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Column Buckling Tables

Critical stress and load of a centrally loaded column about both axes for the six classical end conditions, with the theoretical effective-length factor K of Timoshenko and Gere and the recommended design value of AISC 360-16 Table C-A-7.1 side by side, or an entered K. The Euler stress, the Johnson parabola (transition at Fy/2), the AISC 360-16 Section E3 column curve (0.658F_y/F_e Fy and 0.877 Fe with φc = 0.90 and Ωc = 1.67) and the tangent-modulus (Engesser) stress with a Ramberg–Osgood Et are evaluated at the governing slenderness KL/r, tabulated against KL/r or length in ten steps and drawn with the applied point; an entered load gives the LRFD and ASD demand-to-capacity ratios. The section is typed (A with Ix, Iy or rx, ry) or filled from a saved section, the material from a saved MIL-HDBK-5J material. The default case is Example 13.1 of Hibbeler's Mechanics of Materials (a pinned 12-ft A-36 W8x31: Euler 512 kip, 56.1 ksi above yield, so 329 kip governs).

  • Six end conditions with the theoretical K and the AISC Table C-A-7.1 design K, or an entered K; separate K and length about each axis

  • Euler, Johnson parabola, AISC 360-16 E3 (φc Pn and Pn/Ωc) and the Ramberg–Osgood tangent-modulus iteration at the governing KL/r

  • Table of Fcr and Pcr against KL/r or length in ten steps; chart of the Euler hyperbola, the Johnson parabola, the AISC curve and the tangent-modulus curve with the applied point

  • Demand-to-capacity ratios and margins for an entered load (LRFD and ASD); Euler and inelastic boundary slendernesses

Column Buckling Tables

Unit System: /

Section (typed, or filled from a saved section)

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Material (E, yield strength and the Ramberg–Osgood exponent for the tangent-modulus method)

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Length and End Conditions








Methods Beyond Euler (the Euler stress is always shown)




Applied Load, Table and Chart






End Conditions, Euler, Johnson, AISC E3 and the Tangent Modulus | Method page
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Column Buckling Tables

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, Ix, Iy (or the radii of gyration rx, ry) and a material of modulus E and yield strength Fy 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; Pcr = 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

Fe = π²E/(KL/r)² and Pe = FeA. 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 FyA (Hibbeler Example 13.1: a 12-ft A-36 W8x31 has Fe = 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 Fcr = Fy[1 − Fy(KL/r)²/(4π²E)] starts at Fy for KL/r = 0 and is tangent to the Euler hyperbola at the transition slenderness Cc = √(2π²E/Fy), where both give Fy/2 (Shigley eqs. 4-43 to 4-46). Below Cc the parabola is used, above it the Euler stress. For A-36 steel Cc = 126.1, for Fy = 50 ksi 107.0. The construction is independent of the material class; with a crippling or proportional-limit stress in place of Fy it is the Johnson–Euler curve of aerospace practice.

AISC 360-16 Section E3

For compression members without slender elements the nominal strength is Pn = FcrAg (E3-1) with Fcr = 0.658Fy/Fe Fy when KL/r ≤ 4.71√(E/Fy), which is the same condition as Fy/Fe ≤ 2.25 (E3-2), and Fcr = 0.877 Fe 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 φcPn with φc = 0.90 (LRFD) and the allowable strength Pn/Ωc with Ωc = 1.67 (ASD), Section E1. For Fy = 50 ksi and E = 29,000 ksi the boundary is KL/r = 113.4 and the available stresses are φcFcr = 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 Et = 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/Fy)n the tangent modulus is Et = E/[1 + 0.002 n (E/Fy)(σ/Fy)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 Et depends on the stress sought, Fcr = π²Et(Fcr)/(KL/r)² is solved by Newton's method on g(σ) = σ/E + 0.002 n (σ/Fy)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 Fy, so a solution above Fy is reported as Fy. The curve has no sharp proportional limit either; the tool reports the stress at which Et 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 Fcr,eng agrees with this page to eight digits for the same inputs). Hibbeler's Example 13.4 uses a bilinear curve instead: with Et = 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 Fy, marked *), the Johnson stress and load, the AISC Fcr, Pn, φcPn and Pn/Ωc, and the tangent-modulus ratio Et/E, stress and load. The chart draws the critical stress against KL/r for the same methods, the Euler hyperbola from 1.25 Fy 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/Pcr for the Euler, Johnson and tangent-modulus loads and, for AISC, P/(φcPn) and P/(Pn/Ωc) with the entered load taken as the required strength at each level (Pu for LRFD, Pa 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 (φc) and ASD (Ωc) 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, Fy and n.



References

Euler, L. (1744). Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. Marcum-Michaelem Bousquet, Lausanne.

Engesser, F. (1889). "Ueber die Knickfestigkeit gerader Stäbe." Zeitschrift des Architekten- und Ingenieur-Vereins zu Hannover, Vol. 35, pp. 455–462.

Shanley, F. R. (1947). "Inelastic Column Theory." Journal of the Aeronautical Sciences, Vol. 14, No. 5, pp. 261–268.

Ramberg, W., and Osgood, W. R. (1943). Description of Stress-Strain Curves by Three Parameters. NACA Technical Note 902, National Advisory Committee for Aeronautics, Washington, DC.

Timoshenko, S. P., and Gere, J. M. (1961). Theory of Elastic Stability, 2nd ed. McGraw-Hill, New York.

American Institute of Steel Construction (2016). Specification for Structural Steel Buildings. ANSI/AISC 360-16, Chicago, IL.

American Institute of Steel Construction (2011). Steel Construction Manual, 14th ed. AISC, Chicago, IL.

Hibbeler, R. C. (2014). Mechanics of Materials, 9th ed. Pearson Prentice Hall, Upper Saddle River, NJ.

Gere, J. M., and Goodno, B. J. (2013). Mechanics of Materials, 8th ed. Cengage Learning, Stamford, CT.

Budynas, R. G., and Nisbett, J. K. (2015). Shigley's Mechanical Engineering Design, 10th ed. McGraw-Hill Education, New York.

U.S. Department of Defense (2003). Metallic Materials and Elements for Aerospace Vehicle Structures. MIL-HDBK-5J, 31 January 2003.

Worked Examples

Published examples this tool reproduces; each page lists the inputs to enter and the printed answers.

All worked examples

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