Boundary shear distribution in alluvial channel

In a channel of uniform flow, the average boundary shear stress is $\tau_0 = \gamma R S_0$. It is not uniform around the wetted perimeter. It is largest on the bed near the centre line, falls towards the banks, is a maximum again at about the lower third of the side slope, and is zero at the water surface and at the corners (bottom edges) of the section.

For a trapezoidal channel with side slope 1.5H : 1V and $b/y \ge 4$ (USBR results, $y$ = depth of flow):

        water surface (tau = 0)
   \                               /
    \   0.76 gamma y S0           /   side
     \   (max on side)           /
      \                         /
       \_______________________/
        0     0.97 gamma y S0  0     bed
    corner  (max on bed)     corner

Design values. For simple design, take the maximum shear stress on the bed as $\tau_b \approx \gamma y S_0$ (0.97 rounded up) and on the sides as $\tau_s \approx 0.75\,\gamma y S_0$. For a wide channel ($b \gg y$) the bed shear stress is uniform at $\gamma y S_0$.

For example, with $y = 1$ m and $S_0 = 0.001$: $\gamma y S_0 = 9.81\ \text{N/m}^2$, so the bed carries about $9.5\ \text{N/m}^2$ and the side slope about $7.5\ \text{N/m}^2$. The side stress is lower than the bed stress, but the critical stress on the side is also reduced (by the factor $K$), so the banks often govern the design.