Biaxial column 400x500, equal steel all sides

Data and assumptions

• Column 400 mm (width $b$) × 500 mm (depth $D$); $P_u = 1200$ kN; $f_{ck} = 20$ MPa, $f_y = 415$ MPa. Clear cover 40 mm (IS 456 cl. 26.4.2.1), 8 mm ties.
• Moments: about major axis (x-x) $M_{ux}$ = 120 kN·m; about minor axis (y-y) $M_{uy}$ = 100 kN·m.
• Unsupported length not given; a short column with $L = 3$ m is assumed.
• $d'/D = 0.15$ in both directions: $d' = 0.15 \times 500 = 75$ mm about the major axis and $0.15 \times 400 = 60$ mm 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

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; $M_{ux,min} = P_u e = 27.2$ kN·m
• About y-y: $e_y = 3000/500 + 400/30 = 19.3$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 24.0$ kN·m

Design moments

• Design moments: $M_{ux} = 120.0$ kN·m, $M_{uy} = 100.0$ kN·m (not less than the minimum-eccentricity values).

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{1200000}{20 \times 400 \times 500} = 0.300$ 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), using the given $d'/D = 0.15$ about both axes.

Check

$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 20 \times 197185 + 0.75 \times 415 \times 2815 = 2651$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{1200}{2651} = 0.453$, so $\alpha_n = 1.42$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).

$$\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{120.0}{198.9}\right)^{1.42} + \left(\frac{100.0}{168.6}\right)^{1.42} = 0.963 \le 1.0$$

The interaction condition is satisfied; the section is safe.

Provide 14-16 mm bars ($A_{sc} = 2815$ mm², $p = 1.41\%$), 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

CROSS-SECTION 400 x 500
    +-------------------------+
    |                         |
    |   o     o     o     o   |
    |                         |
    |   o                 o   |
    |                         |
    |   o                 o   |
    |                         |
    |   o                 o   |
    |                         |
    |   o     o     o     o   |
    |                         |
    +-------------------------+
   o = 16 mm bars, ties 8 mm @ 250 c/c

Answer: 400 × 500 mm column with 14-16 mm longitudinal bars ($p = 1.41\%$) and 8 mm ties @ 250 mm c/c.