Find a column's Euler critical buckling load (Pcr) from its modulus of elasticity, moment of inertia, length, and end-support condition, plus critical stress and slenderness ratio.
Results
Calculated
Critical buckling load (Pcr)
—
π²EI / (KL)² — Euler's critical load
Critical stress (σcr)
—
Pcr ÷ cross-sectional area
Slenderness ratio (KL/r)
—
Effective length ÷ radius of gyration
Effective length (KL)
—
K × unsupported length
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How to use this calculator
Enter the column's modulus of elasticity, moment of inertia, cross-sectional area, unsupported length, and end-support condition, then click Calculate to get the Euler critical buckling load. Click Clear to reset all fields to the example values and start a new calculation.
Understanding the inputs
Modulus of elasticity (E) measures the material's stiffness — about 200 GPa for structural steel, 69 GPa for aluminum, 10–13 GPa for softwood. Moment of inertia (I) and cross-sectional area (A) come from the column's cross-section geometry (for a rectangle b×h, I = b·h³/12 and A = b·h). Length (L) is the actual unsupported length between supports, and the end condition sets the K factor that accounts for how much the supports restrain rotation and translation at each end.
Interpreting the results
The critical buckling load (Pcr) is the axial compressive load at which an ideal, initially straight column becomes unstable and buckles sideways. Critical stress (σcr) is that load divided by the cross-sectional area — compare it against the material's yield strength to see whether buckling or yielding governs. The slenderness ratio (KL/r) tells you how reliable the Euler formula is for this column: it applies best to long, slender columns and overestimates capacity for short, stocky ones.
Frequently Asked Questions
What is Euler's buckling formula?
Euler's formula gives the critical load at which an ideal, initially straight column buckles elastically: Pcr = π²EI / (KL)², where E is the modulus of elasticity, I is the moment of inertia of the cross-section, L is the unsupported length, and K is an effective-length factor set by the end supports.
What does the end-condition factor K mean?
K converts the actual column length into an equivalent pinned-pinned length. Theoretical values are K = 1.0 for pinned-pinned, K = 0.5 for fixed-fixed, K = 0.7 for fixed-pinned, and K = 2.0 for fixed-free (cantilever) columns. A larger K means the same physical length buckles at a lower load.
What is the slenderness ratio, and why does it matter?
The slenderness ratio is the effective length divided by the radius of gyration, KL/r, where r = √(I/A). Euler's formula is reliable for slender columns, typically a slenderness ratio above roughly 100–120 for structural steel. Short, stocky columns tend to yield or crush before reaching that load, so the formula overestimates their capacity.
Does buckling depend on material strength?
Euler's critical load depends only on stiffness (E and I) and geometry (L and K), not on the material's yield or ultimate strength. Two columns with the same E, I, length, and end conditions buckle at the same load even if made from steel of different strength grades. Strength only matters when checking whether the column yields before it buckles, which governs for short columns.