GRAVITY DAM Part I: FORCES A gravity dam is a solid structure, made of concrete or masonry, constructed across a river to create a reservoir on its upstream. The section of the gravity dam is approximately triangular in shape, with its apex at its top and maximum width at bottom. The section is so proportioned that it resists the various forces acting on it by its own weight. Most of the gravity dams are solid, so that no bending stress is introduced at any point and hence, they are sometimes known as solid gravity dams to distinguish them from hollow gravity dams in those hollow spaces are kept to reduce the weight. Early gravity dams were built of masonry, but now-a-days with improved methods of construction, quality control and curing, concrete is most commonly used for the construction of modern gravity dams. A gravity dam is generally straight in plan and, therefore, it is also called straight gravity dam. However, in some cases, it may be slightly curved in plan, with its convexity upstream. When the curvature becomes significant, it becomes on arch dam. The gravity dams are usually provided with an overflow spillway in some portion of its length. The dam thus consists of two sections; namely, the non-overflow section and the overflow section or spillway section. The design of these two sections is done separately because the loading conditions are different. The overflow section is usually provided with spillway gates. The ratio of the base width to height of most of the gravity dam is less than 1.0. The upstream face is vertical or slightly inclined. The slope of the downstream face usually varies between 0.7: 1 to 0.8: 1. Gravity dams are particularly suited across gorges with very steep side slopes where earth dams might slip. Where good foundations are available, gravity dams can be built upto any height. Gravity dams are also usually cheaper than earth dams if suitable soils are not available for the construction of earth dams. This type of dam is the most permanent one, and requires little maintenance. The most ancient gravity, dam on record was built in Egypt more than 400 years B.C. of uncemented masonry. Archeological experts believe that this dam was kept in perfect condition for more than 45 centuries. The highest gravity dam in the world is Grand Dixence Dam in Switzerland, which is 285 ill high. The second highest gravity dam is Bhakra Dam in India, which has a height of 226 m.
Basic Definitions 1. Axis of the dam: The axis of the gravity dam is the line of the upstream edge of the top (or crown) of the dam. If the upstream face of the dam is vertical, the axis of the dam coincides with the plan of the upstream edge. In plan, the axis of the dam indicates the horizontal trace of the upstream edge of the top of the dam. The axis of the dam in plan is also called the base line of the dam. The axis of the dam in plan is usually straight. However, in some special cases, it may be slightly curved upstream, or it may consist of a combination of slightly curved RIGHT portions at ends and a central ABUTMENT straight portion to take the best advantages of the topography of the site. 2. Length of the dam: The length of the dam is the distance from one abutment to the other, measured along the axis of the dam at the level of the top of the dam. It is the usual practice to mark the distance from the left abutment to the right abutment. The left abutment is one which is to the left of the person moving along with the current of water. 3. Structural height of the dam: The structural height of the dam is the difference in elevations of the top of the dam and the lowest point in the excavated foundation. It, however, does not include the depth of special geological features of foundations such as narrow fault zones below the foundation. In general, the height of the dam means its structural height. 1
4. Maximum base width of the dam: The maximum base width of the dam is the maximum horizontal distance between the heel and the toe of the maximum section of the Dam Axis M.W.L dam in the middle of the valley. F.R.L. 5. Toe and Heel: The toe of the dam is the downstream edge of the base, and the heel is the upstream edge of the base. When a person moves along with water current, his toe comes first and heel comes later.
Structural Height
Hydraulic Height
6. Hydraulic height of the dam: The hydraulic height of the dam is equal to the difference in elevations of the highest controlled water surface on the upstream of the dam (i. e. FRL) and the lowest point in the river bed.
u/s River Bed
d/s Heel
Toe
Base Width
Forces Acting on Gravity Dam A gravity dam is subjected to the following main forces: 1. Weight of the dam 2. Water pressure
3. Uplift pressure
4. Wave pressure
5. Earth and Silt pressure
6. Ice pressure
7. Wind pressure
8. Earthquake forces
9. Thermal loads.
These forces fall into two categories as: a) Forces, such as weight of the dam and water pressure, which are directly calculable from the unit weights of the materials and properties of fluid pressures; and b) Forces, such as uplift, earthquake loads, silt pressure and ice pressure, which can only be assumed on the basis of assumption of varying degree of reliability. It is in the estimating of the second category of the forces that special care has to be taken and reliance placed on available data, experience, and judgment. It is convenient to compute all the forces per unit length of the dam. 1. Weight of the dam The weight of the dam is the main stabilizing force in a gravity dam. The dead load to be considered comprises the weight of the concrete or masonry or both plus the weight of such appurtenances as piers, gates and bridges. The weight of the dam per unit length is equal to the product of the area of cross-section of the dam and the specific weight (or unit weight) of the material. The unit weight of concrete and masonry varies considerably depending upon the various materials that go to make them. It is essential to make certain that the assumed unit weight for concrete/masonry or both can be W2 obtained with the available aggregates/ stones. The specific weight of the concrete is usually taken as 24 W3 kN/m3, and that of masonry as 23 kN/m3 in preliminary designs. However, for the final design, the specific weight W1 is determined from the actual tests on the specimens of materials. It is essential that the actual specific weight of concrete during the construction of
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the dam should not be less than that considered in the final design. Attempts should be made to achieve the maximum possible specific weight. The factors governing the specific weight of the concrete are water-cement ratio, compaction of concrete and the unit weight of the aggregates. For high specific weight, the aggregates used should be heavy. For convenience, the cross-section of the dam is divided into simple geometrical shapes, such as rectangles and triangles, for the computation of weights. The areas and controids of these shapes can be easily determined. Thus the weight components W1, W2, W3 etc. can be found along with their lines of action. The total weight W of the dam acts at the C.G. of its section. 2. Reservoir and Tailwater loads (Water pressure) The water pressure acts on the upstream and downstream faces of the dam. The water pressure on the upstream face is the main destabilizing (or overturning) force acting on a gravity dam. The tail water pressure helps in the stability. The tail water pressure is generally small in comparison to the water pressure on the upstream face. Although the weight of water varies slightly with temperature, the variation is usually ignored. In case of low overflow dams, the dynamic effect of the velocity of approach may be significant and will deserve consideration. The mass of the water flowing over the top of the spillway is not considered in the analysis since the water usually approaches spouting velocity and exerts little pressure on the spillway crest. If gates or other control features are used on the crest they are treated as part of the dam so far as application of water pressure is concerned. The mass of water is taken as 1000 kg/m3. Linear distribution of the static water pressure acting normal to the face of the dam is assumed. Tailwater pressure adjusted for any retrogression should be taken at full value for non-overflow sections and at a reduced value for overflow sections depending on the type of energy dissipation arrangement adopted and anticipated water surface profile downstream. The full value of corresponding tailwater should, however, be used in the case of uplift. The water pressure intensity p (kN/m2) varies linearly with the depth of the water measured below the free surface y (m) and is expressed as p = γwy
where γw is the specific weight of water (= 9.81 kN/m3 for ρw =1000 kg/m3). For simplification, the specific weight of water may be taken as 10 kN/m3 instead of 9.81 kN/m3. The water pressure always acts normal to the surface. While computing the forces due to water pressure on inclined surface, it is convenient to determine the components of the forces in the horizontal and vertical directions instead of the total force on the inclined surface directly.
D C
h PV1 PH
E
h/3
B PV2
γwh A (a) U/s face vertical: When the upstream face of the dam is vertical, the water pressure diagram is triangular in shape with a pressure intensity of γwh at the base, where h is the depth of water. The total water pressure per unit length is horizontal and is given by PH =
1 γ wh2 2
It acts horizontally at a height of h/3 above the base of the dam.
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(b) U/s face inclined: When the upstream face ABC is either inclined or partly vertical and partly inclined, the force due to water pressure can be calculated in terms of the horizontal component PH and the vertical component PV. The horizontal component is given as earlier and acts horizontal at a height of (h/3) above the base. The vertical component PV of water pressure per unit length is equal to the weight of the water in the prism ABCD per unit length. For convenience, the weight of water is found in two parts PV1 and PV2 by dividing the trapezium ABCD into a rectangle BCDE and a triangle ABE. Thus the vertical component PV = PV1 + PV2 = weight of water in BCDE + weight of water in ABE. The lines of action of PV1 and PV2 will pass through the respective centroids of the rectangle and triangle. 3. Uplift pressure Water has a tendency to seep through the pores and fissures of the foundation material. It also seeps through the joints between the body of the dam and its foundation at the base, and through the pores of the material in the body of the dam. The seeping water exerts pressure and must be accounted for in the stability calculations. The uplift pressure is defined as the upward pressure of water as it flows or seeps through the body of the dam or its foundation. A portion of the weight of the dam will be supported on the upward pressure of water; hence net foundation reaction due to vertical force will reduce. The area over which the uplift pressure acts has been a question of investigation from the early part of this century. One school of thought recommends that a value one-third to two-thirds of the area should be considered as effective over which the uplift acts. The second school of thought, recommend that the effective area may be taken approximately equal to the total area. The code of Indian Standards recommends that the total area should be considered as effective to account for uplift. According to the Indian Standard (IS : 6512-1984), there are two constituent elements in uplift pressure: the area factor or the percentage of area on which uplift acts and the intensity factor or the ratio which the actual intensity of uplift pressure bears to the intensity gradient extending from head water to tail water at various points. Effective downstream drainage, whether natural or artificial, will generally limit the uplift at the toe of the dam to tail water pressure. Formed drains in the body of the dam and drainage holes drilled subsequent to grouting in the foundation, where maintained in good repair, are effective in giving a partial relief to the uplift pressure intensities under and in the body of the dam. The degree of effectiveness of the system will depend upon the character of the foundation and the dependability of the effective maintenance of the drainage system. In any case, observation of the behaviour of the dam will indicate the uplift pressures actually acting on the structure and when the uplift pressure are seen to approach or exceed design pressures, prompt remedial measures 4
should necessarily be taken to reduce the uplift pressures to values below the design pressures. This following criteria are recommended by IS code for the calculating uplift forces : (a) Uplift pressure distribution in the body of the dam shall be assumed, in case of both preliminary and final designs, to have an intensity which at the line at the formed drains exceeds the tailwater pressure by one-third the differential between reservoir level and tailwater level. The pressure gradient shall then be extending linearly to heads corresponding to reservoir level and tailwater level. The uplift shall be assumed to act over 100 percent 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 tailwater pressure by one-third the differential between the reservoir and tailwater heads. The pressure gradient shall then be extended linearly to heads corresponding to reservoir level and tailwater level. The uplift shall be assumed to act over 100 % area. For final designs, the uplift criteria in case of dams founded on compact and unfissured 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 % of the area. (c) In absence of line of drains and for the extreme loading conditions F and G, the uplift shall be taken as varying linearly from the appropriate reservoir water pressure at the u/s face to the appropriate tailwater 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, a 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 vary linearly to the tailwater pressure at the d/s face. The uplift is assumed to act over 100 % 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 tailwater elevation) that may be expected immediately downstream of the structure. (e) It is assumed that uplift pressures are not affected by earthquakes. 4. Earth and Silt Pressures Gravity dams are subjected to earth pressures on the downstream and upstream faces where the foundation trench is to he backfilled. Except in the abutment sections in specific cases and in the junctions of the dam with an earth or rockfill embankment, earth pressures have usually a minor effect on the stability of the structure and may be ignored.
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The present procedure is to treat silt as a saturated cohesionless soil having full uplift and whose value of internal friction is not D C materially changed on account of submergence. Experiments indicate that silt pressure and water pressure exist together in a submerged fill and that the silt pressure on PH1 the dam is reduced in the proportion that the PV1 weight of the fill is reduced by h submergence. IS code recommends that a) Horizontal silt and water pressure is Silt B E assumed to be equivalent to that of a fluid PH2 3 with a mass of 1360 kg/m , and b) Vertical PV2 silt and water pressure is determined as if PV3 silt and water together have a density of A 1925 kg/m3. 5. Ice Pressure The problem of ice pressure in the design of dam is not encountered in India except, perhaps, in a few localities. Ice expands and contracts with changes in temperature. In a reservoir completely frozen over, a drop in the air temperature or in the level of the reservoir water may cause the opening up of cracks which subsequently fill with water and freezed solid. When the next rise in temperature occurs, the ice expands and, if restrained, it exerts pressure on the dam. In some cases the ice exerts pressure on the dam when the water level rises. For ice sheets of wide extent this pressure is moderate but in smaller ice sheets the pressure may be of the same order of magnitude as in the case of extreme temperature variation. Ice is plastic and flows under sustained pressure. The duration of rise in temperature is, therefore, as important as the magnitude of the rise in temperature in the determination of the pressure exerted by ice on the dam. Wind drag also contributes to the pressure exerted by ice to some extent. Wind drag is dependent on the size and shape of the exposed area, the roughness of the surface area and the direction of wind. Existing design information on ice pressure is inadequate and somewhat approximate. Good analytical procedures exist for computing ice pressures, but the accuracy of results is dependent upon certain physical data which have not been adequately determined. These data should come from field and laboratory. Till specific reliable procedures become available for the assessment of ice pressure it may be provided for at the rate of 250 kPa applied to the face of dam over the anticipated area of contact of ice with the face of dam. 6. Wind Pressure Wind pressure does exist but is seldom a significant factor in the design of a dam. Wind loads may, therefore, be ignored. 7. Wave Pressure In addition to the static water loads the upper portions of dams are subject to the impact of waves. Wave pressure against massive dams of appreciable height is usually of little consequence. The force and dimensions of waves depend mainly on the extent and configuration of the water surface, the velocity of wind and the depth of reservoir water. The height of wave is generally more important in the determination of the free board requirements of dams to prevent overtopping by wave splash. An empirical method based upon research studies on specific cases has been recommended by T. Saville for computation of wave height hw (m). It takes into account the effect of the shape of reservoir and also wind
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velocity over water surface rather than on land by applying necessary correction. It gives the value of different wave heights and the percentage of waves exceeding these heights so that design wave height for required exceedance can be selected. Wind velocity of 120 km/h over water in case of normal pool condition and of 80 km/h over water in case of maximum reservoir condition should generally be assumed for calculation of wave height if meteorological data is not available. When maximum wind velocity is known, the same shall be used for full reservoir level (FRL) condition and 2/3 times that for MWL condition. The maximum unit pressure pw in kPa occurs at 0.125 hw, above the still water level and is given by the equation:
p w = 24hw The wave pressure diagrams can be approximately represented by the triangle l-2-3 as in Fig.
The total wave force Pw, (in kN) is given by the area of the triangle, Pw = 20hw2 The centre of application is at a height of 0.375 hw, above the still water level. Sometimes the following Molitor’s empirical formulae are used to estimate wave height hw = 0.032 Vw F + 0.763 − 0.271( F )1 / 4
for F < 32 km
hw = 0.032 Vw F
for F > 32 km
where Vw = wind velocity in km/hr and F = fetch length of reservoir in km. 8. Thermal Loads Measures for temperature control of concrete in solid gravity dams are adopted during construction. Yet it is noticed that stresses in the dam are affected due to temperature variation in the dam on the basis of data recorded from the thermometer embedded in the body of the dam. The cyclic variation of air temperature and the solar radiation on the downstream side and the reservoir temperature on the upstream side also affect the stresses in the dam. Even the deflection of the dam is maximum in the morning and it goes on reducing
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to a minimum value in the evening. The magnitude of deflection is also affected depending on whether the spillway is running or not. It is generally less when spillway is working than when it is not working. While considering the thermal load, temperature gradients are assumed depending on location, orientation, surrounding topography, etc. 9. Earthquake Forces The earthquake sets up primary, secondary, Raleigh and Love waves in the earth's crust. The waves impart accelerations to the foundations under the dam and. causes its movement. In order to avoid rupture, the dam must also move along with it. This acceleration introduces an inertia force in the body of dam and sets up stresses initially in lower layers and gradually in the whole body of the dam. Earthquakes cause random motion of ground which can be resolved in any three mutually perpendicular directions. This motion causes the structure to vibrate. The vibration intensity of ground expected at any location depends upon the magnitude of earthquake, the depth of focus, distance from the epicentre and the strata on which the structure stands. The predominant direction of vibration is horizontal. The response of the structure to the ground vibration is a function of the nature of foundation soil; materials, form, size and mode of construction of the structure; and the duration and the intensity of ground motion. IS:1893 - 1984 code specifies design seismic coefficient for
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structures standing on soils or rocks which will not settle or slide due to loss of strength during vibrations. The seismic coefficients recommended in this standard are based on design practice conventionally followed and performance of structures in past earthquakes. In the case of structures designed for horizontal seismic force only, it shall be considered to act in any one direction at a time. The vertical seismic coefficient shall be considered in the case of structures in which stability is a criterion of design. For the purpose of determining the seismic forces, the country is classified into five zones as shown in Fig. The following assumptions shall be made in the earthquake resistant design of structures: a) Earthquake causes impulsive ground motion which is complex and irregular in character, changing in period and amplitude each lasting for small duration. ‘Therefore, resonance of the type as visualized under steady state sinusoidal excitations will not occur as it would need time to build up such amplitudes. b) Earthquake is not likely to occur simultaneously with wind or maximum flood or maximum sea waves. c) The value of elastic modulus of materials, wherever required, may be taken as for static analysis unless a more definite value is available for use in such condition. Permissible Increase in Stresses: Whenever earthquake forces are considered along with other normal design forces, the permissible stresses in materials, in the elastic method of design, may be increased by one-third. However, for steels having a definite yield stress, the stress be limited to the yield stress; for steels without a definite yield point, the will stress will be limited to 80 percent of the ultimate strength or 0.2 percent proof stress whichever is smaller and that in prestressed concrete members, the tensile stress in the extreme fibres of the concrete may be permitted so as not to exceed 2/3 of the modulus of rupture of concrete. Design Seismic Coefficient for Different Zones: The earthquake force experienced by a structure depends on its own dynamic characteristics in addition to those of the ground motion. Response spectrum method takes into account these characteristics and is recommended for use in case where it is desired to take such effects into account. For design of other structures an equivalent static approach employing use of a seismic coefficient may be adopted. As per IS Code, for dams up to 100 m height, the seismic coefficient method shall be used for the design of the dams; while for dams over 100 m height the response
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spectrum method shall be used. Both the seismic coefficient method (for dams up to 100 m height) and response spectrum method (for dams greater than 100 m height) are meant only for preliminary design of dams. For final design dynamic analysis or detailed investigations are made in accordance with IS: 4967 – 1968. For design of dam using the approach of linear variation of normal stresses across the cross-section, tensile stresses may be permitted in the upstream face up to 2 percent of the ultimate crushing strength of concrete. The basic seismic coefficients (α0) and seismic zone factors (F0) in different zones shall be taken as given in Table 1. The design seismic forces shall be computed on the basis of importance of the structure I (Table 3) and its soil-foundation system β (Table 2).
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In Seismic Coefficient Method the design value of horizontal seismic coefficient (αh) shall be computed as:
α h = β Iα 0 where β = a coefficient depending upon the soil foundation system (Table 2), I = a factor depending upon the importance of the structure (for dams it is 3, see Table 3). In response Spectrum Method the response acceleration coefficient is first obtained for the natural period and damping of the structure and the design value of horizontal seismic coefficient (αh) shall be computed using
α h = β IF0 S a g where Sa/g = average acceleration coefficient as read from Fig for a damping of 5 percent and fundamental period of vibration of the dam corresponding to
T = 5.55
H2 B
γm gEm
where H = height of the dam in m, B = base width of the dam in m, γm = unit weight of the material of dam in N/m3, g = acceleration due to gravity in m/s2, and Em, = modulus of elasticity of the material in N/m2. Where a number of modes are to be considered for seismic analysis αh shall be worked out corresponding to the various mode periods and dampings and
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then design forces shall be computed. In case design response spectra is specially prepared for a structure at a particular site, the same may be used for design directly instead of the above equation. The vertical seismic coefficient (αv) may be taken as half of the horizontal seismic coefficient i.e.
α v = 0.5α h In important structures where there is a possibility of amplification of vertical seismic coefficient, dynamic analysis is preferable. Effect of Horizontal Acceleration: Horizontal· acceleration causes two forces: (1) Inertia force in the body of the dam, and (2) Hydrodynamic pressure of water. Inertia forces: The inertia force acts in a direction opposite to the acceleration imparted by, earthquake forces and is equal to the product of the mass of the dam and the acceleration. For dams up to 100 m height the horizontal seismic coefficient shall be taken as 1.5 times seismic coefficient αh at the top of the dam reducing linearly to zero at the base as shown in Fig. This inertia force shall be assumed to act from upstream to downstream or downstream to upstream to get the worst combination for design. It causes an overturning moment about the horizontal section adding to that caused by hydrodynamic force. For dams over 100 m height the response spectrum method shall be used. The base shear, VB and base moment MB may be obtained by the following formulae:
VB = 0.6Wα h
M B = 0.9W h α h
where W = total weight of the masonry or concrete in the dam in N, and h = height of the centre of gravity of the dam above the base in m. For any horizontal section at a depth y below top of the dam shear force, Vy, and bending moment My, may be obtained as follows
V y = C v' VB
M y = C m' M B
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where C’v and C’m, are given in Fig. Hydrodynamic Effects Due to Reservoir: Due to horizontal acceleration of the foundation and dam there is an instantaneous hydrodynamic pressure (or suction) exerted against the dam in addition to hydrostatic forces. The direction of hydrodynamic force is opposite to the direction of earthquake acceleration. In 1952, Zanger presented formulae for computing the hydrodynamic pressure exerted on vertical and sloping face by horizontal acceleration. The formulae were derived by electrical analogy, based on the assumption that water is incompressible. The pressure variation is elliptical-cumparabolic. The hydrodynamic pressure at depth y below the reservoir surface shall be determined as follows:
pey = C s α h γ w h where pey = hydrodynamic pressure intensity (Pa) at depth y, h = depth of reservoir (m) and Cs = coefficient which varies with shapes of u/s face and depth of water. Approximate values of Cs, for dams with vertical or constant upstream slopes may be obtained as follows: Cs =
Cm 2
y y y y 2 − 2− + h h h h
where Cm = maximum value of Cs, which can be read from Fig. or obtained from
θ C m = 0.7351 − 90 where θ = angle, in degrees the u/s face of the dam makes with vertical. For dams with combination of vertical and sloping faces, an equivalent slope may be used for obtaining the approximate value of Cs,. If the height of the vertical portion of the upstream face of the dam is equal to or greater than onehalf the total height of the dam, analyze it as if vertical throughout. If the height of the vertical portion of the upstream face of the dam is less than one-half the total height of the dam, use the pressure on the sloping line connecting the point of 13
intersection of the upstream face of the dam and the reservoir surface with the point of intersection of the upstream face of the dam with the foundation. The total pressure at depth y may be found by integrating the pressure curve above that plane. Taking the pressure variation to be elliptical-cum-parabolic, the total pressure at depth y will be equal to the average of the areas of the quarter ellipse and semi parabola. Hence
Pey =
1 π 2 p ey y + pey y = 0.727 pey y 2 4 3
Similarly, the moment of pressure about the joint upto which the pressure is taken is given by half the sum of the moments of the quarter ellipse and semi-parabola. Hence
M ey =
1 π 4 2 2 11 4 y + p ey y × y = + pey y 2 = 0.299 pey y 2 p ey y × 2 4 3π 3 5 2 3 15
where Pey = hydrodynamic shear in N/m at any depth y, and Mey = moment in N.m/m due to hydrodynamic force at any depth y. Effect of Horizontal Acceleration on the Vertical Component of Reservoir and Tail Water Load: Since the hydrodynamic pressure ( or suction ) acts normal to the face of the dam, there shall, therefore, be a vertical component of this force if the face of the dam against which it is acting is sloping, the magnitude at any horizontal section being PeV = (Pey 2 − Pey1 ) tan θ
where PeV = increase (or decrease) in vertical component of load due to hydrodynamic force, Pey2 = total horizontal component of hydrodynamic force at the elevation of the section being considered, Pey1 = total horizontal component of hydrodynamic force at the elevation at which the slope of the dam face commences, and θ = angle between the face of the dam and the vertical. The moment due to the vertical component of reservoir and tail water load may be obtained by determining the lever arm from the centroid of the pressure diagram. Effects of Vertical Acceleration: The effect of vertical earthquake acceleration is to change the unit weight of water and concrete or masonry. Acceleration upwards increases the weight and acceleration downwards decreases the weight. Due to vertical acceleration a vertical inertia force F = αVW is exerted on the dam, in the direction opposite to that of the acceleration. When the acceleration is vertically upwards, the inertia force F = αVW acts vertically downwards, thus increasing momentarily the downward weights. When the acceleration is vertically downwards the inertia force F = αVW acts upwards and decreases momentarily the downward weight. For methods of design (seismic coefficient up to 100 m and response spectrum over 100 m) Vertical seismic coefficient (αV) shall be taken as 0.75 times the value of αh (of the respective method) at the top of the dam reducing linearly to zero at the base. Effect of earthquake acceleration on uplift forces: Effect of earthquake acceleration on uplift forces at any horizontal section is determined as a function of the hydrostatic pressure of reservoir and tail-water against the faces of the dam. During an earthquake the water pressure is changed by the hydrodynamic effect. However, the change is not considered effective in producing a corresponding increase or reduction in the uplift force. The duration of the earthquake is too short to permit the building up of pore pressure in the concrete and rock foundations.
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Effect of earthquake acceleration on dead silt loads: It is sufficient to determine the increase in the silt pressure due to earthquake by considering hydrodynamic forces on the water up to the base of the dam and ignoring the weight of the silt. Earthquake Forces for Overflow Sections: The provisions for the dam as given earlier will be applicable to over-flow sections as well. In this case, the height of the dam shall be taken from the base of the dam to the top of the spillway bridge for computing the period as well as shears and moments in the body of the dam. However, for the design of the bridge and the piers, the horizontal seismic coefficients in either direction may be taken as the design seismic coefficient for the top of the dam (1.5αh) and applied uniformly along the height of the pier.
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