Short column 300x400, 1400 kN

Check of the given section 300 × 400 mm

Even with the largest practical steel (for example 14-25 mm bars, $p = 5.7\%$), the interaction ratio is 1.34 > 1 for $P_u = 1400$ kN with the given moments. The given section is therefore too small, and the column must be enlarged.

Data and assumptions

• The section given in the question (300 × 400 mm) is checked first and found inadequate (see above), so a revised section 400 × 500 mm is designed.
• Column 400 mm (width $b$) × 500 mm (depth $D$); $P_u = 1400$ kN; $f_{ck} = 20$ 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) $M_{ux}$ = 200 kN·m; about minor axis (y-y) $M_{uy}$ = 110 kN·m.
• Effective length is taken equal to the unsupported length, 3 m.

Slenderness

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; $M_{ux,min} = P_u e = 31.7$ kN·m
• About y-y: $e_y = 3000/500 + 400/30 = 19.3$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 28.0$ kN·m

Design moments

• Design moments: $M_{ux} = 200.0$ kN·m, $M_{uy} = 110.0$ kN·m (not less than the minimum-eccentricity values).

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{1400000}{20 \times 400 \times 500} = 0.350$ 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

$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 20 \times 195602 + 0.75 \times 500 \times 4398 = 3410$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{1400}{3410} = 0.411$, so $\alpha_n = 1.35$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).

$$\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{200.0}{292.9}\right)^{1.35} + \left(\frac{110.0}{234.0}\right)^{1.35} = 0.958 \le 1.0$$

The interaction condition is satisfied; the section is safe.

Provide 14-20 mm bars ($A_{sc} = 4398$ mm², $p = 2.20\%$), within 0.8% to 4%.

Lateral ties

• Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(20/4, 6) = 6$ mm → provide 8 mm.
• Pitch by IS 456 $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(400, 320, 300) = 300$ mm. For ductile detailing, IS 13920 limits the spacing outside the confining zone to half the least lateral dimension = 200 mm → provide 8 mm ties @ 200 mm c/c outside $l_o$.

Ductile detailing of transverse steel (IS 13920)

• Special confining length at each end: $l_o \ge \max(D,\ L/6,\ 450) = \max(500, 500, 450) = 500$ mm → provide 500 mm from each joint face.
• Spacing of hoops within $l_o$: $\le \min(b/4,\ 6\phi_L,\ 100) = \min(100, 120, 100) = 100$ mm, and not less than 75 mm → 100 mm c/c.
• The bars are about 95 mm apart (more than 75 mm), so the intermediate bars are held by cross-ties in addition to the rectangular hoop.
• Area of one hoop leg: $A_{sh} \ge 0.18\, s\, h\, \dfrac{f_{ck}}{f_y}\left(\dfrac{A_g}{A_k} - 1\right)$, where $h$ is the longer c/c distance between adjacent hoop legs or cross-ties (not more than 300 mm), here $h \approx 120$ mm; $A_k = 320 \times 420 = 134400$ mm² (core measured to the outside of the hoop); $A_g = 200000$ mm². $A_{sh} = 0.18 \times 100 \times 120 \times \dfrac{20}{500} \times \left(\dfrac{200000}{134400} - 1\right) = 42.2$ mm²
• An 8 mm leg gives 50.3 mm² $>$ 42.2 mm², so 8 mm hoops and cross-ties @ 100 mm c/c are adequate. Hoops have 135° hooks with a 6φ (≥ 65 mm) extension.
• Outside $l_o$: 8 mm ties @ 200 mm c/c ($\le b/2 = 200$ mm).

Detailing

CROSS-SECTION 400 x 500
    +-------------------------+
    |   o     o     o     o   |
    |                         |
    |                         |
    |   o                 o   |
    |                         |
    |   o                 o   |
    |                         |
    |   o                 o   |
    |                         |
    |                         |
    |   o     o     o     o   |
    +-------------------------+
   o = 20 mm bars; hoop + cross-ties
   8 mm @ 100 c/c over 500 mm at each end,
   8 mm @ 200 c/c in the middle region

Answer: 400 × 500 mm column with 14-20 mm longitudinal bars ($p = 2.20\%$); 8 mm hoops with cross-ties @ 100 mm c/c over 500 mm at each end, and 8 mm ties @ 200 mm c/c in the middle region.