--- title: "Short column 300x400, 1400 kN" subject: "Design of RCC Structure" chapter: 10 key: "semester_7/design_of_rcc_structure:short-column-300x400-1400-kn" source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/short-column-300x400-1400-kn site: BCT Engineering --- # Short column 300x400, 1400 kN Design of RCC Structure · Chapter 10: Design of compression members: Columns - Asked in: 2076 Asoj, 12 marks - Video: [design of short column | design of column | short column design limit state method | IS456 design](https://www.youtube.com/watch?v=HU7c31UdXyA) — Civil Engineering Exam - 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/short-column-300x400-1400-kn - Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-10 ## Question Determine the longitudinal and transverse reinforcements in a short RC column subjected to factored axial load of 1400 kN and factored moment $M_{ux}$ of 200 kN-m and $M_{uy}$ of 110 kNm. The size of the column is 300 mm x 400 mm and unsupported length of 3 m. Adopt M20 concrete and Fe500 grade (TMT) bars. Also, do the ductile detailing of transverse reinforcements. ## Short answer The 300x400 section is inadequate; revise to 400x500. Use 14-20 mm bars (p=2.20%) and 8 mm ties @200 mm c/c outside the confining zone; provide special confining hoops @75 mm c/c over 500 mm at each end. ## Answer #### Check of the given section 300 × 400 mm Even with the largest practical steel (for example 14-25 mm bars, $p = 5.7\%$), the interaction ratio is 1.34 > 1 for $P_u = 1400$ kN with the given moments. The given section is therefore too small, and the column must be enlarged. #### Data and assumptions - The section given in the question (300 × 400 mm) is checked first and found inadequate (see above), so a revised section 400 × 500 mm is designed. - Column 400 mm (width $b$) × 500 mm (depth $D$); $P_u = 1400$ kN; $f_{ck} = 20$ 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) $M_{ux}$ = 200 kN·m; about minor axis (y-y) $M_{uy}$ = 110 kN·m. - Effective length is taken equal to the unsupported length, 3 m. #### Slenderness | Axis | $l_e$ (mm) | Dimension (mm) | $l_e/$dim | Type | |---|---|---|---|---| | x-x (about major) | 3000 | 500 | 6.00 | short | | y-y (about minor) | 3000 | 400 | 7.50 | 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: $e_x = 3000/500 + 500/30 = 22.7$ mm (min. 20 mm); adopt 22.7 mm; $M_{ux,min} = P_u e = 31.7$ kN·m - About y-y: $e_y = 3000/500 + 400/30 = 19.3$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 28.0$ kN·m #### Design moments - Design moments: $M_{ux} = 200.0$ kN·m, $M_{uy} = 110.0$ kN·m (not less than the minimum-eccentricity values). #### Longitudinal reinforcement $\dfrac{P_u}{f_{ck} b D} = \dfrac{1400000}{20 \times 400 \times 500} = 0.350$ 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 | $p$ (%) | $d'/D$ | $M_{ux1}$ | $M_{uy1}$ | $P_u/P_{uz}$ | $\alpha_n$ | $(M_{ux}/M_{ux1})^{\alpha_n}+(M_{uy}/M_{uy1})^{\alpha_n}$ | |---|---|---|---|---|---|---|---| | 8-25 φ | 1.96 | 0.121 | 283.1 | 210.2 | 0.432 | 1.39 | 1.025 ✗ | | 20-16 φ | 2.01 | 0.112 | 283.5 | 212.0 | 0.428 | 1.38 | 1.022 ✗ | | 14-20 φ | 2.20 | 0.116 | 292.9 | 234.0 | 0.411 | 1.35 | 0.958 ✓ | #### Check $P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 20 \times 195602 + 0.75 \times 500 \times 4398 = 3410$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{1400}{3410} = 0.411$, so $\alpha_n = 1.35$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8). $$ \left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{200.0}{292.9}\right)^{1.35} + \left(\frac{110.0}{234.0}\right)^{1.35} = 0.958 \le 1.0 $$ The interaction condition is satisfied; the section is **safe**. Provide **14-20 mm** bars ($A_{sc} = 4398$ mm², $p = 2.20\%$), within 0.8% to 4%. #### Lateral ties - Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(20/4, 6) = 6$ mm → provide 8 mm. - Pitch by IS 456 $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(400, 320, 300) = 300$ mm. For ductile detailing, IS 13920 limits the spacing outside the confining zone to half the least lateral dimension = 200 mm → provide 8 mm ties @ 200 mm c/c outside $l_o$. #### Ductile detailing of transverse steel (IS 13920) - Special confining length at each end: $l_o \ge \max(D,\ L/6,\ 450) = \max(500, 500, 450) = 500$ mm → provide 500 mm from each joint face. - Spacing of hoops within $l_o$: $\le \min(b/4,\ 6\phi_L,\ 100) = \min(100, 120, 100) = 100$ mm, and not less than 75 mm → 100 mm c/c. - The bars are about 95 mm apart (more than 75 mm), so the intermediate bars are held by cross-ties in addition to the rectangular hoop. - Area of one hoop leg: $A_{sh} \ge 0.18\, s\, h\, \dfrac{f_{ck}}{f_y}\left(\dfrac{A_g}{A_k} - 1\right)$, where $h$ is the longer c/c distance between adjacent hoop legs or cross-ties (not more than 300 mm), here $h \approx 120$ mm; $A_k = 320 \times 420 = 134400$ mm² (core measured to the outside of the hoop); $A_g = 200000$ mm². $A_{sh} = 0.18 \times 100 \times 120 \times \dfrac{20}{500} \times \left(\dfrac{200000}{134400} - 1\right) = 42.2$ mm² - An 8 mm leg gives 50.3 mm² $>$ 42.2 mm², so 8 mm hoops and cross-ties @ 100 mm c/c are adequate. Hoops have 135° hooks with a 6φ (≥ 65 mm) extension. - Outside $l_o$: 8 mm ties @ 200 mm c/c ($\le b/2 = 200$ mm). #### Detailing ```text CROSS-SECTION 400 x 500 +-------------------------+ | o o o o | | | | | | o o | | | | o o | | | | o o | | | | | | o o o o | +-------------------------+ o = 20 mm bars; hoop + cross-ties 8 mm @ 100 c/c over 500 mm at each end, 8 mm @ 200 c/c in the middle region ``` **Answer:** 400 × 500 mm column with 14-20 mm longitudinal bars ($p = 2.20\%$); 8 mm hoops with cross-ties @ 100 mm c/c over 500 mm at each end, and 8 mm ties @ 200 mm c/c in the middle region. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What happens when the given 300 x 400 mm section is checked? A. It is safe with minimum steel B. It is inadequate even with very heavy steel, so 400 x 500 mm is designed C. It fails only because of slenderness D. It needs only more ties Answer: B. Even with 14-25 mm bars the interaction ratio is 1.34 > 1, so the column is enlarged. (Key point: Revised section) 2. Which formula gives the minimum eccentricity used for the minimum moment? A. L/500 + D/30, not less than 20 mm B. L/300 + D/20 C. D/30 only D. L/12 Answer: A. IS 456 cl. 25.4 gives emin = L/500 + D/30 with a minimum of 20 mm, applied about each axis. (Key point: Minimum eccentricity) 3. For 400 x 500 mm column, L = 3000 mm, why is lo = 500 mm? A. It is the largest of D = 500, L/6 = 500 and 450 B. It is L/6 only C. It equals b D. It is twice the hoop spacing Answer: A. lo is the maximum of D, L/6 and 450 mm, which is 500 mm here. 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