Isolated footing 500x500 with moment, M25

Data and assumptions

• Column 500 mm × 500 mm (long side $a=500$ mm along the footing length); footing concrete M25, steel Fe415; column concrete M25; column bars 8-25 mm.
• Service load $P = 1300$ kN, service moment $M = 100$ kN·m (about the other axis also 100 kN·m).
• Allowable (safe) bearing capacity of soil = 220 kN/m² at 1.7 m depth, soil unit weight 18.5 kN/m³.
• Loads are service loads; moments of 100 kN·m act about both axes, so a square footing is used.
• 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

Net safe bearing capacity = $q - \gamma D_f = 220 - 18.5 \times 1.7 = 188.55$ kN/m² (this takes care of the weight of footing and backfill).

$$A_{req} = \frac{P}{q_{net}} = \frac{1300.0}{188.55} = 6.89\ \text{m}^2$$

Eccentricity $e = M/P = 100.0/1300.0 = 0.077$ m (and 0.077 m in the other direction), which is less than $L/6$, so there is no tension under the base. The size is chosen so that

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

Try $L \times B = 3.1 \times 3.1$ m ($A = 9.61$ m²): $q_{max} = 175.6$ kN/m² $\le 188.6$ and $q_{min} = 95.0$ kN/m² $> 0$. Safe.

Factored soil pressure

$P_u = 1.5 \times 1300.0 = 1950.0$ kN, $M_u = 150.0$ kN·m. $p_u = \dfrac{P_u}{BL}\left(1 \pm \dfrac{6e}{L} \pm \dfrac{6e_y}{B}\right)$: $p_{u,max} = 263.3$ kN/m², $p_{u,min} = 142.5$ kN/m² (average 202.9 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 $L$: $(3100 - 500)/2 = 1300$ mm; along $B$: $(3100 - 500)/2 = 1300$ mm. With the maximum pressure assumed over the cantilever (conservative): $M_{u,L} = 689.8$ kN·m, $M_{u,B} = 689.8$ kN·m. Required depth for flexure: $M_{u,lim} = 0.138 f_{ck} b d^2$ 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 $bD$.

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

• Along $L$ side: $V_u = 579.6$ kN, $\tau_v = \dfrac{V_u}{B d} = 0.317$ MPa; $p_t = 0.184\%$, $\tau_c = 0.319$ MPa (IS 456 Table 19, SP 16 formula). $\tau_v < \tau_c$, safe.
• Along $B$ side: $V_u = 579.6$ kN, $\tau_v = 0.317$ MPa; $p_t = 0.184\%$, $\tau_c = 0.319$ MPa. Safe.

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

• Perimeter $b_o = 2[(a + d) + (c + d)] = 2[(500+590) + (500+590)] = 4360$ mm
• $V_u = P_u - p_u (a+d)(c+d) = 1950.0 - 202.9 \times 1.090 \times 1.090 = 1708.9$ kN (average pressure; the effect of moment transfer is neglected).
• $\tau_v = \dfrac{V_u}{b_o d} = 0.664$ MPa; $k_s = 0.5 + \beta_c = 1.00$, $\tau_c' = k_s \times 0.25\sqrt{f_{ck}} = 1.250$ MPa. $\tau_v < \tau_c'$, safe.

Development length

$L_d = \dfrac{0.87 f_y \phi}{4\tau_{bd}}$, $\tau_{bd} = 2.24$ MPa (M25, deformed bars): 16 mm bars (both directions) → $L_d = 645$ mm. Available length from column face (less 50 mm end cover): 1250 mm along $L$, 1250 mm along $B$. Satisfied in both directions.

Bearing stress and transfer of load from column

Column area $A_2 = 500\times500 = 0.250$ m². Bearing stress $= P_u/A_2 = 7.80$ MPa. Permissible bearing stress: column concrete $0.45 f_{ck} = 11.25$ MPa; footing concrete $0.45 f_{ck}\sqrt{A_1/A_2}$ with $\sqrt{A_1/A_2} \le 2$ gives up to 22.50 MPa. The lesser, 11.25 MPa, governs. Capacity of bearing = 2812 kN $\ge$ $P_u = 1950$ kN, so the whole load can be transferred by bearing. Provide minimum dowels of 0.5% of column area = 1250 mm²: use 8 bars of 25 mm (3927 mm²) as dowels, same as column bars. Dowel length in compression $L_{dc} = \dfrac{0.87 f_y\phi}{4 \times 1.25\,\tau_{bd}} = 806$ mm (and the same lap with the column bars). Available depth in footing = 650 - 50 - 32 = 568 mm $< L_{dc}$; dowels are bent at the bottom with 90° bends.

Detailing

PLAN                      ELEVATION
+------------------+      col
| #  #  #  #  #  # |      | |
| #  +------+  #  # |     _|_|__
| #  | col  |  #  # |    |_____|  <- D = 650 mm
| #  +------+  #  # |    bars both ways
| #  #  #  #  #  # |
+------------------+
 L = 3.1 m, B = 3.1 m

• Bottom mesh: 16 mm @ 185 c/c along $L$ (lower layer), 16 mm @ 185 c/c along $B$ (upper layer), 50 mm clear cover; dowels 8-25 mm.

Answer: Footing 3.1 m × 3.1 m × 650 mm thick; 16 mm @ 185 c/c along the length and 16 mm @ 185 c/c along the width.