Find The Center Of The Critical Circle And Toe Circle Failure
the critical circle will be. Note that the value of F is more sensitive to horizontal movements of the circle's centre than to vertical movements.
To determine a reasonable position for the centre of a first trial slip circle is not easy, but a study of the various types of slips that can occur is helpful (it should be remembered, however, that the following considerations apply to homogeneous soils). In the case of soils with angles of shearing resistance that are not less than 3°, the critical slip circle is invariably through the toe as it is for any soil (no matter what its 0 value) if the angle of slope exceeds 53° (Fig. 5.11a). An exception to this rule occurs when there is a layer of relatively stiff material at the base of the slope, which will cause the circle to be tangential to this layer (Fig. 5.11b).
For cohesive soils with little angle of friction the slip circle tends to be deeper and usually extends in front of the toe (Fig. 5.11c); this type of circle can of course be tangential to a layer of stiff material below the embankment which limits the depth to which it would have extended (Fig. 5.1 Id).
In the case of a slope made out of homogeneous cohesive soil it is possible to determine directly the centre of the critical circle by a method that Fellenius proposed in 1936 (Fig. 5.12); the centre of the circle is the intersection of two lines set off from the bottom and top of the slope at angles a and ¡3 respectively (Fellenius's values for a and 8 are given in the table below).
Slope
Angle of slope
Angle a Angle j3
(a) Toe failure
(b) Circle tangential to base
(a) Toe failure
(b) Circle tangential to base
(d) Circle tangential to deep stiff layer
(c) Deep slip circle Fig. 5.11 Types of slip failures.
(d) Circle tangential to deep stiff layer
(c) Deep slip circle Fig. 5.11 Types of slip failures.
This technique is not applicable in its original form to frictional cohesive soils but has been adapted by Jumikis (1962) to suit them, provided that they are homogeneous (Fig. 5.13). It is necessary first to obtain the centre of the Fellenius circle, O], as before, after which a point X is established such that X is 2H below the top of the slope and a distance of 4.5H horizontally away from the toe of the slope (H = the vertical height of the slope). The centre of the critical circle, O, lies on the line XOi extended beyond 01; the distance of O beyond Oi becoming greater as the angle of friction increases.
Such a method can only be used as a means of obtaining a set of sensibly positioned trial slip circles. When the slope is irregular or when there are pore pressures in the soil, conditions are no longer homogeneous and the method becomes less reliable.
Fig. 5.13 Construction for the centre of the critical circle for a c cp soil. EXAMPLE 5.3
The embankment in Fig. 5.14 is made up from a soil with </> = 20° and c = 20 kN/m2. The soil on which the embankment sits has a (j> of 7° and c — 75 kN/m2. For both soils 7 = 19.3 kN/m2. Determine the factors of safety for the two slip circles shown.
Possibl positioi
Possibl positioi
Fig. 5.13 Construction for the centre of the critical circle for a c cp soil. EXAMPLE 5.3
The embankment in Fig. 5.14 is made up from a soil with </> = 20° and c = 20 kN/m2. The soil on which the embankment sits has a (j> of 7° and c — 75 kN/m2. For both soils 7 = 19.3 kN/m2. Determine the factors of safety for the two slip circles shown.
Solution
This example is the classic case of an embankment resting on a stiff layer. The slip circle tangential to the lower layer (Fig. 5.15) will give a lower factor of safety, the example being intended to illustrate this effect.
The sliding sector of soil is conveniently divided into four equal vertical slices. To determine the area of a particular slice its midheight is multiplied by its breadth, and then the weight of the slice is obtained (unit weight x area) and set off as a vector below it. The triangle of forces for the normal and tangential components is then drawn.
The procedure is repeated for each slice, after which the algebraic sum of the tangential forces and the numerical sum of the normal forces is obtained and F evaluated.
The calculations are best set out in tabular form.
Slice 
Area 
Weight 
Normal 
Tangential 
no. 
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