--- title: "Short column 300x450 with ductile detailing" subject: "Design of RCC Structure" chapter: 10 key: "semester_7/design_of_rcc_structure:short-column-300x450-with-ductile-detailing" source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/short-column-300x450-with-ductile-detailing site: BCT Engineering --- # Short column 300x450 with ductile detailing Design of RCC Structure · Chapter 10: Design of compression members: Columns - Asked in: 2074 Chaitra, 14 marks - 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-300x450-with-ductile-detailing - Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-10 ## Question Design a short RC column with following data: Unsupported length = 3.0 m; Factored load, $P_u$ = 1550 kN; Factored moments: $M_{ux}$ = 130 kN.m, $M_{uy}$ = 90 kN.m; Size of column = 300 x 450 mm. Do ductile detailing for transverse steel. ## Short answer Assuming M25 and Fe500, use 10-25 mm bars (p=3.64%) and 8 mm ties @150 mm c/c outside the confining zone; special confining hoops @75 mm c/c over 500 mm at each end. ## Answer #### Data and assumptions - Column 300 mm (width $b$) × 450 mm (depth $D$); $P_u = 1550$ 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) $M_{ux}$ = 130 kN·m; about minor axis (y-y) $M_{uy}$ = 90 kN·m. - Grades of concrete and steel are not given; M25 and Fe500 are assumed. - 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) | 3000 | 450 | 6.67 | short | | y-y (about minor) | 3000 | 300 | 10.00 | 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: $e_x = 3000/500 + 450/30 = 21.0$ mm (min. 20 mm); adopt 21.0 mm; $M_{ux,min} = P_u e = 32.5$ kN·m - About y-y: $e_y = 3000/500 + 300/30 = 16.0$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 31.0$ kN·m #### Design moments - Design moments: $M_{ux} = 130.0$ kN·m, $M_{uy} = 90.0$ kN·m (not less than the minimum-eccentricity values). #### Longitudinal reinforcement $\dfrac{P_u}{f_{ck} b D} = \dfrac{1550000}{25 \times 300 \times 450} = 0.459$ 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 | 2.91 | 0.134 | 203.7 | 114.1 | 0.526 | 1.54 | 1.194 ✗ | | 14-20 φ | 3.26 | 0.129 | 214.7 | 129.0 | 0.497 | 1.50 | 1.056 ✗ | | 10-25 φ | 3.64 | 0.134 | 233.4 | 142.6 | 0.469 | 1.45 | 0.942 ✓ | #### Check $P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 25 \times 130091 + 0.75 \times 500 \times 4909 = 3304$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{1550}{3304} = 0.469$, so $\alpha_n = 1.45$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8). $$ \left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{130.0}{233.4}\right)^{1.45} + \left(\frac{90.0}{142.6}\right)^{1.45} = 0.942 \le 1.0 $$ The interaction condition is satisfied; the section is **safe**. Provide **10-25 mm** bars ($A_{sc} = 4909$ mm², $p = 3.64\%$), within 0.8% to 4%. #### Lateral ties - Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(25/4, 6) = 6.25$ mm → provide 8 mm. - Pitch $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(300, 400, 300) = 300$ mm by IS 456. IS 13920 reduces this to 150 mm outside $l_o$ (see below), so 8 mm ties @ 150 mm c/c are provided. #### Ductile detailing of transverse steel (IS 13920) - Special confining length at each end: $l_o \ge \max(D,\ L/6,\ 450) = \max(450, 500, 450) = 500$ mm → provide 500 mm from each joint face. - Spacing of hoops within $l_o$: $\le \min(b/4, 100) = 75$ mm and not less than 75 mm → 75 mm c/c. - Area of one hoop leg (bar): $A_{sh} \ge 0.18\, s\, h\, \dfrac{f_{ck}}{f_y}\left(\dfrac{A_g}{A_k} - 1\right)$, where $h$ is the longer distance between parallel hoop legs or cross-ties (not more than 300 mm), $A_k = 220 \times 370 = 81400$ mm² (core measured to the outside of the hoop, 40 mm cover) and $A_g = 135000$ mm². - A perimeter hoop alone gives $h = 370$ mm $> 300$ mm, so cross-ties are provided at every intermediate bar (two across the 450 mm direction and one across the 300 mm direction). The leg spacing is then about $370/3 \approx 125$ mm and $220/2 = 110$ mm, so $h \approx 125$ mm. $A_{sh} = 0.18 \times 75 \times 125 \times \dfrac{25}{500} \times \left(\dfrac{135000}{81400} - 1\right) = 55.6$ mm² - One 8 mm leg (50.3 mm²) is not enough, so **10 mm** hoops and cross-ties (78.5 mm² $>$ 55.6 mm²) are used within $l_o$, with 135° hooks and a 6φ (≥ 65 mm) extension. With 10 mm hoops, $d'$ increases by 2 mm and the interaction value becomes about 0.965, still $\le 1.0$. - Outside $l_o$, the spacing must not exceed half the least lateral dimension, $300/2 = 150$ mm, so 8 mm ties @ 150 mm c/c are provided. #### Detailing ```text CROSS-SECTION 300 x 450 +-------------------------+ | | | o o o | | | | | | o o | | | | o o | | | | | | o o o | | | +-------------------------+ o = 25 mm bars within l_o (500 mm at each end): 10 mm hoops + cross-ties @ 75 c/c elsewhere: 8 mm ties @ 150 c/c ``` **Answer:** 300 × 450 mm column with 10-25 mm longitudinal bars ($p = 3.64\%$); 10 mm hoops with cross-ties @ 75 mm c/c over 500 mm at each end, and 8 mm ties @ 150 mm c/c over the rest of the height. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The question gives no grades. What does the answer do? A. Refuses to design B. Assumes M25 concrete and Fe500 steel C. Uses M20 and Fe415 D. Uses mild steel Answer: B. Missing data are stated as assumptions: M25 and Fe500. (Key point: Assumed grades) 2. Why were 8-25φ and 14-20φ rejected? A. Steel exceeded 4% B. Interaction sums 1.194 and 1.056 exceeded 1 C. They are not available sizes D. Pu/Puz was above 1 Answer: B. Both interaction values exceeded 1.0, so more steel (10-25φ giving 0.942) was needed. (Key point: Why trials failed) 3. For b = 300 mm, why is hoop spacing 75 mm within lo? A. b/4 = 75 mm, which is below 100 mm B. It is 100 mm always C. It is L/6 D. It is 16 times bar diameter Answer: A. Spacing is limited to min(b/4, 100) = 75 mm and must not be less than 75 mm. (Key point: Hoop spacing in lo) 4. Ash required is 164.5 mm². What was provided? A. 3 legs of 8 mm = 151 mm² B. 4 legs of 8 mm = 201 mm² C. 2 legs of 8 mm = 101 mm² D. 1 leg of 10 mm = 79 mm² Answer: B. Three legs give only 151 mm², less than 164.5; four legs give 201 mm², which is adequate. 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