Quick Answer
For a Fixed-Pinned column, the effective length factor is K = 0.7 (theoretical) or K = 0.8 (AISC recommended for design).
One end fixed, one end pinned (rotation allowed at pin)
End Condition Diagram
○ (Pin connection)
║
║
║
║
▓▓▓▓▓▓▓▓▓▓▓▓▓ (Fixed wall)
Practical examples: Column with welded base plate and simple beam connection at top, flag pole with rigid base and guy wire at top.
Common in building construction where the base is a heavy foundation and the top is a beam with simple framing connections.
Euler Buckling Formula
Pcr = π²EI / (KL)²
Where: E = modulus of elasticity, I = moment of inertia, K = effective length factor, L = actual column length
The critical buckling stress can also be expressed as:
σcr = π²E / (KL/r)²
Where r = radius of gyration = √(I/A), and KL/r = slenderness ratio
Example Calculation — W10×49, L = 15 ft
| Section: | W10×49 (A = 14.4 in², Ix = 272 in⁴, rx = 4.35 in) |
| Steel: | A992 (E = 29,000 ksi, Fy = 50 ksi) |
| Length: | 15 ft = 180 in |
| K factor: | 0.7 (Fixed-Pinned) |
Step 1: Effective length = KL = 0.7 × 180 = 126 in
Step 2: Slenderness ratio = KL/r = 126 / 4.35 = 29.0
Step 3: Critical buckling load:
Pcr = π²(29,000)(272) / (126)² = 4903.7 kips
Step 4: Critical buckling stress:
σcr = 4903.7 / 14.4 = 341.1 ksi
Comparison: All End Conditions (W10×49, L = 15 ft)
| End Condition | K (theoretical) | K (recommended) | Slenderness | Pcr |
|---|---|---|---|---|
| Fixed-Fixed | 0.5 | 0.65 | 20.7 | 9611.3 kips |
| Fixed-Pinned | 0.7 | 0.8 | 29.0 | 4903.7 kips |
| Pinned-Pinned | 1.0 | 1.0 | 41.4 | 2402.8 kips |
| Fixed-Free (Cantilever) | 2.0 | 2.1 | 82.8 | 600.7 kips |
| Fixed-Guided (Fixed-Slide) | 1.0 | 1.2 | 41.4 | 2402.8 kips |
Buckling Capacity vs. Length (Fixed-Pinned, K = 0.7)
Reference table for a W10×49 column with fixed-pinned end conditions:
| Column Length | KL/r | σcr (ksi) | Pcr (kips) |
|---|---|---|---|
| 8 ft | 15.4 | 1199.3 ksi | 17239.6 kips |
| 10 ft | 19.3 | 767.6 ksi | 11033.4 kips |
| 12 ft | 23.2 | 533.0 ksi | 7662.1 kips |
| 14 ft | 27.0 | 391.6 ksi | 5629.3 kips |
| 15 ft | 29.0 | 341.1 ksi | 4903.7 kips |
| 16 ft | 30.9 | 299.8 ksi | 4309.9 kips |
| 18 ft | 34.8 | 236.9 ksi | 3405.4 kips |
| 20 ft | 38.6 | 191.9 ksi | 2758.3 kips |
| 24 ft | 46.3 | 133.3 ksi | 1915.5 kips |
| 28 ft | 54.1 | 97.9 ksi | 1407.3 kips |
| 30 ft | 57.9 | 85.3 ksi | 1225.9 kips |
⚠️ Important Design Notes
- Euler’s formula applies only to long, slender columns where buckling governs (KL/r > ~60-80 for steel).
- For short, stocky columns, material yielding governs — use AISC Chapter E provisions.
- Always check both axes — buckling occurs about the weaker axis unless braced.
- Use the recommended K values (not theoretical) for design per AISC 360.
- The safety factor (ASD Ω = 1.67) or resistance factor (LRFD φ = 0.9) must be applied to Pcr for design.
Try our interactive Column Buckling Calculator
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