Rectangular column with working-load moment

Data and assumptions

• Column 350 mm (width $b$) × 500 mm (depth $D$); $P_u = 1800$ 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: $M_{ux}$ = 225 kN·m.
• Working loads are converted to ultimate: $P_u = 1.5 \times 1200 = 1800$ kN, $M_u = 1.5 \times 150 = 225$ kN·m. Short column with $L = 3$ m is assumed (not stated). 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: $e_x = 3000/500 + 500/30 = 22.7$ mm (min. 20 mm); adopt 22.7 mm; $M_{ux,min} = P_u e = 40.8$ kN·m
• About y-y: $e_y = 3000/500 + 350/30 = 17.7$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 36.0$ kN·m

Design moments

• Design moments: $M_{ux} = 225.0$ kN·m (not less than the minimum-eccentricity values).

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{1800000}{25 \times 350 \times 500} = 0.411$ 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

$M_{ux} = 225.0$ kN·m $\le M_{ux1} = 252.3$ kN·m. Minimum eccentricity about the minor axis: $P_u e_y = 36.0$ kN·m $\le M_{uy1} = 125.7$ kN·m. Both are satisfied; the section is safe.

Provide 12-20 mm bars ($A_{sc} = 3770$ mm², $p = 2.15\%$), 6 on each 350 mm face, within 0.8% to 4%.

The two layers are $500 - 2 \times 58 = 384$ mm apart, which is more than the 300 mm maximum spacing along the periphery (IS 456 cl. 26.5.3.1), so also provide one 12 mm nominal bar at mid-depth of each 500 mm face (not counted in the design).

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) $= \min(350, 320, 300) = 300$ mm → provide 8 mm ties @ 300 mm c/c.

Detailing

CROSS-SECTION 350 x 500
    +-------------------------+
    |   o   o  o   o  o   o   |
    |                         |
    |                         |
    |                         |
    |                         |
    |   x                 x   |
    |                         |
    |                         |
    |                         |
    |                         |
    |   o   o  o   o  o   o   |
    +-------------------------+
   o = 20 mm bars, x = 12 mm nominal side bars, ties 8 mm @ 300 c/c

Answer: 350 × 500 mm column with 12-20 mm longitudinal bars ($p = 2.15\%$), 6 on each 350 mm face, plus one 12 mm nominal bar on each 500 mm face, and 8 mm ties @ 300 mm c/c.