Cylindrical Shell Buckling

Cylindrical Shell Buckling (structures/fem/cylindrical-shell-buckling)

Use when the task is the elastic stability of a curved, unstiffened circular cylindrical shell (fuselage barrel, pressure-vessel or tank shell, duct) under axial compression or bending. This leaf implements the NASA SP-8007 empirical knockdown method ("Buckling of Thin-Walled Circular Cylinders", 1968 original / 2023 NTRS revision 20205011530, public domain): the axial knockdown factor gamma_axial from the shell radius r and thickness t, the axial critical stress, the bending knockdown factor gamma_bending, the bending critical moment, the cross-section ovalization collapse moment, and the plasticity correction factor eta. It pairs with the flat-panel leaf structures/fem/plate-buckling, which explicitly defers the curved case here, and with structures/fem/buckling-analysis for slender straight members. The SP-8007 knockdowns are empirical fits for thin shells with r/t below about 1500; the classical stress terms are elastic, so the plasticity correction eta is reported separately and applied by the user when the material is beyond its proportional limit.

Domain quick reference

• Curvature parameter: phi = (1/16)*sqrt(r/t). It sets the knockdown sensitivity: very thin shells (large r/t) carry larger knockdowns.
• Axial knockdown factor: gamma_axial = 1 - 0.901*(1 - exp(-phi)), monotonic decreasing in r/t and always in (0, 1).
• Axial critical stress: sigma_cr = 0.605gamma_axialEt/r, where the 0.605 coefficient is the classical value 1/sqrt(3(1-nu^2)) at nu = 0.3.
• Bending knockdown factor: gamma_bending = 1 - 0.731*(1 - exp(-phi)).
• Bending critical moment: M_cr = pi0.605gamma_bendingEt*2r. The axial critical stress acts over the full wall section (area pirt) at the extreme-fiber arm r, giving M = sigma_cr*(pirt)*r.
• Ovalization collapse moment: M_ov = 0.987Er*t2/sqrt(1-nu2), the Brazier-style collapse of the cross section under bending.
• Governing verdict: "bifurcation" when M_cr < M_ov, else "ovalization". SP-8007 notes the bending bifurcation precedes collapse for shells in the validity band.
• Plasticity correction: eta = sqrt(E_sec*E_tan)/E with the secant and tangent moduli at the acting stress; eta = 1 in the elastic range.
• SI units throughout: m, Pa, N*m.
• Validity guard: r/t must be below 1500 for the empirical knockdowns.

Workflow

1. Fix the geometry and material: radius_m, thickness_m, e_mod_pa and the Poisson ratio (default 0.3). Confirm r/t < 1500.
2. Get the knockdown factors: knockdown_axial and knockdown_bending, or read gamma_axial and gamma_bending from shell_buckling_assessment.
3. Compute the axial critical stress with axial_critical_stress (gamma defaults to the internal axial knockdown).
4. Compute the bending critical moment with bending_critical_moment (gamma defaults to the internal bending knockdown).
5. Compute the ovalization collapse moment with ovalization_collapse_moment.
6. When the material may be beyond the proportional limit, pass the secant and tangent moduli to plasticity_correction and scale the elastic margins by eta (SP-8007 stresses are elastic).
7. Run shell_buckling_assessment for the full dict including the governing verdict between bifurcation and ovalization collapse.
8. Confirm the deterministic checks with the contract test scripts/test_cylindrical_shell_buckling.py.

Worked example

Aluminum barrel: E = 70 GPa, r = 1.5 m, t = 0.005 m (r/t = 300), nu = 0.3. Real module outputs:

• curvature_parameter = 1.0825 (sqrt(300)/16).
• gamma_axial = 0.4042, in the 0.35-0.45 bound.
• sigma_cr_axial = 57.06 MPa, in the 50-65 MPa bound.
• gamma_bending = 0.5166, in the 0.45-0.60 bound.
• M_cr_bending = 2.578 MNm, in the 2.2-3.0 MNm bound.
• M_cr_ovalization = 2.716 MNm, in the 2.4-3.1 MNm bound.
• Governing: "bifurcation" (bending bifurcation moment below the ovalization collapse).
• Check geometry r = 1.0 m, t = 0.01 m (r/t = 100): gamma_axial = 0.5813 and gamma_bending = 0.6603; the same barrel at r/t = 1600 raises ValueError on the 1500 validity guard.
• eta_plasticity = 1.0 when E_sec = E_tan = E.

The bending knockdown is higher than the axial one at equal r/t, so the bending critical moment sits above the axial stress level scaled by the full section; the ovalization collapse is the cross-section limit that bifurcation must beat to be governing.

Pitfalls

• Using the knockdowns outside the validity band: the SP-8007 empirical fits hold for r/t below about 1500, and every geometry-dependent function raises ValueError at r/t >= 1500 - including the exact boundary.
• Reading the bending moment as a stress: M_cr scales as gamma_bending * E * t^2 * r over the full wall section while the axial case is a stress 0.605 * gamma * E * t / r; comparing the two critical quantities across units invites a wrong governing verdict.
• Assuming bifurcation always governs: the verdict is "ovalization" when the Brazier collapse moment M_ov falls below the bending bifurcation moment; the worked barrel is "bifurcation", but thicker or softer shells can flip it.
• Applying the plasticity correction twice: the classical stress terms are elastic and eta (sqrt(E_sec E_tan)/E, unity at E_sec = E_tan = E) is reported for the user to apply; scaling the elastic margin by eta once is the intended use.
• Confusing this geometric knockdown with material knockdowns: gamma_axial and gamma_bending come from r/t geometry; the strength reduction factors of composites/cmh17-allowables are material factors for fiber-reinforced laminates.
• Treating the unstiffened barrel as a stiffened fuselage: this leaf covers the curved UNSTIFFENED shell; stiffened pressurized fuselage modeling belongs to vehicle-design/structures-integration/fuselage-skin-stringer.

Verification

• Confirm curvature_parameter(1.5, 0.005) = 1.0825 and gamma_axial in the 0.35-0.45 band, sigma_cr in the 50-65 MPa band, M_cr_bending in the 2.2-3.0 MNm band, M_cr_ovalization in the 2.4-3.1 MNm band, and the governing verdict "bifurcation".
• Confirm gamma_axial(1.0, 0.01) = 0.5813 and gamma_bending(1.0, 0.01) = 0.6603 (r/t = 100).
• Confirm both knockdown factors stay in (0, 1) and decrease monotonically as r/t grows.
• Confirm the identities sigma_cr = 0.605gammaEt/r and M_ov = 0.987Ert2/sqrt(1-nu2) hold exactly, and eta = 1.0 at E_sec = E_tan = E.
• Confirm non-positive radius, thickness or modulus, nu outside (-1, 1), an out-of-range explicit gamma, and r/t >= 1500 all raise ValueError.
• Deterministic: no RNG, identical float results run to run.
• Run the contract test offline: python3 scripts/test_cylindrical_shell_buckling.py (34 tests).

Related leaves

• structures/fem/buckling-analysis: straight slender members under Euler-style compression; this leaf covers the curved shell instead.
• structures/fem/plate-buckling: flat plate and skin panel buckling; it defers the curved unstiffened shell case to this leaf.
• structures/thermal-structures/thermal-buckling: restrained flat panels under thermal load.
• vehicle-design/structures-integration/fuselage-skin-stringer: the stiffened pressurized fuselage model; this leaf covers the unstiffened curved barrel.
• structures/composites/cmh17-allowables: material strength reduction factors for fiber-reinforced laminates; the knockdown here is geometric, that leaf's is material.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_cylindrical_shell_buckling.py

The 34 tests cover the worked example within the spec magnitude bounds, the r/t = 100 knockdown values, unit-interval and monotonicity properties of both knockdown factors, closed-form knockdown identity, ValueError rejection of non-positive geometry/modulus, out-of-range nu, the r/t >= 1500 validity guard (including the exact boundary) on every geometry-dependent function and on the assessment dict, the explicit and default-gamma identities for the axial stress and bending moment, the exact ovalization formula, the plasticity correction (elastic and half-modulus cases plus scale invariance and ValueError), run-to-run determinism, and the assessment dict key and eta contract.

Compliance

• Standards referenced, not reproduced: NASA SP-8007 (public domain; named in the body with the pinned coefficients and the r/t validity band, paraphrased) and FAR-25 airframe loads context per standards-map.yaml. No proprietary handbook text is reproduced.
• compliance: STANDARDS-REF, gated: false.