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NACA Report 1255 Z-stringer with its sheet as a beam-column: exact Timoshenko moments, crippling and Johnson–Euler interaction
| Tool | Beam-Column with Crippling |
| Source | Timoshenko, S. P., and Gere, J. M. (1961). Theory of Elastic Stability, 2nd ed. McGraw-Hill, New York. Chapter 1 (eq. 1-7 and 1-8, uniform lateral load); Semonian, J. W., and Peterson, J. P. (1956). NACA Report 1255, Appendix (the Z-stringer and sheet, crippling stress 27.5 ksi from its reference 25); Gerard, G. (1957). NACA TN 3784 eq. (61); Bruhn, E. F. (1973). Analysis and Design of Flight Vehicle Structures, C7 (Johnson–Euler, stringer interaction). |
| Level | Professional |
Problem
The 2024-T3 Z-stringer of the Report 1255 example (t_w 0.064 in, b_w 2.56 in, b_A 0.512 in, b_F 1.024 in) with its 0.102 in sheet at a 3.06 in pitch spans 19.6 in between frames as a pin-ended strut under 9,000 lb of compression and 20 lb/in of pressure load toward the sheet (ultimate). With E = 10,600 ksi, F_cy = 43.6 ksi, F_tu = 64 ksi and the crippling stress of 27.5 ksi the report took from tests, find the exact beam-column moment and deflection, the fibre stresses, the Johnson–Euler column allowable and the margins by the crippling check and the interaction R_a + R_b = 1.
Inputs to Enter
- Z section (formed, 2024-T3): b_w 2.56 in, b_A 0.512 in, b_F 1.024 in, t 0.064 in; F_cc override 27.5 ksi; sheet t_s 0.102 in, pitch b_s 3.06 in
- E 10.6 Msi, F_cy 43.6 ksi, F_tu 64 ksi; L 19.6 in, c 1.0
- P 9,000 lb, w 20 lb/in toward the sheet, no point load, no end moments, ultimate level
Answers
- Stringer A 0.2621 in², I 0.2362 in⁴; Gerard eq. (61) F_cc 30.43 ksi, entered 27.5 ksi used; 2w = 1.9 t √(E/F_cc) = 3.80 in > b_s (full sheet); column A 0.5743 in², I 0.5515 in⁴, ρ 0.980 in (L/ρ = 20)
- P_e = π² E I/L² = 150,200 lb, α 0.0599; M_0 = wL²/8 = 960.4 lb-in → M = (wL²/8)·2(sec u − 1)/u² = 1,023 lb-in (factor 1.0655; 1/(1 − α) = 1.0637); y 0.00699 in
- f_a 15.67 ksi, free-flange bending 3.585 ksi, f_c 19.26 ksi; crippling M.S. = 27.5/19.26 − 1 = +0.43
- Johnson–Euler F_c 26.78 ksi, P_c 15,377 lb; R_a 0.585 + R_b 0.130 = 0.716; interaction M.S. +0.40 (governs); proportional load factor 1.39
Notes
The tool's default inputs are this case; press Submit. The closed forms are checked against a scipy solution of EI y″ + P y = −M_0(x) and against the site's Beam Column Strength tool (Gittleman), which reproduces them within 0.002 % on five uniform struts (test_beam_column_crippling.php, 184 checks).