Biaxial column 400x600, Fe500

Data and assumptions

• Column 400 mm (width $b$) × 600 mm (depth $D$); $P_u = 1000$ 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.
• Both ends fixed: $l_e = 0.65L = 2.6$ m. $M_{ux}=125$ kN·m acts about the major axis (depth 600 mm), $M_{uy}=200$ kN·m about the minor axis (depth 400 mm).
• 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

Minimum eccentricity (cl. 25.4)

$$e_{min} = \max\left(\frac{L}{500} + \frac{D}{30},\ 20\right)$$

• About x-x: $e_x = 4000/500 + 600/30 = 28.0$ mm (min. 20 mm); adopt 28.0 mm; $M_{ux,min} = P_u e = 28.0$ kN·m
• About y-y: $e_y = 4000/500 + 400/30 = 21.3$ mm (min. 20 mm); adopt 21.3 mm; $M_{uy,min} = 21.3$ 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

$\dfrac{P_u}{f_{ck} b D} = \dfrac{1000000}{25 \times 400 \times 600} = 0.167$ 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

$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 25 \times 236230 + 0.75 \times 500 \times 3770 = 4071$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{1000}{4071} = 0.246$, so $\alpha_n = 1.08$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).

$$\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{125.0}{457.9}\right)^{1.08} + \left(\frac{200.0}{277.6}\right)^{1.08} = 0.950 \le 1.0$$

The interaction condition is satisfied; the section is safe.

Provide 12-20 mm bars ($A_{sc} = 3770$ mm², $p = 1.57\%$), 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) $= \min(400, 320, 300) = 300$ mm → provide 8 mm ties @ 300 mm c/c.

Detailing

CROSS-SECTION 400 x 600
    +-------------------------+
    |   o     o     o     o   |
    |                         |
    |                         |
    |   o                 o   |
    |                         |
    |                         |
    |                         |
    |   o                 o   |
    |                         |
    |                         |
    |   o     o     o     o   |
    +-------------------------+
   o = 20 mm bars, ties 8 mm @ 300 c/c

Answer: 400 × 600 mm column with 12-20 mm longitudinal bars ($p = 1.57\%$) and 8 mm ties @ 300 mm c/c.