stress is lower and whose effective strip would be wider, is slightly conservative.

Inter-rivet buckling

If the rivets are far apart the sheet buckles between them as a wide column of length p whose ends are restrained by the rivet heads (TN 3785 eq. 10): F_ir = c π² η E/(12(1 − ν²)) (t_s/p)². The end-fixity coefficients of TN 3785 Table 5 are 4 for flat-head rivets (Howland), 3.5 for spot welds, 3 for brazier-head rivets and 1 for countersunk rivets, which the handbook also suggests for dimpled rivets. The plasticity factor η is the tangent-modulus ratio of the Ramberg–Osgood curve ε = σ/E + 0.002 (σ/F_cy)ⁿ, and the stress is capped at F_cy. The sheet carries no load beyond this stress, so the tool treats it as a cutoff on the column curve.

Wrinkling (NACA Report 1255)

With moderate rivet spacing the sheet does not separate from the stringer but buckles in a long wave that bends the stringer attachment flange with it; the buckles grow into a cylindrical shape and the stringer, forced to follow, cripples. Semonian and Peterson analysed the sheet at failure as a column on an elastic foundation whose stiffness comes from the attachment flange acting as a cantilever of length f, the effective rivet offset: the distance from the stringer web to the line along which the rivets effectively clamp the flange. They evaluated f from panel tests and gave it in their Figure 8 as a function of the rivet pitch-to-diameter ratio p/d and the geometric offset ratio b_o/t_w; the tool carries that chart digitized from the NACA scan and calibrated on its grid lines (the report's own reading of 6.98 at p/d = 10.7 and b_o/t_w = 5 is reproduced as 6.94). With the rotational stiffness of the flange root fixed at α b_w/D_w = 3, found by trial against the tests, the support stiffness is ψ b_s³/(π⁴ D_s) = (12/π⁴)(3 f/b_w + 1)/[(f β/b_w)³ (3 f/b_w + 4)] with β = (b_w/t_w)/(b_s/t_s), the maximum-stress coefficient after minimizing over the buckle length is k_M = 2 √(ψ b_s³/(π⁴ D_s)) (eq. 6 and 7, the report's Figure 9), and the plate stress is σ_M/η = k_M π² E/(12(1 − ν²)) (t_s/b_s)², which the report also writes in closed form as its eq. (12). The plasticity factor is Stowell's for a long simply supported plate, η = (E_sec/E)[½ + ½ √(¼ + ¾ E_tan/E_sec)] (eq. 4), evaluated on the Ramberg–Osgood curve of the sheet material; the tool solves σ_M such that σ_M/η(σ_M) equals the elastic value. If the stringer crippling stress is above σ_M the stringers are stable and the panel fails at σ_M; otherwise the stringers take only their crippling stress and the panel average is (σ_M b_s t_s + F_cc A_st)/(b_s t_s + A_st) (TN 3785 eq. 16 and 17). Two rivet criteria go with the analysis: the pitch must not exceed 0.9 of the wrinkle half-wave length λ = b_s √(2/k_M) (eq. 8 and 9), or the sheet buckles between rivets first, and each rivet must carry the lateral force R_R = E/(1 − ν²) (t_w/f)³ (3 f/t_w + b_w/t_w)/(3 f/t_w + 4 b_w/t_w) (t_s/5) p (eq. 11, a lateral deflection of one fifth of the sheet thickness at maximum load) in tension, by shank strength or head pull-through.

The chart was established from 2024-T3 and 7075-T6 panels with 2117-T3 rivets, p/d from 3 to 15, rivet diameters of at least 0.9 t_s, and formed Z, channel and hat stringers. For extruded stringers with sharp corners it gives a conservative estimate; for rivets of lower modulus or strength than 2117-T3 aluminum it is unconservative. Gerard found the method optimistic for materials whose E/F_cy is far from the 125 of the aluminum alloys and gives a scaled form (TN 3785 eq. 23).

Monolithic panel crippling (TN 3785)

When the rivets are close and strong enough, the sheet-stringer element cripples like a formed multicorner section and further changes in riveting gain nothing. Gerard's correlation for formed Z-stiffened panels counts c = 3 corners for the element (two for the Z plus the rivet line), replaces c t²/A by c t_w t_s/A with A the stringer area plus b_s t_s, and gives σ_f/F_cy = 1.26 [(c t_w t_s/A)(E/F_cy)^1/2]^0.85 (eq. 1 and 25). Hat-stiffened panels take c = 6 with β_c varying from 1.16 to 1.42 with t_w/t_s and a cutoff at 0.8 F_cy because of suspected wrinkling in the tests. The report uses this value as the ceiling of the wrinkling analysis; the tool lists it as a mode of its own.

Governing mode and the chart

The column and inter-rivet modes act on the stringer plus effective sheet; the wrinkling and monolithic modes are averages over the stringer plus the full sheet. The tool therefore compares loads per stringer pitch, divides the lowest by b_s to obtain the allowable running load, and forms the margin of safety against the applied running load at ultimate (a limit load is multiplied by the factor of safety). The chart draws the Johnson parabola and the Euler curve against L′/ρ with the inter-rivet, wrinkling and monolithic strengths as horizontal lines converted to the column-section basis, the allowable at the panel's slenderness and the applied stress; a second chart shows the digitized effective-rivet-offset curves with the panel's point.

Default example

The default inputs are the numerical example of NACA Report 1255: a 2024-T3 Z-stiffened panel with t_w = 0.064 in, t_s = 0.102 in, b_w/t_w = 40, b_s/t_s = 30, b_A/t_w = 8, b_F/b_w = 0.4, b_o/t_w = 5, 3/32-inch brazier-head 2117-T3 rivets at 1 inch, E = 10,600 ksi, ν = 0.32, F_cy = 43.6 ksi, and the stringer crippling stress of 27.5 ksi the report took from tests. The report's values β = 1.33, p/d = 10.7, A_z = 0.262 in², b_st_s = 0.312 in², f/t_w = 6.98, k_M = 3.64, σ_M/η = 39.2 ksi, σ_M = 34.1 ksi, panel stress 31.1 ksi, allowable pitch 2.04 in and required rivet strength 0.239 kips are reproduced within one percent; with a frame spacing of 19.6 in (L/ρ = 20, the slenderness of the report's tests) the Johnson–Euler column from the test crippling stress governs the allowable running load because the column method credits the sheet only with the stringer stress, whereas the wrinkling analysis credits it with σ_M.

Limits

Flat panels with at least four stringers and simply supported or elastically restrained ends; stringer flanges with b_F about 0.4 b_w (a narrower outstanding flange brings in a torsional-local mode the analysis does not cover); the initial buckling stress of the panel (Report 1255 Figure 6, TN 1482) is not computed; curved panels, bonded or integrally stiffened panels, and tension or shear loads are outside the scope. The centre-line section model ignores bend radii; a section drawn in the Section Properties tool can be loaded instead for its exact properties.