Verification of Slope 2000 for non-circular search

The non-circular search capability of Slope 2000 can be illustrated by 8 test problems. The details of these problems and the corresponding data files are :

1. Nc1.dat : 2 soil and a water table.
2. Nc2.dat : 2 soil, a water table and a horizontal pressure
3. Nc3.dat : 2 soil, a water table and a vertical pressure
4. Nc4.dat : 2 soil, a water table and horizontal and vertical pressure
5. Nc5.dat : 3 soil, a water table and a perch water table
6. Nc6.dat : 3 soil, a water table and a perch water table, horizontal pressure
7. Nc7.dat : 3 soil, a water table and a perch water table, vertical pressure
8. Nc8.dat : 3 soil, a water table and a perch water table, vertical and horizontal pressure

These data will be found when the user installed the demo version of Slope 2000. These data files are however not accepted by the demo version of Slope 2000 as they involve more than 1 type of soil.

A foolish but robust way to check the accuracy of Slope 2000 in locating critical non-circular failure surface is to use pattern search approach. To limit the amount of computer time used, the number of slices is limited to 5 in these studies and the slices are divided evenly. The x-ordinates of the left and right of the failure surfaces are fixed so that only the y-ordinates of the four inter-slices are variables. There are hence actually 4 control variables in the present study. Slope 2000 is used to locate the critical failure surface. Based on the critical result obtained, a pattern search for 0.5m above and below the critical failure surface is tried. For example, for data nc1.dat in Table 1, the y-ordinates of the first and last point (13.5 and 22.0) are not variables while the y-ordinates for the intermediate point are varied within :

Point 2 : from 12.55 to 13.55

Point 3 : from 12.73 to 13.73

Point 4 : from 13.65 to 14.65

Point 5 : from 15.23 to 16.23

The y-ordinates will vary with an increment of 0.01m so that there will be 101x101x101x101 or 10406041 combinations (more than 10 millions combinations) !. Slope 2000 is specially modified to perform this pattern search. All these combinations are tried without considering the requirement of kinematically acceptable mechanism. Using several 733 MHz P III computers running for several days, all these combinations are tried and the results are grouped in tables 1 to 4. The critical results all turn out to be kinematically acceptable and all kinematically unacceptable mechanisms have high factors of safety. The method of analysis used in the present comparison is Janbu’s simplified method as this method is fast in convergence. Other methods are not tried because the purpose of these test problems is to demonstrate the accuracy of the optimization search.

The minimum factors of safety using either optimization search or pattern search correspond to the minimum factors of safety using 5 slices in analysis. The number of slices is not a variable in the optimization analysis. If necessary, the user may change the number of slices and perform the sensitivity analysis. The default number of slices is 10 and is sufficient for most cases. If the number of slices is increased to 20, the change of the factor of safety is usually around 1-2%. The user should also notice that the change may be either positive or negative. In view of the small changes in the factor of safety with number of slices used, the author does not think it is necessary to increase the number of slices unless the horizontal extent of the failure mass exceed 20m.

NC1.dat: No. of trial = 10081, critical solution is at trial 9757

NC2.dat: No. of trial = 10585, critical solution is at trial 9428

Table 1 : Comparison between optimization search and pattern search for nc1.dat and nc2.dat

NC3.dat: No. of trial = 9577, critical solution is at trial 9279

NC4.dat: No. of trial = 10585, critical solution is at trial 10296

Table 2 : Comparison between optimization search and pattern search for nc3.dat and nc4.dat

NC5.dat: No. of trial = 12097, critical solution is at trial 12044

NC6.dat: No. of trial = 13105, critical solution is at trial 12605

Table 3 : Comparison between optimization search and pattern search for nc5.dat and nc6.dat

NC7.dat: No. of trial = 11593, critical solution is at trial 11545

NC8.dat: No. of trial = 12601, critical solution is at trial 12090

Table 4 : Comparisons between optimization search and pattern search for nc7.dat and nc8.dat

The x and y-ordinates of the eight test problems are shown in Tables 1 to 4. The results as shown in Tables 1 to 4 are clear and definite proof that Slope 2000 is able to locate the critical non-circular failure surfaces for relatively complicated cases with the presence of external loads. The minimum factors of safety from more than 10 million combinations and several days computer time are actually the same as that using Slope 2000 with less than 20000 trials and 5 minutes analysis. The slight differences in the location of the critical failure surface can be neglected in actual practice but such differences can be explained in the following way :

1. The default precision for locating the critical failure surface is 0.0001 in Slope 2000. That means, in the various search paths of the solution domain vector spaces, the search along a particular path will stop and a new path will begin if the reduction of the factor of safety is less than 0.0001. For the problems as shown in Table 1 to 4, the author has actually tried to use a tolerance as low as 0.000001 but the reduction in the factor of safety is less than 0.0001 and the change is fifth places behind decimal.
2. In the pattern search, the increment is 10 mm each which is fine enough for all practical purposes. The actual minimum may not fall exactly at the grid point and hence theoretically the solution can still be further improved.
3. The change of the factor of safety is usually very small for a minor change in the profile of the failure surface if the location of the failure surface is close to the critical failure surface. Based on the critical failure surface, the author has tried to vary the y-ordinates by 25 mm and virtually the same factor of safety is obtained with such a change in failure surface profile. In fact, this behaviour is well known and is confirmed once again in the present study.

The author would like to emphasize that any proof of the non-circular failure surface search will automatically verify the circular failure surface search and vice versa. The optimization algorithm in Slope 2000 treats the factor of safety as a general function and its behaviour over the solution domain is estimated with the simulated annealing method. Once the behaviour of the function is known, refined search will be performed to increase the accuracy of the solution. This approach differ greatly from commercial softwares in:

1. In Stabl, random search is used and tremendous trials (nearly same as pattern search) is required for a good accuracy. If a soft band exist in the problem, it is highly unlikely that random search will obtain the critical failure surface.
2. In Stabl, many kinematically unacceptable failure surfaces are generated which is a waste of computer resources.
3. In SLOPE/W and Oasys, the circular pattern search option actually compute all the factors of safety and select the lowest one. The behaviour of the factor of safety function is not assessed.

The critical failures for nc4.dat and nc8.dat are shown in Fig.1 and 2. Failure surfaces for other cases are not shown so as to reduce the length of this manual. The users are strongly advised to try these test cases and view the critical failure surfaces.

Fig. 1 Critical non-circular failure surface for nc4.dat

Fig.2 Critical non-circular failure surface for nc8.dat