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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.

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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 in the middle of the valley.

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.

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.

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

obtained with the available aggregates/ stones. The

specific weight of the concrete is usually taken as 24

kN/m3, and that of masonry as 23 kN/m

3 in preliminary

designs. However, for the final design, the specific weight

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

Structural

Height

Base Width

River Bed

F.R.L.

Dam Axis

Heel Toe

d/s u/s

M.W.L

Hydraulic

Height

W3

W2

W1

3

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

yp wγ=

where γw is the specific weight of water (=

9.81 kN/m3 for ρw =1000 kg/m

3). 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.

(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

2

2

1hP wH γ=

It acts horizontally at a height of h/3 above the base of the dam.

PV1

A

B

C

D

h

γwh

PV2

E PH

h/3

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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

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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

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

the dam is reduced in the proportion that the

weight of the fill is reduced by

submergence. IS code recommends that a)

Horizontal silt and water pressure is

assumed to be equivalent to that of a fluid

with a mass of 1360 kg/m3, and b) Vertical

silt and water pressure is determined as if

silt and water together have a density of

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

PV1

A

B

C D

h

Silt

PV3

E

PH1

PV2

PH2

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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:

ww hp 24=

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,

220 ww hP =

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

4/1)(271.0763.0032.0 FFVh ww −+= for F < 32 km

FVh ww 032.0= 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:

0αβα Ih =

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

gSIF ah 0βα =

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

m

m

gEB

HT

γ2

55.5=

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.

hv αα 5.0=

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:

hBhB hWMWV αα 9.06.0 ==

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

BmyBvy MCMVCV '' ==

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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-cum-

parabolic. The hydrodynamic

pressure at depth y below the

reservoir surface shall be

determined as follows:

hCp whsey γα=

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:

−+

−=h

y

h

y

h

y

h

yCC ms 22

2

where Cm = maximum value of Cs,

which can be read from Fig. or

obtained from

−=90

1735.0θ

mC

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 one-

half 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

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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

ypypypP eyeyeyey 727.03

2

42

1=

+=π

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

22 299.015

4

3

1

2

1

5

2

3

2

3

4

42

1ypypyypyypM eyeyeyeyey =

+=

×+×=π

π

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

( ) θtan12 eyeyeV PPP −=

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.

15

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.