Column 350x350, 1440 kN, ductile detailing

Check of the given section 350 × 350 mm

Even with the largest practical steel (for example 14-25 mm bars, $p = 5.6\%$), the interaction ratio is 1.77 > 1 for $P_u = 1440$ 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 (350 × 350 mm) is checked first and found inadequate (see above), so a revised section 450 × 450 mm is designed.
• Column 450 mm (width $b$) × 450 mm (depth $D$); $P_u = 1440$ 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) $M_{ux}$ = 195 kN·m; about minor axis (y-y) $M_{uy}$ = 180 kN·m.
• Effective length is taken equal to the unsupported length, 3.6 m.

Slenderness

Minimum eccentricity (cl. 25.4)

$$e_{min} = \max\left(\frac{L}{500} + \frac{D}{30},\ 20\right)$$

• About x-x: $e_x = 3600/500 + 450/30 = 22.2$ mm (min. 20 mm); adopt 22.2 mm; $M_{ux,min} = P_u e = 32.0$ kN·m
• About y-y: $e_y = 3600/500 + 450/30 = 22.2$ mm (min. 20 mm); adopt 22.2 mm; $M_{uy,min} = 32.0$ kN·m

Design moments

• Design moments: $M_{ux} = 195.0$ kN·m, $M_{uy} = 180.0$ kN·m (not less than the minimum-eccentricity values).

Longitudinal reinforcement

$\dfrac{P_u}{f_{ck} b D} = \dfrac{1440000}{20 \times 450 \times 450} = 0.356$ 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

$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 20 \times 196610 + 0.75 \times 500 \times 5890 = 3978$ kN $\dfrac{P_u}{P_{uz}} = \dfrac{1440}{3978} = 0.362$, so $\alpha_n = 1.27$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).

$$\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{195.0}{330.6}\right)^{1.27} + \left(\frac{180.0}{330.6}\right)^{1.27} = 0.974 \le 1.0$$

The interaction condition is satisfied; the section is safe.

Provide 12-25 mm bars ($A_{sc} = 5890$ mm², $p = 2.91\%$), within 0.8% to 4%.

Lateral ties

• Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(25/4, 6) = 6$ mm → provide 8 mm.
• Pitch by IS 456 $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(450, 400, 300) = 300$ mm. For ductile detailing, IS 13920 limits the spacing outside the confining zone to half the least lateral dimension = 225 mm → provide 8 mm ties @ 220 mm c/c outside $l_o$.

Ductile detailing of transverse steel (IS 13920)

• Special confining length at each end: $l_o \ge \max(D,\ L/6,\ 450) = \max(450, 600, 450) = 600$ mm → provide 600 mm from each joint face.
• Spacing of hoops within $l_o$: $\le \min(b/4,\ 6\phi_L,\ 100) = \min(112, 150, 100) = 100$ mm, and not less than 75 mm → 100 mm c/c.
• The bars are about 110 mm apart (more than 75 mm), so the two intermediate bars on each face are held by cross-ties: a rectangular hoop plus 2 cross-ties in each direction.
• 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 $h \approx 135$ mm; $A_k = 370 \times 370 = 136900$ mm² (core measured to the outside of the hoop); $A_g = 202500$ mm². $A_{sh} = 0.18 \times 100 \times 135 \times \dfrac{20}{500} \times \left(\dfrac{202500}{136900} - 1\right) = 46.6$ mm²
• An 8 mm leg gives 50.3 mm² $>$ 46.6 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 @ 220 mm c/c ($\le b/2 = 225$ mm).

Detailing

CROSS-SECTION 450 x 450
    +-------------------------+
    |                         |
    |  o      o     o      o  |
    |                         |
    |                         |
    |  o                   o  |
    |                         |
    |  o                   o  |
    |                         |
    |                         |
    |  o      o     o      o  |
    |                         |
    +-------------------------+
   o = 25 mm bars; hoop + 2 cross-ties each way
   8 mm @ 100 c/c over 600 mm at each end,
   8 mm @ 220 c/c in the middle region

Answer: 450 × 450 mm column with 12-25 mm longitudinal bars ($p = 2.91\%$); 8 mm hoops with cross-ties @ 100 mm c/c over 600 mm at each end, and 8 mm ties @ 220 mm c/c in the middle region.