Mechanical & Aerospace
Column Buckling Lab
Why does a long thin column fail long before it crushes? Set the geometry, material and end conditions and the critical buckling load updates, with the operating point shown on the σcr–slenderness design curve. Slender columns follow Euler (Pcr ∝ 1/L²); short stocky ones yield first and follow the Johnson parabola. Fixing the ends shortens the effective length and multiplies the load four-fold.
19.6 kN
Critical load Pcr
28 MPa
Critical stress
267
Slenderness λ
Euler (slender)
Regime
The method, with your numbers
Euler–Johnson column buckling
- 1
Compute the section properties
A = πd²/4 I = πd⁴/64 r = √(I/A)
A = π·30²/4 = 707 mm² I = π·30⁴/64 = 39761 mm⁴ r = 7.5 mm
For a solid circle r = d/4 exactly — the radius of gyration depends only on the shape, not the material.
- 2
Effective length from the end condition
Le = K·L
Le = 1·2000 = 2000 mm (Pinned–pinned)
Pinned–pinned is the textbook baseline (K = 1). Fix an end and Le shrinks — and Pcr grows as 1/Le².
- 3
Slenderness ratio — pick the regime
λ = Le / r λc = √(2π²E / σy)
λ = 2000 / 7.5 = 266.7 λc = 125.7
λ ≥ λc — slender column: it buckles elastically (Euler) before the material can yield.
- 4
Euler critical stress (elastic buckling)
σcr = π²E / λ²
σcr = π²·200000 / 266.7² = 27.8 MPa
Only the stiffness E appears — a slender column buckles long before the material yields, so yield strength is irrelevant here.
- 5
Critical buckling load
Pcr = σcr·A
Pcr = 27.8·707 = 19621 N ≈ 19.6 kN
MPa × mm² gives newtons directly. Double the diameter and Pcr jumps ~16× in the Euler regime — I grows as d⁴.
Drag any control above — every number here recalculates. Want this method for any problem? Step Sheets →
Material
End conditions
Slender columns fail by Euler buckling (Pcr ∝ 1/L²); stocky ones yield first (Johnson). Fixing the ends shortens the effective length and multiplies the load.
How to use this simulation
Why does a long thin column fail long before it crushes? Set the geometry, material and end conditions and the critical buckling load updates, with the operating point shown on the σcr–slenderness design curve. Slender columns follow Euler (Pcr ∝ 1/L²); short stocky ones yield first and follow the Johnson parabola. Fixing the ends shortens the effective length and multiplies the load four-fold.
Everything runs in your browser — no sign-up, no download. Change a value and the result updates instantly, so you can build a feel for how each input shapes the outcome. It pairs with Crameleon's practice exams and step sheets when you want to go from intuition to working the problems.