About Column Buckling Calculator
The column buckling calculator computes the Euler elastic critical buckling load Pcr for a straight, axially loaded, prismatic column. It applies the effective length factor K for four idealised end conditions: pinned-pinned (K = 1.0), fixed-fixed (K = 0.5), fixed-pinned (K = 0.7), and fixed-free, the cantilever case (K = 2.0).
Enter the modulus of elasticity E, the least second moment of area I, the unbraced length L, and the cross-sectional area A in a consistent SI unit set (newtons, millimetres, megapascals). The tool returns the critical load, the effective length, the radius of gyration, the slenderness ratio, and the critical buckling stress, and when a yield strength is supplied it reports whether elastic buckling or yielding governs.
How It Works
- Select the column end condition, which sets the theoretical effective length factor K.
- Enter E (MPa), the least I (mm^4), the unbraced length L (mm), and the area A (mm^2); optionally enter the yield strength Fy (MPa).
- The calculator computes the effective length K*L, then Pcr = pi^2 E I / (K L)^2 and the critical stress Pcr / A.
- It derives the radius of gyration r = sqrt(I/A) and the slenderness ratio K*L / r, and compares the critical stress with the yield strength to decide whether buckling or yielding governs.
Worked Example
A pinned-pinned steel column (K = 1.0) with E = 200,000 MPa, I = 2.0e7 mm^4, length L = 6000 mm, area A = 6000 mm^2, and Fy = 355 MPa. The effective length is K L = 6000 mm. Pcr = pi^2 * 200000 * 2.0e7 / 6000^2 = 1.097e6 N (1097 kN). The radius of gyration r = sqrt(2.0e7 / 6000) = 57.7 mm, so the slenderness ratio K L / r = 6000 / 57.7 = 103.9. The critical stress is 1.097e6 / 6000 = 182.8 MPa, which is below Fy = 355 MPa, so elastic buckling governs.
Formulas
- Euler critical buckling load
Pcr = pi^2 * E * I / (K * L)^2- Effective length factors (theoretical)
K = 1.0 pinned-pinned | 0.5 fixed-fixed | 0.7 fixed-pinned | 2.0 fixed-free- Radius of gyration
r = sqrt(I / A)- Slenderness ratio
lambda = K * L / r- Critical buckling stress
sigma_cr = Pcr / A = pi^2 * E / lambda^2
Standards & References
- Euler buckling theory
- Gere & Timoshenko, Mechanics of Materials
- Theory of Elastic Stability (Timoshenko & Gere)
Frequently Asked Questions
What is the effective length factor K?
K converts the actual unbraced length into the effective length K*L over which the column buckles in a half sine wave. Theoretical values are 1.0 for pinned-pinned, 0.5 for fixed-fixed, 0.7 for fixed-pinned, and 2.0 for fixed-free.
When does the Euler formula stop being valid?
The Euler formula assumes elastic buckling, so it is valid only when the critical stress stays below the yield (or proportional-limit) stress. For stocky columns below the critical slenderness sqrt(pi^2 E / Fy), yielding or inelastic buckling governs instead.
Which moment of inertia should I enter?
Enter the least (minimum) second moment of area, because the column buckles about its weakest axis unless it is braced against that direction. Using the larger I would overestimate the buckling capacity.
Does the calculator include imperfections or a safety factor?
No. It reports the theoretical elastic critical load only. Real designs apply code-based buckling curves, imperfection factors, and partial safety or resistance factors to obtain a reduced design capacity.