Limit-equilibrium benchmarks

Seven published cases are run against OGR Slip2D 0.1.235 as the test suite runs them. The reference of each case is a published value, and its tolerance follows from how precise that source is (Section 4.1.4).

The cases are in the validacion/casos/ folder of the repository. Each holds the model (modelo.ogr), the expected values with their source and tolerance (esperado.json) and a description (caso.md). The numbering is the order in which the cases were added.

4.2.1The seven cases at a glance#

Table 4.2.1 lists the cases and Figure 4.2.1 shows, for every method of every case, the difference from the reference value against the tolerance band of the case. The sections that follow give each case in full.

4.2.2Case 001 · ACADS 1(a): homogeneous slope, total stress#

Problem 1(a) of the survey of Giam & Donald (1989): a homogeneous slope analysed in total stresses, without pore pressure, for which the survey asks the factor of safety and its critical circular surface. Thirty-three programs solved it independently, and the report gives three numbers: a referee value of 1.00, the mean of the 33 programs, 0.991, and the mean Bishop value of 18 programs, 0.993. The case takes the mean of the 33 programs as the reference for both methods it declares. It is the value with statistical support, and it is not tied to the result of any one program. The tolerance is 2.0 %: the referee value and the mean differ by 0.9 % between themselves, and asking for 0.5 % would demand from the source a precision it does not have.

What it checks. The search: the critical circle has to be found, and a search that missed it would return a higher factor of safety. Every coordinate comes from the text of the problem statement and none is read from a figure. Spencer and GLE / Morgenstern–Price are not declared, because the survey gives a mean Bishop value and a mean of all the programs, but no value for a rigorous method.

4.2.3Case 002 · Yamagami & Ueta (1988): homogeneous slope#

A homogeneous slope without pore pressure, from the paper of Yamagami & Ueta (1988) on the search for non-circular slip surfaces by the Morgenstern–Price method. The authors publish a Bishop simplified factor of safety of 1.348 and an Ordinary (Fellenius) one of 1.282, and the case declares both. They are values calculated and published by the authors and not a mean between programs that disagree, which is why its tolerance, 1.5 %, is narrower than the 2.0 % of cases 001 and 007.

What it checks. Two methods at once. The two published values differ by 4.9 % of the Bishop value, so reproducing both fixes also the difference between them, and not only their level. The case runs a grid search of circular surfaces, so it checks the search as well. The coordinates are labelled on the figure that documents the problem, and no dimension had to be scaled off a drawing.

4.2.4Case 003 · ACADS 1(c): three layers#

Problem 1(c) of the same survey (Giam & Donald, 1989), the layered counterpart of case 001. The report gives a referee value of 1.39, the mean Bishop value of 16 programs, 1.406, and the mean of 31 programs, 1.381. The case uses the Bishop mean, which has statistical support and belongs to the method; the tolerance, 1.5 %, covers the 1.1 % that separates it from the referee value.

What it checks. Regions and the assignment of materials: the slope has three, with different strengths. The coordinates are in the text of the problem, and only how the two material lines connect is drawn in a figure. Two checks accompany the case. A critical circle of this problem published in a program verification manual, with centre (34.121, 43.254) and entry and exit points at x = 29.702 and 50.991, has both points at 18.781 from the centre, which settles a radius that the figure reads ambiguously; placed on the geometry of the case, that circle cuts the ground at x = 29.703 and 50.991, to a millimetre. And Bishop's method on that same circle gives a factor of safety within 1 % of the published Bishop value, which depends on the layers the circle crosses.

4.2.5Case 004 · Arai & Tagyo (1985), example 1: a high, steep slope#

Example 1 of Arai & Tagyo (1985): a homogeneous slope without pore pressure, 20 m high at 1V:1.5H, for which the paper gives a Bishop simplified factor of safety of 1.451. Together with case 005 it is the only one with a slope this high and this steep, so an error that grows with the height or with the inclination would show here first. The coordinates are labelled on the figure of the problem.

4.2.6Case 005 · Arai & Tagyo (1985), example 3: the same slope with water#

Example 3 of Arai & Tagyo (1985): the slope of example 1 with a water table, for which the paper gives a Bishop simplified factor of safety of 1.138. The two cases share their external boundary and their material, so any difference between them comes from the pore pressure. The tolerance is 2.5 %, for the same reason as in case 004: the value of 1985 and the reanalyses recorded in the case file differ by 1.9 %.

What it checks. The water. It is the only case with pore pressure, so an error in the computation of $u$ could pass the other six and would show here. The pressure is hydrostatic below the water table and zero above it, with the pore-fluid unit weight of the project, 9.81 kN/m³. The coordinates, the water table included, are labelled on the figure of the problem. The case declares only the Bishop value, the one it takes from the paper.

The note of case 004 applies here as well: the value obtained lies 1.58 % below the published one, within the 2.5 % that the case allows for the known gap between the value of 1985 and modern reanalyses.

4.2.7Case 006 · Reinforcement example (1999): the minimum-depth filter#

The reinforcement back-analysis example of the reference manual of a slope-stability program, published in 1999; the case is called Reinforcement example (1999), as on the Verification page. The example is a cohesionless slope with a distributed load on the crest, and it asks for the reinforcement force that brings the slope to a factor of safety of 1.5, starting from the unreinforced factor, 0.764, found on the circle with centre (−11.41, 35.264) and radius 34.426 m. The statement adds two conditions: only failure surfaces with a minimum depth of 2 m are considered, and a focus point is placed at the toe, so that only circles through it are considered. The source has no entry in the bibliography.

What it checks. That the minimum-depth filter reaches the engine. It is the only case that exercises a surface filter: the others check methods and searches, and none checked that a surface-filter setting arrived at the analysis, which for many versions it did not. The condition is part of the statement and not a detail of the solution: without it the problem does not have the published answer. The case file records that, with the same model and the filter off, the minimum falls on a sliver 0.669 m thick on the face of the slope, with its centre on the edge of the grid. If the filter stopped reaching the engine this case would fail: in version 0.1.235 the grid search of the model with the filter off gives 0.7277, centred at (−20.0, 45.6), which is 4.7 % below the published value and outside the tolerance.

Two of the three numbers of the material are not published. The friction angle is deduced, not assumed: the example publishes a factor $F_{\mathrm{r}} = 1.960$ together with a friction angle of 54.93° for the reinforced material, the two being related by $\tan\phi'_{\mathrm{r}} = F_{\mathrm{r}}\tan\phi'$, and a friction angle of 36° gives $\arctan(1.960\,\tan 36^\circ) = 54.92^\circ$. A second pair of values in the manual, 2.044 and 56.04°, is consistent with it: $\arctan(2.044\,\tan 36^\circ) = 56.04^\circ$. Two independent equations are satisfied by a single friction angle. The unit weight, 20 kN/m³, cannot be deduced and is taken knowingly: with no cohesion the factor of safety hardly depends on it. The case file reports that, on the published circle, a unit weight from 16 to 22 kN/m³ moves the factor between 0.7595 and 0.7668, about 1 % in total, which is the reason why the tolerance is 1.0 % and not narrower.

4.2.8Case 007 · ACADS 2(b): the Talbingo dam on a stated circle#

Problem 2(b) of the survey of Giam & Donald (1989): the Talbingo dam, with four materials, at the end of construction, on a circle that the statement gives: centre (100.3, 291.0) and radius 278.8 m. The report gives a referee value of 2.29, a mean Bishop value of 11 programs, 2.204, and a mean of 24 programs, 2.239. The case compares Bishop with its own mean, 2.204, and Spencer and GLE / Morgenstern–Price with the referee value, 2.29. The tolerance is 2.0 %: the referee value and the mean of the 24 programs differ by 2.2 %, and a narrower tolerance would demand from the source a precision it does not have.

What it checks. The methods alone. The circle is stated, so there is no search and the number measures the method. And the two published references differ by 3.9 %, because Bishop has its own mean and the referee value is the one for the problem: a method of complete equilibrium that returned Bishop's value would fail. It is the only case that can tell a rigorous method from Bishop's against a published number, and the reason why it was added (Section 4.1.5).

The coordinates of the 26 points of the section are tabulated in the problem statement, and how they connect was read from its drawings. Rockfill, transition and filter have the same parameters, so of the five interior lines only the two that bound the core can affect the result, and those are not ambiguous.

The case file does not say where the reference value of Janbu corrected, 2.073, comes from; it is listed as the case declares it.

4.2.9Checks of the methods on stated surfaces#

Besides the seven cases, the test suite checks the methods of equilibrium on surfaces that are stated and not searched, so that the answer does not depend on a search. Two of those checks are described here: a published worked example and a set of exact identities.

A published worked example: Appendix G of EM 1110-2-1902#

The Corps of Engineers methods and the Lowe–Karafiath method share one recursion for the interslice forces (Section 3.2.7). It is checked slice by slice against the worked example of rapid drawdown in Appendix G of USACE (2003), whose Figures G-7a, G-7b and G-9 print the complete table of its twelve slices: the interslice force, the normal force on the base and the factor of safety. The slices have $c' = 0$ and $\phi' = 30^\circ$, with pore pressure on the bases and a water load on the top, in the first stage, and are undrained ($\phi = 0$, with cohesion) in the second. The manual publishes $F = 3.49$ for the first stage and $F = 1.35$ for the second. Because the manual says that its numbers were rounded for the tables, the tolerances are fractions of the scale of each column and not of each value.

The inclination of the interslice forces is the one quantity of the example that the tables do not print, and the check shows that the data pin it down: with horizontal interslice forces the same slices do not close near the published column, and the test requires the residual of the last slice to exceed 20 % of the scale. The rules that give the inclination in each method are checked as plain geometry: for Corps of Engineers #1 it is a constant, the inclination of the chord of the slip surface; for #2 it is the slope of the ground above each slice, so that it is horizontal under a flat crest; both agree with the geometry to 10⁻¹². And the identity named in the manual itself holds: with horizontal interslice forces the recursion reproduces Janbu's simplified method (Janbu, 1973) within 0.5 %.

Exact identities of Spencer and GLE#

The system of interslice forces of the Spencer method (Spencer, 1967) and of the GLE / Morgenstern–Price method (Morgenstern & Price, 1965) has exact anchors, cases in which its answer has to equal that of another method to every figure, and four tests are those anchors. They were written after the defect described in Section 4.1.5, which no factor of safety could reveal because Bishop's value is a plausible one. Two of them follow Fredlund & Krahn (1977): with $\lambda = 0$ the interslice forces are horizontal, there is no interslice shear, and each branch of the solution degenerates into a method the program already has.

The tolerances are relative to the factor of safety of the other method. I3 is the strongest, because the recursion it compares with is the one checked against the worked example above, so Spencer's force branch is anchored to a case solved by hand in a published manual. The model is the geometry and the material of ACADS 1(a) ($c' = 3.0$ kPa, $\phi' = 19.6^\circ$, $\gamma = 20.0$ kN/m³), a circle chosen by hand, with centre (33, 52) and radius 28 m, and 25 slices. Each of I1 to I3 is run dry, with a piezometric line through (20, 23), (45, 27) and (70, 31) and $H_u = 1$, and with a horizontal seismic coefficient of 0.15. A further test shows that $\lambda$ moves the answer, which all the identities, stated at $\lambda = 0$ or between two force branches, could not do alone: the moment branch at $\lambda = 0.5$ differs from the one at $\lambda = 0$ by more than 10⁻³.

4.2.10References#

• Arai, K. & Tagyo, K. (1985). Determination of noncircular slip surface giving the minimum factor of safety in slope stability analysis. Soils and Foundations, 25(1), 43–51.
• Bishop, A.W. (1955). The use of the slip circle in the stability analysis of slopes. Géotechnique, 5(1), 7–17.
• Fredlund, D.G. & Krahn, J. (1977). Comparison of slope stability methods of analysis. Canadian Geotechnical Journal, 14(3), 429–439.
• Giam, S.K. & Donald, I.B. (1989). Example Problems for Testing Soil Slope Stability Programs. Civil Engineering Research Report No. 8/1989, Monash University, Melbourne.
• Janbu, N. (1973). Slope stability computations. In Hirschfeld, R.C. & Poulos, S.J. (eds), Embankment-Dam Engineering: Casagrande Volume. John Wiley & Sons, New York, 47–86.
• Morgenstern, N.R. & Price, V.E. (1965). The analysis of the stability of general slip surfaces. Géotechnique, 15(1), 79–93.
• Spencer, E. (1967). A method of analysis of the stability of embankments assuming parallel inter-slice forces. Géotechnique, 17(1), 11–26.
• U.S. Army Corps of Engineers (2003). Engineering and Design: Slope Stability. Engineer Manual EM 1110-2-1902. Department of the Army, Washington, DC.
• Yamagami, T. & Ueta, Y. (1988). Search for noncircular slip surfaces by the Morgenstern–Price method. Proceedings of the 6th International Conference on Numerical Methods in Geomechanics, Innsbruck, 1335–1340.