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Local buckling of a 2024-T3 flange with one edge free: exact coefficient against TN 3781 Table 7

Tool	Local Buckling Coefficients
Source	Gerard, G., and Becker, H. (1957). Handbook of Structural Stability Part I: Buckling of Flat Plates. NACA Technical Note 3781, National Advisory Committee for Aeronautics, Washington, DC. Eq. (1), Table 7 (long-plate coefficients) and fig. 2 (flange coefficient against Poisson's ratio); Stowell, E. Z. (1948), NACA Report 898, table p. 128.
Level	Undergraduate

Problem

A long 2024-T3 flange 1.5 in wide and 0.064 in thick (E = 10.7 Msi, ν = 0.33, F_cy = 39 ksi, Ramberg-Osgood n = 20) is simply supported along one unloaded edge and free along the other, under uniform compression. Find the buckling coefficient, the elastic critical stress, whether plasticity reduces it, and the margin against a 5 ksi limit stress.

Inputs to Enter

b = 1.5 in, t = 0.064 in, length blank (long plate); loading compression; unloaded edges SS-free; loaded edges simply supported
E = 10.7 Msi, ν = 0.33, F_cy = 39 ksi, n = 20; plasticity correction on; applied σ_c = 5 ksi at limit, factor 1.0

Open the Local Buckling Coefficients tool (loaded with this example)

Answers

k_c = 0.4077: the exact long-flange value 6(1 − ν)/π² = 0.4073 at ν = 0.33 plus (b/λ)² at the 50 b half-wave cap (TN 3781 Table 7 prints 0.43 for ν = 0.3; the report's 0.4256 is the ν = 0.3 limit)
D = E t³/(12(1 − ν²)) = 262.3 lb-in; σ_e = k_c π² E/(12(1 − ν²)) (t/b)² = 7.330 ksi
Stowell case A (long flange, SS edge): η = E_sec/E = 1.000 at 7.33 ksi, so σ_cr = 7.330 ksi, elastic (σ_cr/F_cy = 0.188)
R_c = 5/7.330 = 0.682; M.S. = 1/R_c − 1 = +0.47 at limit

Notes

The tool's default inputs are this flange; press Submit. The coefficient comes from the exact Lévy solution of the plate equation (test_local_buckling.php, 262 checks: TN 3781 Table 7, figs. 14, 17, 22, 23(a) and 26 traced, Stein and Neff 9.34, Bryan's closed form, Stowell's factors, and the NACA ARR 3K04 moment-distribution tables for I, channel, Z and tube sections within 1.3 %). An independent numpy solver (References/Local-Buckling/checks/localbuck_check.py) gives k = 0.40770 and σ_e = 7.3298 ksi.