Tools › Structures › Plates and Shells › Sphere and Cone Buckling › Example
NASA SP-8019 truncated cone: 7075-T6 frustum under axial compression and bending with the SP-8019 correlation factors
| Tool | Sphere and Cone Buckling |
| Source | National Aeronautics and Space Administration (1968). Buckling of Thin-Walled Truncated Cones. NASA Space Vehicle Design Criteria (Structures), NASA SP-8019, Washington, DC. Section 4.2, eq. (1)-(14) and (17)-(18); National Aeronautics and Space Administration (1969). Buckling of Thin-Walled Doubly Curved Shells. NASA SP-8032, Section 4.2.1. |
| Level | Graduate |
Problem
An unstiffened 7075-T6 aluminium truncated cone (compression modulus 10.5 Msi, ν = 0.33, F_cy = 69 ksi, Ramberg-Osgood n = 15) has middle-surface radii of 10 in at the small end and 20 in at the large end, a semivertex angle of 20° and a wall of 0.06 in. It carries an ultimate axial compression of 15,000 lb and an ultimate bending moment of 80,000 lb-in. Find the cone's radii of curvature and lengths, the SP-8019 design buckling load, moment, hydrostatic pressure and torque with the report's correlation factors 0.33, 0.41, 0.75 and 0.67, the applied small-end stresses and the margin of safety by the straight-line interaction R_c + R_b = 1.
Inputs to Enter
- Truncated cone: r_1 = 10 in, r_2 = 20 in, α = 20°, t = 0.06 in; E = 10.5 Msi, ν = 0.33, F_cy = 69 ksi, n_c = 15 (7075-T6 bare sheet, MIL-HDBK-5J)
- Ultimate loads: P = 15,000 lb compression, M = 80,000 lb-in, T = 0, no pressure; correlation factors from SP-8019
Answers
- ρ₁ = r₁/cos α = 10.64 in, ρ₂ = 21.28 in, ρ̄ = (r₁ + r₂)/(2 cos α) = 15.96 in, slant length L = 29.24 in, axial length 27.47 in, taper ratio 0.5
- Eq. (1)-(2): P_cl = 2πEt² cos²α/√(3(1 − ν²)) = 128,270 lb; P_cr = 0.33 P_cl = 42,330 lb, a small-end stress of 11.95 ksi (η = 1, elastic)
- Eq. (5)-(6): M_cr = 0.41 πEt²r₁cos²α/√(3(1 − ν²)) = 262,950 lb-in (extreme-fibre stress 14.85 ksi)
- Eq. (7)-(8): p_cr = 0.75 · 0.92E/[(L/ρ̄)(ρ̄/t)^(5/2)] = 3.426 psi (hoop stress at the large end 1.215 ksi); eq. (11)-(12): radius factor 0.7041 (Figure 1 at a taper of 0.5), r = 13.23 in, T_cr = 0.67 · 52.8 D (t/ℓ)^(1/2)(r/t)^(5/4) = 298,020 lb-in (τ_cr = 7.905 ksi)
- Applied σ_P = P/(2π r₁ t cos α) = 4.234 ksi, σ_M = M/(π r₁² t cos α) = 4.517 ksi; R_c = 0.3544, R_b = 0.3042, R_c + R_b = 0.6586; M.S. = 1/0.6586 − 1 = +0.52
Notes
The tool's default inputs are this cone; press Submit. The case is the tool's own default checked against the SP-8019 equations read page by page (the report prints no worked example) and an independent numpy recomputation (References/Shell-Buckling/checks/spherecone_check.py); test_sphere_cone_buckling.php (170 checks) also verifies the SP-8019 Figure 1 and Figure 2 readings, the SP-8032 cap curve 0.14 + 3.2/λ² and Figure 4, and the equivalent-cylinder limits against SP-8007 (0.92 against 0.9177, 52.8 against 51.25). With 20 psi of internal pressure the same cone carries P_press = 62,290 lb and M_press = 362,780 lb-in (eq. 15-16, Δγ = 0.065) and the margin rises to +1.17.