Braced column 450x350, 4.0 m height

Data and assumptions

• Column 350 mm (width $b$) × 450 mm (depth $D$); $P_u = 700$ kN; $f_{ck} = 25$ 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) $M_{ux}$ = 100 kN·m; about minor axis (y-y) $M_{uy}$ = 80 kN·m.
• Held in position at both ends and restrained against rotation at one end: $l_e = 0.8L = 0.8 \times 4.0 = 3.2$ m (IS 456 Table 28).
• 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: $e_x = 4000/500 + 450/30 = 23.0$ mm (min. 20 mm); adopt 23.0 mm; $M_{ux,min} = P_u e = 16.1$ kN·m
• About y-y: $e_y = 4000/500 + 350/30 = 19.7$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 14.0$ kN·m

Design moments

• Design moments: $M_{ux} = 100.0$ kN·m, $M_{uy} = 80.0$ kN·m (not less than the minimum-eccentricity values).

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{700000}{25 \times 350 \times 450} = 0.178$ 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

$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 25 \times 155087 + 0.75 \times 415 \times 2413 = 2496$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{700}{2496} = 0.280$, so $\alpha_n = 1.13$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).

$$\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{100.0}{194.4}\right)^{1.13} + \left(\frac{80.0}{142.0}\right)^{1.13} = 0.992 \le 1.0$$

The interaction condition is satisfied; the section is safe.

Provide 12-16 mm bars ($A_{sc} = 2413$ mm², $p = 1.53\%$), 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) $= \min(350, 256, 300) = 256$ mm → provide 8 mm ties @ 250 mm c/c.

Detailing

CROSS-SECTION 350 x 450
    +-------------------------+
    |   o     o     o     o   |
    |                         |
    |                         |
    |   o                 o   |
    |                         |
    |                         |
    |                         |
    |   o                 o   |
    |                         |
    |                         |
    |   o     o     o     o   |
    +-------------------------+
   o = 16 mm bars, ties 8 mm @ 250 c/c

Answer: 350 × 450 mm column with 12-16 mm longitudinal bars ($p = 1.53\%$) and 8 mm ties @ 250 mm c/c.