A detailed sketch of a gravity dam is shown in Fig. 13.1. All the predominant forces that act on the dam have been shown in the figure itself.
The forces that act on the dam are the following:
1. Weight of the dam
2. Horizontal hydrostatic pressure due to water
3. Uplift pressure due to water percolated under the dam
4. Earthquake pressure
5. Wind pressure
6. Ice pressure
7. Wave pressure
8. Pressure due to silt deposited on U/S face.
Out of above eight forces, acting on the dam, first three forces are the major forces that are considered in the design. All other forces are not of much significance and are considered only under specific conditions.
Force # 1. Weight of the Dam:
It is the most important force, particularly for gravity dams. Stability of the dam largely depends upon this force. For the design purpose only unit length of the dam is considered. Cubic content of the cement concrete is determined for unit length of the dam. This cubic content when multiplied by the density, gives the total weight (W) of the dam.
The total weight (W) is considered acting at the C.G. of the dam section. The position of C.G. of the dam section can be found out by dividing the dam section into several triangles, rectangles, and trapeziums and by taking moments of these weights about any point at the base of the dam.
Force # 2. Horizontal Hydrostatic Pressure due to Water:
This is the largest external force acting on the dam. It has the largest capacity for disturbing the stability of the dam. It is a horizontal force which acts at the C.G. of the pressure distribution diagram, due to water. The pressure distribution diagram is always triangular with zero value at surface of the water and increasing linearly to maximum at the base of the dam.
The value of maximum horizontal pressure at base of the dam is wh where w is the density of water in kg/m3 and h the depth of water in meters. Since pressure distribution diagram due to water is triangular, the value of the total horizontal pressure (P) due to water, will be area of the triangle.
Force # 3. Uplift Pressure due Seeping Water under the Dam:
The water that seeps through the pores of the material comprising dam and foundation, causes uplift pressure and tries to tilt or topple the dam. A part of the weight of the dam would get neutralized by uplift pressure and thus net foundation reaction due to vertical forces will be reduced.
Intensity of uplift pressure is maximum at U/ S end of the dam and it goes on decreasing towards the D/S end. As some water can see into the concrete dam also, the uplift pressure may occur anywhere in the dam also. It is very difficult to find out the value of uplift pressure accurately. It depends upon the factors like, cut off on U/S side, fissures in the foundation rocks, drain ability of the foundation etc.
There are two schools of thought as to on how much area uplift pressure should be considered as acting. According to one thought one-third to two-third area of foundation should be considered effective. The second thought, propagated by Terzaghi recommends consideration of full area as the effective area.
According to U.S.B.R., intensity of uplift pressure at D/S (Toe) and U/S (heel) is taken equal to the hydrostatic pressure of water. The variation of uplift pressure from heel to toe is linear or straight line. In order to release uplift pressure, drainage galleries are provided in the body of the dam.
The magnitude of the uplift pressure at the face of the gallery is taken as equal to hydrostatic pressure at the Toe plus one third the difference of the hydrostatic pressures at the heel and Toe.
Uplift pressure at heel A = wh
Uplift pressure at toe F = wh’
Uplift pressure at gallery point G = w [h’ +1/3 (h- h’)]
Criteria for Design:
The following criteria are recommended for the calculation of uplift forces:
(a) Uplift pressure distribution in the body of the dam shall be assumed, in case of both preliminary and final design, to have an intensity which at the line at the formed drains exceeds the tail water pressure by one-third the differential between reservoir level and tail water level. The pressure gradient shall then be extending linearly to heads corresponding to reservoir level and tail water level. The uplift shall be assumed to act over 100 per cent of the area.
(b) Uplift pressure distribution at the contact plane between the dam and its foundations and within the foundation shall be assumed for preliminary designs to have an intensity which at the line of drains exceeds the tail water pressure by one-third the differential between the reservoir and tail water heads.
The pressure gradient shall then be extended linearly to heads corresponding to the reservoir level and tail water level. The uplift shall be assumed to act over 100 per cent area. For final designs, the uplift criteria in case of dams founded on compact and un-fissured rock shall be as specified above.
In case of highly jointed and broken foundation, however, the pressure distribution may be required to be based on electrical analogy or other methods of analysis taking into consideration the foundation condition after the treatment proposed. The uplift shall be assumed to act over 100 per cent of the area.
(c) For the extreme loading conditions F and G given later the uplift shall be taken as varying linearly from the appropriate reservoir water pressure at the U/S face to the appropriate tail water pressure at the D/S face. If the reservoir pressure at the section under consideration exceeds the vertical normal stress (computed without uplift) at the U/S face, horizontal crack is assumed to exist and to extend from the U/S face towards the D/S face of the dam to the point where the vertical normal stress (computed on the basis of linear pressure distribution without uplift) is equal to the reservoir pressure at the elevation.
The uplift is assumed to be the reservoir pressure from the U/S face to the end of the crack and from there to varying linearly to the tail water pressure at the D/S face. The uplift is assumed to act over 100 per cent of the area.
(d) No reduction in uplift is assumed at the D/S toe of spillways on account of the reduced water surface elevation (relative to normal tail water elevation) that may be expected immediately downstream of the structure.
(e) It is assumed that uplift pressures are not affected by earthquakes.
Force # 4. Earthquake Pressure:
Primary, secondary, Releigh and Love waves, are set up in the earth’s crust, due to earthquakes. These waves impact acceleration to the foundation under the dam and cause its movement. In order to prevent cracking or rupture, the dam must also move along with the movements of the foundation.
The acceleration generated due to earthquake develops inertia force in the dam, due to which, stresses are first induced in lower layers and then in whole of the dam. The earthquake waves can travel in any direction. They are however resolved in vertical and horizontal directions.
The effects of acceleration in horizontal and vertical directions are given as follows:
(i) Effect in Vertical Direction:
Due to vertical component of acceleration, a vertical inertia force F = αW is developed on the dam. The direction of this force is in opposite direction to the acceleration. In other words inertia force acts downwards when acceleration is upwards, thus increasing the downward weight momentarily.
Similarly when acceleration is vertically downwards the inertia force F = αW acts upwards, thus decreasing the downward weight temporarily. Hence unit weight of the dam and water get altered as follows, due to vertical acceleration.
For dam material = We (1 ± α)
For water = w (1 ± α)
Where we = unit weight of dam material,
α = acceleration.
(ii) Effect in Horizontal Direction.
Two forces are developed due to horizontal effect of acceleration.
(a) Inertia force in dam
(b) Hydrodynamic pressure of water.
(a) Inertia Force:
Inertia force is equal to the product of the mass of the dam and the acceleration.
F = Mass x acceleration
Where F = Inertia force
W = Weight of the dam
α = Acceleration coefficient
g = Acceleration due to gravity.
Mass = W/g
And acceleration = α g.
∴ F = (W/g) x αg = W α
Inertia force acts in a direction opposite to the acceleration imparted by earthquake forces. This force should be considered at the C.G. of the mass regardless of the shape of the cross-section and acts horizontally downstream.
(b) Hydrodynamic Pressure:
Horizontal acceleration towards the reservoir causes a momentary increase in water pressure. The theoretical computations reveal that the distribution of hydrodynamic pressure due to earth on the upstream face of the dam is elliptical-cum-parabolic. The following formula of U.S.B.R. and given by C.N. Zargar may be used to evaluate pressure intensity at a depth y below the maximum water level –
Pey = Intensity of hydrodynamic pressure
a = Acceleration coefficient
w = Unit weight of water
h = Depth of water in the reservoir
Cy = A dimensionless pressure coefficient at a depth y below water level the of Cy is found out by formula –
Where h = Maximum depth of water in the reservoir
y = Depth from the reservoir surface to the elevation in question.
Cm = Maximum value of the pressure coefficient for a given constant slope
θ = Angle in degrees that the slope of the U/S face of the dam makes with the horizontal
The variation of pressure along the depth is shown in Fig. 13.1.
If the U/S face of the dam is inclined, but inclination does not extend to more than half the depth of the reservoir, the face may be considered as vertical for computation of Cm. If slope extends to more than half the depth, overall slope θ should be considered upto full reservoir level for finding the value of Cm as shown in Fig. 13.3.
The total pressure Pe on the portion of the dam upto depth y form top is given by –
Pe = 0.726 Pey x y.
The moment Me about the joint upto which pressure is taken is given by –
Me = 0.3 Pey x y2.
The intensity of the earthquake pressure depends on the acceleration of vibratory motion and is expressed in terms of a fraction of (g) the acceleration due to gravity. Such intensity maps are available for the whole of the country. A detailed map of different zones of earthquake is given in I.S.: 1893-1970.
Following accelerations have been adopted for some projects in India:
If the distribution of hydrodynamic pressure is assumed to be parabolic, the increase in water pressure (Pe) due hydrodynamic effect is given by –
Pe = 0.555 αwh2
This pressure is assumed to be acting at 4h / 3π above the base of the dam.
Rossi-Forel Intensity Rating of Earthquakes:
This rating classifies different intensities of earthquakes according to the severity and extent of damage. The rating also provides the values of acceleration in terms of g for various types of earthquakes. For dams and buildings earthquakes from 7 to 10 rating are important and these are given in Table 13.1.
There are maps available which divide the region into different zones according to the severity of the earthquake.
India has been divided into following five zones:
In this zone earthquake intensity is least. Parts of western Rajasthan and central Deccan plateau are covered by this zone.
Here intensity and frequency of earthquake is greater than zone I. This area consists of the outer periphery of Deccan plateau and consists mostly of Tamil Nadu and Kerala districts. Zone III:
Intensity of earthquake in this region is twice the intensity of zone II. Southern U.P., southern Punjab, southern Bihar, southern Bengal and parts of Tamil Nadu near Vijayawada. Region also includes western ghat except Koyana and Poona region.
The intensity of earthquake is quite severe. Indo-gangetic plains, Delhi, north Punjab, northern U.P. Bihar and Bengal, Runn of Kutch and western ghat near Poona or Koayana are included in this zone.
Intensity of earthquake is maximum in this region. Himalayas on North, Arakan and Naga hills on the East, Mallar range in the West and a strip of foothills is included in this zone.
Basic horizontal seismic coefficients recommended for these zones are as follows:
1983-1975 recommends the seismic coefficient method for dams upto 140 m height and the response spectrum method for dams greater than 100 m height.
For dams upto 100 m height the horizontal seismic coefficient is taken 1.5 times seismic coefficient at top of the dam reducing linearly to zero at the base vertical seismic coefficient is taken 0.75 times the value of h at top reducing linearly to zero at the base.
Force # 5. Wind Pressure:
It is generally not considered in the design. If it has to be considered, it should be considered only on that portion of the superstructure which is exposed to wind. It is normally taken as 100 to 150 kg/m2, acting on the exposed area.
Force # 6. Ice Pressure:
This pressure is considered only when dams are located either at very high altitudes or in very cold regions. In very cold conditions, the reservoir gets covered with a layer of ice. When this layer is subjected to expansion and contraction due to variation in temperature, a force is developed which acts on the dam at level of water in the reservoir.
This force acts linearly along the length of the dam at the reservoir level. The average magnitude of this force may be taken as 5 kg cm2 of the contact area of the ice layer with U/S face of the dam. The thermal expansion of ice is about five times that of concrete.
Force # 7. Wave Pressure:
Waves develop on the surface of the reservoir due to wind. The waves exert pressure on the dam. The magnitude of wave pressure depends upon the height of the waves, developed in the reservoir. The height of the wave as per D. A. Molitor can be found out by formula –
Where hw = Height of wave in metres from trough to crest.
V = Wind velocity in km per hour.
F = Fetch or straight length of water expanse in km.
After having found out the value of hw the pressure intensity due to wages is found out by formula pw = 2.4 w hw (t/m2).
The pw is considered as acting at hw/ 8 metres above the still water surface.
The pressure distribution due to wave pressure is curved but for all design purposes it is considered as triangular as shown in Fig. 13.3.
Hence total pressure pw due to waves is given by –
Force # 8. Pressure due to Silt:
Silt gets deposited at the bottom, along the U/S face of the dam. This force does not act when reservoir is new as no silt is there. But after some months or years silt is deposited at the bottom, which exerts additional horizontal pressure on the dam.
If we is the submerged unit weight of the deposited silt, φ is its angle of internal friction and h is the height to which the silt has deposited, the silt pressure (ps) is calculated as follows –
According to U.S.B.R. recommendation which are mostly followed in the design, pressure due to silt is considered as fluid pressure which develops a horizontal pressure with its density as 360 kg/m3 and a vertical pressure with its density equal to 920 kg/m3, so that additional pressure in horizontal and vertical directions are as follows –
Increase in silt pressure due to earthquake is generally not considered.
According to IS: 6512-1972 the silt pressure and water pressure exist together in submerged silt.
The following criteria are recommended for calculating forces due to silt:
(a) Horizontal ‘silt and water pressure’ is assumed to be equivalent to that of a fluid weighing 1360 kg/m3.
(b) Vertical ‘silt and water pressure’ is found as if silt and water together have a density of 1925 kg/m3.