After studying a lot of dam failure constructed based on Bligh’s theory, Khosla came out with the following;
Following are some of the main points from Khosla's Theory
For case of two dimensional flows under a straight floor where:
Thus for the first flow line AB which touches the outline of the floor, the pressure can be obtained by putting different values of x in equation. Fig shows the pressure distribution diameter both by equation 4 as well as Bligh's Theory.
From the fig the following conclusions can be drawn:
Slope of Pressure diagram: At A and B in infinite, hence the floor at A will be theoretically infinite acting downward and that at B will also be infinite acting upward. This will cause sand boiling and hence the floor should be depressed or cut off should be provided at the downstream end.
The following specific causes of general form were considered.
Method of independent variable:
For the determination of seepage below the foundation of hydraulic structure developed the method of independent variable.
A straight horizontal floor of negligible thickness provided with a sheet pile at the upstream end or a sheet pile at the downstream end.
ii) A straight horizontal floor depressed below the bed, but without any vertical cut-off.
The mathematical solution of the flow-nets of the above profiles have been given in the form of curves. From the curves, percentage pressures at various key points E, C, E1, C1 etc) be determined. The important points to note are:
The percentage pressures at the key points of a simple forms will become valid for any complex profile, provided the following corrections are effected:
Let b1 = distance between the two piles 1 and 2, and
D = the depth of the pile line (2), the influence of which on the neighbouring pile (1) of depth d must be determined
b = total length of the impervious floor
c = correction due to interference.
The correction is applied as a percentage of the head
This correction is positive when the point is considered to be at the rear of the interfering pile and negative for points considered in the forward or flow direction with the interfering pile.
Correction for Floor Thickness
Standard profiles assuming the floors as having negligible thickness. Hence the values of the percentage pressures computed from the curves corresponds to the top levels (E1*, C1*) of the floor. However, the junction points of the floor and pile are at the bottom of the floor (E1, C1)
The pressures at the actual points E1 and C1 are interpolated by assuming a straight line variation in pressures from the points E1* to D1 and from D1 to C1
The corrected pressures at E1 should be less than the computed pressure t E1*. Therefore the correction for the pressure at E1 will be negative. And so also is for pressure at C1.
Correction for Slope of Floor
A correction for a sloping impervious floor is positive for the down slope in the flow direction and negative for the up slope in the direction of flow.
The correction factor must be multiplied by the horizontal length of the slope and divided by the distance between the two poles between which the sloping floor exists.
In the diagram above, correction for slope can be applied only to point E2. As the point E2 is terminating at the descending slope in the direction of flow, the correction will be positive. The value of correction will be:
Where C.F. =correction factor
bs = horizontal length of sloping floor
b1 = horizontal distance between the pile lines
Every particle of water while seeping through the sub-soil, at any position will exert a force f, which will be tangential to the streamline at any point. As the streamlines bend upward, the tangential force f will be having a vertical component f1. Also at that point, there will be a downward force W due to the submerged weight of the soil particle. Thus at that point there will be two forces on the particle; one upward vertical component of f, and the other, the submerged weight. It is evident that if the soil particle is not to be dislodged, then the submerged weight must be greater than the upward vertical component of f. The upward vertical component force at any point is proportional to the water pressure gradient dp/dx.
Hence for stability of the soil and for the prevention of erosion and piping, the seeping water when it emerges at the downstream side, at the exit position, the force f1 should be less than the submerged weight W. In other words the exit gradient at the downstream end must be safe.
If at the exit point at the downstream side, the exit gradient is such that the force f1is just equal to the submerged weight of the soil particle, then that gradient is called critical gradient. Safe exit gradients = 0.2 to 0.25 of the critical exit gradient.
Values of safe exit gradient may be taken as:
For the standard form consisting of a floor of a length b, and a vertical cut-off of depth d, the exit gradient at its downstream end is given by:
Exit gradient GE = (H/d) x