Question 15.3: An embankment at 10 m OD along an east-facing bay is to be p......
An embankment at 10 m OD along an east-facing bay is to be protected against erosion and flooding. The table below gives fetches, wind speeds, and refraction coefficients for obtaining design wave heights:
Table 1
(a) Provide a hydraulic design of sea wall protecting the embankment. The maximum still-water level is 3.5 m OD, the depth of water being 3.5 m.
(b) For the wave conditions and slope of the breakwater of Worked example 15.3, using Van der Meer’s formula determine the mass of the armour units for rough quarry stone and the wave run-up if the design storm duration is 6.0 hours. Assume damage number for failure conditions as low.
Table 1
| Direction | |||||
| N | NE | E | SE | S | |
| Fetch (km) | 400 | 250 | 100 | 80 | 60 |
| Design wind speed ({km}\,{h}^{-1}) | 75 | 75 | 50 | 50 | 75 |
| Refraction coefficient, K_{\mathrm{r}} | 0.2 | 0.35 | 0.8 | 0.7 | 0.3 |
(a) The wave heights and periods are calculated using the JONSWAP spectrum (equation (14.48)), given in the table on p. 636.
E_{ J }(f)=\alpha g^2(2 \pi)^{-4} f^{-5} \exp \left[-\frac{5}{4}\left(\frac{f}{f_{ m }}\right)^{-4}\right] \gamma^q (14.48)
Choose 2.17 m as the significant wave height for the design, the significant wave period being 8.16 s.
Consider the sea wall profile shown in Fig. 15.15. Two layers of rough quarry stone will form the cover layer. The density of quarry stone is 2650 kg m ^{-3} . The design wave does not break before reaching the wall, because H/d = 2.17/3.5 =0.62 < 0.78, the criterion for wave breaking. For minimum wave reflection, the slope of the sea wall (equation (15.16)) should be less than
\tan \beta=\frac{8}{T}\left(\frac{H_1}{2 g}\right)^{1 / 2} (15.16)
\tan \beta=\frac{8}{8.16}\left(\frac{2.17}{2.0 \times 9.81}\right)^{1 / 2}= 0.33 .
Table 2
The slope of the upstream face is chosen to be 1V:3H. From Table 15.3, K_{\mathrm{D}} for rough quarry stone is 1.0, the individual weight being given as (equation (15.27))
W_{ r }=\frac{\rho_{ s } g H^3}{K_{ D }\left(\rho_{ s } / \rho-1\right)^3 \cot \beta} (15.27)
W_{ r }=\frac{2650 \times 9.81 \times 2.17^3}{1.0 \times(2.65-1)^3 \times 3} \simeq 20 \, kNmass of units =\frac{20 \times 10^3}{9.81} \simeq 2 \times 10^3 kg .
The deep-water design wave height H_0=H\left(C_{ g } / C_{ g 0}\right)^{1 / 2}=2.17 \times \sqrt{0.82}= 1.97 m. r in equation (15.20) is assumed to be 0.8 (Table 15.1). The wave run-up, R_{{u}} (equation (15.20)) is
R_{ u } / H_0=1.016 \tan \beta\left(H_0 / L_0\right)^{-0.5} r (15.20)
R_{ u }=1.97 \times 1.016 \times \frac{1}{3}\left(\frac{1.97}{104.0}\right)^{-0.5} \times 0.8=3.88 \,m .The run-up corresponds to 3.5 + 3.88 = 7.38 m OD.
The horizontal part at 8.0 m OD is assumed to be 6 m, which is greater than the crest width given by equation (15.30). The thickness of the cover layer (equation (15.28)) is
b_{\min }=3 K_{ D }^{\prime}\left(W_{ r } / \rho_{ s } g\right)^{1 / 3} (15.30)
t_1=n K_{ D }^{\prime}\left(W_{ r } / \rho_{ s } g\right)^{1 / 3} (15.28)
t_1=2 \times 1.0 \times\left(\frac{20 \times 10^3}{2.65 \times 10^3 \times 9.81}\right)^{1 / 3}= 1.83 m.
The area of the cover layer per unit width of sea wall, A= \left[12^2\left(3^2+1^2\right)\right]^{1 / 2} \simeq 38 \,m ^2 From Table 15.4, the porosity of the cover layer is 37 %.
The number of units per unit width of cover layer (equation (5.29)) is
N=A n\left(1-\frac{p_{ r }}{100}\right)\left(\frac{\rho_{ s } g}{W_{ r }}\right)^{2 / 3} (15.29)
N=38 \times 2 \times\left(1-\frac{37}{100}\right)\left(\frac{2650 \times 9.81}{20 \times 10^3}\right)^{2 / 3}=57 .Figure 15.15 gives the profile of the sea wall, which is to be developed further from model studies and other considerations such as site and economic constraints.
(b) \xi=\frac{1}{3}\left(\frac{2.17}{104.0}\right)^{-0.5}
= 2.3
From Table on p. 636, the peak period T_{ z }= 8.16 s.
N_{z}= 6.0 × 3600/8.16 ≅ 2700.
From Table 15.5, S = 2.0. P = 0.4 from Fig. 15.10.
For \xi = 2.3, using equation (15.32) for run-up on rough slopes
\frac{H_{ s }}{\Delta D_{50}}=1.0 P^{-0.13}\left(\frac{S}{\sqrt{N}}\right)^{0.2} \sqrt{\cot \alpha} \xi^{ P } (15.32)
\frac{2.17}{1.65 D_{50}}=1.0 \times 0.4^{-0.13} \times\left(\frac{2}{\sqrt{2700}}\right)^{0.2} \times \sqrt{3} \times 2.3^{0.4}D_{50} = 0.93 m.
Mean mass of rocks M_{50}=0.93^3 \times 2.65=2.13 \times 10^3 kg .
RUN-UP
Peak Irribarren number at the structure \xi_{ p }=(1 / 3) \times(2.17 / 172.0)^{-0.5}
= 2.7
The correction factor r from Table 15.1 is 0.8.
R_{ u }=2.17 \times(2.11-0.09 \times 2.7) \times 0.8= 3.24 m .
Table 2
| Direction | |||||
| N | NE | E | SE | S | |
| Wind speed, U\left( m s ^{-1}\right) | 20.8 | 20.8 | 13.9 | 13.9 | 20.8 |
| gF/U² | 9070 | 5669 | 5077 | 4062 | 1360 |
| U f_{ m } / g | 0.173 | 0.202 | 0.21 | 0.226 | 0.324 |
| α | 0.01 | 0.0114 | 0.0116 | 0.0122 | 0.0155 |
| 1 / f_{ m } \text { or } T_{ m } | 12.26 | 10.5 | 6.75 | 6.27 | 6.54 |
| T_{ z }=T_{ s }( s ) | 9.53 | 8.16 | 5.24 | 4.87 | 5.08 |
| H_{ s }( m ) | 8.25 | 6.46 | 2.69 | 2.38 | 2.92 (equation (14.53)) |
| L_{0 }( m ) | 141.8 | 104.0 | 42.9 | 37.0 | 40.3 \left(L_0=g T_s^2 / 2 \pi\right) |
| d / L_0 | 0.025 | 0.034 | 0.082 | 0.095 | 0.087 |
| C_{ g } / c_0 | 0.36 | 0.41 | 0.55 | 0.58 | 0.56 |
| C_{ g } / Cg _0 | 0.72 | 0.82 | 1.1 | 1.16 | 1.12 \left(C_{ g 0}=c_0 / 2\right) |
| K_{ s }=\left(C g_0 / C_{ g }\right)^{1 / 2} | 1.18 | 1.12 | 0.95 | 0.93 | 0.94 |
| K_{ r } | 0.2 | 0.3 | 0.8 | 0.7 | 0.3 |
| Design wave height (m) | 1.94 | 2.17 | 2.04 | 1.55 | 0.82 |
H_{ s }=0.552 g \alpha^{1 / 2} / \pi^2 f_{ m }^2 . (14.53)
| Table 15.1 Factor r for various armour units | |
| Armour unit | r |
| Smooth, impervious | 1.0 |
| Concrete slabs | 0.9 |
| Concrete block | 0.85–0.9 |
| Grass on clay | 0.85–0.9 |
| One layer of quarrystone (impervious) | 0.8 |
| Rubble stone placed at random | 0.5–0.8 |
| Two or more layers of rockfill | 0.5 |
| Tetrapods | 0.5 |
| Table 15.3 Values of K_{D} in Hudson’s formula (SPM): no damage and minor overtopping | ||||||
| Armour unit | Number of units in cover layer |
Structure trunk | Structure head | Slope | ||
| Structure trunk | Non-breaking wave | Breaking wave | Non-breaking wave | |||
| Smooth quarrystone | 2 | 1.2 | 2.4 | 1.1 | 1.9 | 1.5–3.0 |
| Smooth quarrystone | >3 | 1.6 | 3.2 | 1.4 | 2.3 | 1.5–3.0 |
| Rough angular quarrystone | 2 | 2.0 | 4.0 | 1.9 | 3.2 | 1.5 |
| 1.6 | 2.8 | 2.0 | ||||
| 1.3 | 2.3 | 3.0 | ||||
| Rough angular quarrystone | >3 | 2.2 | 4.5 | 2.1 | 4.2 | 1.5–3.0 |
| Tribar | 2 | 9.0 | 10.0 | 8.3 | 9.0 | 1.5 |
| 7.8 | 8.5 | 2.0 | ||||
| 6.0 | 6.5 | 3.0 | ||||
| Dolos | 2 | 15.8 | 31.8 | 8.0 | 16.0 | 2.0 |
| 7.0 | 14.0 | 3.0 | ||||
| Tetrapod | 2 | 7.0 | 8.0 | 5.0 | 6.0 | 1.5 |
| 4.5 | 5.5 | 2.0 | ||||
| 3.5 | 4.0 | 3.0 | ||||
| Table 15.4 Layer coefficient K_D^{\prime} and porosity for various armour units | |||
| Armour unit | Number of layers, n | Layer coefficient, K_D^{\prime} | Porosity, P_r (%) |
| Smooth quarrystone | 2 | 1.02 | 38 |
| Rough quarrystone | 2 | 1.0 | 37 |
| Rough quarrystone | >3 | 1.0 | 40 |
| Tetrapod | 2 | 1.10 | 50 |
| Tribar | 2 | 1.02 | 54 |
| dolos | 2 | 0.94 | 56 |
| Table 15.5 Variation in damage number for failure conditions | |||
| Slope | Initial | Damage (S) intermediate | Failure |
| 1:1.5 | 2 | – | 8 |
| 1:2 | 2 | 5 | 8 |
| 1:3 | 2 | 8 | 12 |
| 1:1.4 | 3 | 8 | 17 |
