Contents preface (VII) introduction 1—37

(vi) Wave Pressure
The upper portion of a dam is also subjected to the impact of waves. Wave pressure against massive dams of large height is usually of little importance. Wave pressure is related to wave height h_w as follows (2):
(a) The maximum wave pressure p_w (in kilopascals) occurs at 0.125 h_w above the still water level and is given by the equation

pw = 24 hw

(16.1)

where, h_w is the height of the wave in metres.

(b) The total wave force P_w (in kilonewtons) is given by

P	= 20 h^2	(16.2)
w	w

and acts at 0.375 h_w above the still water level in the downstream direction. (c) The wave height h_w can be calculated using the following relations:

h = 0.032	VF + 0.76 – 0.27 F^1/4 for F < 32 km	(16.3)
w
hw = 0.032	VF for F > 32 km	(16.4)

Here, V is the wind velocity in kilometres per hour and F is the fetch in kilometres.

538 IRRIGATION AND WATER RESOURCES ENGINEERING
The height of the wave and the wind set-up decide the freeboard which is the vertical distance between the top of the dam and the still water level. The wind set-up S (in metres) is estimated by the Zuider Zee formula

S =	V ^2F	(16.5)
62 ,000 D

in which, D is the average depth in metres over the fetch distance F.

4

The minimum freeboard should be equal to wind set-up plus ₃ times wave height above

the normal pool elevation or above maximum reservoir level corresponding to the design flood, whichever gives higher crest elevation for the dam (2). The freeboard shall not, however, be less than 1.0 m above the mean water level corresponding to the design flood.
(vii) Earthquake
Gravity dams are elastic structures which may be excited to resonate by seismic disturbances. Such dams should be designed so that they remain elastic when subjected to the design earthquake. The design earthquake should be determined considering (i) historical records of earthquakes to obtain frequency of occurrence versus magnitude, (ii) useful life of the dam, and (iii) statistical approach to determine probable occurrence of earthquakes of various magnitudes during the life of the dam. A gravity dam should also be designed to withstand the maximum credible earthquake which is defined as the one having a magnitude usually larger than any historical recorded earthquake (3).
Earthquakes impart random oscillations to the dam which increase the water and silt pressures acting on the dam and also the stresses within the dam. An earthquake movement may take place in any direction. Both horizontal and vertical eathquake loads should be applied in the direction which produces the most unfavourable conditions. For a gravity dam, when the reservoir is full, the most unfavourable direction of earthquake movement is upstream (so that the inertial forces acting downstream may result in resultant force intersecting the base of the dam outside middle-third of the base besides increasing the water load and, therefore, the increased overturning moment) and is downward for vertical earthquake movement as it causes the concrete, and water above the sloping faces of the dam to weigh less resulting in reduced stability of the dam. When the reservoir is empty, more unfavourable is the downstream ground motion causing inertial forces to act upstream so that the resultant may intersect the base of the dam outside middle-third of the base. The effect of earthquake forces depends on (i) their magnitude which, in turn, depends on the severity of the earthquake, (ii) the mass of the structure and its elasticity, and (iii) the earthquake effects on the water load. For estimation of earthquake load, knowledge of earthquake acceleration or intensity, usually expressed in relation to acceleration due to gravity g, is useful. This ratio of earthquake acceleration to gravitational acceleration is termed seismic coefficient and is designated as α_h. The value of seismic coefficient for horizontal as well as vertical earthquake accelerations for different zones of the country are different and can be obtained from the Codes. Considering a structure of mass M moving with an acceleration α_h g in the horizontal direction during an earthquake, the horizontal earthquake force acting on the structure, P_e is given as

P_e = M α h g = ^W_g α h g = α hW
where, W is the weight of the structure. The value of α_h has usually been taken as 0.1 in the absence of any other specified value. Similarly, the value of the seismic coefficient in the vertical direction can be taken as 0.05.

GRAVITY DAMS

539

The inertia of water in the reservoir also produces a force on the face of the dam during an earthquake. For dams with vertical or sloping upstream face, the variation of horizontal hydrodynamic earthquake pressure with depth is given by the following equations (5):

and
where,

and

pe = c1 αh ρ gh

yI ^O

(16.6)

c

=

cm

^L y F

−

yI

y F

−

(16.7)

2

M

G 2

J +

G 2

J P

1

M h H

hK

h H

hK P

N

Q

p_e = hydrodynamic earthquake pressure normal to the face, c₁ = a dimensionless pressure coefficient,
α_h = ratio of horizontal acceleration due to earthquake and the gravitational acceleration, i.e., horizontal acceleration factor,
ρ = mass density of water,
g = acceleration due to gravity. h = depth of reservoir,

y = vertical distance from the reservoir surface to the elevation under consideration, c_m = the maximum value of c₁ for a given slope (Fig. 16.3).

Angleoffacefromtheverticalindegrees

90°

80°
70°

60°
50°
40°
30°
20°
10°

0°	0.1	0.2	0.3	0.4	0.5	0.6	0.7
0

Maximum pressure coefficient c_m

Fig. 16.3 Variation of c_m with inclination of the upstream face
Let V_pe (or V_pe′) represent change in horizontal component of reservoir (or tail-water) load on the face above a section due to horizontal earthquake loads and it is computed for each increment of elevation selected for the study and the totals obtained by summation because of the nonlinear response (1). Likewise, M_pe (or M_pe′) which represents the moment of V_pe (or V_pe′) about the centre of gravity of the section, is computed. The inertia forces for concrete in the dam should be computed for each increment of height, using the average acceleration factor for that increment. The inertia forces to be used while considering an elevation in the dam are the summation of all the incremental forces above that elevation and the total of their moments

540 IRRIGATION AND WATER RESOURCES ENGINEERING
about the centre of gravity at the elevation being considered. The horizontal concrete inertia force (V_e ) and its moment (M_e ) can be calculated using Simpson’s rule (1). Alternatively, V_pe and M_pe can be obtained from the following equations (4):

V_pe = 0.726 p_e y

and	M	= 0.299 p	e	y^2
	pe

The effects of vertical accelerations may be determined using the appropriate forces, moments, and the vertical acceleration factor α_v. The forces and moments due to water pressure normal to the faces of the dam and those due to the dead loads should be multiplied by the appropriate acceleration factors to determine the increase (or decrease) caused by the vertical downward (or upward) accelerations. The effect of earthquake on uplift forces is considered negligible.
Dams having upstream face as a combination of vertical and sloping faces are analysed as follows (4):
(a) If the height of the vertical portion of the upstream face of a dam is equal to or greater than one- half the total height of the dam, analyse the dam as if it has a vertical upstream face throughout.
(b) If the height of the vertical portion of the upstream face of a dam is less than half the total height of the dam, use the pressure which would occur if the upstream face has a constant slope (equal to the slope of the sloping portion of the upstream face) from the water surface elevation to the heel of the dam.