Fixed-end column 4 m, 1000 kN

Data and assumptions

• Column 300 mm (width $b$) × 350 mm (depth $D$); $P_u = 1000$ kN; $f_{ck} = 25$ MPa, $f_y = 415$ MPa. Clear cover 40 mm (IS 456 cl. 26.4.2.1), 8 mm ties.
• Moment about major axis: $M_{ux}$ = 100 kN·m.
• Both ends fixed and held in position: $l_e = 0.65L = 2.6$ m. M25 concrete and Fe415 steel are assumed (not stated). The 100 kN·m moment acts about the major axis. Section size is selected by trial.
• 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: $e_x = 4000/500 + 350/30 = 19.7$ mm (min. 20 mm); adopt 20.0 mm; $M_{ux,min} = P_u e = 20.0$ kN·m
• About y-y: $e_y = 4000/500 + 300/30 = 18.0$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 20.0$ kN·m

Design moments

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

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{1000000}{25 \times 300 \times 350} = 0.381$ 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

$M_{ux} = 100.0$ kN·m $\le M_{ux1} = 103.3$ kN·m. Minimum eccentricity about the minor axis: $P_u e_y = 20.0$ kN·m $\le M_{uy1} = 88.5$ kN·m. Both are satisfied; the section is safe.

Provide 14-16 mm bars ($A_{sc} = 2815$ mm², $p = 2.68\%$), 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) $= \min(300, 256, 300) = 256$ mm → provide 8 mm ties @ 250 mm c/c.

Detailing

CROSS-SECTION 300 x 350
    +-------------------------+
    |                         |
    |    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: 300 × 350 mm column with 14-16 mm longitudinal bars ($p = 2.68\%$) and 8 mm ties @ 250 mm c/c.