Column and S-Beam Buckling Calculator

How to use

1. Select Circular column or S-shaped beam.
2. Enter outer diameter D and inner diameter d in mm, requiring 0 ≤ d < D.
3. Enter length L in m and Young’s modulus E in GPa; both must be positive.
4. For a circular column, choose mode I–IV matching the intended idealized coefficient.
5. For an S-shaped beam, choose type I or II and enter non-negative deflection z in m; z changes bending stress only.
6. Select Calculate to update the result. Select Reset to restore the active example.
7. Read critical load first, then section and stiffness metrics. Correct any red warning before use.
8. Do not use this screening model for short or inelastic columns, local shell buckling, imperfect/eccentric members, unsupported beam geometries, safety certification, or final code compliance.

Equations used

SI is used internally: D is outer diameter (m), d is inner diameter (m), L is unsupported or member length (m), z is non-negative transverse deflection (m), and E is Young’s modulus (Pa).

Calculated quantities are cross-sectional area A (m²), second moment of area I (m⁴), axial stiffness S (N/m), critical compressive load F_cr (N), longitudinal stiffness S_x (N/m), lateral stiffness S_z (N/m), and bending-stress magnitude σ_b (Pa). Coefficients are the dimensionless column coefficient C, dimensionless S-beam stiffness coefficients c_xx and c_zz, and dimensionless S-beam buckling coefficient f_k.

Circular column

C = π (mode I), π²/4 (mode II), 4π² (mode III), or 2.046π² (mode IV).

S-shaped beam

Type I: (c_xx, c_zz, f_k) = (1, 12, 4π²). Type II: (3, 72/5, 36π²). Positive F_cr is compression capacity; positive z gives the reported positive stress magnitude. S-beam stiffness and buckling load are independent of z. Type II gives 9× the type-I load and 1.2× the lateral stiffness for equal inputs.

Assumptions and limits: straight homogeneous linear-elastic members, annular circular section, idealized coefficients, and small deformation. Yielding, local buckling, imperfections, eccentricity, residual stress, connection flexibility, and post-buckling are excluded.

References: Anton van Beek, Tribology-ABC formula pages “Buckling of a round column” and “S-shaped beam”; Euler critical load and effective-length idealization.