The pore pressure ratio ru has been used for many years as a parameter to represent the global pore water pressure condition in a slope stability analysis. Nowadays, the use of computer programs often precludes the need for chart based or coefficient based methods of slope stability analysis using this parameter.
Nevertheless, these methods are useful for rapid preliminary assessments and in computer programs many designers still choose to represent the pore pressure conditions using a single ru value. It is shown in this paper that the choice of a particular value is rather arbitrary and therefore can lead to significant errors in the factor of safety obtained.
By determining the ru value related to a given water table level, the factors affecting their relationship have been established and are discussed.
Averaging techniques have been traditionally used to determine a single global pore pressure coefficient or ratio ru based on procedures suggested by a number of authors.
However, these do not always give the same factor of safety for the given water table, as discussed below. An equivalent ru is proposed which provides the same factor of safety as a given water table location. It is shown that when the water table falls to a certain level the critical slip circle lies above the water table so the slope should be treated as a 'dry' slope with ru = 0.
Averaging techniques for ru
Bishop and Morgenstern (1960) expressed the pore pressure u at any point in a cross-section in terms of the pore pressure ratio ru defined by:
ru = u = w hw(1) h h
where h represents the total stress at the point. The pore pressure ratio is not constant in a cross-section or along a slip surface so to obtain a single global value for use in a slope stability analysis they suggested an averaging technique. Using their technique the author has determined single values of average ru for different water table locations represented by the water table parameter hw/H, illustrated in Figure 1. A quasi-steady state seepage condition can be represented reasonably by a water table at toe level beyond the toe, at hw behind the crest and inclined within the slope between these two levels. When hw lies below the toe level the water table is assumed to be horizontal.
Values of average ru obtained from the technique of Bishop and Morgenstern for values of hw/H are presented in Figure 2. Only the sloping section above the toe is relevant with this method. The value of w/ has been taken as 0.49 assuming a unit weight of soil of 20kN/m3. Curves representing other water table configurations could be produced.
A more recent method suggested by Bromhead (1992) distinguishes average ru values for shallow failure modes, general failure modes and foundation failure modes and extends the cross-section below the toe of the slope. Using this method the values of average ru related to the water table parameter hw/H have been determined and are presented in Figure 3. It is found that the depth factor, D to the base of the critical slip circle, as defined in Figure 2, and the failure mode have a significant effect on the value of average ru. However, as shown later, the depth to the base of the critical slip circle, D does not necessarily correspond with the depth to a hard stratum.
Mitchell (1983) suggests that the approximate location of the critical slip circle should first be obtained and then the value of average ru determined along this slip circle, rather than for the whole cross-section. Unfortunately, the location of the critical slip surface is dependent on the ru value adopted so this procedure requires iteration and can be laborious. A value of average ru recommended by Mitchell for a parabolic top flow line (or water table) in a homogeneous slope has also been plotted on Figure 2 for comparison.
The author (Barnes, 1992) has shown that the factor of safety of a slope can be obtained for a given water table parameter, hw/H, using the stability coefficients, a and b, and adopting the expression:
Fw = a + b tan (2)
The method of Bishop and Morgenstern gives the factor of safety of the slope for an average ru value using the stability coefficients, m and n and adopting the expression:
Fr = m - n ru(3)
In order to compare the author's method with the average ru method, values of the pore pressure ratio ru have been determined which give the same factors of safety, ie Fr = Fw, when the 'equivalent' ru is given by:
equivalent ru = m - a - b tan(4) n
Values of equivalent ru have been determined for a range of values of slope angle, cH, hw/H and and these relationships are presented in Figures 4 to 7. It is noted that these charts do not imply complete compatibility between the two approaches since the critical circles for each method are likely to have different locations even for the same factor of safety.
The coefficients m and n produced by the author (Barnes, 1991) have been used since they generally give lower factors of safety. They were obtained for circles passing through the toe of the slope, so there is no need to consider which depth factor to a hard stratum, D to use.
The figures show that the values of equivalent ru do not correspond with a single value of hw/H but depend on the slope angle, c /H and to a lesser extent the value. The relationship is not very sensitive to the value so the values plotted on the figures are for = 30degrees.
Compared with figure 2 it can be seen that ru values obtained from the averaging method of Bishop and Morgenstern can be significantly under- estimated especially for flatter slopes and higher cohesion values. They will then give factors of safety which are too high and, therefore, seriously in error.
The averaging method of Bromhead in figure 3 shows the same trend as the equivalent ru since the cohesion c increases as the slip surface deepens so the general and foundation failure modes could be adopted when the cohesion is significant. However, the average ru from this figure can be significantly overestimated especially for the deep hard stratum assumption (D=2) giving factors of safety which are too low and, therefore, over conservative.
The foregoing refers to soils with a cohesion intercept, c, when the critical slip surface is represented by a circular arc and the mode of failure is rotational. For soils with no cohesion intercept, c=0, the critical slip surface is given by a plane surface parallel to the slope surface (infinite slope case) and the mode of failure is translational. The factor of safety for this case in terms of the pore pressure ratio, assuming a water table parallel to the surface (and with steady seepage) is given by:
F = (1 - ru sec 2) tan (5) tan
This expression suggests that the factor of safety does not depend on the location of the slip surface. However, the value of ru in the equation is the value on the slip surface and not a global or average value for the slope. If H represents the depth to the slip surface and hw is the depth to the water table then at a point on the slip surface, ru is given by:
ru = w (1 - hw)(6) H
This expression is plotted in figure 8 and shows that for a given water table level hw the value of ru depends on the assumed depth to the slip surface, H.
In the author's water table (hw/H) approach the location of the critical slip circle with the lowest factor of safety was determined. Initially as the water table falls and hw/H increases the factor of safety increases and the critical circle cuts below the water table (described as the 'wet' condition).
However, it has been found (Barnes, 1992) that at a certain water table level the critical circle rapidly rises and from then on the lowest factor of safety is given for a critical circle that lies wholly above the water table, described as the 'dry' condition. From then on the minimum factor of safety is not dependent on the location of the water table. An example of this is shown in figure 9. Hence, ru should be taken as zero for a water table that lies below this critical level.
The determination of the wet or dry condition can be usefully assessed by the water table approach but this is not available with the ru approach.
Location of critical circle
The cases considered by Bishop and Morgenstern were extended by O'Connor and Mitchell (1977) and Chandler and Peiris (1989) and these authors gave values of the stability coefficients, m and n for depth factors to a hard stratum, D, of 1.00, 1.25 and 1.50 with the circles touching the hard stratum.
In the analysis carried out by the author a search was made for the circle giving the lowest factor of safety in a homogeneous cross-section without constraining the circles to fixed depth factors. In this way, the actual depth factor, D, for the most critical circle could be obtained. These values are compared for a typical example in
It has been shown that:
The use of an averaging method to obtain a single global ru value for a particular water table condition can give factors of safety significantly in error both on the conservative and unconservative sides.
An equivalent ru giving the same factor of safety as for a particular water table location depends not only on hw/H but also on the slope angle, and c /H.
For the analysis of a plane translational slide in a cohesionless soil the value of ru is that on the slip surface so for a given water table the depth to the slip surface H must be known or assumed; an arbitrary overall value is not appropriate.
The condition when the critical circle lies above the water table ('dry' condition) can be determined from the water table hw/H method. When the water table lies below this critical level it has no effect on the minimum factor of safety and hence ru should be zero.
The stability coefficient method using ru with fixed depth factors to a hard stratum can overestimate the factor of safety.
The use of a single ru value to represent global pore pressure conditions in slope stability analyses is arbitrary and can lead to significant errors in the factor of safety obtained.
Barnes GE, 1991. A simplified version of the Bishop and Morgenstern slope stability charts. Canadian Geotechnical Journal, 28, 630-637.
Barnes GE, 1992. Stability coefficients for highway cutting slope design. Ground Engineering, 26, 4, 26-31.
Bishop AW and Morgenstern NR, 1960. Stability coefficients for earth slopes. Geotechnique, 10, 129-150.
Bromhead EN, 1992. The stability of slopes, Surrey University Press. Blackie and Sons, UK.
Chandler RJ and Peiris TA, 1989. Further extensions to the Bishop and Morgenstern slope stability charts. Ground Engineering, 22, 4, 33-38.
Mitchell RJ, 1983. Earth structures engineering, Allen and Unwin, USA.
O'Connor MJ and Mitchell RJ, 1977. An extension of the Bishop and Morgenstern slope stability charts. Canadian Geotechnical Journal, 14, 144-151.