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Effective width of a 0.040 in 2024-T3 sheet at a stringer stress of 35 ksi: von Kármán, Marguerre, Koiter, Winter and the plasticity factor
| Tool | Effective Sheet Width |
| Source | Gerard, G. (1957). Handbook of Structural Stability, Part IV: Failure of Plates and Composite Elements. NACA Technical Note 3784, National Advisory Committee for Aeronautics, Washington, D.C. (eqs 5, 6, 10, Table 1); Marguerre, K. (1937). The Apparent Width of the Plate in Compression. NACA Technical Memorandum 833 (eqs 7.8, 7.9); Gerard, G. (1957). Handbook of Structural Stability, Part V. NACA Technical Note 3785 (eq. 10, Table 5); von Kármán, T., Sechler, E. E., and Donnell, L. H. (1932). "The Strength of Thin Plates in Compression." Transactions of the ASME, Vol. 54, pp. 53–57; Winter, G. (1947). "Strength of Thin Steel Compression Flanges." Transactions of the ASCE, Vol. 112, pp. 527–554. |
| Level | Professional |
Problem
A 0.040 in 2024-T3 sheet (E_c = 10.7 Msi, F_cy = 39 ksi, Ramberg-Osgood n_c = 20, MIL-HDBK-5J Table 3.2.3.0(b)) is riveted to stringers 4 in apart with flathead rivets at a 0.75 in pitch; the stringers carry 35 ksi in compression. Find the buckling stress of the sheet between stringers, the effective width acting with each stringer by von Kármán's rule and by the large-deflection formulas, the effect of plasticity, the effective sheet area added to a 0.25 in² stringer and whether inter-rivet buckling limits the sheet.
Inputs to Enter
- t 0.040 in, stringer pitch 4 in, simply supported edges (k = 4.0), ν 0.3
- E_c 10.7 Msi, F_cy 39 ksi, n_c 20, plasticity correction on
- Stringer stress 35 ksi (or 8,750 lb on 0.25 in²); flathead rivets (c = 4) at 0.75 in pitch
Answers
- σ_cr = 4 π² E/(12(1 − ν²)) (t/b)² = 3.868 ksi; the whole sheet would be effective only below that stress
- Elastic widths: von Kármán 1.9 t √(E/σ) = 1.329 in (exact constant 1.9014: 1.330 in); Marguerre (σ_cr/σ)^(1/3) b = 1.920 in and 0.19 + 0.81 √(σ_cr/σ) → 1.837 in; Koiter 1.671 in; Winter 1947 1.218 in; AISI ρ = (1 − 0.22/λ)/λ with λ = 3.008 → 1.233 in
- Plasticity: E_s/E = 0.934, E_t/E = 0.416, Stowell η = 0.824 at 35 ksi; design width 1.9 t √(η E/σ) = 1.206 in, effective area 0.0483 in², sheet load 1,689 lb, stringer plus sheet 0.2983 in² carrying 10,440 lb (sheet share 16.2 %)
- Inter-rivet buckling (TN 3785 eq. 10, c = 4, p = 0.75 in): elastic 110 ksi, with E_t/E 35.73 ksi ≥ 35 ksi, so the rivet line does not reduce the width; at a 1 in pitch it would buckle at 34.0 ksi, and countersunk rivets at 1 in buckle at 15.5 ksi and cut the sheet load to 1,237 lb
Notes
The tool's default inputs are this case; press Submit. No handbook prints this exact case: every value is a closed form checked in test_effective_width.php (177 checks) and by the numpy script References/Effective-Width/effwidth_check.py, with the formula constants verified on the NACA pages filed there (TN 3784 eqs 5, 6, 10 and Table 1; TM 833 eqs 7.8 and 7.9 and its Figure 3 test points; TN 3785 eq. 10 and Table 5).