5.5.4 Rapid determination of F for a homogeneous, regular slope with a constant pore pressure ratio

If on a trial slip circle the value of F is determined for various values of ru and the results plotted, a linear relationship is found between F and ru (see Example 5.7). The usual values of ru encountered in practice range from 0.0 to 0.7 and it has been established that this linear relationship between F and ru applies over this range. The factor of safety, F, may therefore be determined from the expression:

F = m — nru in which m is the factor of safety with respect to total stresses (i.e. when no pore pressures are assumed) and n is the coefficient which represents the effect of the pore pressures on the factor of safety. These terms m and n are known as stability coefficients and were evolved by Bishop and Morgenstern (1960); they depend upon c'/7H (the stability number with c' equalling cohesion with respect to effective stress), cot 0 (the cotangent of the slope angle, e.g. a 5:1 slope means 5 horizontal to 1 vertical), and 0' (the angle of friction with respect to effective stresses).

Bishop and Morgenstern prepared charts of m and n for three sets of c'r/H values (0.0, 0.025, 0.05) that are reproduced at the end of this chapter and cover slopes from 2:1 (26^°) to 5:1 (llj°). Extrapolation, within reason, is possible for a case outside this range.

Graphs to cover depth factor values, D, up to 1.5 were produced for c'o'H = 0.05, but for the other two cases D values greater than 1.25 were not calculated as such values are not critical in these instances. As in Taylor's analysis, the effect of tension cracks has not been included. O'Connor and Mitchell (1977) extended the work of Bishop and Morgenstern to include c'/jH — 0.075 to 0.150.

Determination of an average value for ru

Generally ru will not be constant over the cross-section of an embankment and the following procedure can be used to determine an average value.

In Fig. 5.28 the stability of the downstream slope is to be determined. From the centre line of the cross-section divide the base of the dam into a suitable

Fig. 5.28 Determination of average ru value.

Fig. 5.28 Determination of average ru value.

number of vertical slices (a, b, c, d), and on the centre line of each slice determine ru values for a series of points as shown. Then the average pore pressure ratio on the centre line of a particular slice is

The average ru for whole cross-section

where Aa = area of the slice a, and rua = average ru value in slice a.

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