Short column 300x450 with ductile detailing

Data and assumptions

• Column 300 mm (width $b$) × 450 mm (depth $D$); $P_u = 1550$ 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) $M_{ux}$ = 130 kN·m; about minor axis (y-y) $M_{uy}$ = 90 kN·m.
• Grades of concrete and steel are not given; M25 and Fe500 are assumed.
• 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: $e_x = 3000/500 + 450/30 = 21.0$ mm (min. 20 mm); adopt 21.0 mm; $M_{ux,min} = P_u e = 32.5$ kN·m
• About y-y: $e_y = 3000/500 + 300/30 = 16.0$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 31.0$ kN·m

Design moments

• Design moments: $M_{ux} = 130.0$ kN·m, $M_{uy} = 90.0$ kN·m (not less than the minimum-eccentricity values).

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{1550000}{25 \times 300 \times 450} = 0.459$ 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

$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 25 \times 130091 + 0.75 \times 500 \times 4909 = 3304$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{1550}{3304} = 0.469$, so $\alpha_n = 1.45$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).

$$\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{130.0}{233.4}\right)^{1.45} + \left(\frac{90.0}{142.6}\right)^{1.45} = 0.942 \le 1.0$$

The interaction condition is satisfied; the section is safe.

Provide 10-25 mm bars ($A_{sc} = 4909$ mm², $p = 3.64\%$), within 0.8% to 4%.

Lateral ties

• Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(25/4, 6) = 6.25$ mm → provide 8 mm.
• Pitch $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(300, 400, 300) = 300$ mm by IS 456. IS 13920 reduces this to 150 mm outside $l_o$ (see below), so 8 mm ties @ 150 mm c/c are provided.

Ductile detailing of transverse steel (IS 13920)

• Special confining length at each end: $l_o \ge \max(D,\ L/6,\ 450) = \max(450, 500, 450) = 500$ mm → provide 500 mm from each joint face.
• Spacing of hoops within $l_o$: $\le \min(b/4, 100) = 75$ mm and not less than 75 mm → 75 mm c/c.
• Area of one hoop leg (bar): $A_{sh} \ge 0.18\, s\, h\, \dfrac{f_{ck}}{f_y}\left(\dfrac{A_g}{A_k} - 1\right)$, where $h$ is the longer distance between parallel hoop legs or cross-ties (not more than 300 mm), $A_k = 220 \times 370 = 81400$ mm² (core measured to the outside of the hoop, 40 mm cover) and $A_g = 135000$ mm².
• A perimeter hoop alone gives $h = 370$ mm $> 300$ mm, so cross-ties are provided at every intermediate bar (two across the 450 mm direction and one across the 300 mm direction). The leg spacing is then about $370/3 \approx 125$ mm and $220/2 = 110$ mm, so $h \approx 125$ mm. $A_{sh} = 0.18 \times 75 \times 125 \times \dfrac{25}{500} \times \left(\dfrac{135000}{81400} - 1\right) = 55.6$ mm²
• One 8 mm leg (50.3 mm²) is not enough, so 10 mm hoops and cross-ties (78.5 mm² $>$ 55.6 mm²) are used within $l_o$, with 135° hooks and a 6φ (≥ 65 mm) extension. With 10 mm hoops, $d'$ increases by 2 mm and the interaction value becomes about 0.965, still $\le 1.0$.
• Outside $l_o$, the spacing must not exceed half the least lateral dimension, $300/2 = 150$ mm, so 8 mm ties @ 150 mm c/c are provided.

Detailing

CROSS-SECTION 300 x 450
    +-------------------------+
    |                         |
    |    o       o       o    |
    |                         |
    |                         |
    |    o               o    |
    |                         |
    |    o               o    |
    |                         |
    |                         |
    |    o       o       o    |
    |                         |
    +-------------------------+
   o = 25 mm bars
   within l_o (500 mm at each end): 10 mm hoops + cross-ties @ 75 c/c
   elsewhere: 8 mm ties @ 150 c/c

Answer: 300 × 450 mm column with 10-25 mm longitudinal bars ($p = 3.64\%$); 10 mm hoops with cross-ties @ 75 mm c/c over 500 mm at each end, and 8 mm ties @ 150 mm c/c over the rest of the height.