--- title: "Biaxial column 500x600, 1300 kN" subject: "Design of RCC Structure" chapter: 10 key: "semester_7/design_of_rcc_structure:biaxial-column-500x600-1300-kn" source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/biaxial-column-500x600-1300-kn site: BCT Engineering --- # Biaxial column 500x600, 1300 kN Design of RCC Structure · Chapter 10: Design of compression members: Columns - Asked in: 2075 Asoj, 15 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/biaxial-column-500x600-1300-kn - Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-10 ## Question Determine the longitudinal and transverse reinforcement in bi-axially loaded column having the following parameters: Unsupported length of column = 3.10 m; Size of column = 500 mm x 600 mm; Factored moment, $M_{ux}$ = 125 kN.m; Factored load, $P_u$ = 1300 kN; Factored moment, $M_{uy}$ = 200 kN.m. Use M25 concrete and Fe 500 steel. Take reinforcement in four side. Sketch the details. ## Short answer Short column; no slenderness effect. Use 12-16 mm bars (p=0.80%) and 8 mm ties @250 mm c/c; biaxial interaction ratio is 0.796, so safe. ## Answer #### Data and assumptions - Column 500 mm (width $b$) × 600 mm (depth $D$); $P_u = 1300$ 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) $M_{ux}$ = 125 kN·m; about minor axis (y-y) $M_{uy}$ = 200 kN·m. - 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) | 3100 | 600 | 5.17 | short | | y-y (about minor) | 3100 | 500 | 6.20 | 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: $e_x = 3100/500 + 600/30 = 26.2$ mm (min. 20 mm); adopt 26.2 mm; $M_{ux,min} = P_u e = 34.1$ kN·m - About y-y: $e_y = 3100/500 + 500/30 = 22.9$ mm (min. 20 mm); adopt 22.9 mm; $M_{uy,min} = 29.7$ kN·m #### Design moments - Design moments: $M_{ux} = 125.0$ kN·m, $M_{uy} = 200.0$ kN·m (not less than the minimum-eccentricity values). #### Longitudinal reinforcement $\dfrac{P_u}{f_{ck} b D} = \dfrac{1300000}{25 \times 500 \times 600} = 0.173$ 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 | $p$ (%) | $d'/D$ | $M_{ux1}$ | $M_{uy1}$ | $P_u/P_{uz}$ | $\alpha_n$ | $(M_{ux}/M_{ux1})^{\alpha_n}+(M_{uy}/M_{uy1})^{\alpha_n}$ | |---|---|---|---|---|---|---|---| | 12-16 φ | 0.80 | 0.093 | 409.9 | 333.0 | 0.306 | 1.18 | 0.796 ✓ | #### Check $P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 25 \times 297587 + 0.75 \times 500 \times 2413 = 4253$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{1300}{4253} = 0.306$, so $\alpha_n = 1.18$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8). $$ \left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{125.0}{409.9}\right)^{1.18} + \left(\frac{200.0}{333.0}\right)^{1.18} = 0.796 \le 1.0 $$ The interaction condition is satisfied; the section is **safe**. Provide **12-16 mm** bars ($A_{sc} = 2413$ 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) $= \min(500, 256, 300) = 256$ mm → provide 8 mm ties @ 250 mm c/c. #### Detailing ```text CROSS-SECTION 500 x 600 +-------------------------+ | o o o o | | | | | | o o | | | | | | | | o o | | | | | | o o o o | +-------------------------+ o = 16 mm bars, ties 8 mm @ 250 c/c ``` **Answer:** 500 × 600 mm column with 12-16 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. Which moment acts about the major axis in this column? A. Muy = 200 kN·m B. Mux = 125 kN·m, about the axis with depth 600 mm C. Both act about the minor axis D. Mux = 200 kN·m Answer: B. Mux = 125 kN·m acts about the major axis (depth 600 mm) and Muy = 200 kN·m about the minor axis (depth 500 mm). (Key point: Moment axes) 2. Which steel percentage limit governs the choice of 12-16φ (p = 0.80%)? A. Maximum 6% B. Minimum 0.8% of gross area C. Minimum 2% D. Maximum 0.8% Answer: B. IS 456 requires at least 0.8% longitudinal steel in columns, and 12-16φ just meets it. (Key point: Minimum steel) 3. What is Pu/Puz and what does it decide? A. 1300/4253 = 0.306, giving αn = 1.18 B. 4253/1300 = 3.27, giving αn = 3.27 C. 0.173, giving αn = 1.0 D. 0.806, giving αn = 2.0 Answer: A. Pu/Puz = 1300/4253 = 0.306; interpolating between 0.2 and 0.8 gives αn about 1.18. (Key point: Pu/Puz ratio) 4. Why is the tie pitch 250 mm for this column? A. Least lateral dimension 500 mm governs B. 16 x 16 = 256 mm is the smallest limit, so 250 mm is used C. 300 mm is the smallest limit D. It is 8 x bar diameter Answer: B. Pitch must not exceed the least of 500, 16 x 16 = 256 and 300 mm, so 250 mm is adopted. 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