Reinforced Concrete Column Buckling Calculator: Euler Load with the Transformed Section
The elastic buckling load of a reinforced concrete column.
Engineering student calculators
Free textbook calculators with every step shown, so you can check your own working. Pick one, type in your numbers and see the formulas worked out. Or switch to Ask a question for an explained answer. No account needed.
Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.
How this calculator works
Enter the column width and depth, the bar diameter, the cover and the length.
Choose the end conditions, or type your own effective length factor, and enter the modulus of elasticity of the concrete and of the steel.
The calculator finds the moment of inertia of the concrete and of the bars, the transformed moment of inertia, the effective length and the Euler buckling load about the weaker axis.
Formulas used
- m = \frac{E_s}{E_c}
- I_c = \frac{bh^3}{12},\ I_s = 4A_sy^2
- I_t = I_c + (m-1)I_s
- L_e = KL
- P_{cr} = \frac{\pi^2E_cI_t}{L_e^2}
Worked example
With the default numbers in the calculator above:
- Modular ratio: 8.00
- Moment of inertia of the concrete (weaker axis, ignoring the steel): 42187500.00 mm4
- Area of one bar: 201.06 mm2
- Distance from the centroid to the centre of a bar: 42.00 mm
- Moment of inertia of the four bars: 1418692.98 mm4
- Transformed moment of inertia: 52118350.84 mm4
- Effective length: 1500.00 mm
- Buckling load (Euler): 5715.42 kN
- For comparison: the concrete alone: 4626.38 kN
Change the numbers in the calculator to see the full working, every step, for your own case.
A second worked example
A different case, with these numbers: End conditions Both ends pinned (K = 1).
- Modular ratio: 8.00
- Moment of inertia of the concrete (weaker axis, ignoring the steel): 42187500.00 mm4
- Area of one bar: 201.06 mm2
- Distance from the centroid to the centre of a bar: 42.00 mm
- Moment of inertia of the four bars: 1418692.98 mm4
- Transformed moment of inertia: 52118350.84 mm4
- Effective length: 3000.00 mm
- Buckling load (Euler): 1428.85 kN
- For comparison: the concrete alone: 1156.59 kN
Common questions
What is the transformed section?
The steel is replaced by an equivalent area of concrete, found by multiplying it by the modular ratio. The column is then treated as one material, so the normal beam and column formulas apply.
Why (m - 1) and not m?
Because the concrete that the bars replace was already counted in the moment of inertia of the whole section. The bars add only the extra stiffness.
Is the Euler load the strength of the column?
No. It is the load at which a perfectly straight, elastic column would buckle. A short, stocky column crushes long before it gets there, and a real column is designed to the concrete standard (AS 3600 in Australia).
What effective length factor should I use?
0.5 for both ends fixed, 0.7 for one end fixed and one pinned, 1.0 for both ends pinned and 2.0 for one end fixed and one free. These are the theoretical values.
Related calculators
Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any reinforced concrete or structural mechanics text (Euler buckling and the transformed section). AS 3600 for concrete structures.
Need a calculator we do not have?
Embed this calculator on your own page
Copy this code into your page. The calculator runs inside the frame and links back to Drafting Central.
Need this drawn, designed or checked?
Drafting Central is a mechanical engineering and drafting business working with customers across Australia and around the world. A calculator gives you the numbers; we can turn them into drawings and designs.
Need help with a real project?
Talk to Drafting Central →