Cylindrical Shell Buckling: Method

Cylindrical Shell Buckling: Method, Conventions and Limits

What the tool does

The tool estimates the buckling strength of a thin, unstiffened, isotropic circular cylindrical shell under axial compression, bending, external (lateral or hydrostatic) pressure and torsion, alone or combined, and the increase of the compression and bending strengths produced by internal pressure. It follows the recommended practice of NASA SP-8007 (1968), Buckling of Thin-Walled Circular Cylinders, Section 4.2, which is still the basis of preliminary shell design in aerospace: the classical buckling coefficients of Donnell's shell theory, reduced by empirical lower-bound correlation (knockdown) factors, and a linear interaction of the load ratios. The results page prints every equation with its SP-8007 number, the substituted values, the design allowables, the applied stresses, the interaction sum and the margin of safety, and three charts: the correlation factors against r/t, the interaction line with the load point, and the buckling-coefficient curves of the report's Figures 1, 4 and 5 with this cylinder's points.

Why a knockdown factor

Linear shell theory gives the classical buckling stress of a cylinder in axial compression, σ_cl = E t / (r √(3(1 − ν²))) = 0.605 E t/r for ν = 0.3 (Lorenz 1908, Timoshenko 1910, Southwell 1914). Tests from the 1930s on gave as little as a fifth of it. The reason, understood from Donnell, von Kármán and Tsien in the 1930s and 40s and made rigorous by Koiter (1945), is that the axially compressed cylinder has a steeply falling post-buckling path, so geometric imperfections of a fraction of the wall thickness, together with load and boundary non-uniformities, trigger collapse far below the bifurcation load. Because the imperfections of a real structure are unknown at the design stage, the design practice became to multiply the classical value by a correlation factor γ that bounds the test data from below. SP-8007 collected the curves that had been fitted at that time (Weingarten, Morgan and Seide 1965 for compression and for internal pressure; Seide, Weingarten and Morgan 1960 for bending) and states them as equations in r/t.

The curvature parameter and the buckling coefficients

Batdorf (NACA TN 1341, 1947) showed that Donnell's equation for a simply supported cylinder collapses onto one parameter, Z = L² √(1 − ν²) / (r t), so that every buckling load can be written as a plate-type formula N = k π² D / L², D = E t³ / (12(1 − ν²)), with the coefficient k a function of Z alone (and of r/t only for very long cylinders that buckle into two circumferential waves). The tool computes these coefficients rather than reading the report's charts:

Axial compression (SP-8007 eq. 1–2): k_x = m²(1 + β²)² + 12 γ² Z² / (π⁴ m² (1 + β²)²), minimised over the number of axial half-waves m and the buckle aspect ratio β. The minimum is k_x = 4√3 γZ / π² (eq. 3) whenever γZ exceeds π² / (2√3) = 2.849, which gives back σ = γ E t / (r √(3(1 − ν²))) (eq. 4) for all but very short cylinders, and 1 + 12 γ² Z² / π⁴ below it, tending to the wide-column value k_x = 1 as Z → 0 (the report's Figure 1). The factor γ² in the second term is where the report inserts the correlation factor, so that the design curve is the theoretical one shifted along the Z axis.

External pressure (eq. 13–14): N_y = p r = k_y π² D / L² with k_y = [(1 + β²)² + 12 γ² Z² / (π⁴ (1 + β²)²)] / β² minimised over β; for hydrostatic pressure, which also loads the ends, β² in the denominator becomes β² + ½. The tool minimises numerically (the report's Figure 4): k_y → 4 for lateral and 2 for hydrostatic pressure as Z → 0, and k_y → 1.04 √(γZ) for γZ above about 100 (eq. 15), which is the familiar p = 0.855 E √γ / [(1 − ν²)^(3/4) (r/t)^(5/2) (L/r)] (eq. 16). Long cylinders buckle into an oval (two circumferential waves) at the pressure p = γ E (t/r)³ / (4(1 − ν²)) of eq. (18)–(19), the Bresse–Bryan collapse pressure of a ring; this branch lies above the main curve and governs beyond γZ ≈ 11.8 (r/t)² (1 − ν²), where Donnell's continuous-β minimum would correspond to fewer than two waves. The report's correlation factors are √γ = 0.75 on the main curve (eq. 20, so γ = 0.5625 multiplies Z) and γ = 0.90 on the two-wave branch (eq. 21).

Torsion (eq. 25–28): the buckle is skewed, so no closed form exists. Batdorf, Stein and Schildcrout (NACA TN 1344, 1947) solved Donnell's equation by Galerkin's method with w = Σ sin(mπx/L)[a_m sin(ny/r) + b_m cos(ny/r)], which gives the coupled equations K_p a_p − c Σ G_pm b_m = 0, K_p b_p + c Σ G_pm a_m = 0 with K_p = (p² + β²)² + 12 γ² Z² p⁴ / (π⁴ (p² + β²)²), G_pm = m p / (p² − m²) for m + p odd and c = 8 β k_xy / π. The tool solves the 16-term symmetric eigenvalue problem and minimises over β (the report's Figure 5 and TN 1344 Table 1: k_xy = 5.34 for the flat plate at Z = 0, 7.52 at Z = 10, 27.5 at 100, 150 at 1000, 850 at 10⁴). SP-8007's straight-line form k_xy = 0.85 (γZ)^(3/4), valid for 50 < γZ < 78 (r/t)² (1 − ν²), is printed beside it, as is the two-wave form τ = γ E (t/r)^(3/2) / [3√2 (1 − ν²)^(3/4)] of eq. (27), which governs for longer cylinders. The correlation factor is γ^(3/4) = 0.67 (eq. 28), inserted as γZ in the solution.

Bending (eq. 9–10): the critical extreme-fibre stress is the axial-compression value with the correlation factor 1 − 0.731(1 − e^−φ) in place of 1 − 0.901(1 − e^−φ), φ = √(r/t)/16, both stated for r/t < 1500. Bending is less imperfection-sensitive because only the most compressed part of the wall is near its critical stress; the moment follows from M = π r² t σ for a thin wall.

Plasticity

When the elastic design stress approaches the yield strength the modulus must be reduced. SP-8007 gives plasticity factors η that replace E by η E: for compression and bending √(E_sec E_tan) / E for moderately long cylinders and E_tan / E as Z → 0 (eq. 7–8; the tool interpolates linearly in γZ between them up to the straight-line transition at 2.849); for torsion E_sec / E with the secant modulus read at twice the critical shear stress (eq. 29); for external pressure the three expressions of eq. (22)–(24) by the ranges of γZ. The moduli come from the Ramberg-Osgood curve ε = σ/E + 0.002 (σ/F_cy)ⁿ, with the MIL-HDBK-5J compressive yield strength and shape exponent (a saved material fills both). Because η depends on the stress it reduces, the critical stress is the fixed point σ_cr = η(σ_cr) σ_e, found by bisection. The tool also caps every allowable at F_cy (F_cy/√3 in shear, hoop stress F_cy in pressure); with F_cy entered and no exponent only this cutoff is applied, and with neither the results are purely elastic.

Internal pressure

Internal pressure helps a compressed cylinder twice: the pressure thrust p π r² (and in bending the membrane moment, counted at 80 % of p π r³) must be overcome before the wall is compressed, and the hoop tension smooths the imperfections and suppresses the diamond pattern, so that at high pressure the shell buckles axisymmetrically near the classical stress. SP-8007 expresses the second effect as an increase Δγ of the buckling-stress coefficient read from its Figure 6 against (p/E)(r/t)², a lower bound to the Mylar and metal tests of the early 1960s; the tool interpolates a digitized copy of that curve (Δγ = 0.089 at 0.1, 0.22 at 1.0, 0.26 at 5 and above). The pressurized allowables are P_press = 2π E t² [γ/√(3(1 − ν²)) + Δγ] + p π r² and M_press = π r E t² [γ_b/√(3(1 − ν²)) + Δγ] + 0.8 p π r³ (eq. 33 and 35), used as loads in the interaction. The report gives no pressurized torsion allowable, so the torsion ratio keeps the unpressurized value.

Combined loads and the margin

SP-8007 recommends a straight-line interaction for every pair it treats (R_c + R_b = 1, R_c + R_p = 1, R_c + R_t = 1, eq. 30–32) and states that in general the sum of the critical load ratios is unity; the tool sums every ratio present, R_tot = R_c + R_b + R_p + R_t, and reports M.S. = 1/R_tot − 1, the common factor by which all applied loads may grow before buckling. Loads entered at limit level are multiplied by the ultimate factor first; internal pressure is never factored, because a load that alleviates buckling must not be increased (SP-8007 Section 3.2). The applied membrane stresses are σ = P/(2π r t), σ_b = M/(π r² t) and τ = T/(2π r² t); hydrostatic pressure's axial component p r/2t is carried by the hydrostatic coefficient and is not entered again.

Conventions

r is the radius to the mid-surface, L the length between supports that hold the wall round (ring frames, bulkheads), t the wall thickness; the ends are simply supported (the classical Donnell boundary conditions, which SP-8007 adopts because the imperfection scatter hides the difference between end conditions). The axial load is positive in compression; a tensile load is entered negative and is not credited. The bending moment and torque enter as magnitudes. Pressure is a positive magnitude whose sense (external lateral, external hydrostatic, internal) is chosen in the select. Lengths, forces, moments, stresses and pressures go through the unit dictionary (USCS in, lb, lb-in, ksi, psi; SI mm, kN, N-m, MPa, kPa) and every computed quantity is stored in SI and displayed in the chosen system.

Reading the results

Each load section prints the correlation factor, the abscissa γZ, the buckling coefficient, the elastic design stress, the plasticity factor when it applies and the design allowable in bold, followed by the buckling load, moment, pressure or torque. The report's closed-form approximations (eq. 4, 16, 26) and two-wave branches (eq. 19, 27) are printed beside the computed values with a note on whether they apply, so the page can be checked against a hand calculation. The summary table lists the design allowable, the applied ultimate stress and the ratio of every load; the margin box carries the interaction margin. In Figure 2 a cylinder far to the right (large r/t) has a small γ and depends more on manufacturing quality; in Figure 3 the margin is the distance from the load point to the line along the ray from the origin; in Figure 4 a point on the straight parts of the curves means the closed forms apply, a point on the curved parts near Z → 0 means the cylinder is short enough to behave as a wide plate.

Limits

The method is for unstiffened isotropic walls of uniform thickness with r/t above about 10 (the thin-shell theory) and below 1500 (the correlation data); the tool warns beyond 1500 and refuses r/t below 5. The correlation factors are lower bounds to the test population of the 1930s to 1960s; the 2020 revision of SP-8007 (Hilburger) keeps them as a conservative baseline and offers analysis-based factors for well-characterized manufacturing, which may be entered in the knockdown select when they have been established. SP-8007 cautions that the axial-compression correlation has not been verified for L/r above 5, and a long cylinder must also be checked as a column; long tubes in bending ovalize (Brazier) before they buckle, which the method does not include. Stiffened, sandwich and orthotropic shells, cutouts, elastic end restraint, thermal stresses and the combined internal pressure with torsion are outside the tool. For conical shells see NASA SP-8019.