In the present study, a number of interesting features of the SRM were high-lighted that are important for a proper analysis of a slope. Although most research has concentrated on the FOS between the LEM and SRM, the present works have compared the locations of the critical failure surfaces from these two methods. In a simple and homogeneous soil slope, the differences in the FOS and locations of the critical failure surfaces from the Figure 4.12 Global and local minima by LEM.
SRM and LEM are small and both methods are satisfactory for engineering use. It is found that when the cohesion of the soil is small, the difference in the FOS from the two methods is greatest for higher friction angles. When the cohesion of the soil is large, the difference in the FOS is greatest for lower friction angles. With regard to the flow rule, the FOS and locations of the crit-ical failure surface are not greatly affected by the choice of the dilation angle (which is important for the adoption of the SRM in slope stability analysis).
When an associated flow rule is assumed, the critical slip surfaces from the SRM2 appear to be closer to those from the LEM than those from the SRM1.
The use of the SRM requires Young’s modulus, Poissons’ ratio and the flow rule being defined. The importance of the flow rule has been discussed in the previous section. Cheng et al. (2007a) have also tried different combinations of Young’s modulus and Poissons’ ratio and found these two parameters to be insensitive to the results of analysis.
For the SRM, the effects of the dilation angle, the tolerance for nonlinear equation analysis, the soil moduli and the domain size (boundary effects) are Figure 4.13 (a) Global and local minimum factors of safety are very close for a
slope. (b) FOS = 1.327 from SRM.
(b) (a)
usually small but still noticeable. In most cases, these factors cause differ-ences of just a few per cent and are not critical for engineering use of the SRM. Because the use of different LEM methods will also give differences in the FOS of several per cent, the LEM and SRM can be viewed as similar in performance for normal cases.
Drastically different results are determined from different computer programs for the problem with a soft band. For this special case, the FOS is very sensitive to the size of the elements, the tolerance of the analysis and the number of iterations allowed. It is strongly suggested that the LEM be used to check the results from the SRM. This is because the SRM is highly sensi-tive to the nonlinear solution algorithms and flow rule for this special type of problem. The SRM has to be used with great care for problems with a soft band of this nature.
The two examples with local minima for the LEM illustrate another limi-tation of the SRM in engineering use. With the SRM, there is strain localiza-tion during the solulocaliza-tion and the formalocaliza-tion of local minima is unlikely. In the LEM, the presence of local minima is a common phenomenon, and this is a major difference between the two methods. Thus, it is suggested that the LEM should be preformed in conjunction with the SRM as a routine check.
Through the present study, two major limitations of the SRM have been established: (1) it is sensitive to nonlinear solution algorithms/flow rule for some special cases and (2) it is unable to determine other failure surfaces that may be only slightly less critical than the SRM solution but still require treatment for good engineering practice. If the SRM is used for routine analy-sis and design of slope stabilization measures, these two major limitations have to be overcome and it is suggested that the LEM should be carried out as a cross-reference. If there are great differences between the results from the SRM and LEM, great care and engineering judgement should be exercised in assessing a proper solution. There is one practical problem in applying the SRM to a slope with a soft band. When the soft band is very thin, the number of elements required to achieve a good solution is extremely large, so that very significant computer memory and time are required. Cheng (2003) has tried a slope with a 1 mm soft band and has effectively obtained the global mini-mum FOS by the simulated annealing method. If the SRM is used for a prob-lem with a 1 mm thick soft band, it is extremely difficult to define a mesh with a good aspect ratio unless the number of elements is huge. For the SRM with a 500 mm thick soft band, about 1 hr of CPU time for a small problem (sev-eral thousand elements) and sev(sev-eral hours for a large problem (more than 10,000 elements) were required for the Phase program, whereas the program Flac3D required 1–3 days (for small to large meshes). If a problem with a 1 mm thick soft band is to be modelled with the SRM, the computer time and memory required will be huge and the method is not applicable for this special case. The LEM is perhaps better than the SRM for these cases.
For the SRM, there are further limitations that are worth observing.
Shukha and Baker (2003) have found that there are minor but noticeable
differences in the factors of safety from Flac using square elements and distorted elements. The use of distorted elements is however unavoidable in many cases. Furthermore, when both the soil parameters c′and φ′are very small, it is well known that there are numerical problems with the SRM. The failure surface in this case will be deep and wide and a large domain is required for analysis. It has been found that the solution time is extremely long and a well-defined critical failure surface is not well established from the SRM. For the LEM, there is no major difficulty in estimating a FOS and the critical failure surface under these circumstances.
The advantage of the SRM is the automatic location of the critical failure surface without the need for a trial and error search. With the use of modern global optimization techniques, the location of critical failure surfaces by a simulated annealing method, a genetic algorithm or other methods as dis-cussed in Chapter 3 is now possible and a trial and error search with the LEM is no longer required. Although the LEM suffers from the limitation of an interslice shear force assumption, the SRM requires a flow rule and suffers from being sensitive to the nonlinear solution algorithm/flow rule for some special cases.
Griffith and Lane (1999) have suggested that a non-associated flow rule should be adopted for slope stability analysis. As the effect of flow rule on the SRM is not negligible in some cases, such as those involving a soft band, the flow rule is indeed an issue for a proper slope stability analysis. It can be con-cluded that both the LEM and SRM have their own merits and limitations, and the use of the SRM is not really superior to the use of the LEM in rou-tine analysis and design. Both methods should be viewed as providing an esti-mation of the FOS and the probable failure mechanism, but engineers should also appreciate the limitations of each method when assessing the results of their analyses.
Although 2D SRM is available in several commercial programs, there are still various difficulties with 3D SRM and the authors have tested two commercial softwares. For simple and normal problems, there is no major problem with the 3D SRM, and the results are also close to the 3D LEM. There are, however, various difficulties with the 3D SRM for complicated non-homogeneous prob-lems with contrasting soil parameters. More importantly, many strange results may appear when soil nails are present, and there is a lack of good termination criteria for the FOS determination in this case. The authors have also found that the reliance on the default setting for 3D SRM programs may not be adequate for many cases, and there is a lack of a clear and robust method for the FOS determination when a soil nail is present. The authors are still working on this issue in various aspects, and, in general, the authors’ view is that the 3D SRM is far from being mature for ordinary engineering use.