Buckling Calculator

Effective length, K L: The column length after the end-condition factor K is applied. A larger effective length makes buckling easier.

Radius of gyration about x, r_x: A size measure found from the x-axis stiffness and area. Larger r_x means better resistance to buckling about x.

Radius of gyration about y, r_y: The same idea for the y axis. The smaller of r_x and r_y usually controls.

Slenderness ratio about x, KL/r_x: Effective length divided by radius of gyration about x. Higher slenderness means more column-like buckling behavior.

Slenderness ratio about y, KL/r_y: Effective length divided by radius of gyration about y. The larger axis slenderness often governs.

Transition slenderness, C_c: The boundary used to decide when Johnson or Euler behavior is more appropriate in auto mode.

Critical buckling stress, sigma_cr: The predicted compressive stress at the start of buckling for the governing axis.

Critical buckling load, P_cr: The theoretical axial compression load where an ideal straight column first buckles [1].

Governing axis: The axis with the lower buckling resistance, usually tied to the smaller radius of gyration.

Governing method: The formula that produced the reported result: Euler, Johnson, or the method you forced in advanced options.

Mistake: Leaving "Cross-sectional area, A (mm^2)" filled in direct mode but forgetting "Second moment of area about x, I_x (mm^4)" or "Second moment of area about y, I_y (mm^4)".
Fix: In direct mode, enter all three direct-property fields so the calculator can compare both axes and report the weaker one.

Mistake: Typing shape dimensions into "Rectangle width, b (mm)" or "Outer diameter, D (mm)" while "Section input mode" is still set to direct.
Fix: Switch "Section input mode" to shape before using "Cross-section shape" and the visible dimension fields.

Mistake: Entering "Unsupported column length (m)" in mm instead of meters.
Fix: Convert first and enter meters only in "Unsupported column length (m)". For example, 1500 mm should be entered as 1.5.

Mistake: Using a hollow tube but making "Inner diameter, d (mm)" equal to or larger than "Outer diameter, D (mm)".
Fix: For a hollow circle, make sure "Outer diameter, D (mm)" is greater than "Inner diameter, d (mm)".

Mistake: Assuming the bigger inertia always controls and not checking "Governing axis".
Fix: Read "Governing axis" and compare "Radius of gyration about x, r_x" with "Radius of gyration about y, r_y". The smaller radius usually buckles first.

Mistake: Entering 0 or a negative value for "Safety factor, N" and expecting "Allowable load with safety factor" to work.
Fix: Leave "Safety factor, N" blank to hide allowable load, or enter a positive value greater than 0.

• This tool assumes a straight, centrally loaded column with ideal end conditions and no initial crookedness.
• It does not include eccentric loading, residual stress, local buckling, connection flexibility beyond the chosen K factor, or code-based design reduction factors.
• Euler results are most appropriate for slender columns; if Euler stress comes out above yield strength, elastic buckling may not be the best model.
• Johnson results are intended for the intermediate range; if "Formula policy" is set to Johnson only above the transition slenderness, treat the result as a forced comparison rather than the usual preferred model.
• If the Johnson equation gives zero or negative stress, the calculator will not report a usable critical load because that geometry is outside the practical Johnson range for this simple model.
• In direct mode, accurate results depend completely on the entered "Cross-sectional area, A (mm^2)", "Second moment of area about x, I_x (mm^4)", and "Second moment of area about y, I_y (mm^4)".
• Reported "Allowable load with safety factor" is only P_cr divided by N. It is not a full code-check or final design capacity.

Calculation flow

The calculator first finds section properties. In direct mode it uses your entered area and second moments of area. In shape mode it calculates them from the chosen cross-section.

A = b * h; I_x = b * h^3 / 12; I_y = h * b^3 / 12

A = pi * D^2 / 4; I = pi * D^4 / 64

A = pi * (D^2 - d^2) / 4; I = pi * (D^4 - d^4) / 64

Next it applies the selected end-condition factor K to the actual unsupported length to get effective length.

K L = K * L

Then it computes radius of gyration and slenderness for both principal axes, using mm for section geometry and converting length from m to mm.

r = sqrt(I / A)

λ (slenderness ratio) = K * L / r

The transition slenderness marks the usual boundary between Johnson and Euler behavior.

C_c = sqrt(2 * pi^2 * E / sigma_y)

For each axis, the calculator evaluates Euler stress and, when needed, Johnson stress.

sigma_cr,Euler = pi^2 * E / (K * L / r)^2

sigma_cr,Johnson = sigma_y * (1 - (sigma_y / (4 * pi^2 * E)) * (K * L / r)^2)

In auto mode, Johnson is used when the axis slenderness is below C_c, and Euler is used when it is at or above C_c. In forced modes, the selected formula is used even if it is outside its usual range, and the result note explains that choice.

Critical load is found from stress times area, or equivalently from the Euler load form. The tool checks both x and y axes and reports the smaller load as the governing result because that axis buckles first [1].

P_cr = sigma_cr * A

P_cr = pi^2 * E * I / (K * L)^2

If you enter a positive "Safety factor, N", the allowable load is reported as a simple division.

P_allow = P_cr / N

Worked mini-example

Suppose L = 2 m, K = 1.0, E = 200 GPa, sigma_y = 250 MPa, A = 1000 mm^2, and I_x = I_y = 83333 mm^4. Then r_x = r_y = sqrt(83333 / 1000) = 9.129 mm, so KL/r is about 2000 / 9.129 = 219.1. The transition slenderness is about 125.7, so auto mode picks Euler. Euler stress is about pi^2 * 200000 / 219.1^2 = 41.13 MPa, and P_cr = 41.13 * 1000 = 41125 N = 41.13 kN.

Interpretation

A higher "Critical buckling load, P_cr" means the ideal column can carry more axial compression before buckling. A larger slenderness ratio means buckling is more likely to control. If the x and y results are very different, the smaller radius of gyration is the weak direction to watch.

Assumptions used in the math

The formulas are standard ideal-column relations for straight members under concentric compression. Real columns can buckle earlier because of crookedness, eccentricity, local wall buckling, imperfect end restraint, or design-code reduction factors, so field or code values may be lower than the calculator result [1].