Pick the section (bar, tube, rectangle, box or I-shape), the material, the supports on each axis and the bracing, and get the critical buckling load, the slenderness, the effective length factor K from finite elements, the buckled shape and the safety factor by Euler, Johnson, AISC 360 or the secant formula for eccentric loads.
Euler applies to long columns (slenderness above Cc = π√(2E/Fy)); intermediate columns use Johnson's parabola.
The length of an equivalent pinned column: Le = K·L, where K depends on the supports (0.5 fixed–fixed, 1 pinned–pinned, 2 fixed–free).
A slender column in compression does not usually fail because the material crushes, but because it suddenly bends sideways: this is buckling, a loss of stability that can happen at stresses well below yield. This guide explains Euler's critical load, the effective length factor, when Johnson must be used instead of Euler, and works an example.
For a perfectly straight column of elastic material with a centered load, the load at which it stops being stable is
Pcr = π² · E · I / (K · L)²
E is the modulus of elasticity, I the moment of inertia of the section about the axis it buckles around, L the length and K the effective length factor, which depends on the supports. Le = K·L is the length of an equivalent pinned column.
These are theoretical values. Design codes such as AISC recommend somewhat larger values for fixed ends, because in practice no fixed end is perfect. The calculator finds K by solving the stability problem with finite elements using the actual supports and braces, and with the "recommended K" option it uses the AISC design values for the classic cases.
Slenderness is Le/r, where r = √(I/A) is the radius of gyration. Dividing Pcr by the area gives the critical stress σcr = π²E/(Le/r)². The column buckles about the axis with the largest Le/r: for an I-shape that is almost always the weak axis y–y, unless it is braced. A brace at mid-height halves Le about that axis and quadruples its critical load.
Euler only holds if the column buckles elastically, that is, if σcr stays below the yield strength Sy. The transition slenderness used by Shigley is
Cc = √(2π²E / Sy)
If Le/r > Cc, the column is long and Euler applies. If it is smaller, the material starts to yield before elastic buckling and the Johnson parabola is used:
σcr = Sy − (Sy · Le / (2π r))² / E
For structural steel, the AISC 360 specification defines its own curve, and for an eccentric load the secant formula is used; the calculator offers all four methods.
A solid round A36 steel bar (E = 200 GPa, Sy = 250 MPa), d = 50 mm, L = 2 m, pinned at both ends, with a 50 kN load.
In the calculator, with these data and the "Euler + Johnson" method, you get the same slenderness, the same critical load and the same safety factor.
R. G. Budynas and J. K. Nisbett, Shigley's Mechanical Engineering Design (McGraw-Hill); J. M. Gere and B. J. Goodno, Mechanics of Materials (Cengage); for structural steel, the ANSI/AISC 360 specification. Simuveliq is not affiliated with these authors, publishers or AISC.
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