--- title: "Buckling behaviour of columns" subject: "Design of Steel and Timber Structure" chapter: 6 key: "semester_6/design_of_steel_and_timber_structure:buckling-behaviour-of-column" source: https://www.bctengineer.com/q/semester_6/design_of_steel_and_timber_structure/buckling-behaviour-of-column site: BCT Engineering --- # Buckling behaviour of columns Design of Steel and Timber Structure · Chapter 6: Compression Members - Asked in: 2076 Bhadra, 6 marks - Video: [Understanding Buckling](https://www.youtube.com/watch?v=21G7LA2DcGQ) — The Efficient Engineer - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/design_of_steel_and_timber_structure/buckling-behaviour-of-column - Whole chapter: https://www.bctengineer.com/md/semester_6/design_of_steel_and_timber_structure/chapter-6 ## Question Explain about Buckling Behaviour of column. ## Short answer A column under axial compression buckles suddenly sideways at the critical load, losing stability. Euler's critical load is Pcr = π²EI/(KL)². Real columns have imperfections (initial crookedness, residual stresses) reducing strength; the Perry-Robertson formula uses an imperfection factor α based on buckling class. Effective length factor K accounts for end restraints. ## Answer #### Buckling of columns A column under axial compression stays straight until a critical load is reached; beyond this it suddenly bends sideways (buckles) with a large lateral deflection, even though the stress may be well below the yield stress. Buckling is a loss of stability, not a material failure. #### Euler's behaviour of an ideal (perfect) column Euler's critical load for a perfectly straight, elastic, pin-ended column: $$ P_{cr}=\frac{\pi^2EI}{(KL)^2},\qquad f_{cc}=\frac{\pi^2E}{\lambda^2},\ \ \lambda=\frac{KL}{r} $$ - The load is directly proportional to $EI$ and inversely proportional to the square of the effective length $KL$. - A slender column (large $\lambda$) buckles elastically at $f_{cc} lambda ``` #### Behaviour of real columns Real columns differ from Euler's ideal column because of: 1. **Initial crookedness** (out-of-straightness), usually about $L/1000$. 2. **Eccentricity of load** and non-uniform end restraint. 3. **Residual stresses** from rolling/welding, which cause early yielding of parts of the section. 4. **Material non-linearity**: yielding in the inelastic range at intermediate slenderness. 5. **Local buckling** of thin flanges or webs, and **torsional / flexural-torsional buckling** of open sections (angles, channels, tees). For intermediate $\lambda$ the strength falls below both the yield load and the Euler load, so a smooth curve is used (Perry-Robertson): $$ f_{cd}=\frac{f_y/\gamma_{m0}}{\phi+\sqrt{\phi^2-\lambda_n^2}},\quad \phi=0.5[1+\alpha(\lambda_n-0.2)+\lambda_n^2],\ \ \lambda_n=\sqrt{f_y/f_{cc}} $$ The imperfection factor $\alpha$ depends on the buckling class: a (0.21), b (0.34), c (0.49), d (0.76), chosen from IS 800:2007 Table 10 according to the section type, axis and flange thickness. #### Effective length Support conditions change the buckled shape (IS 800:2007 Table 11, theoretical value with recommended design value in brackets): pinned-pinned $K=1.0$ (1.0); fixed-fixed $0.5$ (0.65); fixed-pinned $0.7$ (0.8); fixed-free $2.0$ (2.0). A column that is free to sway has a larger $KL$ (e.g. both ends fixed in rotation but free to sway: 1.0, design 1.2). #### Remedies Reduce $KL$ (bracing, intermediate restraints), increase $r$ (use a deeper, hollow or built-up section), use stocky compact sections, and provide stiff end connections. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What best describes buckling of a column? A. Yielding of material at the yield stress B. Sudden lateral deflection due to loss of stability, possibly below yield stress C. Fracture from fatigue D. Crushing of concrete Answer: B. Buckling is a loss of stability, not material failure, and can happen at stress well below yield. (Key point: Nature of buckling) 2. How does Euler's critical load depend on effective length? A. Proportional to KL B. Inversely proportional to (KL)^2 C. Independent of KL D. Proportional to (KL)^2 Answer: B. P_cr = pi^2 EI/(KL)^2, so doubling the effective length reduces the load to a quarter. (Key point: Euler's critical load) 3. Which factor makes real columns weaker than the ideal Euler column? A. Perfect straightness B. Initial crookedness, load eccentricity and residual stresses C. Higher modulus of elasticity D. Very short length Answer: B. Imperfections such as crookedness (about L/1000), eccentricity, residual stresses and non-linearity reduce strength. (Key point: Why real columns differ) 4. In the Perry-Robertson formula, how does the imperfection factor alpha vary with buckling class? A. a = 0.21, b = 0.34, c = 0.49, d = 0.76 B. a = 0.76, b = 0.49, c = 0.34, d = 0.21 C. All classes use 0.49 D. a = 0.1, b = 0.2, c = 0.3, d = 0.4 Answer: A. IS 800 Table 10 gives alpha of 0.21, 0.34, 0.49 and 0.76 for classes a to d; larger alpha means lower strength. (Key point: Buckling classes) ## Related in Design of Steel and Timber Structure - [Classification of steel sections by buckling](https://www.bctengineer.com/question.md?key=semester_6%2Fdesign_of_steel_and_timber_structure%3Aclassify-steel-local-buckling) — Design of Steel and Timber Structure, chapter 1, asked 2× - [Timber columns by slenderness ratio](https://www.bctengineer.com/question.md?key=semester_6%2Fdesign_of_steel_and_timber_structure%3Atimber-column-types-slenderness) — Design of Steel and Timber Structure, chapter 11 - [Built-up column with battens, 4 m, 1000 kN](https://www.bctengineer.com/question.md?key=semester_6%2Fdesign_of_steel_and_timber_structure%3Abuiltup-battens-1000kn-4m) — Design of Steel and Timber Structure, chapter 6, asked 2× - [Braced built-up column, 3 m, 1600 kN](https://www.bctengineer.com/question.md?key=semester_6%2Fdesign_of_steel_and_timber_structure%3Abraced-builtup-1600kn-3m) — Design of Steel and Timber Structure, chapter 6, asked 2× - [Box Sal column 5 m, 400 kN](https://www.bctengineer.com/question.md?key=semester_6%2Fdesign_of_steel_and_timber_structure%3Atimber-box-column-sal-5m) — Design of Steel and Timber Structure, chapter 11 ## Related in other subjects - [Beam deflection used for column buckling load](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ass-beam-deflection-to-column-buckling) — Strength of Materials, chapter 8 - [Euler's buckling load for columns](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Aeuler-crippling-load-pinned-column) — Strength of Materials, chapter 10 - [Types of timber columns and column bases](https://www.bctengineer.com/question.md?key=semester_5%2Fdesign_of_timber_and_masonry_structures%3Atimber-column-types-and-bases) — Design of Timber and Masonry Structures, chapter 3 - [Buckling load, effective length, Euler hinged](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aeuler-both-ends-hinged-define-buckling) — Strength of Materials, chapter 8