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Thin 7075-T6 cylinder under axial compression and bending: NASA SP-8007 lower-bound buckling stresses and the interaction margin

Tool	Cylindrical Shell Buckling
Source	National Aeronautics and Space Administration (1968). Buckling of Thin-Walled Circular Cylinders. NASA Space Vehicle Design Criteria (Structures), NASA SP-8007, Washington, DC. Section 4.2, eq. (1)-(6), (9)-(10), (25)-(28) and (30); Batdorf, S. B., Stein, M., and Schildcrout, M. (1947). Critical Stress of Thin-Walled Cylinders in Torsion. NACA Technical Note 1344, Table 1.
Level	Graduate

Problem

An unstiffened 7075-T6 aluminium cylinder (compression modulus 10.5 Msi, ν = 0.33, F_cy = 69 ksi, Ramberg-Osgood n = 15) has a mid-surface radius of 20 in, a length of 40 in between ring frames and a wall of 0.05 in (r/t = 400). It carries an ultimate axial compression of 10,000 lb and an ultimate bending moment of 200,000 lb-in. Find the Batdorf parameter, the SP-8007 lower-bound correlation factors, the design buckling stresses in compression, bending and torsion, the applied membrane stresses and the margin of safety by the SP-8007 interaction R_c + R_b = 1.

Inputs to Enter

Radius 20 in, length 40 in, wall thickness 0.05 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 = 10,000 lb compression, M = 200,000 lb-in, T = 0, no pressure; correlation factors from SP-8007

Open the Cylindrical Shell Buckling tool (loaded with this example)

Answers

Z = L² √(1 − ν²) / (r t) = 1,510; φ = √(r/t)/16 = 1.25; γ_c = 1 − 0.901(1 − e^−1.25) = 0.3571 (SP-8007 Figure 2 at r/t 400: 0.36), γ_b = 1 − 0.731(1 − e^−1.25) = 0.4784
Classical stress σ_cl = E t / (r √(3(1 − ν²))) = 16.05 ksi; design stresses σ_c,cr = γ_c σ_cl = 5.734 ksi (k_x = 378.7 from eq. 2), σ_b,cr = 7.681 ksi; P_cr = 36,030 lb, M_cr = 482,600 lb-in; plasticity factor η = 1.0 (far below F_cy)
Torsion allowable from the TN 1344 Galerkin solution: γZ = 885, k_xy = 137.2 (the 0.85 (γZ)^(3/4) line gives 138.1), τ_cr = 2.077 ksi, T_cr = 261,000 lb-in
Applied σ_P = P/(2π r t) = 1.592 ksi, σ_M = M/(π r² t) = 3.183 ksi; R_c = 0.2776, R_b = 0.4144, R_c + R_b = 0.6920; M.S. = 1/0.6920 − 1 = +0.45

Notes

The tool's default inputs are this cylinder; press Submit. Every coefficient is recomputed from the Donnell solutions rather than read from the SP-8007 charts: k_x from eq. (2) minimised (exactly 4√3 γZ/π² here), k_y from eq. (14) minimised and k_xy from the 16-term eigenvalue problem of NACA TN 1344 eq. (8), whose Table 1 values (5.34 at Z = 0, 7.545 at 10, 27.86 at 100, 153 at 1,000, 851.9 at 10,000) the library reproduces to within the table's own 2-3-term truncation; the SP-8007 straight-line forms (eq. 16, 26) and two-wave branches (eq. 19, 27) are printed beside the computed values. The numbers above and the numpy cross-check References/Shell-Buckling/checks/cylbuck_check.py are in test_cylinder_buckling.php (193 checks).