# Khosla’s Theory

# KHOSLA’S THEORY

- Some siphons on the upper Chenab canal, which were design according to Bligh’s theory gave trouble.
- Khosla along with his associates was ask by the government to investigate the causes of trouble and to suggest remedial measures.
- Khosla and his associates insert some pipes on the D/S side of the weir through impervious aprons of some of the trouble giving works.
- The pipes insert to verify whether the pressures below the impervious apron were in accordance with Bligh’s theory or not.

**Based on his investigations A.N. Khosla drew the following conclusions: **

- The outer faces of the end piles are much more effective than their inner faces and also more effective than the horizontal length of the impervious apron.
- Intermediate sheet piles, if equal to or smaller in length than the outer piles, are almost ineffective and do not provide any additional creep length.
- They may only cause a local redistribution of pressures.
- Piping or undermining of the impervious floors starts from the D/S end of the pucca impervious floor.
- If the exit gradient at the D/S end was more than the critical gradient for the soil underlying the foundation.
- The soil particles will get lifted up and carried away with seeping water.
- This process if one starts progresses continuously towards the U/S side and ultimately a cavity is form and failure of the weir becomes imminent.
- It is very essential to have a deep vertical cut off at the D/S end.
- This measure prevents undermining to a large extent.
- In 1929 Panjnad weir was designed according to Khosla’s theory.
- The pipes were inserted in its D/S floor, to verify the pressures of seeping water at various points.
- The pressures were found as they should have been as per Khosla’s theory.
- This gave wide recognition to Khosla’s theory and since then most of the irrigation works in the world are being design according to this theory.
- Khosla proved that seeping water through permeable soils follows parabolic streamlines and not along with the underside profile of the impervious floor envisag by Mr. Bligh.
- This is the fundamental difference between the two theories.
- The flow of water through permeable soils, as assume by Mr. Khosla.
- Since seeping water flows in parabolic streamlines their theoretical solution is possible.
- The flow of seeping water takes place according to the Laplace equation.

\[\frac{d^{2}\phi }{dx^{2}}+\frac{d^{2}\phi }{dy^{2}}= 0\]

- This equation can be solved graphically or mathematically and a graph of equipotential lines and flow lines in form of a flownet show in Fig. 15.6 can be prepare.
- After having prepare the flownet, uplift pressure at any point below the weir’s impervious floor, can be easily found out.

**Khosla’s method of analysis point out following more points of difference from Bligh’s method. **

- D/S half of the floor of the weir is subject to more uplift pressure than that according to Bligh’s method.
- Slope of pressure diagram as found out by Khosla is infinite at entrance and exit points.
- An infinite force would be acting downwards at entrance point and upwards at exit point.
- The infinite upward force at exit point would cause boiling of sand.

- In order to prevent sand from boiling a deep cut-off or a depress floor is essentially provid at the D/S end.
- The distribution of uplift pressure below the impervious floor of the weir for both Khosla’s and Bligh’s methods
- In order to calculate the uplift pressure and exit gradient, Khosla consider flown as the pattern of flow of seeping water.
- Mathematical solutions were evolv by breaking up the complex profile into a number of simple standard forms.

** The following may be the most useful simple standard forms of break up profiles. **

(i) Horizontal straight floor of negligible thickness with a sheet pile at U/ S end and a sheet pile at D/S end.

(ii) Horizontal straight floor depress below the bed and having no vertical cut off at all.

(iii) Horizontal straight floor of negligible thickness with a cut off vertical sheet pile at some intermediate point.

- All these profiles were analyzed by Mr. Khosla’s team with the help of Schwarz Christoffel transformation.
- They prepare three plates, Plate 15.1, 15.2, and 15.3. Plate 15.1 gives the values of pressures at key points C, D, and E, when the sheet pile is at neither of the ends but at some intermediate point.
- Plate 15.2 gives uplift pressures at key points for the case when the sheet pile is at the D/S end.
- Plate 15.3 gives the values of safe exit gradients.
- Plates 15.1 and 15.2 give pressures at key points C, D and E.
- Pressures at intermediate points can be linearly interpolate
- The method of use of the plates is given on the plates themselves, just to illustrate, examples have been given here..