Column Buckling Tables: Method

End Conditions, Euler, Johnson, AISC E3 and the Tangent Modulus

What the tool does

A centrally loaded column of length L with cross-section properties A, I_x, I_y (or the radii of gyration r_x, r_y) and a material of modulus E and yield strength F_y is checked for flexural buckling about both axes. The end conditions give the effective-length factor K, the slenderness KL/r follows for each axis, and the larger slenderness governs because every column curve falls with KL/r. At that slenderness the tool evaluates the Euler stress, the Johnson parabola, the AISC 360-16 column curve and, when a Ramberg–Osgood exponent is entered, the tangent-modulus stress; it then tabulates the same quantities against KL/r (or length) in ten steps and draws the curves with the applied stress. The margin analysis of the Column Stability tool (Johnson–Euler with a crippling cut-off, Engesser with a material curve, load types and factors) is not repeated here; this page is the end-condition and table companion.

Effective length and the six classical cases

The buckled shape of a pinned–pinned column is a half sine wave of length L. For other end conditions the deflected shape between inflection points is still a half sine wave, of length KL, so the critical load is π²EI/(KL)² and the slenderness is KL/r with r = √(I/A). The theoretical K values follow from the characteristic equations of Timoshenko and Gere (Theory of Elastic Stability, §2.1–2.2): fixed–fixed 0.5, fixed–pinned tan kL = kL with kL = 4.4934 (K = π/4.4934 = 0.699, printed as 0.7; P_cr = 2.046 π²EI/L² = 20.19 EI/L²), pinned–pinned 1.0, fixed–guided (the top held against rotation but free to sway) 1.0, fixed–free 2.0 and pinned–guided 2.0. Real connections never reach full fixity, so AISC 360-16 Commentary Table C-A-7.1 prints beside them the recommended design values 0.65, 0.80, 1.0, 1.2, 2.1 and 2.0; the tool lists both and uses the one selected. A K from an alignment chart or a frame analysis is entered directly. Ticking "Separate y Axis" gives the weak axis its own end condition and unbraced length, the usual case of a column braced at mid-height about its weak axis (Hibbeler Example 13.2) or framed differently in the two planes (Hibbeler Example 13.3).

Euler

F_e = π²E/(KL/r)² and P_e = F_eA. The derivation assumes a perfectly straight, centrally loaded, linearly elastic column, so the Euler stress is only an upper bound: when it exceeds the yield strength the column yields first and the tool cuts the Euler row off at F_yA (Hibbeler Example 13.1: a 12-ft A-36 W8x31 has F_e = 56.1 ksi, above 36 ksi, so P = 36 × 9.13 = 329 kip); when it exceeds about half the yield strength the real column is already inelastic and the three methods below apply.

Johnson parabola

J. B. Johnson's empirical parabola F_cr = F_y[1 − F_y(KL/r)²/(4π²E)] starts at F_y for KL/r = 0 and is tangent to the Euler hyperbola at the transition slenderness C_c = √(2π²E/F_y), where both give F_y/2 (Shigley eqs. 4-43 to 4-46). Below C_c the parabola is used, above it the Euler stress. For A-36 steel C_c = 126.1, for F_y = 50 ksi 107.0. The construction is independent of the material class; with a crippling or proportional-limit stress in place of F_y it is the Johnson–Euler curve of aerospace practice.

AISC 360-16 Section E3

For compression members without slender elements the nominal strength is P_n = F_crA_g (E3-1) with F_cr = 0.658^F_y/F_e F_y when KL/r ≤ 4.71√(E/F_y), which is the same condition as F_y/F_e ≤ 2.25 (E3-2), and F_cr = 0.877 F_e otherwise (E3-3). The inelastic branch is calibrated to the residual stresses and the initial out-of-straightness (L/1500) of rolled and welded steel members; the 0.877 of the elastic branch is the same out-of-straightness allowance. The design strength is φ_cP_n with φ_c = 0.90 (LRFD) and the allowable strength P_n/Ω_c with Ω_c = 1.67 (ASD), Section E1. For F_y = 50 ksi and E = 29,000 ksi the boundary is KL/r = 113.4 and the available stresses are φ_cF_cr = 40.0, 37.5, 21.7, 10.0 and 5.65 ksi at KL/r = 40, 50, 100, 150 and 200, the numbers tabulated in the Manual's available-critical-stress table. Section E2 suggests KL/r ≤ 200 for compression members; the tool notes when the governing slenderness exceeds it.

Tangent-modulus (Engesser) theory

Engesser (1889) replaced E in the Euler formula by the tangent modulus E_t = dσ/dε at the critical stress; Shanley (1947) showed that the tangent-modulus load is the load at which bending of a straight inelastic column begins, so it is the accepted lower bound of the buckling load and the reduced-modulus load is an upper bound that is never reached. With the Ramberg–Osgood curve ε = σ/E + 0.002(σ/F_y)ⁿ the tangent modulus is E_t = E/[1 + 0.002 n (E/F_y)(σ/F_y)ⁿ⁻¹] (MIL-HDBK-5J, Section 9.3.2; the exponent n is tabulated for aerospace alloys and can be filled from a saved material). Because E_t depends on the stress sought, F_cr = π²E_t(F_cr)/(KL/r)² is solved by Newton's method on g(σ) = σ/E + 0.002 n (σ/F_y)ⁿ − π²/(KL/r)² = 0, a monotone convex function whose root is reached in a handful of iterations from a start above it; the iterations are tabulated. The Ramberg–Osgood curve keeps rising past F_y, so a solution above F_y is reported as F_y. The curve has no sharp proportional limit either; the tool reports the stress at which E_t has fallen to 0.99 E and the slenderness at which the Euler formula reaches it, above which the two methods differ by less than 1 %. The same iteration is the Engesser solution of the Column Stability tool (its F_cr,eng agrees with this page to eight digits for the same inputs). Hibbeler's Example 13.4 uses a bilinear curve instead: with E_t = 120 GPa and KL/r = 80 the tangent-modulus stress is 185.1 MPa against the Euler 231.3 MPa, which lies above the 150 MPa proportional limit.

Table, chart and ratios

The table runs over ten equal steps of slenderness up to the entered maximum (or of length up to the entered maximum, converted with the governing axis's K and r) and lists for each step the length, the Euler stress and load (cut off at F_y, marked *), the Johnson stress and load, the AISC F_cr, P_n, φ_cP_n and P_n/Ω_c, and the tangent-modulus ratio E_t/E, stress and load. The chart draws the critical stress against KL/r for the same methods, the Euler hyperbola from 1.25 F_y down, with the yield line, the critical stresses at the governing slenderness and the applied stress P/A. For an entered load the demand-to-capacity ratios are FS × P/P_cr for the Euler, Johnson and tangent-modulus loads and, for AISC, P/(φ_cP_n) and P/(P_n/Ω_c) with the entered load taken as the required strength at each level (P_u for LRFD, P_a for ASD); the margins of safety 1/(D/C) − 1 are shown for the two AISC checks.

Units and conventions

Lengths, areas, moments of inertia, stresses and loads go through the unit dictionary (in / mm, in² / mm², in⁴ / mm⁴, ksi / MPa, kip / kN); K, KL/r, n, the factor of safety and the ratios are plain numbers. The Load Level radio says whether the entered load is ultimate or limit: the Euler, Johnson and tangent-modulus critical loads are strength allowables, so their margins and demand-to-capacity ratios are written at ultimate load (ULT), an ultimate load used as entered and a limit load multiplied by the factor of safety FS (default 1.5); the AISC ratios keep the code's own LRFD (φ_c) and ASD (Ω_c) convention with no further factor. Results are stored in SI and shown in the entered length unit and the system's stress and load units. The x axis is the strong axis (larger I) by convention, but nothing in the tool depends on it: each axis is checked with its own K, L and r and the larger slenderness governs.

Limits

Flexural buckling only: torsional and flexural-torsional buckling (AISC E4), slender-element sections with the Q or effective-area reductions (E7), beam-columns and eccentric loads (secant formula, Chapter H interaction), crippling of thin-walled sections (Column Stability tool), tapered or stepped columns, and columns of composite or non-metallic materials are outside the scope. The AISC curve is calibrated for structural steel; for aluminium alloys the Aluminum Design Manual's straight-line inelastic formulas apply and for aerospace alloys the Johnson–Euler or Engesser methods with the handbook properties. The Johnson parabola and the tangent-modulus stress are material-independent constructions on the entered E, F_y and n.