1.

Earth Dams

1.1 Introduction

Embankment dams are of two types:

(i) Earthfill or Earth Dams
(ii) Rockfill or earth- rock Dams

The bulk of mass in an earthfill dam consists of soils while in the rockfill dam it consists
of rock materials.

Rockfill dams: The designation ‘rockfill embankment’ is appropriate where over 50%
of the fill material may be classified as rock pieces. It is an embankment which uses
large size rock pieces to provide stability and impervious membrane to provide water
tightness.

Modern practice is to specify a graded rockfill heavily compacted in relatively thin layers
by heavy plants. The constructions method is essentially similar to that of Earthfill Dams.
Materials used for membrane are earth, concrete, steel, asphalt or wood. The
impervious membrane can be placed either at the upstream face of the dam or as a core
inside the embankment. Such a construction therefore becomes similar to diaphragm
type. Rockfill embankments employing a thin u/s membrane are referred to as decked
rockfill dams.

1.2 Types of Earth-fill Dams

Depending upon the method of construction, earth dam can be divided in two categories:
(i) Rolled-fill Dam
(ii) Hydraulic-fill Dam

In the Rolled fill Dam, the embankment is constructed in successive, mechanically

compacted layers. The suitable materials are transported from borrow pits to the
construction site by suitable earth moving machineries. It is then spread by Bulldozers,
and sprinkled to form layers of limited thickness having proper water content. They are
then thoroughly compacted and bonded with the preceding layer by means of power
operated rollers of proper design and weight.

In the case of Hydraulic fill dam the materials are excavated, transported and placed by
Hydraulic fill method. In this method the flumes are laid at a suitable falling gradient
along the outer edge of the embankment. The material mixed with water at borrow pits,
is pumped into these flumes. The slush is discharged through the outlets in the flume, at
suitable interval along their length. The slush thus flows towards the center of the bank.
The course material of the slush settles at the outer edge while finer material settles at
the center. No compaction is done.

Rolled fill earth dams can further be subdivided into the following types:
(i) Homogeneous embankment type
(ii) Zoned embankment type
(iii) Diaphragm embankment type

CENG 6606- Hydraulic Structures II Page 1

 Embankment Dam

Earth Dam Rockfill Dam Composite

Type

According to design
According to method of
Constructuion

Homogenous Zoned Diaphriagm Rolled Hydraulic Semi Hydraulic

fill type fill type fill type

(1) Homogeneous Earth Dams: are constructed entirely or almost entirely of one
type of earth material (exclusive of slope protection). A homogeneous earth dam is
usually built when only one type of material is economically available and/or the
height of dam is not very large.

Figure 1.1 Homogeneous

a b) With rock toe

a) With horizontal blanket
Figure 1.2 Modified homogeneous

(2) Zoned Earth Dam: Contains materials of different kinds in different parts of the
embankment. The most common type of an earth dam usually adopted is the zoned earth
dam as it leads to an economic & more stable design of the dam. In a zoned earth dam,
there is a central impervious core which is flanked by zones of more pervious material. The
pervious zones, also known as shells, enclose, support and protect the impervious core.
The U/s shell provides stability against rapid drawdown of reservoirs while the downstream
shell acts as a drain to control the line of seepage and provides stability to the dam during
its construction and operation. The central impervious core checks the seepage.

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 Figure 1.3 Zoned Sections

(3) Diaphragm type embankment dams: In this the bulk of the embankment is
constructed of pervious material and a thin diaphragm of impermeable material is provided
to check the seepage. The diaphragm may be of impervious soils, cement concrete,
bituminous concrete or other material and may be placed either at the centre of the section
as a central vertical core or at the u/s face as a blanket.

Figure 1.4: Diaphragm type embankment section

1.3. Causes of Failure of Earth Dams

In spite of taking great care in construction of earth dams, some failures have occurred in the
past. However, knowledge of the principal lessons learnt from failures and damages in the past
is an essential part of the training of earth dam designer.

On the basis of investigation reports on most of the past, failures are categorized into three
main classes:
1. Hydraulic failures : 40%
2. Seepage failures : 30%
3. Structural failures: 30%

Hydraulic Failures: Hydraulic failures include the following:

(i) Overtopping
(ii) Erosion of u/S face
(iii) Erosion of d/S face
(iv) Erosion of d/S toe
(v) Dam failure due to rising of water level

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Seepage failures: Seepage failures may be due to,
(a)
Piping through the body of the dam
(b)
Piping through the foundation of the dam
(c)
Conduit leakage
(d)
Sloughing of downstream toe.
Structural Failures: Structural failures may be due to the following reasons:
(i) Slides in embankments (u/s & d/s slope failures)
(ii) Foundation slides (spontaneous liquefaction)
(iii) Liquefaction slides
(iv) Failure by spreading
(v) Failure due to earthquake
(vi) Slope protection failures
(vii) Failure due to damage (or holes) caused by burrowing animals
(viii) Damage caused by water soluble materials

Figure 1.5

1.4. Criteria for Safe Design of Earth Dam

An earth dam must be safe and stable during all phases of construction and operation of the
reservoir. The practical criteria for the design of earth dams may be stated briefly as follows.
1. No overtopping during occurrence of the inflow design flood;
a. Appropriate design flood
b. Adequate spillway
c. Sufficient outlet works
d. Sufficient free board
2. No seepage failure;
a. Phreatic (seepage) line should exit the dam body safely without sloughing
downstream face.
b. Seepage through the body of the dam, foundation and abutments should be
controlled by adapting suitable measures.
c. The dam and foundation should be safe against piping failure.
d. There should be no opportunity for free passage of water from u/S to d/S both
through the dam and foundation.
3. No Structural failure;
a. Safe u/s & d/s slope during construction
b. Safe u/s slope during sudden drawdown condition.
c. Safe d/s slope during steady seepage condition
d. Foundation shear stress within the safe limits.
e. Earthquake resistant dam

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 4. Proper slope protection against wind & rain drop erosion.
5. Proper drainage
6. Economic section

1.4.1 Overtopping Protection of Dams Used as Spillways

Overtopping protection remains a reasonable, safe, and the most economical solution.
Both concrete and embankment dams can be provided with overtopping protection.
Advances in technology, particularly roller compacted concrete (RCC) and precast
concrete blocks, have made this approach more workable and reliable.

The design of any type of overtopping protection is intimately related to many site-
specific issues such as the design flood, consequences of failure, downstream impact,
site topography, existing embankment design, and materials etc. An overtopping
protection system for embankment dams greater than 30 m in height require detailed
analyses to ensure stability for high unit discharge and velocity.

Various design needs are discussed in the case of embankment overtopping with
reference to Figure 1.6 that helps visualizing various problems that may be
encountered. As the flow lines from the upstream reach approach near crest, part of the
upstream slope may be subjected to flow velocities, which otherwise would have been a
stagnant zone. In most cases, the standard riprap protection already provided in this
portion may be adequate to withstand these velocities.

Figure 1.6 Schematic of embankment overtopping.

The flow then becomes nearly parallel to the crest, and a transformation of the sub-
critical flow to critical state occurs and flow velocities increase. Protection against
erosion may be necessary in this portion. As the flow passes down the downstream
corner, separation followed by negative pressures takes place over the edge and
immediate downstream slope. Generally, RCC crests block with rounded shape,
extending on the upstream slope and with a cutoff as shown in Figure 1.8 is provided.
Special care is necessary to prevent seepage of the flow into the underlying material.

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The flow on the slope remains in a supercritical state with velocities increasing as it
descends further down. Any protective layer that is provided in this portion should be
designed to resist the velocities and uplift forces, both hydrodynamic and due to
seepage. Seepage flow may be influenced by the flow through the embankment and by
infiltration into the downstream slope by the flow down the slope. Filters and pressure
relief drains are provided to drain out the seepage flow.

The protective lining on the slope may be in the form of cast-in-place concrete slab,
RCC lining, or precast concrete blocks. RCC or concrete blocks facilitate provision of
steps as compared to the cast-in-place concrete lining, for which special formwork is
needed.

Provision of an energy dissipator is required at the toe. In the case of protective lining
composed of steps, a part of energy is dissipated on the slope itself and the energy
dissipator could be designed for the residual energy leaving the toe. The most common
form of an energy dissipator for the embankment overflow is the hydraulic jump stilling
basin.

Slope protection lining

The most common form of protective lining for the embankment slope are: cast-in-place
concrete slabs, RCC lining, and precast concrete blocks.

1. Cast-in-place Concrete
Cast-in-place concrete on the downstream face is preferred for dams of heights up to
about 75 m and depths of overflow up to about 8 m. A continuous concrete slab, with or
without reinforcement, is placed over the entire downstream face of the dam similar to
the upstream facing currently used on concrete faced rockfill dam. The continuous
reinforcement provides monolithic behavior, controls cracks and creates a nearly
impervious barrier without projections in the flow. Other design features include sub-
surface drainage, an overflow crest, and accommodations for fluctuating pressures in
the hydraulic jump region at the toe. Existing dams protected with cast-in-place concrete
facing have heights up to 75 m, depth of overflow up to 8 m, and unit discharge up to 30
m3/s/m.

2. Roller Compacted Concrete (RCC)

In recent years, the development of roller compacted concrete (RCC) technology has
provided a method of erosion protection of embankment dams, which is proved to be
cost-effective while affording a number of advantages. RCC protection is normally very
rapid with minimal project disruption. In most cases construction is limited to the dam
crest and downstream slope with no requirement of reservoir restrictions. The
performance of the dams provided with RCC that have been subjected to overtopping
flows has been excellent.

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Figure 1.7 Stepped overlay of RCC on an existing embankment dam.

The entire crest of the embankment, the downstream face, and the area downstream of
the toe are covered with roller compacted concrete placed in 30–60 cm thick horizontal
lifts starting at the toe and proceeding up the slope of the embankment. The minimum
width for the horizontal lift is specified as 2.5 m. Each lift could be made into a form of
steps that may also help energy dissipation of the flow passing down the slope. A
drainage system consisting of drainage blanket and filter along with perforated pipe
drains is necessary. The existing dams provided with this type of protection have been
designed for heights up to 45 m, depth of overflow up to 6 m, and unit discharge up to
30cumec/m. Figure 1.7 shows typical cross-section with relevant details.

3. Precast Concrete Block System

Perhaps the most innovative of alternatives to provide protection to the embankment
dams during overtopping are precast concrete block systems. Such innovative systems
require the greatest attention to avoid problems of possible failure. Based upon
research and case history data during the past decade in the USA, Great Britain, and
former Soviet Union, cost-effective methods for overflow protection for small to medium
size embankment dams have been developed. There are two types of systems
consisting of concrete blocks;
(i) Cellular concrete mats
(ii) Wedge shaped concrete blocks
Cellular concrete mats provide a viable option for protecting embankment slopes from
catastrophic erosion that could occur during overtopping. These are prefabricated mats
of precast concrete blocks tied together by cables and anchored in place. Cellular
concrete mats have been successfully adopted for dams up to 12 m in height, 1.2 m
depth of overflow, and maximum flow velocity of up to 8m/s.

The development of wedge shaped blocks relies on prevention of uplift by overlapping

part of the blocks. Flow over the steps separates from the block surface and reattaches

CENG 6606- Hydraulic Structures II Page 7

some distance along the next step downstream. Within the separation zone, the flow
rotates and forms a low or negative pressure zone. By providing drainage holes through
each block in the low-pressure zone, it is possible to control the buildup of seepage
pressure and hold the block onto the underlying embankment. Figure 1.8 shows the
details. The blocks are inherently stable as the curved streamlines provide a stabilizing
down thrust (Figure 1.9).

Figure 1.8 Wedge shaped concrete blocks.

The roughness of the stepped surface provides additional energy dissipation and
reduces the stilling basin requirement at the toe. The concrete crest is constructed as a
broad crested or slightly curved overflow weir. The upstream portion of the concrete
crest is terminated in a low velocity area to avoid erosion by the approach flow. The
embankment dams, protected with wedge shaped blocks are up to 35m in height, 12m
of overtopping depth, unit discharge up to 60 m3/s/m, and flow velocity up to 25m/s.

Figure 1.9 Stability of wedge shaped blocks

1.4.2. Section of an Earth Dam

The preliminary design of an earth dam is done on the basis of past experience and on the
basis of the performance of the dams built in the past. We shall discuss here the preliminary
selection of the following terms:
1) Top width

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 2) Free board
3) Casing or outer shells
4) Central impervious core
5) Cut-off trench
6) Downstream drainage system.
1) Top width: The crest width of an earth dam depends on the following considerations:
 Nature of the embankment materials and minimum allowable percolation distance
through the embankment at the normal reservoir level.
 Height of the structure
 Importance of the structure
 Width of highway on the top of the dam
 Practicability of construction
 Protection against earthquake forces.

Following are some of the empirical expressions for the top width b of the earth dam, in terms of
the height H of the dam:
H
b 3 For very low dam (H<10m)
5
b=0.55H1/2 + 0.2H For medium dam (10m<H<30m)
b=1.65(H+1.5)1/3 For large dam (H>10m)

2) Free board: Free board is the vertical distance between the horizontal crest of the
embankment and the reservoir level. Normal free board is the difference in the level
between the crest or top of the embankment and normal reservoir level. Minimum free
board is the difference in the elevation between the crest of the dam and the maximum
reservoir water surface that would result and spillway function as planned. Sufficient free
board must be provided so that there is no possibility whatsoever of the embankment
being overtopped.

U.S.B.R suggests the following free boards:

Table 1.1: U.S.B.R practice for free board

Nature of spillway Height of dam Free Board
Free Any Minimum 2m and maximum 3m over
the maximum flood level
Controlled Less than 60 m 2.5 above the top of gates
Controlled Over 60m 3m above the top gates

3) Shoulder or outer shells: The function of shoulder or outer shells is to impart stability
and protect the core. The relatively pervious materials, which are not subjected to
cracking on direct exposure to atmosphere, are suitable for shell. Table 1.2 (a) gives
recommendations for suitability of soils used for earth dams as per IS: 8826-1978.

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Table 1.2 (a) Suitability of Soils for Construction of Earth Dams
Homogenous Zoned earth dam
Relative Suitability section Pervious casing Impervious core
1. very suitable GC SW,GW GC
2. Suitable CL,CI GM CL,CI
3. Fairly suitable SP, SM,CH SP,GP CM,GC,SM SC,CH
4. Poor - - ML,MI,MH
3. Not suitable - - OL, NI, OH ,Pt

The design slopes of the upstream and downstream embankments may vary widely, depending
on the character of the materials available, foundation conditions and the height of the dam. The
slopes also depend up on the type of the dam (i.e. homogeneous, zoned or diaphragm).

The upstream slope may vary from 2:1 to as flat as 4:1 for stability. A storage dam subjected to
rapid drawdown of the reservoir should have an upstream zone with permeability sufficient to
dissipate pore water pressure exerted outwardly in the upstream part of the dam. If only
materials of low permeability are available, it is necessary to provide flat slope for the rapid
drawdown requirement. However, a steep slope may be provided if free draining sand and
gravel are available to provide a superimposed weight for holding down the fine material of low
permeability. The usual downstream slopes are 2:1, where embankment is impervious.

Table 1.2(b): Side slopes for earth dams according to Terzaghi

Type of material Upstream slope Downstream slope
Homogeneous well graded material 2:1 2:1
Homogeneous coarse silt 3:1 2.5:1
Homogeneous silty clay or clay
H less than 15 m 2.5:1 2:1
H more than 15 m 3:1 2.5:1
Sand or sand and gravel with clay core 3:1 2.5:1
Sand or sand and gravel with R.C core wall 2.5:1 2:1

1.4.3. Seepage Analysis

Seepage analysis: is used,

 To determine the quantity of water passing through the body of the dam and foundation.
 To obtain the distribution of pore water pressure.

Assumptions to be made in seepage analysis

 The rolled embankment and the natural soil foundation of the earth dam are
incompressible porous media. The size of the pore spaces do not change with time,
regardless of water pressure (Isotropic).
 The seeping water flows under a hydraulic gradient which is due only to gravity head
loss, or Darcy’s law for flow through porous medium is valid.
 There is no change in the degree of saturation in the zone of soil through which the
water seeps and the quantity flowing into any element of volume is equal to quantity
which flows out in the same length of time (Steady flow).
 The hydraulic boundary conditions at entry and exit are known.

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1.4.4. Laplace equation for two dimensional flows
In earth dams, the flow is essentially two dimensional. Hence we shall consider only two
dimensional flows. V
Vy  y
y

y

∆y
Vx Vx
∆x Vx  x
x
Vy

Consider an element of soil is size x, y and of unit thickness perpendicular to the plane of the
paper. Let Vx and Vy be the entry velocity components in x and y direction. Then
 v 
 v x  x x  and
 x 
 v 
 v y  y y 
 y 
will be the corresponding velocity components at the exit of the element. According to
assumption 3 stated above, the quantity of water entering the element is equal to the quantity of
water leaving it. Hence, we get
 v   v y 
vx y.1  v y x.1   vx  x x y.1   v y  x.1
 x   y 
From which
𝜕𝑣𝑥 𝜕𝑣𝑦
+ =0 … (i)
𝜕𝑥 𝑑𝑦
This is the continuity equation.
According to assumption 2:
h
vx  K xix  K x * … (ii)
x
h
And VY  k Y i Y  Ky …(iii)
y
Where h = hydraulic head under which water flows.
Kx and Ky are coefficient of permeability in x and y direction.
Substituting (ii) and (iii) in (i), we get
𝜕 2 𝐾𝑥 ℎ 𝜕 2 𝐾𝑦 ℎ
+ =0 … (1.1)
𝜕2𝑥 𝜕2𝑦
For an isotropic soil,
Ky = Kx = K
Hence we get from eq. (3.1)
 2h  2h
 0
x 2 y 2
Substituting velocity potential =  = K*h, we get

CENG 6606- Hydraulic Structures II Page 11

  2  2
 0 … (1.2)
x 2 y 2

This is the Laplace equation of flow in two dimensions. The velocity potential  may be defined
as a scalar function of space and time such that its derivative with respect to any direction gives
the fluid velocity in that direction.
This is evident, since we have
=Kh
 h
K  K .i x  v x
x x
 h
Similarly , K  K .i y  v y
y y
The solution of Eq. 1.2 can be obtained by
i) analytical methods
ii) graphical method
iii) experimental methods
The solution gives two sets of curves, know as equipotential lines and stream lines (or flow
lines), mutually orthogonal to each other, as shown in the Figure below. The equipotential lines
represent contours of equal head (potential). The direction of seepage is always perpendicular
the equipotential lines. The paths along which the individual particles of water seep through the
soil are called stream lines or flow lines.

Figure 1.10: Flow net

. 1.4.5. Computation of rate of seepage from flow net

A network of equipotential lines and flow lines is known as a flow net. Fig.1.10 shows a portion
of such a flow net. The portion between any two successive flow lines is known as flow channel.
The portion enclosed between two successive equipotential lines and successive flow lines is
known as field.

Let: b and l be the width and length of the field.

h = head drop through the field.
q = discharge passing through the flow channel.
H = total head causing flow
= difference between upstream and downstream heads

Then, from Darcy’s law of flow through soils:

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 h
q  K . (bx1) … (i) (Considering unit thickness)
l
If Nd = total number of potential drops in the complete flow net,
h
Then h 
Nd
h b
 q  K   … (ii)
Nd l
Hence the total discharge through the complete flow net is given by
h b Nf b
q  q  k .  .N f  kh .
Nd l Nd l
Where Nf = total number of flow channels in the net. The field is square and hence b=l
Nf
Thus, q  kh
Nd
This is the required expression for the discharge passing through a flow net, and is valid only for
isotropic soils in which
k x  k y  k.

1.4.6. Seepage discharge for anisotropic soil

Let us now consider the case of an anisotropic flow medium in which kx  ky
 2h  2h
For such a case, the flow equation (3.1) becomes kx  k 0
x 2 y 2
y

This is not a Laplace equation. Hence flow net can not be drawn directly. Rewriting, it we get
k x 2h  2h
 0
k y x 2 y 2

ky
Let us put xn  x
kx
Where xn is the new co-ordinate variable in the x - direction.
Then the above equation becomes,
 2h  2h
 0 … (1.3)
xn2 y 2
This is in Laplace form.

Figure 1.11

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To plot of the flow for such a case, the cross-section through anisotropic soils is plotted to a
natural scale in the y-direction, but to a transformed scale in the x-direction; all dimensions
ky
parallel to x- axis being reduced by multiplying by the factor . The flow net obtained for this
kx
transformed section will now be constructed in the normal manner as if the soil were isotropic.
The actual flow net is then obtained by re- transforming the cross- section including the flow net,
kx
back to the natural scale by multiplying the x- coordinates by factor . The actual flow net
ky
thus will not have orthogonal set of curves. As shown in figure 1.11, field of transformed section
will be a square one, while the field of actual section (retransformed) will be a rectangular one
Kx
having its length in x direction equal to times the width in y direction.
Ky
Let kx = permeability coefficient in x- direction, of the actual anisotropic soil field.
K’ = equivalent permeability of the transformed field.
Then, for the transformed section
h
q  k '' (lx1) … (a)
l
For the actual field,
h
q  k x (lx1) … (b)
kx
(l )
ky
Since the quantity of flow is the same,
h h
k' (l )  k x (l )
l kx
l
ky … (1.4)
ky
Hence k '  kx  kxk y
kx
Nf Nf
Hence the discharge is given by q  k 'h  Kxky h (1.5)
Nd Nd

1.4.7. Phreatic Line in Earth Dam

Phreatic line (seepage line) or saturation line is the line at the upper surface of the seepage
flow at which the pressure is atmospheric.

Figure 1.12 Phreatic line in Earth dam

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 Phreatic line for a homogeneous Earth dam with horizontal Drainage blanket
Figure below shows a homogeneous earth dam with horizontal drainage blanket FK at its toe.
The phreatic line in this case coincides with the base parabola ADC except at the entrance. The
basic property of the parabola which is utilized for drawing the base parabola is that the
distance of any point p from the focus is equal to the distance of the same point from the
directrix. Thus;

Distance PF = Distance PR where, PR is the horizontal distance of P from the Directrix

EG.

Figure 1.13

Graphical method

Steps:
 Starting point of base parabola is at A; AB = 0.3L
 F is the focal point
 Draw a curve passing through F; center at A
 Draw a vertical line EG which is tangent to the curve
 EG is the directrix of the base parabola
 Plot the various points P on the parabola in such a way that PF = PR

Analytical method
PF = PR

x 2  y 2  x  yo
From point A (known), x = b and y = h

 yo  b 2  h 2  b

𝑥2 + 𝑦2 = 𝑥 + 𝑏 2 + ℎ2 − 𝑏 Equation of parabola … (1.6)

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Discharge through the body of Earth dam
v  k *i
q  v * A  k *i * A
dy
qk y *1
dx

From parabola equation, y  2 xy o  y 0

2

d ( y 0  2 xy o )
2

qk ( y o  2 xy o )
2

dx

𝑦0
𝑞=𝑘 𝑦02 + 2𝑥𝑦0 = 𝑘 𝑦0 …………. (1.7)
𝑦02 + 2𝑥𝑦0

Phreatic line for a dam with no filter

General solution by Casagrande
Figure below shows a homogeneous dam with no horizontal drainage filter at the d/s side.
The focus in this case will be the lowest point F of the d/s slope.

Fig 1.14: Dam with no drainage filter.

And the base parabola BKC will evidently cut the d/s slope at K and extend beyond the
limits of the dam, as shown by dotted line. However, according to exit conditions, the
phreatic line must emerge out at some point M, meeting the d/s face tangentially at J. The
portion JF is then known as discharge face and always remains wet. The correction a,
a
by which the parabola is to be shifted downwards, is found by the value of given by
a  a
Casagrande for various values of the slope  of the discharge face. The slope angle  can
even exceed the value of 900. Thus we observe that
a
= value found from table … (i)
a  a

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 a+ a=KF from Fig 1.14 . (ii)

Solving (i) and (ii), the value a and a can be found.

a
Table for the value of with slope angle 
a  a

 a
a  a
300 0.36
600 0.32
900 0.26
1200 0.18
1350 0.14
1500 0.10
1800 0.0

Discharge through the body of Earth dam

Figure 1.15

a. Analytical Solution of Schaffernak and Van Iterson for < 300

In order to find the value of a analytically, Schaffernak and Van Iterson assumed that the
energy gradient
dy
i  tan   . This means that the gradient is equal to the slope of the line of seepage, which
dx
is approximately true so long as the slope is gentle (i.e. <300).

For the vertical section through J, i.e. JJ1

dy
qK y
dx
dy
but  i  tan 
dx
and y= JJ1= a sin 
Substituting in (i), we get

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 q = k (a sin) (tan) … (1.8)

This is the expression for discharge.

Again
dy
qk y  k (a sin  )(tan  )
dx
a( sin  ) (tan  )dx  ydy

Integrating between the limits:

x= a (cos ) to x = b
y= a (sin  ) to y = h , we get

b h
a sin  tan   dx
a cos
   ydy
a sin

and
h 2  a 2 sin 2 a
 a sin  tan  (b  a cos  ) 
2

From which, we obtain, after simplification,

𝑏 𝑏2 ℎ2
𝑎= 𝑐𝑜𝑠 𝛼
− 𝑐𝑜𝑠 2 𝛼
− 𝑠𝑖𝑛 2 𝛼 … (1.9)

b. Analytical solution of Casagrande for 300< <600

It will be observed that the previous solution gives satisfactory results for slope < 300. For
steeper slopes, the deviation from correct values increases rapidly beyond tolerable limits.
Casagrande suggested the use of sin  instead of tan. In other words, it should be taken as
(dy/ds) instead of (dy/dx), where s is the distance measured along the phreatic line.

Figure 1.16

dy
Thus q  kiA  k A (1.10)
ds
dy
At J, s= a and y = a sin  then,  sin 
ds
Where s = distance measured along the curve.
Substituting in (1.10), we get

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 q = k. (sin) (a sin ) = k(a sin2) …(1.11)

This is the expression for the discharge.

dy
Again q  k y  ka sin 2 
ds
 a (sin2 ) ds = ydy

Integrating between the following limits (s = a to s =S)

Where S = total length of the parabola
And (y = a sin to y=h), we get

S h
a sin 2   ds   ydy
a a sin 

h 2  a 2 sin 2 
a sin 2  .( S  a) 
2
h2
or a  2aS 
2
0
sin 2 

ℎ2
From which 𝑎 =𝑆− 𝑆2 − 𝑠𝑖𝑛 2 𝛼
…. (1.12)

Taking S (h2+b2)1/2 we get

h2
a  h b  h b 
2 2 2 2

sin 2 

𝑎= 𝑏 2 + ℎ2 − 𝑏 2 − ℎ2 𝑐𝑜𝑡 2 𝛼 … [1.13]

Phreatic line for homogenous Earth dam with rock toe

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 Figure 1.17

Phreatic line for zoned Earth dam with central core

Figure 1.18

1.4.8. Characteristics of Phreatic line (Seepage line)

Based on the above discussions, the characteristics of the phreatic line may be summarized
below:
1. At the entry point, the phreatic line must be normal to the upstream face since
the upstream face is a 100% equipotential line. For other entry condition
(Fig.1.19), the phreatic line starts tangentially with the water surface.

Fig 1.19 Entry conditions of phreatic line

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 2. The pressure along the phreatic line is atmospheric. Hence the only change in
the head along it is due to drop in the elevation of various points on it. Due to
this, the successive equipotential lines will meet it at equal vertical intervals.
3. The focus of the base parabola lies at the break out point of the bottom flow line,
where the flow emerges out from relatively impervious medium to a highly
pervious medium.
4. When horizontal filter or drainage toe is provided, the phreatic line would tend to
emerge vertically.
5. In the absence of any filter, the seepage line will cut the downstream slope at
some point above the base. The location of this point, and the phreatic line itself,
is not dependent on the permeability or any other property, so long as the dam is
homogeneous. The geometry of the dam alone decides these.
6. The presence of pervious foundation below the dam does not influence the
position of phreatic line.
7. In the case of a zoned dam with central impervious core, the effect of outer shells
can be neglected altogether. The focus of the base parabola will be located at
the downstream toe of the core (Fig. 1.18)

1.5. Stability Analysis

Stability analyses under the following four heads are generally needed:
1. Stability analysis of downstream slope during steady seepage.
2. Stability of upstream slope during sudden drawdown condition.
3. Stability of upstream & downstream slopes during and immediately after construction.
4. Stability of foundation against shear.

1. Swedish Circle Method of Slope Stability

It is one of the most generally accepted methods of checking slope stability. In this method the
potential surface is assumed to be cylindrical, and the factor of safety against sliding is defined
as the ratio of average shear strength, as determined by Coulomb’s equation S = C +  tan to
the average shearing stress determined by statics on the potential sliding surface. In order to
test the stability of the slope, a trial slip circle is drawn, and the soil material above assumed slip
surface is divided into a convenient number of vertical strips or slices. The trial sliding mass (i.e.
the soil mass contained with in the assumed failure surface) - is divided in to a number (usually
5 to 8) of slices which are usually, but not necessarily, of equal width. The width is so chosen
that the chord and arc subtended at the bottom of the slice passes through material of one type
of soil.

The forces between the slices are neglected and each slice is assumed to act independently as
a column of soil of unit thickness and width b. The weight W of each slice is assumed to act at
its centre. If the weight of each slice is resolved into normal (N) and tangential (T) components,
then the normal component will pass through the center of rotation (O), and hence does not
cause any driving moment on the slice. However, the tangential component T cause a driving
moment which is expressed as T*r, where r is the radius of the slip circle. The tangential
components of the few slices at the base may cause resisting moment; in that case T is
considered negative.

If c is the unit cohesion and  L is the curved length of each slice, then resisting force from
Column’s equation is = c  L + N tan 

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For the entire slip surface AB, we have
Driving moment Md = rT
Resisting moment Mr = [ cL  tan N  ] r
Where T = sum of all tangential components
N = sum of all normal components
2 r
L= L= length AB of slip circle
360 0

Hence factor of safety against sliding is

M r cL  tan  N  Shear Strength available

Fs   = …. (1.14)
Md T shear Strength required for Stability

Figure 1.20 A portion of

slip surface for slices

Method of locating center of critical slip circle

For location of the most critical slip circle, a number of trial slip surfaces are assumed and the
factors of safety are found. The circle, which gives the minimum factor of safety, is the most
critical circle. To reduce the number of trials, the Fellenius line is usually drawn (Figure 1.21).
Fellenius has shown that for a homogeneous slope, the center of most critical circle lies on line
AB, called the Fellenius Line.

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The center of most critical circle may lie anywhere on the line AB or its extension. However, its
exact position can be obtained only after conducting the stability analysis for different slip
surfaces. The centers of the trial circles are marked as O1, O2, etc. on the line AB. The
corresponding factors of safety F1, F2, etc. are plotted at the corresponding centers as
perpendicular ordinates on the line AB. The curve of factor of safety is obtained by joining the
ends of these ordinates. The center O corresponding to the minimum factor of safety is the
center of the most critical circle.

The above construction is for a c-Φ soil. For a purely cohesive soil (Φ = 0), the point A itself
represents the center of the most critical circle.

Figure 1.21 Location of center of critical slip circle

Stability of down stream slope during steady seepage

Critical condition for d/s slope occurs when the reservoir is full and percolation is at its maximum
rate. The directions of seepage forces tend to decrease stability. In other words, the saturated
line reduces the effective stress responsible for mobilizing shearing resistance.

cL  tan  ( N  U )
F .S .  ….. (1.15)
T
When U is the total pore pressure on the slope surface

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 Fig 1.22 Stability of down stream slope during steady seepage

The pore-water pressure at any point is represented by the piezometric head (hw) at that point.
Thus the variations of pore water pressure along a likely slip surface is obtained by measuring
at each of its intersections with an equipotential line, the vertical height from that intersection to
the level at which the equipotential line cuts the phreatic line. The pore pressure represented by
vertical height so obtained are plotted to scale in a direction normal to the sliding surface at the
respective point of intersection. The distribution of pore water pressure on the critical slope
surface during steady seepage is shown hatched in Fig.1.22.The area of U- diagram can be
measured with help of a planimeter.

In the absence of a flow net, the F.S of the d/s slope can approximately be from the equation
cL  tan  N '
F .S .  … (1.16)
T
The following unit weights may be used for the calculation of  N' and T when pore
pressure are otherwise not included in the stability analysis; however the Phreatic line needs to
be drawn.

Location Driving force Resisting force

Below phreatic surface Saturated weight Submerged weight
Above phreatic surface Moist weight Moist weight

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Examples (Stability Analysis):
1. Swedish circle method: Check the stability of the upstream face slope of the dam for the
assumed surface shown. The properties of the shell, core and foundation materials are as follows:

Shell Core Foundation

Saturated unit weight (kN/m³) 22 21 22
Submerged unit weight (kN/m³) 12 11 12
Moist unit weight (kN/m³) 21 20 21
Angle of internal friction (°) 30 20 30
Unit cohesion (kN/m²) 10 60 10

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Solution:

The soil column above the slip surface is divided into 7 slices. The slices have been taken such that the
base of each slice is in one type of material. The entire soil mass has been assumed to be saturated.
Actually, a small portion above the phreatic line under steady seepage conditions will not be saturated.
The computations for the stability analysis are shown in the table below. The weight of each slice is
computed from its area and the corresponding unit weight. The values of Φ and c are taken
corresponding to the material through which the base of that slice passes. The value of the angle θ, which
the vertical lines make with the normal, are measured and written in Col. (7).

S. Weights N’ = T=
No Area of slice (m²) (kN) Θ W’ W sin Φ N’ c b cb
Shell Core Found Saturat Subm (°) cos θ θ (°) tan Φ sec θ sec
(S) (C) (F) ed erged θ
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)
1 2040
3740 (=170x
80 - 90 (=170x22) 12) -23 1878 -1461 30 1084 10 21.7 217
2 240 - 220 10120 5520 -15 5332 -2619 30 3078 10 20.7 207
3 400 - 200 13200 7200 0 7200 0 30 4157 10 20.0 200
4 560 - 200 16720 9120 14 8849 4045 30 5109 10 20.6 206
5 720 - 80 17600 9600 25 8700 7438 30 5023 10 22.1 221
8180
(=480x
15180 12+
(=480x22 220x11
6 480 220 - +220x21) ) 36 6618 8923 20 2409 60 24.7 1482
7 40 300 - 7180 3780 55 2168 5882 20 789 60 34.9 2094
∑= 40745 22208 21650 4627

Taking full pore pressure, the factor of safety is computed as

 N  U  cL
tan  tan  N   cL a
Fs 
a
Fs  →
T T
21650  4627
Fs   1.18  1.50 , unsafe
22208

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