Short column 300x500, 2000 kN, M30

Data and assumptions

• Column 300 mm (width $b$) × 500 mm (depth $D$); $P_u = 2000$ kN; $f_{ck} = 30$ 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) $M_{ux}$ = 190 kN·m; about minor axis (y-y) $M_{uy}$ = 95 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

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; $M_{ux,min} = P_u e = 45.3$ kN·m
• About y-y: $e_y = 3000/500 + 300/30 = 16.0$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 40.0$ kN·m

Design moments

• Design moments: $M_{ux} = 190.0$ kN·m, $M_{uy} = 95.0$ kN·m (not less than the minimum-eccentricity values).

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{2000000}{30 \times 300 \times 500} = 0.444$ 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 $d' = 40 + 8 + \phi/2$.

Check

$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 30 \times 144973 + 0.75 \times 500 \times 5027 = 3842$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{2000}{3842} = 0.521$, so $\alpha_n = 1.53$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).

$$\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{190.0}{299.9}\right)^{1.53} + \left(\frac{95.0}{148.7}\right)^{1.53} = 0.999 \le 1.0$$

The interaction condition is satisfied; the section is safe.

Provide 16-20 mm bars ($A_{sc} = 5027$ mm², $p = 3.35\%$), within 0.8% to 4%.

Lateral ties

• Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(20/4, 6) = 6$ mm → provide 8 mm.
• Pitch $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(300, 320, 300) = 300$ mm → provide 8 mm ties @ 300 mm c/c.

Detailing

CROSS-SECTION 300 x 500
    +-------------------------+
    |    o   o   o   o   o    |
    |                         |
    |                         |
    |    o               o    |
    |                         |
    |    o               o    |
    |                         |
    |    o               o    |
    |                         |
    |                         |
    |    o   o   o   o   o    |
    +-------------------------+
   o = 20 mm bars, ties 8 mm @ 300 c/c

Answer: 300 × 500 mm column with 16-20 mm longitudinal bars ($p = 3.35\%$) and 8 mm ties @ 300 mm c/c.