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Pinned 12-ft A-36 W8x31 column: Euler load above the yield load, with the Johnson and AISC E3 values at the same slenderness (Hibbeler Example 13.1)

ToolColumn Buckling Tables
SourceHibbeler, R. C. (2014). Mechanics of Materials, 9th ed. Pearson Prentice Hall, Upper Saddle River, NJ. Example 13.1 (Euler load and yield check); American Institute of Steel Construction (2016). Specification for Structural Steel Buildings. ANSI/AISC 360-16, Chicago, IL. Sections E1 and E3, Commentary Table C-A-7.1.
LevelUndergraduate

Problem

An A-36 steel W8x31 (A = 9.13 in², I_x = 110 in⁴, I_y = 37.1 in⁴, E = 29,000 ksi, σ_Y = 36 ksi) is used as a 12-ft pin-ended column. Find the largest axial load it can carry before it buckles or yields, and compare the Euler value with the Johnson parabola and the AISC 360-16 E3 column strength at the same slenderness; for an applied 200 kip give the LRFD and ASD demand-to-capacity ratios.

Inputs to Enter

  • A 9.13 in², I_x 110 in⁴, I_y 37.1 in⁴; E 29,000 ksi, F_y 36 ksi; L 144 in, pinned–pinned (theoretical K = 1.0; AISC recommended 1.0)
  • Johnson parabola and AISC 360-16 E3 selected; tangent-modulus method off (no Ramberg–Osgood n); P 200 kip, FS 1.0; table against KL/r up to 200
Open the Column Buckling Tables tool (loaded with this example)

Answers

  • r_y = (37.1/9.13)^0.5 = 2.016 in, KL/r_y = 71.43 governs (KL/r_x = 41.49)
  • Euler: F_e = π² (29,000)/71.43² = 56.1 ksi, P_cr = 512 kip (book 512 kip); F_e > σ_Y = 36 ksi, so the column yields first and P = σ_Y A = 329 kip (book 329 kip)
  • Johnson: C_c = (2π² E/F_y)^0.5 = 126.1, F_cr = 36 [1 − 36 (71.43)²/(4π² 29,000)] = 30.22 ksi, P_cr = 275.9 kip
  • AISC E3: 4.71 (E/F_y)^0.5 = 133.7 > 71.43, F_cr = 0.658^(36/56.09) × 36 = 27.52 ksi, P_n = 251.2 kip, φ_c P_n = 226.1 kip, P_n/Ω_c = 150.4 kip
  • P = 200 kip: LRFD D/C = 0.884 (M.S. +0.13), ASD D/C = 1.33 (M.S. −0.25); Euler (yield cut-off) D/C 0.608, Johnson 0.725

Notes

The tool's default inputs are this example; press Submit. The book's example asks only for the Euler load and the yield check; the Johnson and AISC values are the tool's closed-form evaluations at the same slenderness, checked against numpy in test_column_buckling_tables.php (References/Column-Buckling/column_check.py). The table and chart run to KL/r = 200.