Design and Construction of Concrete Gravity Dams (Part - 4) - Civil Engineering (CE) PDF Download

Free board 

Free board is the vertical distance between the top of the dam and the sill water level. IS:6512-1984 recommends that the free board shall be wind set-up plus 4/3 times wave height above normal pool elevation or above maximum reservoir level corresponding to design flood, whichever gives higher crest elevation. Wind set-up is the shear displacement of water towards one end of a reservoir by wind blowing continuously – or in repeated regular gusts – from one direction. The Zuider Zee formula (Thomas, 1976) and recommended by IS: 6512-1984 may be used as a guide for the estimation of setup(S): 

S = V²FCosA / kD          (14)

Where 

S = Wind set-up, in m        

V = Velocity of wind over water in m/s        

F = Fetch, in km          

D = Average depth of reservoir, in m, along maximum fetch        

A = Angle of wind to fetch, may be taken as zero degrees for maximum set-up        

K = A constant, specified as about 62000 

Set-up of the reservoir will depend upon the period of time over which the wind blows, that is, at least 1hour, for a fetch of 3km or 3hours for a fetch of 20km. On a 80km fetch, a wind speed of 80 km/hour must last for at least 4hours, whereas for a wind speed of 40km/hour it must last around 8hours for maximum set-up. 

The free-board shall not be less than 1.0m above Maximum Water Level (MWL) corresponding to the design flood. If design flood is not same as Probable Maximum Flood (PMF), then the top of the dam shall not be lower than MWL corresponding to PMF. 

Stability analysis of gravity dams  

The stability analysis of gravity dams may be carried out by various methods, of which the gravity method is described here. In this method, the dam is considered to be made up of a number of vertical cantilevers which act independently for each other. The resultant of all horizontal and vertical forces including uplift should be balanced by an equal and opposite reaction at the foundation consisting of the total vertical reaction and the total horizontal shear and friction at the base and the resisting shear and friction of the passive wedge, if any. For the dam to be in static equilibrium, the location of this force is such that the summation of moments is equal to zero. The distribution of the vertical reaction is assumed as trapezoidal for convenience only. Otherwise, the problem of determining the actual stress distribution at the base of a dam is complicated by the horizontal reaction, internal stress relations, and other theoretical considerations. Moreover, variation of foundation materials with depth, cracks and fissures which affect the resistance of the foundation also make the problem more complex. The internal stresses and foundation pressures should be computed both with and without uplift to determine the worst condition.

The stability analysis of a dam section is carried out to check the safety with regard to 

1. Rotation and overturning
2. Translation and sliding
3. Overstress and material failure

Stability against overturning

Before a gravity dam can overturn physically, there may be other types of failures, such as cracking of the upstream material due to tension, increase in uplift, crushing of the toe material and sliding. However, the check against overturning is made to be sure that the total stabilizing moments weigh out the de-stabilizing moments. The factor of safety against overturning may be taken as 1.5. As such, a gravity dam is considered safe also from the point of view of overturning if there is no tension on the upstream face. 

Stability against sliding 

Many of the loads on the dam act horizontally, like water pressure, horizontal earthquake forces, etc. These forces have to be resisted by frictional or shearing forces along horizontal or nearly-horizontal seams in foundation. The stability of a dam against sliding is evaluated by comparing the minimum total available resistance along the critical path of sliding (that is, along that plane or combination of plans which mobilizes the least resistance to sliding) to the total magnitude of the forces tending to induce sliding. 

Figure. 37: Stability against sliding along concrete dam - rock base interface. Good rock is assumed to exist below

Sliding resistance is also a function of the cohesion inherent in the materials at their contact and the angle of internal friction of the material at the surface of sliding. The junction plane between the dam and rock is rarely smooth. In fact, special efforts are made during construction to keep the interface as rough as possible. There may, however be some lower plane in the foundation where sliding is resisted by friction alone especially if the rock is markedly stratified and horizontally bedded. Figure 37 shows a typical dam profile with the bed-rock and foundation interface inclined at an angle α. Factor of Safety against sliding (F) along a plane may be computed from the following formula: 

        (15) 

Where F_φ and F_c are the Partial Factor of Safety in respect of friction and Partial Factor of Safety of cohesion. IS: 6512-1984 recommends these values to be as given in the following table

The value of cohesion and internal friction may be estimated for the purpose of preliminary designs on the basis of available data on similar or comparable materials. For final designs, however, the value of cohesion and internal friction has to be determined by actual laboratory and field tests, as specified in the Bureau of Indian Standards code IS: 7746-1975 “Code of practice for in-situ shear test on rocks”. 

In the presence of a horizon with low shear resistance, for example, a thin clay seam or clay infill in a discontinuity (Figure 38), then it would be advisable to include downstream passive wedge resistance P, as a further component of the total resistance to sliding which can be mobilized. In this case, the Factor of Safety along sliding has to be found along plane B-B computing the net shear force and net cohesive force along this plane. The net shear force would now be equal to: 

(WCosα + ∑ ΗSinα ) tanφ    (16)

Where W is the weight of the wedge; αis the assumed angle of sliding failure, ∑Η is the net destabilizing horizontal moment; and φ is the internal friction within the rock at plane B-B. The net cohesive force along plane B-B is determined as equal to C.A_B-B. Here, C is the cohesion of material and A_B-B, the area, along plane B-B. 

Figure.33: Sliding against presence of weak seams resisted by passive wedge resistance

Failure against overstressing

A dam may fail if any of its part is overstressed and hence the stresses in any part of the dam must not exceed the allowable working stress of concrete. In order to ensure the safety of a concrete gravity dam against this sort of failure, the strength of concrete shall be such that it is more than the stresses anticipated in the structure by a safe margin. The maximum compressive stresses occur at heel (mostly during reservoir empty condition) or at toe (at reservoir full condition) and on planes normal to the face of the dam. The strength of concrete and masonary varies with age, the kind of cement and other ingredients and their proportions in the work can be determined only by experiment.

The calculation of the stresses in the body of a gravity dam follows from the basics of elastic theory, which is applied in a two-dimensional vertical plane, and assuming the block of the dam to be a cantilever in the vertical plane attached to the foundation. Although in such an analysis, it is assumed that the vertical stresses on horizontal planes vary uniformly and horizontal shear stresses vary parabolically, they are not 

strictly correct. Stress concentrations develop near heel and toe, and modest tensile stresses may develop at heel. The basic stresses that are required to be determined in a gravity dam analysis are discussed below:

Normal stresses on horizontal planes

On any horizontal plane, the vertical normal stress (σ_z) may be determined as: 

               (17) 

Where 

∑V = Resultant vertical load above the plane considered  

T = Thickness of the dam block, that is, the length measured from heel to toe    

e = Eccentricity of the resultant load                

y = Distance from the neutral axis of the plane to the point where () is being determined 

At the heel, y = -T/2 and at toe, y = +T/2. Thus, at these points, the normal stresses are found out as under: 

           (18) 

            (19) 

The eccentricity e may be found out as: 

e = Net moment / Net vertical force            (20) 

Naturally, there would be tension on the upstream face if the overturning moments under the reservoir full condition increase such that e becomes greater than T/6. The total vertical stresses at the upstream and downstream faces are obtained by addition of external hydrostatic pressures. 

Shear stresses on horizontal planes    

Nearly equal and complimentary horizontal stress (τ_zy) and shear stresses (τ_yz) are developed at any point as a result of the variation in vertical normal stress over a horizontal plane (Figure 39). The following relation can be derived relating the stresses with the distance y measured from the centroid: 

             (21)

Where  

τ_yz D = (σ_zD – p_D) tanφ_D, the shear stress at downstream face 

τ_yz U = - (σ_zU – pU) tanφD, the shear stress at upstream face 

H = the height of the dam 

The shear stress is seen to vary parabolically from τ_yz U at the upstream face up to τ_yz D at the downstream face. 

Normal stresses on vertical planes 

These stresses, σ_y can be determined by consideration of the equilibrium of the horizontal shear forces operating above and below a hypothetical element within the dam (Figure 39). The difference in shear forces is balanced by the normal stresses on vertical planes. Boundary values of σ_y at upstream and downstream faces are given by the following relations: 

σ _yU = P_u + (σ _zU - P_y) tan²φ U

 σ _yD = P_y + (σ _zD - P_y) tan²φ D

Figure.39: State of stress in a concrete gravity dam 

Principal stresses 

These are the maximum and minimum stresses that may be developed at any point within the dam. Usually, these are denoted as σ₁ and σ₃respectively, and are oriented at a certain angle to the reference horizontal or vertical lines. The magnitude of σ₁ and σ₃may be determined from the state of stress σ_z ,σ_y and τ_yz at any point by the following formula: 

            (24) 

The maximum and minimum shear stress is obtained from the following formula: 

              (25) 

The upstream and downstream faces are each planes of zero shear, and therefore, are planes of principal stresses. The principal stresses at these faces are given by the following expressions: 

(26) 

                                            (27)

    (28) 

                           (29) 

Permissible stresses in concrete 

According to IS: 6512-1984, the following have to be followed for allowable compressive and tensile stresses in concrete:

Compressive strength of concrete is determined by testing 150mm cubes. The strength of concrete should satisfy early load and construction requirements and at the age of one year, it should be four times the maximum computed stress in the dam or 14N/mm², whichever is more. The allowable working stress in any part of the structure shall also not exceed 7N/mm^2. No tensile stress is permitted on the upstream face of the dam for load combination B. Nominal tensile stresses are permitted for other load combinations and their permissible values should not exceed the values given in the following table: 

Where f_c is the cube compressive strength of concrete. 

Small values of tension on the downstream face is permitted since it is improbable that a fully constructed dam is kept empty and downstream cracks which are not extensive and for limited depths from the surface may not be detrimental to the safety of the structure.