Footing, 450x500 column, 6#20, uniaxial moment

Data and assumptions

• Column 500 mm × 450 mm (long side $a=500$ mm along the footing length); footing concrete M20, steel Fe500; column concrete M20; column bars 6-20 mm.
• Given loads are factored: $P_u = 1100$ kN, $M_u = 120$ kN·m. Service loads = factored / 1.5: $P = 733.3$ kN, $M = 80.0$ kN·m.
• Allowable (safe) bearing capacity of soil = 100 kN/m² at 1.5 m depth.
• Column concrete grade assumed M20 (same as footing). Moment acts about the minor axis so that it varies the pressure along the 500 mm side.
• Clear cover 50 mm for the footing (IS 456 cl. 26.4.2.2); effective depth $d = D - 60$ mm (cover + half bar). Load factor 1.5.

Size of footing

Weight of footing and backfill is taken as 10% of the column load.

$$A_{req} = \frac{1.1\,P}{q_a} = \frac{1.1 \times 733.3}{100} = 8.07\ \text{m}^2$$

Eccentricity $e = M/P = 80.0/733.3 = 0.109$ m, which is less than $L/6$, so there is no tension under the base. The size is chosen so that

$$q_{max} = 1.1\frac{P}{BL}\left(1 + \frac{6e}{L}\right) \le q_a$$

Try $L \times B = 4.1 \times 2.3$ m ($A = 9.43$ m²): $q_{max} = 99.2$ kN/m² $\le 100.0$ and $q_{min} = 71.9$ kN/m² $> 0$. Safe.

Factored soil pressure

$P_u = 1.5 \times 733.3 = 1100.0$ kN, $M_u = 120.0$ kN·m. $p_u = \dfrac{P_u}{BL}\left(1 \pm \dfrac{6e}{L}\right)$: $p_{u,max} = 135.3$ kN/m², $p_{u,min} = 98.0$ kN/m² (average 116.6 kN/m²). Self-weight does not cause bending in the footing, so it is excluded.

Bending moments and reinforcement

Cantilever projections from the column face: along $L$: $(4100 - 500)/2 = 1800$ mm; along $B$: $(2300 - 450)/2 = 925$ mm. Along $L$ (critical section at the column face on the side of maximum pressure, linear pressure): $M_{u,L} = B\left[p_f\dfrac{x^2}{2} + (p_{max}-p_f)\dfrac{x^2}{3}\right]$, with $p_f = 118.9$ kN/m² at the face and $x = 1.8$ m: $M_{u,L} = 483.7$ kN·m. Along $B$ (average pressure): $M_{u,B} = 204.6$ kN·m. Required depth for flexure: $M_{u,lim} = 0.133 f_{ck} b d^2$ is far larger than the moment for $d = 590$ mm, so the depth is governed by shear (one-way or punching). Trial $D = 650$ mm, $d = 590$ mm, found by the shear checks below.

$A_{st} = \dfrac{0.5 f_{ck}}{f_y}\left[1 - \sqrt{1 - \dfrac{4.6 M_u}{f_{ck} b d^2}}\right] b d$; minimum steel = 0.12% of $bD$.

For the rectangular footing, the short-direction steel is placed in a central band of width $B = 2300$ mm in the proportion $\dfrac{2}{\beta+1} = \dfrac{2}{1.78+1} = 0.719$ (IS 456 cl. 34.3.1), where $\beta = L/B$.

• Central band: $0.719 \times 3198 = 2299$ mm² → 12 mm @ 105 mm c/c (21 bars).
• Each outer strip (900 mm wide): remaining steel 450 mm² or minimum 0.12% = 702 mm², whichever is more → 12 mm @ 125 mm c/c (7 bars).

Check for one-way shear (at distance $d$ from the column face)

• Along $L$ side: $V_u = 361.2$ kN, $\tau_v = \dfrac{V_u}{B d} = 0.266$ MPa; $p_t = 0.147\%$, $\tau_c = 0.288$ MPa (IS 456 Table 19, SP 16 formula). $\tau_v < \tau_c$, safe.
• Along $B$ side: $V_u = 160.2$ kN, $\tau_v = 0.066$ MPa; $p_t = 0.137\%$, $\tau_c = 0.288$ MPa. Safe.

Check for two-way (punching) shear (at $d/2$ from the column face)

• Perimeter $b_o = 2[(a + d) + (c + d)] = 2[(500+590) + (450+590)] = 4260$ mm
• $V_u = P_u - p_u (a+d)(c+d) = 1100.0 - 116.6 \times 1.090 \times 1.040 = 967.8$ kN (average pressure; the effect of moment transfer is neglected).
• $\tau_v = \dfrac{V_u}{b_o d} = 0.385$ MPa; $k_s = 0.5 + \beta_c = 1.00$, $\tau_c' = k_s \times 0.25\sqrt{f_{ck}} = 1.118$ MPa. $\tau_v < \tau_c'$, safe.

Development length

$L_d = \dfrac{0.87 f_y \phi}{4\tau_{bd}}$, $\tau_{bd} = 1.92$ MPa (M20, deformed bars): 12 mm bars (both directions) → $L_d = 680$ mm. Available length from column face (less 50 mm end cover): 1750 mm along $L$, 875 mm along $B$. Satisfied in both directions.

Bearing stress and transfer of load from column

Column area $A_2 = 500\times450 = 0.225$ m². Bearing stress $= P_u/A_2 = 4.89$ MPa. Permissible bearing stress: column concrete $0.45 f_{ck} = 9.00$ MPa; footing concrete $0.45 f_{ck}\sqrt{A_1/A_2}$ with $\sqrt{A_1/A_2} \le 2$ gives up to 18.00 MPa. The lesser, 9.00 MPa, governs. Capacity of bearing = 2025 kN $\ge$ $P_u = 1100$ kN, so the whole load can be transferred by bearing. Provide minimum dowels of 0.5% of column area = 1125 mm²: use 6 bars of 20 mm (1885 mm²) as dowels, same as column bars. Dowel length in compression $L_{dc} = \dfrac{0.87 f_y\phi}{4 \times 1.25\,\tau_{bd}} = 906$ mm (and the same lap with the column bars). Available depth in footing = 650 - 50 - 24 = 576 mm $< L_{dc}$; dowels are bent at the bottom with 90° bends.

Detailing

PLAN                      ELEVATION
+------------------+      col
| #  #  #  #  #  # |      | |
| #  +------+  #  # |     _|_|__
| #  | col  |  #  # |    |_____|  <- D = 650 mm
| #  +------+  #  # |    bars both ways
| #  #  #  #  #  # |
+------------------+
 L = 4.1 m, B = 2.3 m

• Bottom mesh: 12 mm @ 130 c/c along $L$ (lower layer); along $B$ (upper layer), 12 mm @ 105 c/c in the central 2.3 m band and 12 mm @ 125 c/c in each outer strip, 50 mm clear cover; dowels 6-20 mm.

Answer: Footing 4.1 m × 2.3 m × 650 mm thick; 12 mm @ 130 c/c along the length; along the width, 12 mm @ 105 c/c in the central 2.3 m band and 12 mm @ 125 c/c in the outer strips.