Meaning of shear stress reduction factor K

Physical meaning of $K$

$K$ is the ratio of the critical shear stress needed to start motion of a particle on the side slope to the critical shear stress needed on the level bed:

$$K = \frac{\tau_{cs}}{\tau_{cb}} = \sqrt{1 - \frac{\sin^2\theta}{\sin^2\phi}} < 1$$

where $\theta$ is the side slope angle and $\phi$ the angle of repose of the material. It tells us by what fraction the bed critical stress must be reduced to get the safe stress for the side slopes.

Why $K$ is needed in design

1. Gravity helps the flow on the sides. A particle on a side slope has a component of its weight down the slope in addition to the drag of the flow. It therefore moves at a lower shear stress than a particle on the flat bed.
2. The sides are the critical part of the section. Without $K$ a designer would size the channel using the bed critical stress only, and the banks would erode first, even though the shear stress on the banks (about $0.75\gamma yS_0$ for trapezoidal channels) is lower than on the bed (about $\gamma yS_0$).
3. It links the side slope to the stability. $K$ decreases as the side slope becomes steeper and falls to zero when $\theta = \phi$. By checking the side-slope stress against $K\tau_c$ the designer chooses a flatter side slope or a different cross-section so that both bed and banks are stable.
4. It gives a uniform safety. Stable-channel design needs the largest of $\tau_b/\tau_{cb}$ and $\tau_s/\tau_{cs}$ to be below 1; $K$ lets both be compared on the same basis.