Thickness of launching apron, 7 m shank

A launching apron of loose stone is laid at the toe of the guide bund. When scour occurs it launches down a 2H:1V slope. The laid thickness is found by equating the stone volume before and after launching.

Data

$Q = 8000$ m³/s, flood depth $h = 5$ m, $d = 0.3$ mm, laid width of apron $= 7$ m (shank portion).

Step 1. Silt factor and scour depth

$$\begin{aligned} f &= 1.76\sqrt{d_{mm}} = 1.76\sqrt{0.3} = 0.964 \\ R &= 0.47\left(\frac{Q}{f}\right)^{1/3} = 0.47\left(\frac{8000}{0.964}\right)^{1/3} = 9.52\ \text{m} \end{aligned}$$

Maximum scour at the shank: $1.5R = 14.27$ m below HFL. Scour below the bed: $D' = 1.5R - h = 14.27 - 5 = 9.27$ m.

Step 2. Pitching and launched thickness

$$t = 0.06\,Q^{1/3} = 0.06\times20.00 = 1.20\ \text{m}, \qquad 1.5t = 1.80\ \text{m}$$

Step 3. Thickness of the 7 m apron before launching

Slope length of launched apron $= \sqrt5\,D' = 2.236\times9.27 = 20.74$ m.

$$\begin{aligned} T\times7 &= 1.5t\times\sqrt5 D' = 1.80\times20.74 \\ T &= 5.33\ \text{m} \end{aligned}$$

Answer: thickness of the 7 m wide apron as laid $T = 5.33$ m (it launches to 1.80 m thick on the 2:1 slope). A 7 m apron is short against a required width of $1.5D' = 13.91$ m, so a thick apron is needed; it may be better to widen it to $13.9$ m, when $T = 2.68$ m.

  pitching
  ___\__ T = 5.33 m
 bed __|<-- 7 m -->|___ river bed
   after launch: 2H:1V, 1.80 m thick, D' = 9.27 m