Shear reinforcement and punching shear in footings

Procedure for design of shear reinforcement (beam)

1. Compute factored shear $V_u$ at the critical section (distance $d$ from the support face).
2. Nominal shear stress $\tau_v=\dfrac{V_u}{bd}$; check $\tau_v\le\tau_{c,max}$.
3. Find $\tau_c$ from IS 456 Table 19 using $p_t=\dfrac{100A_{st}}{bd}$ and $f_{ck}$.
4. If $\tau_v\le\tau_c$, provide minimum stirrups: $\dfrac{A_{sv}}{b\,s_v}\ge\dfrac{0.4}{0.87f_y}$.
5. If $\tau_v>\tau_c$, shear carried by steel $V_{us}=V_u-\tau_cbd$, then $$s_v=\frac{0.87f_yA_{sv}d}{V_{us}}\le\min(0.75d,\ 300\ \text{mm})$$
6. Provide closer spacing near supports and wider spacing where shear is small.

Punching (two-way) shear in isolated footings

A column punches through the footing along a truncated pyramid. IS 456 (cl. 31.6) checks it at a section at $d/2$ from the face of the column (critical perimeter $b_0$), in addition to one-way shear at $d$ from the face.

   plan                         section
  +--------------+              col
  |  +--------+  |           ____|__|____
  |  | col    |  |   d/2         \  /  45
  |  +--------+  |   |<--->|____/__\____
  |  critical    |    critical perimeter
  +--------------+

Steps

1. Factored upward soil pressure $q_u=P_u/(\text{footing area})$ (net, excluding self weight).
2. Punching shear force: $V_u=q_u\left[BL-(a+d)(b+d)\right]$ for a column $a\times b$.
3. Critical perimeter: $b_0=2[(a+d)+(b+d)]$.
4. Nominal punching shear stress: $\tau_v=\dfrac{V_u}{b_0\,d}$.
5. Permissible punching shear stress: $\tau_c'=k_s\tau_c$, where $k_s=0.5+\beta_c\le1$ ($\beta_c$ = short side/long side of column) and $\tau_c=0.25\sqrt{f_{ck}}$.
6. Condition: $\tau_v\le\tau_c'$. If not satisfied, increase the depth $d$ of the footing (shear reinforcement is not normally used in footings).
7. Also check one-way (beam) shear at distance $d$ from the column face with $\tau_c$ from Table 19 for the footing steel percentage.