Khosla's Theory
Lecture-03
Khosla's Theory
After studying a lot of
hydraulic structures failures constructed based on Bligh’s theory, Khosla came
out with the following results.
Khosla's theory
demonstrated that the real uplift pressures were substantially different from those
calculated by Bligh's theory and derived the subsequent conclusions:
· The outer faces of the end sheet piles are
much more effective than the inner ones and the horizontal length of the floor.
· The intermediated piles of smaller length
than the outer piles are ineffective except for local redistribution of
pressure.
·
Undermining of floor started from tail
end.
· It was absolutely essential to have a
reasonably deep vertical cut off at the downstream end to prevent undermining.
· Khosla and his associates took into
account the flow pattern below the impermeable base of hydraulic structure to
calculate uplift pressure and exit gradient.
· Starting with a simple case of a
horizontal flow with negligibly small thickness, various cases were analyzed
mathematically.
· Seeping water below a hydraulic structure
does not follow the bottom profile of the impervious floor as described by
Bligh instead each particle tracks its path via a sequence of streamlines.
Khosla’s Theory and Concept of Flow Nets
The seepage water does
not creep along the bottom contour of pucca flood as started by Bligh, but on
the other hand, this water moves along a set of stream-lines. This steady
seepage in a vertical plane for a homogeneous soil can be expressed by Laplacian
equation:
Where, ∅ = Flow potential = Kh;
K = the co-efficient of
permeability of soil as defined by Darcy’s law, and
h is the residual head at
any point within the soil.
The above equation
represents two sets of curves intersecting each other orthogonally. The
resultant flow diagram showing both of the curves is called a Flow Net.
Stream Lines
The streamlines represent
the paths along which the water moves through the sub-soil. Every particle
entering the soil at a specific location upstream of the work, will trace out
its own path and will represent a streamline. The initial streamline follows
the bottom contour of the works and is the same as Bligh’s path of creep. The
remaining streamlines follows smooth curves transiting slowly from the outline
of the foundation to a semi-ellipse, as shown below.
Equipotential Lines
An equipotential line, a
line of constant head. Water enters the subsoil at the upstream end and has a
head H. As it moves from the upstream to the downstream, there is a loss of
head. When it emerges at the downstream end, the head becomes zero, because
there is no tail water. At the upstream end, the water has a head of H which is
completely lost through the passage of flow. At the intermediate of its path,
the water has a certain residual head h still to be dissipated in the remaining
seepage length up to the downstream end.
This fact is applicable
to every streamline, and hence, there will be points on different streamlines
having the same value of residual head h. If such points are joined together,
the curve obtained is called an equipotential line.
Exit Gradient
The seepage water exerts
a force at each point in the direction of flow and tangential to the
streamlines as shown in figure above. This force (F) has an upward component
from the point where the streamlines turn upward.
For soil grains to remain
stable, the upward component of this force should be counterbalanced by the
submerged weight of the soil grain. This force has the maximum disturbing
tendency at the exit end, because the direction of this force at the exit point
is vertically upward, and hence full force acts as its upward component.
For the soil grain to
remain stable, the submerged weight of soil grain should be more than this
upward disturbing force. The disturbing force at any point is proportional to
the gradient of pressure of water at that point.
This gradient of pressure
of water at the exit end is called the exit gradient. In order
that the soil particles at exit remain stable, the upward pressure at exit
should be safe. In other words, the exit gradient should be safe.
The exit gradient, GE can
be calculated from the following equation:
b = horizontal length of
the floor
d = depth of the
downstream cut-off
H = U/S water level – D/S
water level
Or using chart
Safe exit gradients of
different soil types are given in Table
Table: Safe exit gradient
for three types of soil
Critical Exit Gradient
This exit gradient is
said to be critical, when the upward disturbing force on the grain is just equal
to the submerged weight of the grain at the exit. When a factor of safety equal
to 4 to 5 is used, the exit gradient can then be taken as safe. In other words,
an exit gradient equal to ¼ to 1/5 of the critical exit gradient is ensured, so
as to keep the structure safe against piping.
The submerged weight (Ws)
of a unit volume of soil is given as:
Where, 
w
= unit weight of water.
Ss = Specific
gravity of soil particles
n = Porosity of the soil
material
For critical conditions
to occur at the exit point
F = Ws
Where F is the upward
disturbing force on the grain
Force F = pressure
gradient at that point = dp/dl = w
×dh/dl
Under critical
conditions, the critical exit gradient is equal to (1-n) (Ss-1). For
most of the river sands Ss=2.65 and n=0.4 then the value of critical
exit gradient = (1-0.4)(2.65-1) =
1.
Hence an exit gradient
equal to ¼ to 1/5 of the critical gradient means that an exit gradient equal to
¼ to 1/5 has to be provided for keeping the structure safe against piping.
Khosla’s Method of independent variables for determination of pressures and
exit gradient
To understand hydraulic
structure seepage below the foundation, plotting the flow net is necessary.
This can be done through mathematical solutions, electrical analogy methods, or
graphical sketching. However, these methods are complex and time-consuming.
Khosla has developed a simple, quick, and accurate approach called Method of
Independent Variables for designing hydraulic structures like weirs, barrages,
or previous foundations. This method allows for better understanding of seepage
and its effects on the structure.
In this method, a complex
profile like that of a weir is broken into a number of simple profiles; each of
which can be solved mathematically. Mathematical solutions of flow nets for
these simple standard profiles have been presented in the form of equations and
curves given in Plate which can be used for determining the percentage
pressures at the various key points. The simple profiles which hare most useful
are:
(i) A straight horizontal floor of negligible
thickness with a sheet pile line on the u/s end and d/s end. Fig (a).
(ii) A straight horizontal floor depressed
below the bed but without any vertical cut-offs. Fig: (b)
(iii) A straight horizontal floor of negligible
thickness with a sheet pile line at some intermediate point. Fig:(c)
Fig:
(a)
Fig:
(b)
Fig:
(c)
Figure:
Khosla's chart for Depressed Floor and pile at End.
Figure:
Khosla’s Chart for Intermediate pile
In an actual profile, the
above assumptions are not satisfied, the following corrections are needed:
a) Correction
for the mutual interference of piles.
b) Correction
for the thickness of floor.
c) Correction
for the slope of the floor.
Figure:
A typical cross section of a hydraulic structure.
Correction for the Mutual interference of Piles:
The correction C to
be applied as percentage of head due to this effect, is given by
Where,
b′
= The distance between two pile lines.
D
=
The depth of the pile line, the influence of which has to be determined on the
neighboring pile of depth d. D is to be measured below the level
at which interference is desired.
d
=
The depth of the pile on which the effect is considered
b
=
Total floor length
The
correction is positive for the points in the rear of back water, and
subtractive for the points forward in the direction of flow. This equation does
not apply to the effect of an outer pile on an intermediate pile, if the
intermediate pile is equal to or smaller than the outer pile and is at a
distance less than twice the length of the outer pile.
Suppose
in the above figure, we are considering the influence of the pile no (2) on
pile no (1) for correcting the pressure at C1. Since the point C1 is in the
rear, this correction shall be positive. While the correction to be applied to
E2 due to pile no (1) shall be negative, since the point E2 is in the forward
direction of flow. Similarly, the correction at C2 due to pile no (3) is
positive and the correction at E2 due to pile no (2) is negative.
Correction for the thickness of floor:
In
the standard form profiles, the floor is assumed to have negligible thickness.
Hence, the percentage pressures calculated by Khosla’s equations or graphs
shall pertain to the top levels of the floor. While the actual junction points E
and C are at the bottom of the floor. Hence, the pressures at the
actual points are calculated by assuming a straight-line pressure variation.
Since
the corrected pressure at E1 should be less than the calculated pressure at
E1′, the correction to be applied for the joint E1 shall be negative.
Similarly, the pressure calculated C1′ is less than the corrected pressure at
C1, and hence, the correction to be applied at point C1 is positive.
For
different locations of piles, the corrections to be applied are as follows:
a.
A straight horizontal floor of negligible thickness with a sheet pile at
the u/s end. Corrected pressure at point C1:
b.
A
straight horizontal floor of negligible thickness with a sheet pile at some
intermediate point. Corrected pressure at point E1:
c.
A
straight horizontal floor of negligible thickness with a sheet pile at the d/s
pile. Corrected pressure at point E1
Where: ∅C1, ∅D1, ∅E1 are
uplift pressures at points C1, D1, E1, and d1, d2,
and d3 are depth of piles, t1, t2, t3 are floor
thickness respectively.
Equations to find the uplift pressure (∅) at E, C & D:
For U/S & D/S piles
In
terms of the percentage pressure
For floor with D/S sheet
pile (above figure)
For floor with U/S sheet
pile (above figure)
The values of ∅D and ∅E can also be obtained
from the chart.
For intermediate pile:
The values of ∅E, ∅C, and ∅D can also be obtained
from the charts.
Correction for the slope of the floor
A
correction is applied for a sloping floor, and is taken as positive (+ve) for
the down and negative (-ve) for the up slope following the direction of flow.
The slope correction is applicable to the key point of pile line fixed at the beginning
or the end of the slope. The correction factor given below in Table
is to be multiplied by the horizontal length of the slope and divided by the
distance between the two pile lines between which the sloping floor is located.
Values
of correction for standard slopes such as 1:1, 2:1, 3:1, etc. are tabulated in
table
Table:
Correction factor for slope of the floor
The
above table can be represented by a figure as shown below:
Referring
to above Figure, this correction is applicable only to point E2. Since the
slope is down at point E2 in the direction of flow, hence, the correction shall
be (+ve) and will be equal to the correction factor for this slope multiplied
by bs/b1, where bs and b1 are shown in Figure. The slope correction is given in
the following equation:
where,
b1 = distance between two piles which the sloping floor is located and bs =
horizontal length of slope (see Figure), Cs = slope correction, and C
= coefficient due to slope from table and Figure of determining Cs.