Design of Impervious Floor of the Weir | Geography

In this article we will discuss about the design of impervious floor of the weir for sub surface flow.

Khanki weir failed in 1895. An expert’s committee under the Chairmanship of Lt. Col. Caliborn, then Principal of Thomson College, Roorkee was set up to investigate the causes of failure of this weir, which was founded on sandy soil. The committee confirmed the Darcy’s law of seepage through sandy soils for low heads.

Heads are generally low in case of weirs. Later Mr. W.G. Bligh put forward his own theory known by his name “Bligh’s creep theory” for sub-surface flows. Later on Mr. Lane put forward is own theory in 1932 after having analysed a large number of dams. This theory is known as ‘Lane’s weighted creep theory’.

A most elaborate scientific study based on facts was later carried out in 1936 by Pavlovsky and A.N. Khosla. This theory is kwnown as “Khosla theory of seepage”.

Mr. Khosla carried out most of his investigations in Punjab. He published the results of his analysis in Publication No. 12 of Central Board of Irrigation and Power, New Delhi. Bligh’s creep theory, Lane’s weighted creep theory, and Khosla’s creep theories, are being explained in details one after the other.

Bligh’s Creep Theory:

This theory is based on the assumption that seeping water through the soil below the weir, follows the path along the contact of the base, with the underlying sub-soil. The length of the path of seeping water from the point of entrance into the sub-soil from the U/S of the impervious apron, to the point at the D/S end of the impervious apron is known as creep length.

Bligh also assumed that loss of head of the seeping water is proportional to the length of its travel irrespective whether length of travel is in horizontal or vertical direction. He also assumed that unless cut-off sheet piles extend to the impervious subsoil strata, no amount of sheet piling could stop the flow of percolating water.

See Fig. 15.4 (a) AB is the length of impervious apron I and H is the head of water filled upto the top of the weir CD and there is no water on D/S side. According to Bligh’s theory L = I where L is the total creep length. If vertical cut-offs are provide below the impervious apron as shown in Fig. 15.4 (b), the total creep length will be as follows –

L = I + 2d₁ + 2d₂.

Length of vertical cut-off has been taken double because vertical cut-off provides the creep length equivalent to twice the length of the cut-off, as seeping water once goes down and then comes up along the cut-off.

If H is the total head causing seepage or total loss of head, and L the total creep length, the loss of head per unit length of creep (C) is given by –

Loss of head per unit length of creep (C) is known as percolation coefficient. The reciprocal of percolation coefficient is known as coefficient of creep (C).

Safe values of coefficient of creep for different soils are given below:

Weir Design by Bligh’s Theory:

According to Bligh’s theory, two design criteria are to be considered.

(i) Safety against Undermining or Piping:

To safeguard the weirs against failure by piping, the creep length should be provided according to the following formula:

L = CH, where C is coefficient of creep.

Such a length would provide safe hydraulic gradient and seeping water emitting from D/S end of the impervious apron, will not be having sufficient uplift pressure to dislodge the soil particles. This will avoid boiling action of soil at the D/S end of the apron.

(ii) Safety against Uplift Pressure:

To counter balance the force of uplift, sufficient floor thickness should be provided, specially on the D/S side of the weir.

Let at distance L₁ creep length from U/S end of the impervious apron h is the resultant uplift head, the net uplift pressure wh can be computed as follows –

The thickness of the impervious apron on the U/S side of the weir wall is provided as nominal thickness. This is because uplift pressure U/S of the weir wall always remains counter balanced by weight of water, which always remains filled up over it. It is also advantageous if more length of impervious apron is provided on U/S side.

This will reduce the intensity of uplift pressure on the D/ S side of the weir wall. However some minimum length of impervious apron has to be provided on the D/S side also.

Example:

A weir on a sandy soil is shown in Fig. 15.5. Find out the Uplift pressures at points 4, 8, and 12 m from the U/S end of the pucca apron. Also find out the thickness of the floor at these points considering specific gravity y of floor material as 2.24. Adopt Bligli’s theory for analysis and weight of water as one tonne per cubic metre of water.

Solution:

Total length of creep = 2 x 4 +16 + 2 x 6 = 36 m

Hydraulic gradient or percolation coefficient

= H/L = 4/36 = 1/9

I. Uplift at Point A:

I. Uplift at Point B:

III. Consider Point C

Limitations of Bligh’s Theory:

Following are some of the limitations of this theory:

1. He did not distinguish vertical cut-off from horizontal impervious floor as regards to creep length.

2. Bligh did not make any mention of exit gradient. The safety against piping or undermining cannot be guaranteed simply by providing flat gradient. The exit gradient on D/S end of the apron should be maintained well below the critical value for the soil laying underneath the pucca impervious apron.

3. This theory holds good so long as distance between subsequent piles is maintained substantial. The horizontal distance between subsequent piles should not be allowed to be smaller than twice the depth of the pile.

4. Bligh did not make any difference in effectiveness of outer and inner faces of U/S and D/S end piles and intermediate piles. By investigations it has been established that outer faces of both the end piles are much more effective than inner ones.

The intermediate piles, if provided and their lengths are smaller than outer piles, they are ineffective and do not provide any additional creep length. They may, however, cause local redistribution of pressure.

5. Bligh did not feel necessity of providing a sheet pile at D/S end. It has been found absolutely essential on investigations, to have a deep cut off at D/S end. It is considered as most effective measure for preventing the undermining of weirs.

6. Bligh considered loss of head, as linear along the creep length whereas it is not so. Distribution of uplift pressure is also not linear but follows sine curve.

This theory was very well accepted as the basis for design of irrigation structures in earlier days. Bhim Goda weir near Haridwar and some other works constructed according to this theory are doing very well and did not give any trouble. Some of the works designed according to this theory had however failed.

When causes of such failures were investigated, it was found that only those works have failed, below the impervious aprons of which very deep cut off piles were provided, very close to each other. As deep as 12 m deep piles were found in some works. In the case of deep cut offs, the creep length is not twice tire depth of the pile but much smaller than that.

This fact has been proved that flow of seepage neither follows the path along the vertical cut-off piles and nor along the underside of the horizontal impervious apron. Flow of seepage follows parabolic stream lines. From this fact it follows that intermediate piles do not provide any additional seepage length if their lengths are equal to or slightly more than the length of the end piles. But Bligh considered all the piles of equal effectiveness and this fact actually led to the failure of some of the weirs designed according to his theory.

Lane’s Weighted Creep Theory:

Mr. Lane gave weights to vertical creep length and horizontal creep lengths. Horizontal length is given a weight of one and vertical length provided by cut off piles a weight of 3. Total creep length is finally found out by adding total weighted horizontal length and weighted vertically offsets. This theory is Empirical and does not provide any background, as to why vertical cut off should be given weight of three. This theory did not gain any ground and thus is not used anywhere.

Khosla’s Theory:

Some siphons on upper Chenab canal, which were designed according to Bligh’s theory gave trouble. Khosla along with his associates, were asked by the government to investigate the causes of trouble and to suggest remedial measures. Khosla and his associates inserted some pipes on the D/S side of the weir through impervious aprons of some of the trouble giving works.

The pipes were inserted to verify whether the pressures below the impervious apron were in accordance with Bligh’s theory or not. The pressures measured in the pipes were not as they should have been according to Bligh’s.

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

1. The outer faces of the end piles are much more effective than their inner faces and also more effective than horizontal length of the impervious apron.

2. 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 local redistribution of pressures.

3. Piping or undermining of the impervious floors starts from the D/S end of the pucca impervious floor. If the exit gradient at 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 once starts progresses continuously towards the U/S side and ultimately a cavity is formed and failure of weir becomes imminent.

4. 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 theory and since then most of the irrigation works in the world are being designed according to this theory.

Khosla proved that seeping water through permeable soils follows parabolic stream lines and not along the underside profile of the impervious floor as envisaged by Mr. Bligh. This is the fundamental difference between the two theories. Flow of water through permeable soils, as assumed by Mr. Khosla, is shown in Fig. 15.5. Since seeping water flows in parabolic stream lines their theoretical solution is possible. Flow of seeping water takes place according to Laplace equation.

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 prepared. After having prepared the flownet, uplift pressure at any point below the weir’s impervious floor, can be easily found out.

Khosla’s method of analysis pointed out following two more points of difference from Bligh’s method:

1. D/S half of the floor of the weir is subjected to more uplift pressure than that according to Bligh’s method.

2. 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 boiling a deep cut off or a depressed floor is essentially provided at the D/S end.

Distribution of uplift pressure below the impervious floor of the weir for both Khosla’s and Bligh’s methods are shown in Fig. 15.7.

In order to calculate the uplift pressure and exit gradient, Khosla considered flownet as the pattern of flow of seeping water. Mathematical solutions were evolved 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. See Fig. 15.8.

(i) Horizontal straight floor of negligible thickness with a sheet pile at U/S end and a sheet pile at D/S end. Fig.15.8 (a) and (b).

(ii) Horizontal straight floor depressed below the bed and having no vertical cut off at all. [Fig. 15.8 (c)].

(iii) Horizontal straight floor of negligible thickness with a cut off vertical sheet pile at some intermediate point. Fig. 15.8 (b).

All these simple profiles are shown in Fig. 15.8. All these profiles were analysed by Mr. Khosla’s team with the help of Schwarz Christoffel transformation. They prepared 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 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 sheet pile is at 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 interpolated. The method of use of the plates is given on the plates themselves, just to illustrate, examples have been given here-

Example:

Length of horizontal floor is 15 m and 3 m deep vertical sheet pile is attached at its D/S end. Find the uplift pressures at key points.

Head of water is 4 m.

Solution:

(i) At D/S End:

From Plate 15.2 for α = 0.2 we get values of φ_D and φ_E as follows:

φ_D = 27% or pressure at D P_D= 0.27 H = 0.27 x 4

φ_E = 40% or pressure at E P_E = 0.40 H = 04 x 4 = 1.6 m.

(ii) If Pile is Provided at U/S End:

In this case pressures at key points C₁, D₁ can be found out by first finding φ_D and φ_E for the case when pile is at D/S end and then by following relation –

Example:

Thickness of the floor is 3 m and length of the floor is 15 m. Find the uplift pressures at bottom points of D/S and U/S ends of the floor. Water fill is 5 m above the floor.

Solution:

Exit Gradient (G_e):

At each point under the floor, there is a certain residual uplift pressure and a certain rate at which the head is being lost, indicating a pressure gradient.

The pressure gradient at the exit point is called exit gradient. The purpose of design of Weir’s impervious aprons is to restrict the exit gradient below the critical value so that soil particles are not disturbed. For floor length b and vertical cut off of depth d at the D/S end, the exit gradient (G_E) is found out from formula –

The uplift pressure must be kept as low as possible consistent with the safety at the exit, so as to keep the floor thickness to the minimum.

Method of Independent Variables:

This break up shows the theoretical profiles. Actually, the usual weir section consists of a combination of all the three elementary profiles. In addition to this, floor also has some thickness. Khosla solved the actual profile of the weir by an empirical method known as the method of independent variables.

According to this method, actual complex profile is broken into a number of simple profiles known as elementary profiles. Each elementary profile is then treated independent of the others. Each elementary profile is independently amenable to mathematical treatment. The pressures at key points are read from the Khosla’s curves.

The key points are the junction points of the floor and pile and bottom points of different piles. The pressures read from the Khosla’s curves are true for individual elementary profiles. But when all the profiles are combined into one complex form, which is actually the condition, corrections for pressures will have to be applied.

When different elementary profiles are combined, they influence the pressures due to mutual interference of the piles. Thickness and slope of floor also affect the pressures at key points. In order to find out the pressures, at various key points for the weir as a whole, following corrections for the pressures will have to be applied.

They are:

1. Correction for the thickness of the floor.

2. Correction for the mutual interference of piles.

3. Correction for the sloping floor.

1. Correction for the Thickness of the Floor:

Figure 15.8 shows the details of key points for profiles one each at upstream end, downstream end, and intermediate point. Assume thickness of the floor negligibly small. E and C are the key point at the top of the floor. E point is on U/S side and C on D/S side of the pile.

Let E₁ and C₁ be the corresponding key points at the bottom of the floor.

Pressures at points E₁ and C₁ for each profile are computed as follows:

(i) Pile at the U/S end – For point E no correction is required as pressure at this point is not going to interfere with pressure system of any other pile.

(ii) Pile at D/S end – Correction at E₁

(iii) Pile at intermediate point – This will require corrections at both the points E₁ and C₁

2. Correction for the Mutual Interference of Piles.

Correction is applied as follows:

Where C = correction to be applied in percentage of head.

D = the depth of the pile, the influence of which is required to be determined on the neighboring pile of depth d.

d = the depth of the pile on which the effect of another pile of depth D is required to be found out.

b’ = the distance between the two piles.

b = total length of impervious pucca floor.

The sign of this correction is positive for the points in U/S direction, i.e. against the flow, and negative for D/S direction i.e. in the direction of the flow. This equation does not apply to the effect of an outer pile on an intermediate pile, if intermediate pile is equal to or smaller in length than the outer pile and also when distance between them is less than twice the length of the outer pile.

Interference of any pile is only for the faces of the adjacent piles, which lie towards the interfering pile. For example pile No. 2 will interfere with the D/S face of pile No. 1 and U/S face of pile No. 3. Various dimensions marked in Fig. 15.9 are, for the point C on pile No. 1. U/S face of pile No. 2 will be influencing the pressure at point C of pile No. 1. Pile No. I will have no influence on D/S face of pile No. 2 and on pile No. 3. Mutual interference mainly extends upto a distance equal to the depth of the pile. Beyond this, it gradually falls off to negligible in a length of about twice the length of the pile.

Fig. 15.9. Correction for mutual interferences.

3. Correction for Slope:

Correction for slope has to be applied. The correction is plus for the down and minus for the up slopes, following the direction of flow. The corrections are given in the table below for various slopes.

The correction is applicable to the key point of the piles fixed at the start or the end of the slope. In Fig. 15.9 slope correction is applicable only to point E of the pile No. 2. The percentage correction given in the above table is to be further multiplied by the proportion of the horizontal length of the slope to the distance between the two pile lines, in between which the sloping floor is situated. The correction to be applied at point E of pile No. 2, will be obtained by multiplying the appropriate figure from the above table by a factor b_s / b’.

Cut-Off Mathematical Solution of Khosla Theory:

It is an impervious vertical membrane provided at the bottom of the impervious floor of the weir. Water cannot seep through it. Water can only cross it by taking round of its bottom end. It may be made of masonry, concrete, R.C.C., or steel. Cut-offs are provided so as to reduce the uplift pressure below the weir foundation.

They are very helpful in preventing the failure of the weirs by undermining. In the case of all the irrigation structures, the impervious floors must be provided with deep cut-offs, at least one each at D/S and U/S ends. D/ S cut-off in very effective to reduce the exit gradient and also to prevent undermining failure of the weirs. U/S cut-off reduces the effect of uplift pressure throughout the length of the floor of the weir.

Intermediate piles or cut-offs, can also be used but they are not found much effective. Depths of U/S and D/S cut-offs are dependent upon the scour depth. The cuts-offs should be carried deeper than the scour depth.

Scour depth can be found out by following formula-

Where R = Scour depth in metres.

q = Intensity of discharge per metre length of the weir

f = Silt factor.