--- title: "Column 400x500, 1600 kN, 4.0 m" subject: "Design of RCC Structure" chapter: 10 key: "semester_7/design_of_rcc_structure:column-400x500-1600-kn-4-0" source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/column-400x500-1600-kn-4-0 site: BCT Engineering --- # Column 400x500, 1600 kN, 4.0 m Design of RCC Structure · Chapter 10: Design of compression members: Columns - Asked in: 2080 Baisakh, 10 marks - Video: [Complete manual calculation of RCC column design by using IS 456 2000 code | structural design |](https://www.youtube.com/watch?v=ozWFZs3Cato) — Civil Engineering by Shravan - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/column-400x500-1600-kn-4-0 - Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-10 ## Question Design column 400 mm x 500 mm having unsupported length of 4.0 m with both ends effectively held and restrained against rotation at one end with the following data: Factored axial load = 1600 kN; Factored moments = 150 kN-m and 50 kN-m. Use M25 concrete grade and Fe500 steel grade. ## Short answer Provide 8-16 mm bars (0.80%) and 8 mm ties @ 250 mm c/c. The biaxial interaction ratio is 0.852 ≤ 1.0, meeting strength requirements. ## Answer #### Data and assumptions - Column 400 mm (width $b$) × 500 mm (depth $D$); $P_u = 1600$ kN; $f_{ck} = 25$ MPa, $f_y = 500$ MPa. Clear cover 40 mm (IS 456 cl. 26.4.2.1), 8 mm ties. - Moments: about major axis (x-x) $M_{ux}$ = 150 kN·m; about minor axis (y-y) $M_{uy}$ = 50 kN·m. - Held in position at both ends and restrained against rotation at one end: $l_e = 0.8L = 3.2$ m (IS 456 Table 28). The 150 kN·m moment acts about the major axis and the 50 kN·m moment about the minor axis. - Sizes are given as $b \times D$ with $D \ge b$; moments $M_{ux}$ act about the major axis (depth $D$) and $M_{uy}$ about the minor axis (depth $b$). #### Slenderness | Axis | $l_e$ (mm) | Dimension (mm) | $l_e/$dim | Type | |---|---|---|---|---| | x-x (about major) | 3200 | 500 | 6.40 | short | | y-y (about minor) | 3200 | 400 | 8.00 | short | Both ratios are below 12, so the column is **short**; no additional moment is required. #### Minimum eccentricity (cl. 25.4) $$ e_{min} = \max\left(\frac{L}{500} + \frac{D}{30},\ 20\right) $$ - About x-x: $e_x = 4000/500 + 500/30 = 24.7$ mm (min. 20 mm); adopt 24.7 mm; $M_{ux,min} = P_u e = 39.5$ kN·m - About y-y: $e_y = 4000/500 + 400/30 = 21.3$ mm (min. 20 mm); adopt 21.3 mm; $M_{uy,min} = 34.1$ kN·m #### Design moments - Design moments: $M_{ux} = 150.0$ kN·m, $M_{uy} = 50.0$ kN·m (not less than the minimum-eccentricity values). #### Longitudinal reinforcement $\dfrac{P_u}{f_{ck} b D} = \dfrac{1600000}{25 \times 400 \times 500} = 0.320$ Bars are chosen so that $p$ lies between 0.8% (minimum) and 4% (practical maximum; IS 456 cl. 26.5.3.1 allows 6%). The uniaxial capacities $M_{ux1}$, $M_{uy1}$ at $P_u$ are read from the SP-16 interaction charts for rectangular sections; here they are computed by strain compatibility with the IS 456 stress block and the design stress-strain curve of steel (same basis as the charts), with effective cover $d' = 40 + 8 + \phi/2$. | Trial | $p$ (%) | $d'/D$ | $M_{ux1}$ | $M_{uy1}$ | $P_u/P_{uz}$ | $\alpha_n$ | $(M_{ux}/M_{ux1})^{\alpha_n}+(M_{uy}/M_{uy1})^{\alpha_n}$ | |---|---|---|---|---|---|---|---| | 8-16 φ | 0.80 | 0.112 | 191.3 | 147.6 | 0.564 | 1.61 | 0.852 ✓ | #### Check $P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 25 \times 198392 + 0.75 \times 500 \times 1608 = 2835$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{1600}{2835} = 0.564$, so $\alpha_n = 1.61$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8). $$ \left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{150.0}{191.3}\right)^{1.61} + \left(\frac{50.0}{147.6}\right)^{1.61} = 0.852 \le 1.0 $$ The interaction condition is satisfied; the section is **safe**. Provide **8-16 mm** bars ($A_{sc} = 1608$ mm², $p = 0.80\%$), within 0.8% to 4%. #### Lateral ties - Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(16/4, 6) = 6$ mm → provide 8 mm. - Pitch $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(400, 256, 300) = 256$ mm → provide 8 mm ties @ 250 mm c/c. #### Detailing ```text CROSS-SECTION 400 x 500 +-------------------------+ | o o o | | | | | | | | | | o o | | | | | | | | | | o o o | +-------------------------+ o = 16 mm bars, ties 8 mm @ 250 c/c ``` **Answer:** 400 × 500 mm column with 8-16 mm longitudinal bars ($p = 0.80\%$) and 8 mm ties @ 250 mm c/c. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. With L = 4.0 m, both ends held and one end restrained against rotation, what are le and the column type? A. le = 3.2 m, short B. le = 4.0 m, slender C. le = 2.6 m, slender D. le = 3.2 m, slender Answer: A. le = 0.8L = 3.2 m. Ratios are 6.4 and 8.0, below 12, so the column is short. (Key point: Effective length and slenderness) 2. How is the minimum eccentricity about the x-x axis calculated here? A. emin = le/500 + D/30 = 3200/500 + 500/30 B. emin = L/500 + D/30 = 4000/500 + 500/30 = 24.7 mm C. emin = 20 mm always D. emin = D/30 only Answer: B. IS 456 cl. 25.4 uses the unsupported length L: 8 + 16.7 = 24.7 mm, giving Pu*e = 39.5 kN·m, less than the applied 150 kN·m. (Key point: Minimum eccentricity) 3. With Pu/Puz = 0.564, αn = 1.61. What does the result 0.852 mean? A. The section fails because it is below 1 B. The sum of ratios is below 1, so the section is safe C. The steel exceeds 4% D. The column is slender Answer: B. The condition (Mux/Mux1)^αn + (Muy/Muy1)^αn <= 1 is satisfied with 0.852, so 8-16φ (p = 0.80%) is adequate. (Key point: Biaxial check) ## Related in Design of RCC Structure - [Column 35x40 cm, not restrained against rotation](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Acolumn-35x40-cm-not-restrained-against) — Design of RCC Structure, chapter 10 - [Short column 400x500, 1600 kN](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Ashort-column-400x500-1600-kn) — Design of RCC Structure, chapter 10 - [Fixed-end column 4 m, 1000 kN](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Afixed-end-column-4-m-1000) — Design of RCC Structure, chapter 10 - [Column design, 1350 kN, 3.4 m](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Acolumn-design-1350-kn-3-4) — Design of RCC Structure, chapter 10 - [Braced column 450x350, 4.0 m height](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Abraced-column-450x350-4-0-m) — Design of RCC Structure, chapter 10 - [Slender column 400x500, single curvature](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aslender-column-400x500-single-curvature) — Design of RCC Structure, chapter 10 ## Related in other subjects - [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× - [Built-up column, battens, 6 m unsupported, 1200 kN](https://www.bctengineer.com/question.md?key=semester_6%2Fdesign_of_steel_and_timber_structure%3Abuiltup-battens-1200kn-6m-restrained) — Design of Steel and Timber Structure, chapter 6 - [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× - [Brick column design for 80 kN axial load](https://www.bctengineer.com/question.md?key=semester_5%2Fconcrete_technology_and_masonry_structure%3Abrick-column-80kn-roof) — Concrete Technology and Masonry Structure, chapter 8