Unbraced column 400x350, 6.0 m

Data and assumptions

• Column 350 mm (width $b$) × 400 mm (depth $D$); $P_u = 600$ 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) $M_{ux}$ = 100 kN·m; about minor axis (y-y) $M_{uy}$ = 50 kN·m.
• End conditions are not stated. The column is assumed to be held in position and restrained against rotation at one end, and restrained against rotation but free to sway at the other, so $l_e = 1.2L = 7.2$ m (IS 456 Table 28).
• The 100 kN·m moment acts about the major axis and the 50 kN·m moment about the minor axis; being design moments, they are taken as the larger end moments.
• 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: $e_x = 6000/500 + 400/30 = 25.3$ mm (min. 20 mm); adopt 25.3 mm; $M_{ux,min} = P_u e = 15.2$ kN·m
• About y-y: $e_y = 6000/500 + 350/30 = 23.7$ mm (min. 20 mm); adopt 23.7 mm; $M_{uy,min} = 14.2$ kN·m

Design moments

Additional moment due to slenderness: $M_a = \dfrac{P_u D}{2000}\left(\dfrac{l_e}{D}\right)^2$ (38.9 kN·m about x-x, 44.4 kN·m about y-y), reduced by $k = \dfrac{P_{uz}-P_u}{P_{uz}-P_b} \le 1$.

• For the selected steel: $P_{uz} = 2719$ kN, $P_{bx} = 316$ kN, $k_x = 0.88$, $M_{ax} = 34.3$ kN·m; $P_{by} = 10$ kN, $k_y = 0.78$, $M_{ay} = 34.8$ kN·m.
• $P_{by}$ is close to zero because, for bending about the minor axis, two-thirds of the heavy steel (the middle and far bars) lies in the tension zone at the balanced strain; its tensile force almost cancels the concrete compression ($0.36 f_{ck} D x_{u,max} \approx 395$ kN).
• In an unbraced column the additional moment is added to the end moment: $M_{ux} = 100 + 34.3 = 134.3$ kN·m, $M_{uy} = 50 + 34.8 = 84.8$ kN·m (not less than the minimum-eccentricity values).

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{600000}{20 \times 350 \times 400} = 0.214$ 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

$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 20 \times 135175 + 0.75 \times 415 \times 4825 = 2719$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{600}{2719} = 0.221$, so $\alpha_n = 1.03$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).

$$\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{134.3}{275.2}\right)^{1.03} + \left(\frac{84.8}{169.2}\right)^{1.03} = 0.965 \le 1.0$$

The interaction condition is satisfied; the section is safe.

Provide 6-32 mm bars ($A_{sc} = 4825$ mm², $p = 3.45\%$), within 0.8% to 4%.

Lateral ties

• Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(32/4, 6) = 8$ mm → provide 8 mm.
• Pitch $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(350, 512, 300) = 300$ mm → provide 8 mm ties @ 300 mm c/c. The middle bar on each 350 mm face is more than 75 mm from the corner bars, so it is held by a cross-tie.

Detailing

CROSS-SECTION 350 x 400
    +-------------------------+
    |                         |
    |    o       o       o    |
    |                         |
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    |                         |
    |                         |
    |                         |
    |                         |
    |    o       o       o    |
    |                         |
    +-------------------------+
   o = 32 mm bars, ties 8 mm @ 300 c/c

Answer: 350 × 400 mm column with 6-32 mm longitudinal bars ($p = 3.45\%$) and 8 mm ties @ 300 mm c/c.