Tractive force method of channel design

The tractive force method (Lane, USBR) designs a mobile boundary channel by ensuring the shear stress (tractive force) of the flowing water on the bed and banks is less than the critical shear stress of the boundary material.

Basic ideas

• Tractive force on the boundary: $\tau_0 = \gamma R S_0$ (average). For a trapezoidal channel the maximum values are about $\tau_b \approx \gamma y S_0$ on the bed and $\tau_s \approx 0.75\gamma y S_0$ on the sides.
• Critical shear stress on a level bed, $\tau_c$: from the Shields diagram, $\tau_c = \tau_*(\gamma_s - \gamma)d$ (about $0.06(\gamma_s - \gamma)d$ for coarse material), or from Lane's charts for a given $d_{75}$.
• On the side slope, the critical shear stress is reduced to $\tau_{cs} = K\tau_c$, with $K = \sqrt{1 - \sin^2\theta/\sin^2\phi}$.

Design procedure

1. From soil data find the grain size $d$ (or $d_{75}$) and the angle of repose $\phi$.
2. Select the side slope $z$ (with $\theta = \tan^{-1}(1/z) < \phi$) and compute $K$.
3. Find $\tau_c$ on the level bed; apply a reduction for sinuosity or a safety factor if required.
4. Find the permissible depth from the side-slope condition and the bed condition:
   • bed: $\gamma y S_0 \le \tau_c$
   • sides: $0.75\gamma y S_0 \le K\tau_c$ and take the smaller $y$.
5. With that $y$, use Manning's equation ($n$ from the material) to find the base width $b$ that carries $Q$:

$$Q = \frac{1}{n}A R^{2/3}S_0^{1/2},\qquad A = (b + zy)y$$

6. Check the Froude number (the flow should be subcritical) and add freeboard. If $b/y$ is far from the assumed range, repeat with a modified side slope.