Column Buckling Calculator Formula
Understand the math behind the column buckling calculator. Each variable explained with a worked example.
Formulas Used
Euler Critical Load (Pcr)
critical_load = pow(pi, 2) * e_pa * i_m4 / pow(le, 2) / 1000Effective Length
slenderness = leVariables
| Variable | Description | Default |
|---|---|---|
modulus | Elastic Modulus (E)(GPa) | 200 |
inertia | Moment of Inertia (I)(cm^4) | 1000 |
length | Column Length (L)(m) | 4 |
k_factor | Effective Length Factor (K) | 1 |
e_pa | Derived value= modulus * 1e9 | calculated |
i_m4 | Derived value= inertia * 1e-8 | calculated |
le | Derived value= k_factor * length | calculated |
How It Works
Euler Column Buckling
A slender column under axial compression can buckle laterally before the material yields.
Formula
Pcr = pi^2 E I / (K L)^2
K is the effective length factor that accounts for end conditions: K = 1.0 for pinned-pinned, K = 0.7 for fixed-pinned, K = 0.5 for fixed-fixed, and K = 2.0 for fixed-free (cantilever). The formula assumes perfectly straight, elastic behaviour.
Worked Example
A 4 m pinned-pinned steel column with I = 1000 cm^4 and E = 200 GPa.
- 01Effective length Le = 1.0 x 4 = 4 m
- 02E = 200 x 10^9 Pa, I = 1000 x 10^-8 m^4 = 1 x 10^-5 m^4
- 03Pcr = pi^2 x 200 x 10^9 x 1 x 10^-5 / 4^2
- 04Pcr = 9.8696 x 2 x 10^6 / 16 = 1,233,700 N = 1233.7 kN
Frequently Asked Questions
What does the K factor represent?
K adjusts the physical length to an effective length based on how the column ends are restrained. Pinned-pinned is K=1, fixed-fixed is K=0.5, fixed-pinned is K=0.7, and cantilever is K=2.
When does Euler buckling not apply?
Euler buckling applies only to long, slender columns where failure is elastic. For short, stocky columns the material yields before buckling, and inelastic buckling formulas or direct strength checks are needed.
How do I find the moment of inertia for standard sections?
Steel section property tables (such as the AISC Steel Manual or equivalent) list I values for all standard W, HSS, and channel shapes.
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