Column Buckling Analysis Results
Critical Buckling Load (Pcr)
Euler's critical buckling load
Buckling Stress (σcr)
Stress at critical buckling load
Slenderness Ratio (λ)
Effective length / radius of gyration
Safety Factor
Yield strength / buckling stress
The graph shows the relationship between column length and critical buckling load for the given parameters.
Calculation Details
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Column Buckling Guide
Column buckling is a failure mode that occurs when a slender column under axial compression becomes unstable and bends or "buckles" out of its straight position. This happens when the compressive load reaches a critical value known as the buckling load.
Buckling is particularly important for long, slender columns where the failure occurs due to instability rather than material strength.
The critical buckling load (Pcr) for a long, slender column is calculated using Euler's formula:
Pcr = π²EI / (KL)²
Where:
- E = Modulus of elasticity of the material
- I = Moment of inertia of the cross-section
- L = Length of the column
- K = Effective length factor (depends on end conditions)
The effective length factor (K) accounts for different end conditions:
| End Condition | K Value | Description |
|---|---|---|
| Pinned-Pinned | 1.0 | Both ends free to rotate (most common) |
| Fixed-Fixed | 0.5 | Both ends fully restrained |
| Fixed-Free | 2.0 | One end fixed, other end free (cantilever) |
| Fixed-Pinned | 0.7 | One end fixed, other end pinned |
Euler's buckling formula has some limitations:
- Only valid for long, slender columns (high slenderness ratio)
- Assumes perfectly straight columns with axial loading
- Doesn't account for initial imperfections or eccentric loading
- For short columns, material yielding occurs before buckling
For practical applications, always consult relevant design codes (e.g., AISC, Eurocode) which incorporate safety factors and more comprehensive design approaches.