U N I 4~ GRAVITY DAMS

1 Introduction

Design Requirements
Design of an Arbitrary Profile
Stability Analysis

Construction Joints
Transverse and Longitudinal Joints

arnt about the various types of dams, their classification, and their selection.
ere told of the various forces that a dam has to resist and the treatment done
on for enabling the stresses to be trar~smittedsafely. In this unit, you will
stability analysis of a conaete gravity dam.

this unit, you will be able to

discuss the design requirements of a gravity dam,
design an arbitrary section,
carry out the stability analysis of a gravity dam,
explain the purpose of the varioqs joints provided in a dam, and
explain the need for temperature control of concrete.

4.2 D ~ $ I G NREOUIREMENTS
erives its stability from the forces of gravity of the materials within the body
ence the name. In order to withstand the forces and the overturning
by the water stored in the reservoir, the gravity darn shoulC have sufficient
foundations are required for a gravity dam since it transfers the loads to
cantilever a~lion.There are two types of forces:
1) 1I Forces causing stability
Self weight of the darn, and
thrust of tail water.
causing instability :
pressure from reservoir,
) Ijplift pressure,
 c) Forces due to waves on the surface of reservoir,
d) Ice pressure,
e) Temperature stresses,
f) Silt pressure,
g) Wind pressure, and
h) Earthquake forces.
Clasdncatlon of Loadlng for Design
For purposes of design the loads are taken in certain combinations, and the d m section
should be designed so as to develop stresses within the permissible limits. The combinations
are:
Normal Loads
They are those, under the combined action of which the dam section should have adequate
stability, and the permissible stresses and factors of safety should not be exceeded. These
loads are:
a) Weight of the dam and the structure over it,
b) Water pressure corresponding to full reservoir level, and
c) Uplift.
Abnormal Loads
Tbese are loads which in combination with nonnal loads encroach upon the factor of safety
and increase the allowable stresses yet remaining lower than the higher emergency stress
limits. They include:
a) Higher water pressure during floods,
b) Wave pressure,
c) Silt pressure,
d) Ice pressure, and
e) Earthquake force.
Load Combinations
The designs should be based on the most adverse combination of "probable" load
conditions, but should include only those loads having reasonable probability of
simultaneous occurrence. The USBR specifies "Standard" and "Extreme" load
combinations as under:
Standard Load Combinations
a) Normal reservoir level, ice and silt (if applicable), and normal uplift.
b) Normal water level, earthquake, silt (if applicable) and normal uplift.
c) Maximum flood water surface elevation, silt (if applicable), and normal uplift.
Extreme Load Combinations
a) Maximum flood water elevation, silt (if applicable), and extreme uplift (i.e.
drains choked).
Reservoir Empty Condition
a) Empty reservoir (without earthquake) should be computed for design of
reinforcement or grouting studies.
b) Construction stage reservoir empty, earthquake considered but no wind load.

SAQ 1
I) What art: ~ h tdesign
' requlrenlenlc i)la gravity dam'
In) What ,it1 by ~omhinadonol Ioiids ul the tlesign 01 a praklt)
you w~clers~and
dam"
 Gravity Dams

itions required to be met for a gravity dam, subjected to only its self
ue t9 water pressure, P ,and uplift force, U ,can be satisfied by a simple
ar section (Figure 4.1). with its vertex at the reservoir water level, and
y wide at the base where the water pressure is maximum. Such a section
be an elementary section of a gravity dam. For the reservoir empty condition, the
e acting on the dam is its self weight whose line of action will meet the base at b/3
ase width) from the heel of the darn and thus satisfy the stability requirements of no
idth of the arbitrary section is determined for satisfying no tension and
a as given below, and the higher of the two base widths is selected for the
For the section shown in Figure 4.1. (assuming width of the dam as one
!f
lane of the paper) one considers that the resultant, R, of all the three
s, Wc (= 0.5 swbh), W l(= 0.5 wh ) and
0.5 wbhd) (here s = specific gravity of concrete and cJ is a correction factor for uplift
) passes through the downstream middle third point (D), one gets

I
(0.5 swbh) x (i)- (0.5 wh2) x-I!(
Figure 4.1 :Elementary PmBle of a Gravity Dam

(0.5 wbhd) x ($1 =0

h
cf = 1 , b =
rn c'
d if uplift is neglected, that is, = 0,

/?orno sliding to occur. (Wc - 0) = P

It is clear that for satisfying the requirement of stability, the arbitrary section of a gravity
dam should have minimum base width equal to the higher of the base widths obtained from
no-sliding and no-tension criteria.
Again, for an arbitrary section,

zw = (W,- 0 )
,.
"
;

W = 0.5 wbh (S - cf)

 b
For no tension in the dam, e = -
6

Therefore, at the toe of the dam

and at the heel of the dam

Accordingly, the principal stress, OIL) = ~2 cpu

o ysec

Similarly,

The principal and the shear stresses at the heel are, obviously, zero.
Similarly, when the reservoir is empty,

W = 0.5 w b h

2c w ...(4.11)
dl LT = Oyu = -
b
- whs

Sometimes, depending upon whether or not the compressive stress at the toe O ~ exceeds
D
the maximum permissible stress, a,, for Ule material of the dam, a gravity dam is called a
"high or a "low" dam. On this basis, the limiting height, hl, is obtained by equating the
~ a
expression for 0 1 with ,
. Thus,
 ight of a gravity dalv is less than hi, it is a low darn, otherwise it is a high dAm.

4.41
B I
STABILITY ANALYSIS

i
11
s analysis of gravity dams can be carried out by using either of the three methods:
a)
b)
The gravity method,
The trial load method, and
I C) The finite element method.
first method alone will be discussed here as it is the simplest. The gravity method of
is is applicable to the general case of a gravity dam when its blocks are not
monolithic by keying and grouting the joints between them. All these blocks of the
act independently and the load is transmitted to the foundation by cantilever
resisted by the weight of the cantilever. The following assumptions are made in
gravity method of analysis:
i) The concrete in the dam is a homogeneous, isotropic, and uniformly elastic
material,
ii) No differential movements occur at the site of the dam due to the water loads
on the walls and base of the reservoir,
iii) All loads are transmitted to the foundation by the gravity action of vertical,
parallel cantilevers which receive no support from the adjacent cantilever
elements on either side,
iv) Normal stresses on horizontal planes vary uniformly as a straight line from the
upstream face to the downstream face, and
I
V) Horizontal shear stresses have a parabolic variation across horizontal planes
stream face to the downstream face of the dam.
I (v) above are substantially correct, except for horizontal planes
where the effects of foundation yielding affect the stress
Such effects are, however, usually small in dams of low or medium
r, y be significant in high dams in which case stresses near the
should be checked by other suitable methods of stress analysis.

i
As shown in Figure 4.2, Wand H represent, respectively, the sum of all the resultant
vertical and horizontal forces acting on a horizontal plane (represented by the section PQ)of
I a gravity dam. The resultant R of Wand H intersects the section PQ at 0' while 0

R E S E R V O I R F U L L CONDITION
b) RESERVOIR EMPTY CONOIT~ON
represents the centroid of the plane under consideration. The distance between 0 and 0' is
called the eccentricity of loading, e. When e is not equal to zero, the loading on the plane is
eccentric and the normal stress oyxat any point x on the section PQ away from the centroid
0 is given by

where A is the area of the plane PQ and I is the moment of inertia of the plane PQ about an
axis passing through its centroid and parallel to tbe length of the dam. It should be noted
that whereas the direct stress (= x
WIA) at every point of the section PQ is always
compressive, the nature of the bending stress (= x-
we,
I
) depends on the location of 0'
with respect to 0. If 0' lies between 0 and Q, there will be compressive bending stress for
any point between 0 and Q and tensile sltess for any point between 0 and P. Accordingly,
when the reservoir is full, one sholild use the positive sign in Eq. (4.13) for all points
between 0 and Q and the negative sign for all points between 0 and P.Similarly, when the
x
reservoir is empty (in wHich case H may be an earthquake force acting in the upstream
direction), and 0' lies between 0 and P, one should use the positive sign for all points
between 0 and P and the negative sign for all points between 0 and Q.
Considering unit length of the dam and the horizontal distance between the upstream edge P
T ' Thus,
and the downstream edge Q of the plane PQ as T,one can write A = T,and I =
Eq. (4.13) reduces to

One can use this equation for determining the nonnal stress on the base of the dam BB'
(Figure 4.3). If the width of the base BB' is b, Eq. (4.14) for the base of the dam reduces to

Figon4 3 :Normal Strrsscs on the Base d a Cranty Dam

For the toe (B') and the heel (B) of the dam, x = b/2. Hence, the normal stresses at the toe
(oyn) and the heel (ayu)of the dam are as follows:
When the reservoir is full,
oyD = [xf) [I +
 I

Gravity Dmns

~ h e n H reservoir
e is empty,

at if e is less than or equal to b/6, the stress is compressive all

is greater than b/6 there can be tensile stresses on the base. The
rent values of e and when the reservoir is full have been shown in
if there has to be no tension at any point of the base of the dam,
ns of loading must meet the base within the middle-third of the

stresses act is known as a principal plane. Shear stresses are

cordiigly, the upstream and the downstream faces of a
principal planes as the only force acting on these surfaces
which acts normal to these surfaces. Further, at any point in
are mutually perpendicular. Therefore, other principal planes
stream and the downstream faces of a gravity dam. In an
QR at the toe of a gravity dam (Figure 4.4), the plane QR

les to the downstream face PQ. Hence PQ and QR are the principal planes and
of the base of the dam. The stresses acting on the principal planes PQ and QR
ly, p' (tailwater pressure) and U ~ as
D shown in Figure 4.4 and are the
esses. The normal and the tangential stresses acting on PR are a y D and ('tyx)~,
Since the element is vety small, the stresses can be considered to be acting at a
&ring the equilibrium of the element PQR, the algebraic sum of all the forces
irection should be zero. If one considers the unit length of the dam, then

or (PR) cae2 b + pr (PR) sin2 $I) - a Y(PR)

~ =o
Therefor

~ b - p' t m 2 ~
a lo = a ysec2 J4.20)

Thus,kn ingp' and a yfrom ~ Eq. (4.16), one can obtain the principal stress U ~ at
D the toe
of the da Usually, p' is either zero (no tailwater) or very small in comparison to q D .
Therefor is the major principal stress and p' is the minor principal stress. When p' is
zero, Eq. .20) reduces to

the hydrodynamic pressure pre, due to earthquake acceleration (towards the

e effectiveminor principal stress'becomes p' - pre, and Eq. (4.20) becomes
When there is no tailwater, both p' andp', are zero and Eq. (4.21) is used for the
calculation of alu.
Similarly, considering an infinitesimal element at the heel of the dam (Figure 4.4), one can
obtain expression for 01~1 as follows:

011-1= oyusec2 $U - @' +pte) tan2 ~IIJ

For,the condition of empty reservoir, p = p, = 0 and hence
2 ...(4.24)
olu = oyu sec $U

When the reservoir is full, the intensity of water pressure p is usually higher than the normal
stress olu. Therefore, at the heel, y is the major principal stress and olrr is the minor
principal stress. For vertical upstream face, $u = 0 and therefore, olrj equals ~ , L J .
Again, resolving the forces acthg on the infinitesimal element PQR in the horizontal
direction and equating their algebraic sum to zero for the equilibrium condition, one gets

which gives
(~~x)1)
= (011) - P') sin $D . cos $D

2
~ $D - y' tan2 (D - pf) sin $D . cos (D
= ( o y sec

Therefore,

Sinlilarly, considering the equilibrium of the element at the heel of the darn,

Including the effects of earthquake acceleration, Eqs. (4.25) and (4.26) reduce to

In this way, one can calculate the principal stresses at the upstream and the downstream
faces of the dam at any horizontal section by considering only the forces acting above the
section.
An arbitrary section is only an ideal profile which has to be modified for adoption in actual
practice. Modification would mean providing a finite crest width, adequate freehard, batter
in the lower part of the water face and a flatter downstream face. The design of a gravity
dam involves assuming its tentative section and then dividing it into a number of zones by
horizontal planes for stability analysis at the level of each dividing horizontal pl'me. The
analysis can be either two-dimensional or three-dimensional. The following example shows
the two-dimensional method of stability analysis of gravity darns. The three-dimensional
analysis being complicated is best done with the help of computers.
Example 4.1
For the section of the gravity dam shown in Figure 4.5, compute principal stresses
for normal loading and vertical stresses for extreme loading at the heel and toe of
the base of the dam. Also determine factors of safety against overturning and
sliding as well as shear-friction factors for safety for "drains operating" and
"drains not operating conditions". Other data are as follows: -
Sediment deposited to a height of 15 m in the reservoir.
Coefficient of shear friction, p = 0.7 (normal loading)
= 0.85 (extreme loading)
 Gravity Dams
ear strength at concrete-rock interface, C = 150 t/ m2
ight density of concrete = 2.4 t/m3
ight density of water = 1 t/m3
oefficient of horizontal acceleration due to earthquake, a h = 0.1
oefficient of vertical acceleration due to earthquake, a,= 0.05.

lbl U R l F T WESSUT1E YELO DIPCRAM

-
Figure 4.5 :ProBle and Uplift Pressure Diagram for the Gravity Dam of Example 4.1

/ Consider a 1 m wide strip of the dam. The computations are shown in Table 4.1.
Table 4.1 : Computations of Forces and Moments
------

Magnitude of Forces

L
SI. No. l j p e of Load Force (t) Lever Moment about Toe
Vertical Horizontal Arm (m) (Clockwise -ve)
(Downward (Upstream
t ve) (t) + ve) (t)
(3) (4) (5) (6)
I 1. / Dead load (W.)
1 I ABC
BDEH
FGH

Water load
Vertical
Reservoir (Ww)
KJCL
ACL
Tailwater (Ww)

(ii) Horizontal

Tailwater (W1)
 Magnitude of Forces
SI. No. 'Ippe of load Force (t) Lever Moment about Toe
Vertlcal Horizontal Arm (m) (Clockwieo -ve)
@ownward (Upstream (W
+ ve) (t) + ve) (t)
(1) (2) (3) (4) (5) (6) (I)
3. Uplift force, U
0) Drains
operative
(a) PQW 1 ~ 3 8 ~ 4 ..O8 ~ 1- 182.40 73.85 - 13470.24
(b) UVW -
l~S8x4.8~0.5xl.O 139.20 74.65 - 10391.28
(4 QRST lx9x71.45xl.O - 643.05 35 73 - 22976.18
(d) STU -
1~29X71.45~0.5~1.0 1036.03 47.63 - 49345.87
- 2000,68 - 96183.57
(ii) Drains
inoperative
(a) PRSX . -
1 ~ 9 ~ 7 6 . 2 5 ~ 1 . 0 686.25 38.13 - 26166.71
@) SWX lx0.5~87~76.25~1,O3316.88 - 50.83 - 168597.01

- 4003.13 - 194763.72
4. Silt load (Ws)
(a) Excess vertical 1~0.5~15~2.25~
pressure (1.925 - 1.00) + 15.61 + 2357.11
(b) Excess 1~0.5~15~15~
horizontal -
(1.36 1.00) - 40.50 - 405.00
pressure
-
+ 15.61 - 40.50 + 1952.11
5. Earthquake
forces
(i) Inertial
horizontal
force due to
weight of the
dam
- 162.00
(a)
@)
ABC
BDEH
162x0.1
1920XO.1 ---650.25
16.20
192.00
10.00
50.00
28.33
- 9Ci00.00
- - 18421.58
(c) FOH 6502.5~0.1
- 858.45 - 28183.58
(ii) Hydrodynamic
force
Reservoir
- -
At the base. r rm
0.73 (for $U = 0)
Vp, -0.726 (0.73~
(a)
0 . 1 ~ 1 . 096)~96
~ - 488,43 .
Mpr= 0.299 (0,73X
(b) Toilwater
- -
0.1x1.0~96)~96~
At the base, c rm
0.47 (for QD =
- 19311.13
tan-' (0.75)
Vpr 10.726 (0.47~ - 2.76
O.lxl.Ox9)x9
M p -0.299 (O.$~X
0.1~1.Ox9)x9 - 10.25
- 491.19 - 19321.38
Copputatton of Stresses
a) Normal loading combination (normal design reservoir level with appropriate dead
, loads, uplift (with drains operative),silt, ice, tailwater and thennal load8 correspnding
to normal temperature):
 x
Resultant vertical force = W = sum of vertical forces at S1. Nos. 1,2(a),
3(a) and 4 (a) of Table 4.1.

Resultant horizontal force = H = sum of horizontal forces at Sl. Nos. 2 (ii)

and 4(a) of Table 4.1.

x
Moment about toe of the dam at the base = M = sum of moments at S1.
Nos. 1,2,3(a) and4 of Table4.1.
= (418302.75 + 27091.99 - 147334.50- 96183.57 + 1952.11)
= 203828.78 trn

Distance of the resultant from the toe, yt =

CM
-
Cw

Therefore, eccentricity, e = -
76 25
2
- 29.14 = 8.985 m - 8.99 m.

(The resultant passes through the downstream of the centre of the base.)
Using Eqs. (4.16) and (4.17),

= 26.84 t / m 2
Using Eq.(4.20), the major principal stress at the toe, u l ~
= uYo sec2 (D - p' tan2 $0

U~ingEq.(4.251, shear stress at the toe, (Z~%)D= ( u y-~p ')~M$I)

= (156,62- 9) x 0.75 = 110.72 t/m2)
Using Eq.(4,23), with p, = 0, the minor principal stress at the heel,

Using Eq.(4.26), shear stress at the heel,

-
(7,)~~= (oYu .
- p) tan $, = - (26.84 - 96) x 0.15 = 10.37 t/m2.
Further, Major principal stress at the heel = p = 96 t/m2
and Minor principal stress at the toe = p '= 9 t/m2.
Extreme loading combination (normal loading combination and the loadlng
1 due to earthquake):
The inertial and hydrodynamic forces and correspndlng moments due to
horizontal earthquake have been computed as shown in Table 4,l. The effect
of vertical earthquake can be included in stability computations by
multiplying the forces and moments by (1 - av) and (1 + av) for upward and
downward accelerations, respectively. Since the computation of
hydrodynamic force involved the use of unit weight of water, the
hydrodynamic force will also be modified by vertical acceleration due to
earthquake. Further, the effect of earthquake on uplift forces is considered
negligible.
Resultant vertical force with downward acceleration

Resultant vertical force with upward acceleration

Resultant horizontal force with downward acceleration

= (4567.50 + 40.50 + 858.45 + 491.19) x 1.05

~esultanthorizontal force with upward acceleration

= (4567.50+ 40.50 + 858.45 + 491.19) x 0.95

Resultant moment about the toe with downward acceleration

Resultant moment about the toe with upward acceleration

NOW,^^ =
CM -
- 168844'19 = 22.68 m with downward acceleration
7444.06

' = 21.94 m with upward acceleration.

Therefore, Eccentricity, e = 76.2512 - 22.68 = 15.45 m with downward
acceleration
- 76'25
21.94 = 16.19 m with upward acceleration.
- 2
The resultant in both cases, passes through the downstream side o f the centre
of the base.
Therefore, the vertical stresses at the toe and the heel with downward
acceleration,
 = - 21.06 dm2.
Similarly, the vertical stresses at the toe and the heel with upward acceleration,

= - 23.52 t/m2

It should be noted that the upward acceleration causes higher tensile stresses
at the heel and is, therefore, more critical.
Factors of Safety under Normal Loading Combination

a) Factor of safety against overturning = Stabilising moment

Overturning moment

4608.00 - 0.66a
b) Sliding factor = -- ------ -
Cw- 6994.31
I c)
i) Shear-friction factor of safety (with drains operative).

1 ii) Shear-friction factor of safety (with drains inoperative),

- (150 x 76.25 + 0.7 x 499136) = 3.24.

4608.00
Factors of Safety under Extreme Loading Combination

a) Sliding factor (downward acceleration) = -

6255'52 - 0.84.
7444.06

Sliding factor (upward acceleration) = -

5659'76-
- 0.865.
6544.56
Shear-frictionfactor of safety (downward acceleration),
 d) Shear-friction factor of safety (upward acceleration),

4.5 TOP WIDTH

The top width of a gravity dam is generally fixed by the requirement of a roadway or for
access to gate-operating mechanism and the parapet walls. The economical top width of a
dam is around 14 percent of the height of the dam. The usual widths provided vary from 6 -
10 m.

The free board for the dam should he adequate to avoid overtopping of the dam during
maxi~numflood coupled with waves. Generally, a freeboard allowance of 1.5 times h,
(where h, is the height of waves calculated by Eqs. 3.1 or 3.2 in Unit 3), is made. The
economical freeboard is around 5 percent of the height of the dam.

4.7 CONSTRUCTION JOINTS

Concrete in a dam is placed in lifts which are generally 1.5 m high. To develop a proper
bond between the lifts, the top surface of the lower surface is freed of all coatings, laitance,
stains, defective concrete and all foreign material and the surface is roughened. The
construction joint is the surface of the previously placed concrete upon or against which
new concrete is to be placed. Besides permitting subsequent placing of concrete. these joints
facilitate construction, reducc shrinkage stresses and permit installation of embedments
Suitable measures are adopted to ensure proper bond between the previously placed
concrete and the new concrete. They are done by annlving a high-velocilv sand blast and a
 t. A thin mortar layer is sometimes placed on the surface before placing the new
. Construction joints do not require water stops but the lop surface is provided with
ile the concrete is still fresh.

te dams are sub-divided into a number of blocks to relieve the thermal stresses
ent cracking in the body of the dam. The blocks are formed by transverse and
joints.
Tranljerse Joints
transverse joints normal to the dam axis are provided. These joints are 12-18 m
ally spacing being 15 m. These joints are introduced to allow the concrete to
either side of the joint to relieve thermal stresses. Figure 4.6 shows the
d longitudinaljoints in Bhakra dam. The transverse joints are vertical and
nd from the foundation to the top of the dam. Reinforcement bars should not
these joints. The edges of the transverse or contractionjoints at the face are
ed to give a pleasing appearance and to avoid spalling. Such chamfers are
4 cm on the non-overflow blocks and 2 cm x 2 cm on the downstream face of

~on~iJJdina1
Joints
ht of the dam increases, the base width approaches a limiting dimension beyond
itions favouring vertical cracking parallel to the dam axis are created. To prevent
cracks, longitudinal joints are provided. They serve the same purpose in one
am as the transverse joints in the dam as a whole. Spacings of these joints vary
(Figure 4.6). Where the longitudinal joints approach the downstream face of
the joint is turned nonnal to the face to avoid feather-edging of concrete. A gap is
vided at the inclined'portion of the joint which is later dry packed. Extension of
in the upstream face is undesirable and should be terminated at a
of 4 - 5 m from the face. These joints are staggered in adjacent blocks.

'CONTRACTION JOINTS

Figure 4.6 :Trensveme.and hngitudinal Joints in Bhnkra Darn

4.9 KEY-WAYS
Figure 4.7 shows typical keys for joints in gravity darn. Vertical keys in transverse joints
and horizontal keys in longitudinal joints are provided in a dam to increase the shearing
resistance between the adjacent concrete blocks. The resulting structure has better stability
due to the transfer of load from one block to another through the keys. The keys also
increase the percolation path through the joints and thus reduce water leakage. They also
hasten the sealing of the joints with sediment deposits. Shear keys provided in longitudinal
joints improve the stability of the dam by increasing the resistance to vertical shear.

Fig- 4.7 :Typical Keys for Joints in Gravity Dam

4.10 WATERSTOPS AT JOINTS

Figure 4.8 shows waterstop installations in a dam. The waterstops are provided in transverse
joints for stopping the flow of water into the joint and for stopping the flow of grout outside
it. In longitudinal joints the waterstop only retain the grout.

WITIAL FINAL
METAL WATERSTOP

RLBBER WATERSTOP

I
Figore 4.8 :Waterstop lrrptallatioos
The usual practice is to provide two waterstops of copper or monel (an alloy of nickel and
copper) with an asphalt seal in between. In some cases polyvinyl chloride (PVC) and rubber
waterstops have been used. The longitudinal joints are provided with Z type while the
transverse joints are provided with U or M type seals. Construction joints are sometimes
provided with A or Z type seal to prevent seepage along the joint when some opening or
gallery is located close to the face. The distance of the f i s t seal from the upstream face in
ungrouted contractionjoints is about 60 cm.
The pipes inside the asphalt seal are installed for melting the asphalt and adding more
asphalt at a later date.Another metal seal is provided downstream of the asphalt seal to limit
 Gravity Drrs
1of asphalt along the joint between the two seal's. Further downstream, open drains
med drains) about 15 - 20 cm in diameter at 3 m centre to centre are provided.
shqws some of the waterstops.

ible to use large consuuction blocks which result in rapid and economical construction.

. i . 1

a gravity dam to be safe against overturning, the dimensions of the dam should be such
the resultant of all forces intersects the base of the dam within its middle-third.
sider the base of a gravity dam or any horizontal section and the resultant of all the
es acting on the dam above the section. If the line of action of this resultant passes
ide the toe, the dam would overturn. But, if the section of a gravity dam is such that the

a) no tension occurs at the heel,

b) adequate resistance to sliding is available, and
The horizontal forces acting on a dam above any horizontal plane cause failure of the dam
due to sliding if these actuating forces are more than the resistance to sliding on the plane.
The resistance to sliding is due to frictional resistance and the shearing strength of the
material along the plane under consideration. The shear-friction factor of safety, Fs, which is
a measure of stability agauat sliding or shearing, is given by

where,
C = unit cohesion,
A = area of the plane considered,
xw = sum of all vertical forces acting on the plane,
p = coefficient of internal friction, and
xH = sum of driving shew forces.
The shear-friction factor of safety can be used to determine the stability against sliding or
shearing at any horizontal section within the dam,its contact with the foundation or through
the foundation along any plane of weahess. The minimum allowable factor of safety, Fs,
for gravity dams are 3.0,2.0 and 1.0 for the usual, unusual and extreme loading conditions,
respectively. The value of Fs for any plane of weakness within the foundation should not be
less than 4.0,2.7 and 1.3 for the usual, unusual and extreme loading conditions, respectively.
The maximum allowable compressive stress for concrete in a gravity dam should be less
than the specified compressive strength of the concrete divided by 3.0,2.0 and I .O for the
usual, unusual and extreme loading conditions, respectively. The compressive stress should
not exceed 1035 N/sq cm and 1550 N/sq cm for the usual, and unusual loading conditions,
respectively.
The maximum allowable compressive stress in the foundation should be less th'm the
specified compressive strength of the foundation divided by 4.0.2.7 and 1.3 for the usual,
unusual and extreme loading conditions, respectively. These values of factor of safety are
higher than those for concrete so as to provide for uncertainties in estimating the foundation
properties.

4.13 SUMMARY
The design of gravity dams involves the determination of the normal and principal stresses
at the heel and the toe with combination of forces considered for normal loading and for
earthquake conditions. Factors of safety against sliding and the shear friction factors are also
to be determined taking h e arbitrary profile as the preliminary section. The provision of
construction joints, transverse and longitudinal joints and waterstops at the joints are
essential for the safety of the dam and to reduce seepage losses. Temperature stresses are ,

detrimental to the dam and measures have to be adopted to minimise these stresses These
aspects have been covered in this unit.
4.14 I ~ WORDS
Y Gravity Dams

They include higher water pressure due to tloods, wave

pressure, silt pressure, ice pressure and earthquake force.
Concrete in a dam is laid in lifts. These lifts have a
nearly flat surface on the top. The subsequent lift of
concrete is poured on the previously laid concrete layer.
The joint formed at the jdnction of the two lifts is called
a construction joint.
When an earthquake occurs the structure is subject to
additional forces for which the dam section should be
designed to withstand these earthquake forces.
It includes extreme uplift corresponding to drains
choked, water surface at the maximum tlood level and
silt pressure.
Forces {{using Instability : These include uplift pressure, water pressure from the
reservoir, earthquake forces, forces due to waves on the
water surface, ice pressure, temperature stresses, silt
pressure and wind pressure.
Wind causes waves to rise on the water surface of the
reservoir. These waves exert a force on the upstream
face of the dam.
It is the extra height provided in a dam above the
maximum flood level together with waves as a
protection against overtopping.
Water on the downstream face of the dam exerts a
horizontal forces in the upstream direction.
In cold climates the top surface of the reservoir freezes
to form sheets of ice. As the rim of the reservoir and the
upstream face of the dam restrict the ice from
expanding, a pressure is exerted on the dam which is
called ice pressure.
Adjacent concrete blocks of a dam are likely to slide
due to shear. Keys increase the shearing resistance
between the dam blocks. Transverse joints are provided
with vertical keys and longitudinal joints with
horizontal keys.
To prevent uncontrolled cracking of a dam block,
longitudinal joints are provided in each block normal to
the direction of flow. The joints are staggered in
adjacent blocks.
They include the weight of the dam and the structure at
its top, water pressure with full reservoir and uplift
pressure.
Self Weight of the Dam : The weight of the concrete dam and the gate and bridge
structure constitute the self weight of the dam.
The sediment trapped in the reservoir is confined to the
lowest portions of the reservoir. This silt and sediment
exert a pressure on the face of the dam called the silt
pressure.
The checking of the dam section for safety against all
destabilising forces is stability analysis.
This is the load combination recommended by USBR as
occurring simultaneously with a reasonable probability
of occurrence.
Temperature Control : Tensile stresses occur in a dam section when the
temperature falls leadqng to the cracking of concrete.
Temperature control is necessary to reduce the chances
of such cracking.
Temperaturn Stresses : Stresses in concrete caused as a result of temperature
falling or rising are termed as temperature stresses.
TO^ Wi.dth : It is the width of the dam at the top required for a
roadway, gate operating mechanism and the parapet
walls.
Wansverse Joints : The joints provided normal to the dam axis to allow the
concrete to contract on either side of the joint to relieve
the thermal stresses are transverse joints. These joints
are provided between two adjacent dam blocks.
Differential settlement of adjacent blocks can occur
independent of each other due to the transverse joints.
Uplifc Pressure : Water seeping through the foundation of a dam will
exert an upward pressure on the base of the dam. Such
pressure causes instability of the dam structure.
Water Pressure from : Water in the reservoir will exert a horizontal pressure on
Reservoir the upstream face of the dam. The pressure varies from
zero at the top to a maximum at the bottom and equal to
the product of the depth of water and the unit weight of
water.
Waterstops at Joints : In order to prevent the loss of water through the joint
between adjacent blocks, water stop's are necessary.
These are provided on the upstream face of the dam.
Wind Pressure : Wind pressure is neglecting in most cases.

4.15 ANSWERS TO SAQs

SAQ 2
ii) 0,
Low when dam height is less than h 1 =
[wh (S - C' + I)] and high when dam
Om
height is more than hl =
[wh ( S - C' + I)]'
SAQ 3
i) The concrete in the dam is a homogeneous, isotropic, and uniformly elastic
material,
ii) No differential movements occur at the site of the dam due to tl~ewater loads
on the walls and base of the reservoir,
iii) All loads are transmitted to the foundation by the gravity action of vertical,
parallel cantilevers which receive no support from the adjacent cantilever
elements on either side,
iv) Normal stresses on horizontal planes vary uniformly as a straight line from the
upstream face to the downstream face, and
v) Horizontal shear stresses have a parabolic variation across horizontal planes
from the upstream face to the downstream face of the dam.
SAQ 4
The economical top width of a dam is around 14 percent of the height of the dam.
The usual widths provided vary from 6 - 10 m.
SAQ 5
The economical freeboard is around 5 percent of the height of the dam. Generally,
a freeboard allowance of 1.5 times h , (where h, is the height of waves calculated
by Eqs. 3.1 or 3.2 in Unit 3), is made.