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Curved 2024-T3 fuselage skin panel in compression: NACA TN 3783 curvature parameter, transition-curve coefficient and margin

Tool	Curved Panel Buckling
Source	Gerard, G., and Becker, H. (1957). Handbook of Structural Stability, Part III – Buckling of Curved Plates and Shells. NACA Technical Note 3783, National Advisory Committee for Aeronautics, Washington, DC. Pages 27-29 (eq. 28, 30), 53-59 (eq. 65), Figure 7 and Figure 49(a), Appendix A; Gerard, G., and Becker, H. (1957). Handbook of Structural Stability, Part I – Buckling of Flat Plates. NACA Technical Note 3781, Figure 14 and Figure 22.
Level	Graduate

Problem

A fuselage skin panel of 2024-T3 sheet (E_c = 10.7 Msi, ν = 0.33, F_cy = 39 ksi, Ramberg-Osgood n = 20) is 6 in wide between stringers, 20 in long between frames, 0.040 in thick and curved to a 40 in radius, with all edges simply supported. It carries an ultimate compression flow of 60 lb/in along the stringers. Find the curvature parameter, the flat-plate and curved-plate buckling coefficients by TN 3783's transition curve with the Figure 7 knockdown coefficient, the elastic buckling stress, the cylinder, classical and SP-8007 reference stresses, the shear buckling stress from Figure 49 and the margin of safety.

Inputs to Enter

b = 6 in, a = 20 in, r = 40 in, t = 0.040 in; all edges simply supported; E = 10.7 Msi, ν = 0.33, F_cy = 39 ksi, n_c = 20
Loading: axial compression, N_x = 60 lb/in at ultimate level; no internal pressure

Open the Curved Panel Buckling tool (loaded with this example)

Answers

r/t = 1,000; a/b = 3.33; Z_b = (b²/(r t)) √(1 − ν²) = 22.5 × 0.944 = 21.24 (eq. 65)
Flat plate (TN 3781 Figure 14): k_pl = (3/3.333 + 3.333/3)² = 4.045 (three half waves); σ_flat = 1.775 ksi
TN 3783 Figure 7 at r/t = 1,000: C = 0.154 (chart reading 0.16; SP-8007 would give γ = 0.224, 0.605 γ = 0.135); tangent point Z_t = k_pl/(0.581 C) = 45.2 > Z_b, so eq. (28): k_c = 4.045 + (0.581 × 0.154 × 21.24)²/4.045 = 4.939
σ_e = k_c π² E/(12(1 − ν²)) (t/b)² = 2.168 ksi (elastic, η = 1); N_cr = 86.7 lb/in; reference stresses: C E t/r = 1.649 ksi, classical 0.6 E t/r = 6.544 ksi, SP-8007 lower bound 1.465 ksi
Shear (TN 3783 Figure 49(a), a/b 3.33 between the 3.0 and infinite curves): k_s = 1.632 × 5.70 = 9.30, τ_cr = 4.083 ksi (eq. 87 line 4.27 ksi, outside its Z_b > 30 range)
Applied σ = 60/0.040 = 1.50 ksi; R_c = 1.500/2.168 = 0.692; M.S. = 1/0.692 − 1 = +0.45

Notes

The tool's default inputs are this panel; press Submit. The report gives no worked numerical example for curved plates, so the case is the tool's own default checked against the report's equations and digitized Figures 7 and 49, closed forms (the SS coefficient, the tangent-point identity k_c = 2 k_pl, the cylinder-line limit 1.0018 C E t/r) and the numpy cross-check References/Curved-Panel/checks/curvedpanel_check.py (test_curved_panel_buckling.php). With 5 psi of cabin pressure and 40 lb/in of shear added the allowables grow by √(1 + 5/0.687) = 2.88 (TN 3783 eq. 70 and 88, p_cr,ext = 0.687 psi from Batdorf's k_y = 17.4 for the 40 in cylinder between the frames) and the interaction R_c + R_s² = 1 (eq. 89) gives λ = 3.74.