7By Robert Pyke Ph.D., G.E.
Slope stability analyses are among the most basic of geotechnical analyses and they have remained essentially unchanged for over 50 years. To be sure, as computer technology has advanced the available computer programs have come to offer more and more options and nicer graphics, but the underlying methodology has not changed. Part of the justification for this is that slopes designed using these programs that meet standard factors of safety usually seem to work. In fact, the writer knows of few if any cases where a slope stability calculation itself has led to a failure. Failures do occur, but these are almost always cases where analyses have not been run or key factors including the geologic structure and water conditions have been overlooked entirely or assessed incorrectly.
What is unknown, however, is how much unnecessary conservatism there is in standard practice because of both the choice of conservative shear strength parameters and conservatisms in the analyses themselves. This question is becoming increasing critical as the complexity and cost of many projects increases, and is especially critical, for instance, in the design of open pit mine slopes, where many millions of dollars are at stake. But if there are aspects of conventional analyses which are unconservative, this also needs to be known so that analyses and designs can be optimized.
Recent work has clarified the impact of two issues that have not been addressed in conventional slope stability analyses but can and should be included in modern slope stability analyses. These are 3D effects and the effect of seepage forces. 3D effects normally increase the computed factor of safety over that obtained from any 2D cross section, but in unusual cases can cause a small decrease in the computed factor of safety. In something of a surprise, however, seepage forces, which reduce the computed factor of safety if they are included, are not included in conventional limit equilibrium analyses.
Several 3D effect examples which are included in this article, can be found on LinkedIn:
These examples include one instance, the failure of the Kettleman Hills landfill in California, where 3D analyses show a small decrease in the computed factor of safety because of the back slopes having a configuration that gave an extra push to the landfill which sat on a slippery liner system. But what is more usual and more dramatic is that the 3D factor of safety of a landfill in a bottleneck canyon at Puente Hills, also in California, is 60% greater than the 2D factor of safety for a cross-section through the mouth of the canyon.
Even more surprising, however, is that 3D analyses can make a difference even for slopes that have a constant cross section. There are two reasons that this happens. One is that for cohesive slopes there are end-effects, which, if considered, increase the factor of safety. For a simple cohesive slope this can increase the computed factor of safety by as much as 40% for the inner ellipsoid shown in Figure 2.
However, for cohesionless slopes there is a small decrease or no change in the factor of safety because the strengths fall off towards the sides of the potential failure surface as the heights of the columns used to model the potential sliding mass decrease.
The second reason that slopes with a constant cross-section may have a higher factor of safety in a 3D analysis is that if there is a wall or revetment, such as is common in remedial construction for instance, the 3D potential failure surface has to pass through the wall or revetment twice, once at each end of the potential failure, whereas in a 2D analysis it can dive underneath the wall or revetment. Figure 3 shows a 2D cross-section through a Greensteep retaining wall which has been designed to have a factor of safety greater when 1.1 when both a surcharge load and a seismic coefficient are applied. The static factor of safety without the surcharge and seismic loads is 1.88.
However, a 3D spherical slip surface through the wall, as shown in Figure 4, has a factor of safety of 2.53, so that a more economic design might be possible when a 3D analysis is used.
However, if a uniform surcharge load of 200 psf is applied behind the top of the slope, as shown in Figure 4, the 2D factor of safety only falls to 1.67, whereas the 3D factor of safety falls to 1.35 because there is a relatively large increase in the driving forces for the shorter columns, and for cohesive materials such as the wall, show no increase in strength. While the engineer should still be wary of taking the calculated factors of safety at precisely their face value, this is a classic example of how an improved analysis can provide much greater insight to a problem.
Slope failures are often said to be due to water, or rather the failure to recognize the correct water conditions, and application of the pore pressures normal to the bases of the slices in conventional slope stability analyses may give the impression that this accounts for seepage forces in non-hydrostatic conditions. However, this is not correct. The seepage forces that one assumes might be applied by using total unit weights and specifying the pore pressures along the slip surface do not actually make their way into the analysis. This can easily be checked by running an analysis of a cohesive slope with varying phreatic surfaces. Regardless of how steep the phreatic surface, it will make no difference to the computed factor of safety. The reason that a cohesive slope, or a slope in which all the strengths are specified as fixed quantities, as with undrained shear strengths, must be used is that the strength of frictional material will vary with the normal effective stress so that changing the phreatic surface will make a difference, but it does not make a difference to the limit equilibrium problem.
An example of the effect of seepage forces is included in a separate article posted on LinkedIn: tinyurl.com/pyke01
And the more technical details, including the reason that seepage forces are not included in limit equilibrium analyses, are discussed on LinkedIn: tinyurl.com/pyke02
Briefly, the example illustrates the application of seepage forces in the Ordinary Method of Columns (OMC) or non-application of seepage forces in Spencer’s Method using an idealized levee section that is based on real levees in the Sacramento – San Joaquin Delta of California.
Both the OMC and Spencer’s Method give a factor of safety of 1.27 for this case. When the phreatic surface is raised to flood stage, however, the calculated factor of safety by Spencer’s Method remains 1.27, but the factor of safety calculated by the OMC drops to 0.98, suggesting that failures in slopes of this kind are not seen more frequently because they are either designed for healthy factors of safety or using conservative shear strength parameters.
While the levee section in this example is somewhat idealized, the drop in the phreatic surface is real because the water surface is controlled by pumping from a drainage ditch at the landward toe. More commonly, seepage forces will not make this much difference because the head loss and seepage forces will be smaller relative to the gravity forces and the resistance to sliding provided by the soil, suggesting that an increase in the total weights, elimination of negative pore pressures in partially saturated soils, a change from drained to undrained loading conditions and more general softening of soils might be bigger factors in why landslides often appear to be triggered by water. But, in critical cases the engineer needs to use a method of analysis such as the OMC, which does include seepage forces, as a check on standard limit equilibrium calculations.
The same conclusion applies to 3D effects. Sometimes they may not be that great but if you don’t have a tool that you can use to check 3D effects, you’ll never know how much your conventional 2D analyses might be in error.
The program TSLOPE used to perform the calculations reported in this editorial is available for a free trial at www.tagasoft.com