Curved Panel Buckling: Method

Curved Panel Buckling: method, conventions and limits

What the tool does

The skin of a fuselage or a wing is a set of curved panels, each bounded by two stringers (along the axis) and two frames or ribs (around the circumference). Under axial compression or shear a panel buckles elastically at a stress that lies between the flat-plate value and the value of the complete cylinder of the same radius and thickness. The tool computes that stress for one panel by the method of the NACA Handbook of Structural Stability: Part III (Gerard and Becker 1957, NACA TN 3783) for the effect of curvature and Part I (NACA TN 3781) for the flat-plate limit and the plasticity-reduction factors. It reports the compression and shear allowables, the increase from internal pressure, the interaction of compression and shear and a margin of safety, and it draws the design curves with the panel marked on them.

Geometry and the curvature parameter

The panel has the circumferential width b (the arc between the stringers), the axial length a (between the frames), the radius r of the skin mid-surface and the thickness t. The compression acts along the generators, so the loaded edges are the frame lines and the stringers are the unloaded edges. Everything is governed by the curvature parameter of TN 3783 eq. (65),

Z_b = (b² / (r t)) √(1 − ν²),

which is the plate's form of Batdorf's Z for cylinders. Z_b = 0 is the flat plate; large Z_b is a plate so wide or so sharply curved that it behaves as a cylinder. The report's Figure 4 shows the ranges: narrow, transition and wide plates, each of which may be short, of transition length or long. The buckling stress is written in the plate form σ_cr = k π² E / (12 (1 − ν²)) (t/b)², with the whole effect of curvature in the coefficient k.

Flat-plate limit (NACA TN 3781)

At Z_b → 0 the coefficient is the flat-plate value of TN 3781 Figure 14. The tool computes it rather than reading the chart: for unloaded edges simply supported k_c = (m b/a + a/(m b))², minimised over the number of half waves m (4.00 for a long plate); for clamped unloaded edges it solves the Lévy characteristic equation of the plate strip, p₁ sinh(p₁/2) cos(p₂/2) + p₂ cosh(p₁/2) sin(p₂/2) = 0 with p₁² = g² + π g √k, p₂² = π g √k − g² and g = m π b / a, again minimised over m (6.98 for a long plate, 7.69 for a square one). The loaded edges are taken as simply supported in both cases, as in TN 3783's shear curves; TN 3781 notes that their restraint matters little beyond a/b = 4. In shear the flat value is Timoshenko's form of Figure 22, k_s = 5.34 + 4 (b/a)² for simply supported and 8.98 + 5.6 (b/a)² for clamped edges, with b the shorter side.

Compression of curved plates (TN 3783, pp. 53–59 and Appendix A)

Linear theory (Redshaw, Stowell, Batdorf) predicts that a curved plate's coefficient rises from the flat value along a parabola and becomes tangent to the classical cylinder line k_c = 4√3 Z_b/π² = 0.702 Z_b, which is σ_cr = 0.6 E t / r. Tests on cylinders and curved plates fall far below that line, because thin shells are sensitive to initial imperfections. Batdorf, Schildcrout and Stein (NACA TN 1343) replaced the 0.6 by an empirical coefficient C that falls with r/t, and TN 3783 Figure 7 plots that curve together with the Donnell–Wan imperfection theory for three values of the unevenness factor U; the curve for U = 0.00025 merges with the empirical one. The report then fits the curved-plate test data (Crate and Levin, Cox and Clenshaw, Jackson and Hall, Schuette) with the cylinder transition curve of its eq. (28),

k_c = k_pl + (0.581 C Z_b)² / k_pl,

a parabola through the flat value k_pl that is tangent, at Z_t = k_pl / (0.581 C), to the straight line k_c = 1.162 C Z_b; that line is σ_cr = C E t / r, the knocked-down cylinder stress of eq. (30) and (67). The tool follows the parabola up to Z_t and the line beyond it, and reads C from a digitized copy of Figure 7 (the Donnell–Wan U = 0.00025 curve below r/t = 450 and the empirical curve from 500 on; below r/t = 11 the curves lie inside the heavy 0.6 line of the figure and C is held at the last traced value, 0.55; above r/t = 3000 the last slope is extrapolated and the result is flagged). The report built its Figure 38(b) with k_pl = 5.7, the average of the simply supported and clamped values, because the test edges were in between; the tool uses the value of the edge condition chosen, as the report says the scheme "is perfectly general and applies to any set of edge restraints". The report also found that the data in the transition range lie above the curve, with peaks at Z_b of about 5, 50 and 500, but could not reduce them to a design rule; the tool stays on the curve, which is conservative there.

Three reference values are printed beside the result: the knocked-down cylinder stress C E t / r that the panel approaches when b²/(r t) exceeds about 100 (the appendix's criterion), the classical 0.6 E t / r, and the lower bound of NASA SP-8007, γ · 0.605 E t / r with γ = 1 − 0.901 (1 − e^−φ), φ = √(r/t)/16, which is the modern counterpart of C and lies 10–15 % below the TN 3783 curve at r/t near 1000.

Shear of curved plates (TN 3783, pp. 65–69)

In shear the linear theory of Batdorf, Stein and Schildcrout (NACA TN 1346–1348, solutions of Donnell's equation with and without clamping) agrees with the tests, so the report recommends its curves directly. Figure 49 gives k_s against Z_b for long plates (the long side along the generators, (a) simply supported and (b) clamped) and for wide plates (the long side along the arc, (c) and (d)), for a/b = 1, 1.5, 2, 3 and the infinite plate, or 1, 1.5, 2 and the complete cylinder; b is the shorter side in every case. The tool holds a digitized copy of the four panels (61 points per curve from Z_b = 1 to 1000, overlays of the traces in the project's References folder) and interpolates log k_s along the curves and linearly in b/a between them; the curve is used as the ratio k_s(Z_b)/k_s(Z_b → 1) multiplying the TN 3781 flat value, so the flat limit is exact and the curvature increase is the report's. For long plates with Z_b > 30 the report gives the closed form of eq. (87), τ_cr = 0.37 √Z_b τ_cr,flat, valid for simply supported and clamped edges alike; it is printed for comparison. Beyond Z_b = 1000 the last slope of the curve is extrapolated and the result is flagged.

Plasticity

When the elastic buckling stress approaches the yield strength the modulus in the buckling equation must be reduced. With the compressive yield strength and the Ramberg–Osgood exponent n entered (ε = σ/E + 0.002 (σ/F_cy)ⁿ), the tool solves σ_cr = η(σ_cr) σ_e by bisection. The factors are those of the handbook: TN 3781 eq. (A3) for a long simply supported plate, η = (E_s/E) (1 − ν_e²)/(1 − ν²) [0.500 + 0.250 √(1 + 3 E_t/E_s)], and (A4) for a clamped one (0.352 + 0.324 √(…)); TN 3783 eq. (68) for the long curved plate behaving as a cylinder, η = (E_s/E) √[(1 − ν_e²)/(1 − ν²) E_t/E_s]; and TN 3781 eq. (A5), the secant-modulus factor η = (E_s/E)(1 − ν_e²)/(1 − ν²), for shear, with the curve entered at the equivalent compressive stress √3 τ. The plastic Poisson's ratio is ν = 0.5 − (0.5 − ν_e) E_s/E (TN 3781 eq. A1). The handbook states the plate factor for b²/(r t) ≤ 1 and the cylinder factor for b²/(r t) ≥ 100 but gives nothing between; the tool weights the two linearly in log(b²/(r t)) across that range, a choice of its own that is stated on the page. The allowable is cut off at F_cy (F_cy/√3 in shear). With F_cy and no exponent only the cutoff is applied; with neither, the results are elastic.

Internal pressure

Pressure on the concave face (cabin pressure) raises the buckling stress of a curved panel. TN 3783 correlates the tests of Rafel and Sandlin with R_c² + R_p = 1 for compression (eq. 70, Appendix eq. A3) and R_s² + R_p = 1 for shear (eq. 88), where R_p is the ratio of the applied internal pressure to the external pressure that would buckle the cylinder of which the panel is a section. An internal pressure is a negative external pressure, so R_p is negative and the allowable grows by √(1 + p / p_cr,ext). The tool takes that cylinder as the one of radius r between the frames (length a) under lateral pressure and computes its buckling pressure from Batdorf's coefficient, N_y = k_y π² D / a² with k_y = min over β of [(1 + β²)² + 12 Z_L²/(π⁴ (1 + β²)²)] / β² (NACA TN 1341, the theoretical curve of TN 3783 Figure 35(a), which the report finds confirmed by the external-pressure tests). The factor is applied to the elastic stresses before the plasticity correction, and the pressure is never multiplied by the ultimate factor.

Combined loads and the margin of safety

For a curved plate under shear and axial compression TN 3783 recommends the parabola of Batdorf, Schildcrout and Stein, eq. (89), R_s² + R_c = 1, with the empirical compression allowable in R_c; Figure 51 confirms it with tests over 10 ≤ Z_b ≤ 100 and 1 ≤ a/b ≤ 3, and the page says when a panel lies outside that range. The ratios are the applied stresses over the allowables of each load alone. The load factor λ to buckling under proportional loading solves λ R_c + λ² R_s² = 1, λ = [−R_c + √(R_c² + 4 R_s²)] / (2 R_s²), and the margin of safety is λ − 1. Loads entered at limit level are multiplied by the ultimate factor first. Buckling of a skin panel is a stiffness and appearance criterion rather than a strength failure: the buckled skin carries shear in diagonal tension and the stringers carry compression beyond it (Part IV of the handbook and the Diagonal Tension and Effective Sheet Width tools), so the margin here says whether the skin stays unbuckled at the chosen load level.

Limits

The method covers isotropic sheet panels with r/t ≥ 10 and the two edge conditions of Figure 49; intermediate rotational restraint is not offered (TN 3783 suggests TN 3781 Figure 23(a) for shear). It does not cover external pressure on the panel, bending across the panel, the transition-range magnification peaks, post-buckling strength, cutouts, stiffened or sandwich skins, or panels subtending more than half the circumference, which should be treated as cylinders. The compression knockdown C is read from a chart (about ±0.005 in C); the shear curves are chart readings of about 2–3 %.

How to read the results

The page prints each equation with its numbers and the report's equation number: the curvature parameter and the flat-plate coefficient, the C of Figure 7, the transition or cylinder-line coefficient k_c and the elastic stress, the cylinder, classical and SP-8007 reference stresses, the shear coefficient from Figure 49 with the eq. (87) line, the pressure factor and the plasticity factor when they apply, the applied stresses, the ratios and the margin. The summary table lists the flat, curved, pressurized and allowable stresses beside the applied ones. Figure 2 shows where the panel sits between the flat value and the cylinder line, Figure 3 its position on the shear curves, and Figure 4 the load point against the interaction parabola, with the proportional load path to buckling.