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# Embankment lecture 7

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STABILITY OF SLOPES SEEPAGE CONTROL MEASURES AND SLOPE PROTECTION
a finite slope AB, the stability of which is to be analyzed.
The method Consists of assuming a number of trial slip circles, and finding the factor of safety of each.
The circle corresponding to the minimum factor of safely is the critical slip circle.
Let AD be a trial slip circle, with r as the radius and O as the centre of rotation
Let W be the weight of the soil of the wedge ABDA of unit thickness, acting through the centroid G.
The driving moment MD will be equal to W x, where x, is the distance of line of action of W from the vertical line passing through the centre of rotation O.
if cu is the unit cohesion, and l is the length of the slip arc AD, the shear resistance developed along the slip surface will be equal to cu • l, which act at a radial distance r from centre of rotation O.

When slip is imminent in a cohesive soil, a tension crack will always DevelOP by the top surface of the slope along which no shear resistance can develop,
The depth of tension crack is given by
The effect of tension crack is to shorten the arc length along which shear resistance gets mobilised to AB' and to reduce the angle δ to δ'.
The length of the slip arc to be taken in the computation of resisting force is only AB', since tension crack break the continuity at B'.
The weight of the sliding wedge is weight of the area bounded by the ground surface, slip circle arc AB' and the tension crack.

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### Embankment lecture 7

1. 1. SWEDISH SLIP CIRCLE METHOD
2. 2. SWEDISH SLIP CIRCLE The following two cases are considered : ( 1 ) Analysis of purely cohesive soil (Ф = 0 soil) (2) Analysis of a cohesive frictional soil (c — Ф soil)
3. 3. (1) Ф = 0 Analysis (Purely cohesive soil) :
4. 4. a finite slope AB, the stability of which is to be analyzed. The method Consists of assuming a number of trial slip circles, and finding the factor of safety of each. The circle corresponding to the minimum factor of safely is the critical slip circle. Let AD be a trial slip circle, with r as the radius and O as the centre of rotation
5. 5. Let W be the weight of the soil of the wedge ABDA of unit thickness, acting through the centroid G. The driving moment MD will be equal to W x, where x, is the distance of line of action of W from the vertical line passing through the centre of rotation O. if cu is the unit cohesion, and l is the length of the slip arc AD, the shear resistance developed along the slip surface will be equal to cu • l, which act at a radial distance r from centre of rotation O.
6. 6. Hence the resisting moment MR will be equal to cu • I • r. The length of the slip surface AD is given by Driving moment MD = WX Factor of safety = F= cu • I • r. WX
7. 7. Effect of Tension Cracks When slip is imminent in a cohesive soil, a tension crack will always DevelOP by the top surface of the slope along which no shear resistance can develop, The depth of tension crack is given by
8. 8. The effect of tension crack is to shorten the arc length along which shear resistance gets mobilised to AB' and to reduce the angle δ to δ'. The length of the slip arc to be taken in the computation of resisting force is only AB', since tension crack break the continuity at B'. The weight of the sliding wedge is weight of the area bounded by the ground surface, slip circle arc AB' and the tension crack.
9. 9. Further, water will enter in the crack, exerting a hydrostatic pressure Pw actjng on the portion DB’ at a height Z0 /3 from B’ . Hence an additional driving moment due to horizontal hydrostatic pressure PW For depth Z0 Must be taken into account .
10. 10. Effect of Submergence : Slopes of embankment dams, canal banks, etc. may be submerged partly or fully at different times. Fig. shows the cross section of such a slope. It can be seen from the figure that the moment about O of the body of water in the half segment CEH balances that of the water contained in the other half segment FEH.
11. 11. Since the moments of the water pressure balance each other, the net driving moent can be obtained by using bulk unit weight of the soil y above the level of the water surface and. submerged unit weight y' below the water surface. If the slope is fully submerged, the submerged unit weight (y‘) is used for the entire wedge section. In addition the moment due to the water resting on the slope will be an additional resisting moment and increase the factor of safety.
12. 12. Effect of Submergence :
13. 13. (2) c - Ф Analysis (Cohesive frictionai Soil) : • In order to test the stability of the slope of a cohesive frictional soil (c - Ф soil), trial slip circle is drawn. The material above the assumed slip circle is divided into a convenient number of vertical strips or slices as shown in Fig. • The forces between the slices are neglected and each slice is assumed to act independently as a column of soil of unit thickness and of width b.
14. 14. • 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, the normal component will pass through the centre of rotation O, and hence do hot cause any driving moment on the slice. • However, the tangential component T causes a driving moment MD = T x r
15. 15. where r is the radius of the slip circle. The tangential components of a few slices the base may cause resisting moment, in that case T is considered negative.  c is unit cohesion and Δ L is the curved length of each slice then the resisting force from Coulomb's equation is equal to (cΔL + N tanФ).
16. 16. A number of trial slip circles are chosen and factor of safety of each is computed. The slip circle which gives the minimum factor of safety is the critical slip circle.
17. 17. Methods of calculating ε N and ε T: Method-1 : • The value of W, εN and εT may be found by tabulating the values for all slices as indicated below :
18. 18. Method-2 : • ε N and ε T may also be obtained graphically . vertical line drawn through the centre of gravity of the slice and intersecting the top and bottom surfaces of the slice may be assumed to represent the weight of the slice.
19. 19. • This may be resolved graphically into normal and tangential components. These components for all the slices are plotted separately as ordinates on two horizontal base lines. • The plotted points are joined by smooth curves as shown in Fig. They are called N-curve and T-cnrve respectively. • The areas under these curves represent EN and ET, The areas under these curves are measured by planimeter and multiplied by the unit weight of the soil to obtain E N and ET.
20. 20. Method-3 : • A simplified rectangular plot method has been suggested by Singh (1962) to determine E N and E T. • In this method the end ordinate of each slice is assumed to represent As weight of the slice and it is resolved into normal and tangential components as shown Fig. 2.37.
21. 21. • The normal components N, N2 N3 etc. and the tangential components T1, T2, T3 etc. are plotted to form the base of N-rectangle and Trectangle as shown in fig. (b) and (c). • The width of both rectangles being equal to the width of the slices, in this method all the slice should be of the same width. However, if the width of the last slice is not same as that of the other slice, but is less say mb, where m is multiplying factor and b is the width of the each of other slices, then the last N and T is f l + m) • components are reduced by multiplying with the factor ~ before being plotted z J the rectangles. • The area of N and T rectangles multiplied by the unit weight of the soil gives E N E T respectively.
22. 22. Method of Locating Centre of Critical Slip Circle : • In order to reduce the number of trials to find the centre of critical slip circle, Fellinious has given a method of locating the locus on which the probable centre may lie. Felliaious Method of Locating Centre of Critical Slip Circle
23. 23. • For a homogeneous c - Ф soil, the centre of slip circle lie on a line PQ, in which [point Q has its co-ordinates downwards from toe and 4.5H horizontally away as in Fig. 2.38. The point P is located at the intersection of the two lines, one drawn i the toe at an angle a with the slope, and other drawn from the top end of the slope i angle p with the horizontal. The angles a and P are known as the directional angles their values depend on the slope angle i. The values of a and p for different values angle i are given in table below :
24. 24. Values of directional angles a and P for different values of i
25. 25. • According to Fellinious for purely cohesive soils (Ф = 0) the centre of critical circle is located at point P, and for c – Ф soils, the centre of critical slip circle lies at point P on the line QP produced. When the line PQ is obtained, a number of trial centre O1, 0 2 , 0 3 , etc. are selected above point P on the line QP produced. • For each of selected trial centers slip circle is drawn and factor of safety is computed. These factors of safety so obtained are plotted as ordinates on the corresponding centers a smooth curve is obtained. The centre corresponding to the lowest factor of safety the critical circle centre.