Column Buckling Calculator: Euler, Johnson & Effective Length

For the selected bending plane, $r=\sqrt{I/A}$, $L_e=KL$ and $\lambda=L_e/r$. The ideal Euler eigenvalue is

For metals, the Johnson–Euler transition and the concentric curve are

The branches are tangent at $F_0/2$; do not take the minimum of an extrapolated Johnson curve and Euler at all slendernesses. See USAF Stress Analysis Manual (1986), §2.3.1.11.4 (attached link s6), and NIOSH/Gearhart (2007), attached s2 p.2. For timber the separate screen is $\min(F_0,\sigma_E)$, with its crossing at $\pi\sqrt{E/F_0}$.

Ideal factors are 1, 0.5 and 2 for pinned–pinned, fixed–fixed and fixed–free ends. Fixed–pinned uses $K=\pi/q=0.6991556596$, where $q=4.4934094579$ solves $\tan q=q$; 0.7 is the common rounded value. The four mode shapes are derived here from the uniform Euler–Bernoulli beam-column equation and its end conditions. Their amplitudes are free in the eigenvalue problem. The playing input changes only a qualitative magnification and the drawing, not the engineering capacity calculations.

For pinned ends with equal end eccentricity in one bending plane,

These expressions require $0\le P<P_E$. A 52-iteration bisection on $P/P_E$ finds the first load reaching the strength ceiling. With eccentric loading, the screening limit is the smaller of that elastic load and the concentric column estimate. The chosen safety factor divides the load limit, not the calculated fibre stress. See Jonkoping Applied Mechanics §8.19 and USAF §2.3.1.7.

Steel section properties are six original AISC Shapes Database v16.0 records, transcribed in attached s4 p.1 and converted explicitly from inches. Steel presets use rounded nominal SI values consistent with ANSI/AISC 360-22 and ASTM A992/A36: 200 GPa; 345/250 MPa. McKechnie rev.6 (2023), attached s3 p.1, supplies aluminium compression E=69.7 GPa and minimum tensile proof strength 240 MPa, used as a preliminary strength ceiling. USDA Wood Handbook FPL-GTR-282 (2021), chapter 5, Table 5-3a, printed p.5-7 / chapter PDF p.7, supplies Coast Douglas-fir E=13.4 GPa and crushing strength 49.9 MPa at 12% moisture. The unadjusted mean bending modulus is used as a stiffness approximation.

A straight, homogeneous, constant-section member; axial compression through the centroid for the main model; small bending strains; uniform material properties; idealized restraints in the selected plane; no intermediate lateral loads or self-weight variation. A free end carries an axial dead load. The section is stable against local buckling, and the entered second moment is about a principal axis. Check both planes using their actual unbraced lengths and restraints.

The optional secant case has equal end eccentricities producing single-curvature bending at pinned ends. No initial crookedness is added to the numerical model. The animation uses a small illustrative shape amplitude rather than an imperfection specified for design. The project-material override keeps the selected preset's metal/timber model family.

Timber properties are species means for clear dry wood parallel to grain, not characteristic or allowable strengths. Aluminium tensile proof strength is an approximation to a compression-strength ceiling; confirm the actual product's compression properties. The Johnson material strength parameter must be justified for the product being checked.

Custom sections may be asymmetric about the checked principal axis. The entered c is the compression-side extreme-fibre distance for the selected direction of eccentric bending. Check the opposite bending direction and any tensile stress separately.

This is a mechanics and preliminary sizing tool, not a structural design approval. Johnson is a semi-empirical intermediate-column model; it is not the ANSI/AISC 360 column curve. ANSI/AISC 360-22 Chapter E also requires section classification, applicable instability modes and resistance/allowable-strength factors, which are not implemented. The AISC property database is v16.0 (2023). Reference editions and data were reviewed on 26 October 2026; check against the standard in force.

Real end fixity, frame stability, residual stress, initial crookedness, local buckling, torsion, biaxial bending, connection failure, lateral loads, shear deformation, creep, load duration, moisture, temperature, fire and dynamic effects need separate assessment. A strong-axis result alone cannot establish member capacity. Stocky members need compression and bearing checks.

The elastic secant solution loses validity at or above the Euler load; those result cards are withheld by an input condition. Before that point, large displacement or stress above the proportional limit can invalidate an elastic prediction. The smaller-of-two-loads eccentric screen does not reproduce an inelastic beam-column design curve. The 2% displacement notice is an author-selected prompt for nonlinear review, not a code acceptance criterion.

The published USAF secant trial is rounded and contains a 10590/10950 psi transposition. Its reference case states the discrepancy and uses a rounding tolerance without altering the equations. The bamboo study reference tests the reported dimensionless Johnson tangent only; it is not evidence for the timber preset's design strength.