Columns and struts are essential structural members in civil engineering. They transfer loads safely from beams/slabs to foundations. For ECET 2026, questions often test Euler’s theory, Rankine’s formula, and effective length concepts. Mastering this topic gives you a clear edge because it connects theory with real-world design.
📘 Concept Notes
1. Column vs Strut
- Column: Vertical compression member, carries load along axis.
- Strut: Compression member not necessarily vertical (can be inclined/short).
2. Types of Columns (Based on Slenderness Ratio)
- Short Column: Fails by crushing.
- Long Column: Fails by buckling.
- Intermediate Column: Both crushing + buckling considered.
Slenderness Ratio (λ):
where LeL_eLe = effective length, rrr = radius of gyration.
3. Euler’s Buckling Theory (for Long Columns)
Critical load:
where
- EEE = Young’s modulus
- III = least moment of inertia
- LeL_eLe = effective length
Effective Length LeL_eLe:
- Both ends hinged:
- Both ends fixed:
- One end fixed, other free:
- One end fixed, other hinged:
4. Rankine’s Formula (for all columns)
Simplified:
where
- σc\sigma_cσc = crushing stress
- AAA = area of cross-section
- α\alphaα = Rankine’s constant
5. Examples
Example 1: A steel column hinged at both ends, length = 3 m, I=6×106 mm4I = 6 \times 10^6 \, mm^4I=6×106mm4, E=2×105 N/mm2E = 2 \times 10^5 \, N/mm^2E=2×105N/mm2. Find Euler’s critical load.
Example 2: A column has σc=550 MPa\sigma_c = 550 \, MPaσc=550MPa, A=2000 mm2A = 2000 \, mm^2A=2000mm2, α=1/7500\alpha = 1/7500α=1/7500, Le/r=60L_e/r = 60Le/r=60.
⚙️ Formulas
🔟 10 MCQs
Q1. Slenderness ratio is defined as:
a) L/r
b) A/L
c) r/L
d) σ/E
Q2. Euler’s formula is valid for:
a) Short columns
b) Long columns
c) Both short and long columns
d) None
Q3. For a column with both ends fixed, effective length is:
a) L
b) 2L
c) L/2
d) L/√2
Q4. A steel column (E = 2×10^5 N/mm², I = 4×10^6 mm⁴, L = 2 m hinged-hinged). Find Euler load.
a) 987 kN
b) 1974 kN
c) 494 kN
d) 1234 kN
Q5. Rankine’s formula is a combination of:
a) Euler + Hooke’s law
b) Crushing + Buckling theories
c) Stress + Strain law
d) Bernoulli + Lacey
Q6. A column has crushing load 1000 kN, Euler load 500 kN. Rankine load = ?
a) 333.3 kN
b) 250 kN
c) 500 kN
d) 750 kN
Q7. Which fails by buckling?
a) Short column
b) Long column
c) Strut only
d) RCC beams
Q8. If radius of gyration increases, slenderness ratio:
a) Increases
b) Decreases
c) Constant
d) Becomes infinity
Q9. For one end fixed and other free, effective length = ?
a) L
b) 2L
c) L/2
d) L/√2
Q10. Rankine’s constant α depends on:
a) Shape
b) Material
c) Length
d) Load
✅ Answer Key
| Q | Answer |
|---|---|
| 1 | a |
| 2 | b |
| 3 | c |
| 4 | a |
| 5 | b |
| 6 | a |
| 7 | b |
| 8 | b |
| 9 | b |
| 10 | b |
🧠 Explanations
- Q1: λ = L/r → (a).
- Q2: Euler applies to long slender columns → (b).
- Q3: Both ends fixed → Le = L/2 → (c).
- Q4:
→ (a). - Q5: Rankine = Crushing + Buckling → (b).
- Q6:
→ (a). - Q7: Long columns fail by buckling → (b).
- Q8: λ = Le/r. If r ↑, λ ↓ → (b).
- Q9: Fixed-free → Le = 2L → (b).
- Q10: α is material constant → (b).
🎯 Motivation / Why Practice Matters
In ECET 2026 Civil, column and strut questions are guaranteed.
- 1–2 direct formula-based MCQs (Euler/Rankine).
- 1 numerical involving effective length.
By daily practice, you’ll solve these in <1 minute during the exam → gaining a competitive speed advantage.
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