Contents preface (VII) introduction 1—37

15.2.3. Freeboard
All embankment dams must have sufficient extra vertical distance between the crest of the embankment dam and the still water surface in the reservoir. This distance is termed the freeboard (7) and must be such that the effects of wave action, wave run-up, wind set-up, earthquake, and settlement of embankment and foundation do not result in overtopping of the dam. Normal freeboard is measured with respect to the full reservoir level while the minimum freeboard is measured with respect to the maximum water level in the reservoir (7). An adequate freeboard is the best guarantee against the failure of an embankment dam due to overtopping.
Wave action for freeboard computations is best represented by the wave height and wave length both of which depend on fetch and wind velocity. Fetch is defined as the maximum straight line distance over open water on which the wind blows (7). Effective fetch is weighted average fetch of water spread, covered by 45° angle on either side of trial fetch, assuming the wind to be completely non-effective beyond this area. The effective fetch is calculated (7) by drawing fifteen radials on the reservoir contour map at an interval of 6° from a selected point located on the periphery at which fetch is required to be determined (Fig. 15.4). The central radial is drawn in the direction of wind. It should be noted that after the construction of reservoir, the funnelling action of the valley may direct wind towards the dam. As such, the wind direction may preferably be assumed along the maximum fetch line. Each radial runs the full length of the water surface at a given pool elevation. Effective fetch f_e is calculated from

f		= ^∑	Ri cos α i	(15.1)
e
		∑ cos α i

494 IRRIGATION AND WATER RESOURCES ENGINEERING

		Wind
		direction
341.0		6 km
		5
		α = 6°
		4
	Centralradial	3	45°
			2
		1	341.0
Xi
Section A			Dam

Fig. 15.4 Radials for computations of effective fetch
Here, R_i is the effective length of the i^th radial and α_i is the angle between the i^th radial and the central radial. Such values of effective fetches are calculated for 2 or 3 different trial fetches and the maximum value of these effective fetches is used for further computations of wave height H_s and the wave period T_s which are, respectively, given as (7)

gHs

L gfe O

0.47

2

= 0.0026

M

P

(15.2)

V

2

N V

Q

gTs

L gfe O^0.28

and

= 0.45 M

P

(15.3)

V

2

N V

Q

The wave length, Ls (in metres) is obtained from

L = 1.56 T ^2

(15.4)

s

s

Here, H _s and T_s are the wave height (in metres) and wave period (in seconds) of a significant wave, g the acceleration due to gravity (m/s²), V the wind velocity (m/s) over the water surface, and f_e is the effective fetch (in metres).
The wind velocities over the water surface are higher than the wind velocities over the land surface and the difference between the two depends upon the fetch and the surrounding terrain conditions. The ratio of the wind velocity over the water surface to the wind velocity over the land surface is given in Table 15.1. For computation of minimum freeboard, the wind velocity is taken as half to two-thirds the wind velocity adopted for calculating normal freeboard
(7). For calculation of the normal and minimum freeboard, the wave length and design wave height (H₀) are taken as L_s and 1.67 H_s, respectively (7).

EMBANKMENT DAMS						495
	Table 15.1 Ratio of the wind velocity over water surface to the wind
		velocity over land surface (7)
	Effective fetch (km)	1	2	4	6	8	10 and above
	The ratio	1.1	1.16	1.24	1.27	1.3	1.31

Wave run-up is the vertical difference between the maximum elevation attained by wave run-up on a slope and the water elevation on the slope excluding wave action. The wave run-up, R for a smooth surface can be obtained from Fig. 15.5. Wave run-up on a rough surface would be less and, hence, the values of R, obtained from Fig. 15.5, are multiplied by a correction factor obtained from Table 15.2.

Table 15.2 Surface roughness correction factor for wave run-up (7)

	Type of pitching	Recommended correction factor
Cement concrete surface	1.00
Flexible brick pitching	0.80
Hand-placed riprap
(i)	laid flat	0.75
(ii)	laid with projections	0.60
Dumped riprap	0.50

0 R/Hup,

-run waveRelative

2.8

2.4

2.0

1.6

1.2

0.8

0.4

0

0

							and		below
					. 02
				0
			=			. 03
	/L	s				.04
				0				ve
H	o								0	. 05
									0	.06		abo
										0
										.07	and
										0
										.08
										0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Embankment slope

Fig. 15.5 Relative run-up of waves

496 IRRIGATION AND WATER RESOURCES ENGINEERING
If the wave run-up is less than the design wave height H₀, the freeboard is governed by the design wave height H₀ (7).
Wind set-up is the result of piling up of the water on one end of the reservoir on account of the horizontal driving force of the blowing wind. The magnitude of the rise of water surface above the still water surface is called the wind set-up or the wind tide (7).
Consider sections 1 and 2, Fig. 15.6, in a reservoir. Let the distance between these sections be dF. Let the depth at section 1 be D. The wind exerts shear stress τ_w on the water surface (assumed horizontal) as a result of which the depth at section 2 is D + dS. The shear on the bed of the reservoir is, say τ_b. Applying momentum principle to the control volume between sections 1 and 2, one gets

2

₁ ^τw

D + dS

D
^τb

dF
Fig. 15.6 Wind set-up

τ_w dF – τ_b dF + ₂¹ ρgD² − ₂¹ ρg (D + dS)² = ρQ (U₂ – U₁)

Here, ρ is the mass density of water, Q the discharge, and U₁ and U₂ are the velocities of flow at sections 1 and 2, respectively. Since the depth in the reservoir is large, the velocities U₁ and U₂ are negligible and, therefore, the shear stress τ_b too is negligible. If one drops the term containing square of dS (as dS is small), the above equation reduces to

τ_w dF = ρgDdS

∴	dS =	τ w dF
ρgD

Since, τ_w is proportional to the square of the wind velocity V, the wind set-up S can be written as

_{S ∝} V ² F

D
In practice, the wind set-up, S is calculated from the Zuider Zee formula (7):

S =	V ^2 F	(15.5)
62,000D

where, V is the velocity of the wind over the water surface (in km/hr), F the fetch in km, and D is the average depth of water in metres along the maximum fetch line.

The freeboard is thus the sum of the height (or wave run-up) and the wind set-up. In order to account for the uncertain effects of seiches (periodic undulations in the reservoir

EMBANKMENT DAMS

497

water surface believed to be on account of earthquake, intermittent wind, varying atmospheric pressures, and irregular inflow and outflow of water), and vertical settlement of the embankement and foundation, an additional margin of safety is always added to the computed freeboard. The freeboard (normal as well as minimum) should not, however, be less than 2 m

(7). The elevation of the top of the embankment dam is obtained by adding normal freeboard to the full reservoir level and the minimum freeboard to the maximum water level in the reservoir. Obviously, the higher of the two elevations is to be adopted as the elevation of the top of the dam.
Example 15.1 Compute freeboard and the level of top of the dam for the following data:

Full reservoir level		= 340.00 m
Maximum water level		= 342.20 m
Effective fetch
for normal freeboard		= 3.66 km
for minimum freeboard		= 4.00 km
Wind velocity over land for normal freeboard	= 150 km/hr
Average depth of reservoir
for normal freeboard		= 29.0 m
for minimum freeboard		= 31.2 m
Embankment slope		= 2.5 (H) : 1(V)
The upstream face is covered with hand-placed stone pitching.
Solution:
Quantity	For normal		For minimum	Remarks
	freeboard		freeboard
Effective fetch (km)	3.66		4.00
Wind velocity over land (km/h)	150.00		75.00**
Wind coefficient		1.226		1.240	From Table 15.1
Wind velocity over water,
V(km/hr)		183.90		93.00
V(m/s)		51.083		25.83
Significant wave height, Hs (m)	2.37		1.20	From Eq. (15.2)
Wave period, Ts	(s)	4.88		3.71	From Eq. (15.3)
Wave length, Ls	(m)	37.15		21.47	From Eq. (15.4)
Design wave height, H0 (m)	3.96		2.00	H0 = 1.67 Hs,
Wave steepness, H0/Ls	0.1066		0.093
Relative wave run-up, R/H0	1.6		1.6	From Fig. 15.5
Wave run-up, R (m)	6.336		3.20
Surface roughness correction	0.75		0.75	From Table 15.2
factor for wave run-up
Corrected wave run-up (m)	4.752		2.4
Wind set up, S (m)	0.069		0.018	From Eq. (15.5)
Freeboard required (m)	4.752 + 0.069		2.4 + 0.018*
		= 4.821		= 2.418
Elevation of top of dam (m)	340.0 + 4.821		342.2 + 2.418
		= 344.821		= 344.618

498 IRRIGATION AND WATER RESOURCES ENGINEERING

• 
  Assuming wind velocity for minimum freeboard to be half the wind velocity for normal freeboard (7).
  

• 
  Wave run-up is higher than design wave height (H₀)
  

Therefore, adopt elevation of top of the dam as 344.821 or, say, 345.00 m.