A special problem with a soft band has been constructed by Cheng et al.
(2007a) as it appears that similar problems have not been considered previ-ously. The geometry of the slope is shown in Figure 4.6 and the soil properties are shown in Table 4.2. It is noted that c′is zero and φ′is small for soil layer 2 that has a thickness of just 0.5 m. The critical failure surface is obviously controlled by this soft band, and slope failures in similar conditions have actually occurred in Hong Kong.
To consider the size effect (boundary effect) in the SRM, three different numerical models are developed to perform the SRM using Mohr–Coulomb analysis, and the widths of the domains are 28 m, 20 m and 12 m, respec-tively (Figure 4.7). In these three SRM models, various mesh sizes were tried until the results were insensitive to the number of elements used for the analy-sis. For example, when the domain size is 28 m, the FOS was found to be 1.37 (Table 4.3) with 12,000 elements, 1.61 with 6000 elements and 1.77 with 3000 elements using SRM1 analysis and the program Phase.
Because the FOS for this special problem have great differences from those found using the LEM, Cheng et al. (2007a) have tried several well-known com-mercial programs and obtained very surprising results. The locations of the crit-ical failure surfaces from the SRM for solution domain widths of 12 m, 20 m and 28 m are virtually the same. The local failures from the SRM, shown in Figure 4.8b, range from x = 5 m to x = 8 m, and the failure surfaces are virtu-ally the same for the three different solution domains. A majority part of the critical failure surface lies within layer 2, which has a low shear strength, and is far from the right boundary. It is surprising to find that different programs pro-duce drastically different results (Table 4.3A) for the FOS, even though the loca-tions of the critical failure surface from these programs are very similar. For the cases shown in Figure 4.1, and other cases in a latter part of this study, the results are practically insensitive to the domain size, whereas the cases shown in Figure 4.6 are very sensitive to the size of domain for the programs Flac3D (SRM1 and SRM2) and Phase (SRM2). Results from the Plaxis program appear not to be sensitive to the domain size but are quite sensitive to the dilation angle (which is different from the previous example). The SRM1 results from the program Phase are also not sensitive to the domain size for SRM1, but results from SRM2 behave differently. The FOS from Flac3D appear to be overesti-mated when the soil parameters for the soft band are low, but the results from this program are not sensitive to the dilation angle that is similar to all the other Table 4.2 Soil properties for Figure 4.6
Soil name Cohesion (kPa) Friction angle Density Elastic Poisson (degree) (kN/m3) modulus (MPa) ratio
Soil1 20 35 19 14 0.3
Soil2 0 25 19 14 0.3
Soil3 10 35 19 14 0.3
examples in the present study. For the SRM1, the results from Phase and Plaxis appear to be more reasonable as the results are not sensitive to the domain sizes, whereas for the SRM2, Cheng et al. (2007a) take the view that the results from
Figure 4.7 Mesh plot of the three numerical models with a soft band.
Table 4.3A FOS by SRM from different programs when c′for soft band is 0. The values in each cell are based on SRM1 and SRM2, respectively (min.
FOS =0.927 from Morgenstern–Price analysis)
Program/FOS 12 m domain 20 m domain 28 m domain
Flac3D 1.03/1.03 1.30/1.28 1.64/1.61
Phase 0.77/0.85 0.84/1.06 0.87/1.37
Plaxis 0.82/0.94 0.85/0.97 0.86/0.97
Flac2D No solution No solution No solution
Plaxis may be better. It is also surprising to find that Flac2D cannot give any result for this problem, even after many different trials, but the program worked properly for all the other examples in this chapter.
Table 4.3B FOS by SRM from different programs when φ′ =0 and c′ =10 kPa for soft band. The values in each cell are based on SRM1 and SRM2, respectively (min. FOS =1.03 from the Morgenstern–Price analysis)
Program/FOS 28 m domain
Flac3D 1.06/1.06
Phase 0.99/1.0
Plaxis 1.0/1.03
Flac2D No solution
Figure 4.8 Locations of critical failure surfaces from the LEM and SRM for the frictional soft band problem. (a) Critical solution from LEM when soft band is frictional material (FOS = 0.927). (b) Critical solution from SRM for 12m width domain.
(b) (a)
There is another interesting and important issue when the SRM is adopted for the present problems. For the problem with a 12 m domain, Phase cannot pro-vide a result with the default settings and the default settings are varied (including the tolerance and number of iterations allowed) until convergence is achieved. The results of analysis for a 12 m domain with Phase are shown in Tables 4.4 and 4.5.
Table 4.4 FOS with non-associated flow rule for 12 m domain
Element Tolerance Maximum number FOS
number (stress analysis) of iterations
1500 0.001 100 0.8
2000 0.001 100 No result
2000 0.003 100 No result
2000 0.004 100 No result
2000 0.005 100 No result
2000 0.008 100 0.81
3000 0.001 100 No result
3000 0.003 100 0.79
3000 0.004 100 0.8
3000 0.005 100 0.8
3000 0.01 100 0.84
3000 0.001 500 0.77
Table 4.5 FOS with associated flow rule for 12 m domain
Element Tolerance Maximum number FOS
number (stress analysis) of iterations
1000 0.001 100 1.03
1200 0.001 100 1
1500 0.001 100 No result
1500 0.003 100 No result
1500 0.004 100 1
1500 0.005 100 1.39
1500 0.01 100 2.09
1500 0.001 500 0.86
1500 0.003 500 0.98
3000 0.001 100 No result
3000 0.003 100 No result
3000 0.004 100 No result
3000 0.005 100 No result
3000 0.01 100 No result
3000 0.001 500 0.85
3000 0.003 500 0.89
3000 0.004 500 0.9
3000 0.005 500 1.5
3000 0.01 500 2.09
It is observed that the number of elements used for the analysis has a very signif-icant effect on the FOS, which is not observed for the cases in Table 4.1. The tol-erance used in the nonlinear equation solution also has a major impact on the results for this case. This is less obvious for other cases considered in this chapter.
Besides the special results shown above, the FOS from the 28 m domain analysis appear to be large for Flac3D and Phase when the strength parame-ters for the soil layer 2 are low. In fact, it is not easy to define an appropriate FOS from the SRM analysis for this problem. If the cohesive strength of the top soil is reduced to zero, the FOS can be estimated as 0.57 from the rela-tion tanφ/tanθ, where q is the slope angle. It can be seen that, for the LEM, the cohesive strength 20 kPa for soil 1 helps to bring the FOS to 0.927 and a high FOS for this problem is not reasonable. Without the results from the LEM for comparison, it may be unconservative to adopt the values of 1.64 (1.61) from the SRM based on Flac3D.
When the soil properties of the soft band are changed to c′= 10 kPa and φ′= 0, the results of analyses are shown in Table 4.3B. It is found that the crit-ical failure will extend to a much greater distance so that a 28 m wide domain is necessary. The FOS from the different programs are virtually the same, which is drastically different from the results in Table 4.3A (the same meshes were used for Tables 4.3A and 4.3B).
If the soil properties of soils 2 and 3 are interchanged so that the third layer of soil is the weak soil, the FOS from the SRM2 are 1.33 (with all programs) for all three different domain sizes. The corresponding FOS from the LEM is 1.29 from the Spencer analysis. The locations of the critical failure surface from the SRM and LEM for this case are also very close, except for the initial por-tion shown in Figures 4.9a and 4.9b. It appears that the presence of a soft band with frictional material, instead of major differences in the soil parameters, is the actual cause for the difficulties in the SRM analysis. Great care is required in the implementation of a robust nonlinear equation solver for the SRM.
The problems shown in Table 4.3A may reflect the limitations of commer-cial programs rather than the limitations of the SRM, but they illustrate that it is not easy to compute a reliable FOS for this type of problem using the SRM. The results are highly sensitive to different nonlinear solution algo-rithms that are not clearly explained in the commercial programs. Great care, effort and time are required to achieve a reasonable result from the SRM for this special problem and comparisons with the LEM are necessary. It is not easy to define a proper FOS from the SRM alone for the present problem, as the results are highly sensitive to the size of domain and the flow rule. In this respect, the LEM appears to be a better approach for this type of problem.