Slope stability software for soft soil engineering. D-Geo Stability. Verification Report

Transcription

1 Slope stability software for soft soil engineering D-Geo Stability Verification Report

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3 D-GEO STABILITY Slope stability software for soft soil engineering Verification Report Version: 16.2 Revision: December 2016

4 D-GEO STABILITY, Verification Report Published and printed by: Deltares Boussinesqweg HV Delft P.O MH Delft The Netherlands telephone: fax: info@deltares.nl www: For sales contact: telephone: fax: sales@deltaressystems.nl www: For support contact: telephone: fax: support@deltaressystems.nl www: Copyright 2016 Deltares All rights reserved. No part of this document may be reproduced in any form by print, photo print, photo copy, microfilm or any other means, without written permission from the publisher: Deltares.

5 Contents Contents List of Figures List of Tables v vii Introduction 1 1 Group 1: Benchmarks from literature (exact solution) Vertical slope in strictly cohesive material Slope 1:1 with ϕ = Distributed load on horizontal surface Submerged slope 1:1 with ϕ = Phreatic water along slope 1:2 with ϕ = Group 2: Benchmarks from literature (approximate solution) Verification search algorithm Verification search algorithm with external load Verification shear stress restriction Verification shear stress restriction Verification deep circle Verification uniform distributed load Verification water treatment Group 3: Benchmarks from spreadsheets Verification Bishop stress-dependent (linear) Verification Bishop stress dependent (non-linear) Verification geotextiles Earthquake forces Functioning of tree on slope Functioning of the reference level for ratio S = S u /σ y (Su-calculated model) Zone areas acc. to zone plot method Deterministic calculation using design values c-phi model Deterministic calculation using mean values c-phi model Deterministic calculation using design values Stress table model Deterministic calculation using mean values Stress table model Deterministic calculation using design values Cu-calculated model Deterministic calculation using mean values Cu-calculated model Deterministic calculation using design values Cu-measured model Deterministic calculation using mean values Cu-measured model Probabilistic calculation c-phi model 6 variables: cohesions, degree of cons. and hydraulic pressures with a normal distribution Probabilistic calculation c-phi model 2 variables: cohesion and model factor with a normal distribution Probabilistic calculation c-phi model 1 variable: friction angle with a normal distribution Probabilistic calculation c phi model 1 variable: degree of cons. with a logarithmic distribution Probabilistic calculation c-phi model 2 variables: cohesion and model factor with a logarithmic distribution Probabilistic calculation c-phi model 1 variable: friction angle with a logarithmic distribution Probabilistic calculation c-phi model 2 variables: cohesion with a normal and logarithmic distribution Deltares iii

6 D-GEO STABILITY, Verification Report 3.23 Probabilistic calculation Cu-measured model 2 variables: und. cohesion and model factor with a normal distribution Probabilistic calculation Cu-measured model 2 variables: und. cohesion and model factor with a logarithmic distribution Probabilistic calculation Cu-measured model 2 variables: und. cohesions with a normal and logarithmic distribution Probabilistic calculation Cu-measured model 2 variables: und. cohesions (non-uniform distribution) with a normal distribution Probabilistic calculation Su-calculated model 5 variables: ratio S, hydraulic press., D.O.C. and model factor with a normal distribution Probabilistic calculation Su-calculated model 3 variables: ratio S, degree of cons. and model factor with a logarithmic distribution Probabilistic calculation Su-calculated model 2 variables: ratio S with a normal and a logarithmic distribution Probabilistic calculation Stress table model 2 variables: tangential stress and model factor with a normal distribution Probabilistic calculation Stress table model 4 variables: degree of cons. and hydraulic pressures with a normal distribution Probabilistic calculation Stress table model 2 variables: tangential stress and model factor with a log. distribution Probabilistic calculation Stress table model 1 variable: degree of cons. with a logarithmic distribution Probabilistic calculation Stress table model 2 variables: tangential stresses with a normal and a logarithmic distribution Probabilistic calculation using all shear strength models Probabilistic calculation of the design water level from three external water cases (D-GEO STABILITY and MProStab) MProStab 2 variables: cohesion and model factor with a log. distribution MProStab 1 variable: friction angle with a log. distribution MProStab 1 variable: hydraulic pressure with a normal distribution MProStab 1 variable: D.O.C. with a normal distribution Horizontal balance method Pseudo characteristic shear strength model (local measurements) Pseudo characteristic shear strength model (global measurements) Deterministic calculation using design values Uplift Van model Deterministic calculation using mean values Uplift Van model Probabilistic calculation All shear strength models Uplift Van model Verification of the functioning of nails Verification of the shear strength model Su-calculated with yield stress Group 4: Benchmarks generated by the program Verification Cu-measured method Verification MSeep results with piezometric lines Comparison between D-Geo Stability and MProStab (without autocorrelation) Comparison between D-Geo Stability and MProStab (with autocorrelation) Comparison between Uplift Van and Uplift Spencer Comparison between Genetic Algorithm and Grid search methods (Spencer) Simple case Comparison between Genetic Algorithm and Grid search methods (Spencer) Complex case Group 5: Benchmarks compared with other programs Simple homogeneous slope without pore water Simple homogeneous slope with pore water iv Deltares

7 Contents 5.3 A simple slope from Fredlund and Krahn A simple slope from Fredlund and Krahn, with both pore water and weak layer A complex slope with various layers Su-calculated model with yield stress measurements Verification of the reference line for the undrained shear strength models Verification of the dilatancy References 129 Deltares v

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9 List of Figures List of Figures 1.1 Geometry of benchmark Geometry of benchmark 1-2, slope 1:1 with ϕ = Geometry of benchmark Geometry of benchmark Geometry of benchmark Geometry of benchmark Geometry of benchmark Geometry of benchmark Geometry of benchmark Geometry and geotextiles of benchmark Calculation of angle α (bm3-3b and bm3-3c) Geometry of benchmark Geometry of benchmark Geometry of benchmark 3-6, dike and load at the top of a clay layer Geometry of benchmark Division of the slip circle in 8 slices of 1 meter Configuration of benchmark 3-16: slice 2 divided into 2 layers Sigma-Tau curves for slice 1 and bottom of slice Geometry of benchmark 3-41, check of horizontal balance of a dike retaining water Geometry of benchmark Geometry of benchmark Geometry of benchmark Geometry of benchmark 4-4, load on left side Geometry of benchmark Geometry of benchmark Geometry of benchmark Geometry of benchmark 5-1 with slip plane Geometry of benchmark 5-2 with slip plane and water Geometry of benchmark Geometry of benchmark Geometry of benchmark 5-6a Geometry of benchmark 5-6b Geometry of benchmark 5-6c Geometry of benchmark 5-6d Geometry of benchmark 5-6e Results of benchmark Geometry of benchmark Geometry of benchmark Deltares vii

10 D-GEO STABILITY, Verification Report viii Deltares

11 List of Tables List of Tables 1.2 Grid properties Results of benchmark Grid and tangent line properties Results of benchmark Grid and tangent line properties Results of benchmark Grid and tangent line properties Results of benchmarks 1-4a and 1-4b Grid and tangent line properties Results of benchmarks 1-5a and 1-5b Soil properties (bm2-1) Grid and tangent line properties (bm2-1) Results of benchmark Grid and tangent line properties (bm2-2) Results of benchmark Grid and tangent line properties (bm2-3) Results of benchmark Soil properties (bm2-4) Grid and tangent line properties (bm2-4) Results of benchmark Grid and tangent line properties (bm2-5) Results of benchmark Grid and tangent line properties (bm2-6) Results of benchmark Grid and tangent line properties (bm2-7) Results of benchmark Soil properties (bm3-1) Grid and tangent line properties (bm3-1) Results of benchmark Sigma-Tau curve (bm3-2) Results of benchmark Results of benchmark Soil properties (bm3-4) Grid and tangent line properties (bm3-4) Results of benchmark Tree on slope properties Soil properties Grid and tangent line properties Results of benchmark Properties of the soil materials (bm3-6) Grid and tangent line properties (bm3-6) Results of benchmark Zone area for safety Slip circle definition (bm3-7) Results of benchmark Unit weight of soil and water Grid and tangent line properties Probabilistic soil properties (bm3-8) Geometrical data s of the different slices Results for the design values (bm3-8) Deltares ix

12 D-GEO STABILITY, Verification Report 3.32 Results of benchmark Results for the mean values (bm3-9) Results of benchmark Probabilistic soil properties (bm3-10) Sigma-Tau curve (bm3-10) Results of benchmark 3-10a Results of benchmark 3-10b Results of benchmark 3-10c Results of benchmark 3-11a Results of benchmark 3-11b Results of benchmark 3-11c Probabilistic soil properties (bm3-12) Results for the design values (bm3-12) Results of benchmark Results for the mean values (bm3-13) Results of benchmark Probabilistic soil properties (bm3-14) Results for the design values(bm3-14) Results of benchmark Results for the mean values (bm3-15) Results of benchmark Probabilistic soil properties (bm3-16) Results of benchmark Probabilistic soil properties (bm3-17) Results of benchmark Probabilistic soil properties (bm3-18) Results of benchmark Probabilistic soil properties (bm3-19) Results of benchmark Probabilistic soil properties (bm3-20) Results of benchmark Probabilistic soil properties (bm3-21) Results of benchmark Probabilistic soil properties (bm3-22) Results of benchmark Probabilistic soil properties (bm3-23) Results of benchmark Probabilistic soil properties (bm3-24) Results of benchmark Probabilistic soil properties (bm3-25) Results of benchmark Probabilistic soil properties (bm3-26) Results of benchmark Probabilistic soil properties (bm3-27) Results of benchmark Probabilistic soil properties (bm3-28) Results of benchmark Probabilistic soil properties (bm3-29) Results of benchmark Probabilistic soil properties (bm3-30) Sigma-Tau curve of slice 1 (bm3-30b) Results of benchmark 3-30a (using a coefficient of variation) Results of benchmark 3-30b (using a characteristic curve) Results of benchmark x Deltares

13 List of Tables 3.86 Sigma-Tau curve of slice 1 (bm3-32a) Sigma-Tau curve of slice 1 (bm3-32b) Results of benchmark 3-32a (using a coefficient of variation) Results of benchmark 3-32b (using a characteristic curve) Sigma-Tau curve (bm 3-19) Results of benchmark Sigma-Tau curve (bm3-34) Results of benchmark Probabilistic soil properties (bm3-35) Results of benchmark Reliability index (bm3-36) Results of benchmark 3-36a Results of benchmark 3-36b Correlation parameters (bm3-37) Unit weight of soil (bm3-37) Grid and tangent line properties (bm3-37) Cohesion and friction angle MProStab, slice 1 (bm3-37) Results of benchmark Correlation parameters (bm3-38) Unit weight of soil (bm3-38) Grid and tangent line properties (bm3-38) Cohesion and friction angle MProStab, slice 1 (bm3-38) Results of benchmark Probabilistic soil properties (bm3-39) Correlation parameters (bm3-39) Results of benchmark Correlation parameters (bm3-40) Unit weight of soil (bm3-40) Grid and tangent line properties (bm3-40) Probabilistic soil properties for slice 1 (bm3-40) Results of benchmark Soil properties (bm3-41) Calculation area (bm3-41) Calculation of the resisting force (bm3-41) Results of benchmark Soil properties (bm3-42) Sigma-Tau curves (bm3-42) Analytical results of benchmark Results of benchmark Analytical result of benchmark Results of benchmark Results for the design values (bm3-44) Results of benchmark Results for the mean values (bm3-45) Results of benchmark Results of benchmark Results of benchmark 3-47a Results of benchmark 3-47b Material properties (benchmark 3-48) Pre-consolidation stress at different positions (benchmark 3-48) Results of benchmark bm Safety factor and moments Soil properties, Cu-measured model (bm4-1) Soil properties, c-phi model (bm4-1) Deltares xi

14 D-GEO STABILITY, Verification Report 4.3 Results of benchmark Results of benchmark Correlation parameters (bm4-3) Results of benchmark Unit weight of soil (bm4-4) Grid and tangent line properties (bm4-4) Probabilistic soil properties for MProStab (bm4-4a) Correlation parameters (bm4-4a) Probabilistic soil properties for D Geo Stability (bm4-4b) Results of benchmark Soil properties (bm4-6) Results of benchmarks 4-5a and 4-5b Soil properties (bm4-6) Results of benchmarks 4-6a and 4-6b Results of benchmarks 4-7a and 4-7b Results of benchmark Results of benchmark Soil properties (bm5-3) Results of benchmark Soil properties (bm5-4) Results of benchmark Soil properties (bm5-5) Results of benchmark Soil properties for benchmarks Results of benchmark Soil properties for benchmarks Variations of the dilatancy angle in the calculations Results of benchmark xii Deltares

15 Introduction Deltares Systems commitment to quality control and quality assurance has leaded them to develop a formal and extensive procedure to verify the correct working of all of their geotechnical engineering tools. An extensive range of benchmark checks have been developed to check the correct functioning of each tool. During product development these checks are run on a regular basis to verify the improved product. These benchmark checks are provided in the following sections, to allow the users to overview the checking procedure and verify for themselves the correct functioning of D-GEO STABILITY. The benchmarks for Deltares Systems are subdivided into five separate groups as described below. Group 1 (chapter 1) Benchmarks from literature (exact solution) Simple benchmarks for which an exact analytical result is available. Group 2 (chapter 2) Benchmarks from literature (approximate solution) More complex benchmarks described in literature, for which an approximate solution is known. Group 3 (chapter 3) Benchmarks from spreadsheets Benchmarks which test program features specific to the program using spreadsheets. Group 4 (chapter 4) Benchmarks generated by the program Benchmarks for which the reference results are generated by the program. Group 5 (chapter 5) Benchmarks compared with other programs The benchmarks in this chapter have no exact solution, but are compared with other programs (using the same method). The number of benchmarks in group 1 will probably remain the same in the future. The reason for this is that they are very simple, using only the most basic features of the program. The number of benchmarks in group 2 may grow in the future. The benchmarks in this chapter are well documented in literature. There are no exact solutions for these problems available, however in the literature estimated results are available. When verifying the program, the results should be close to the results found in the literature. Groups 3, 4 and 5 of benchmarks will grow as new versions of the program are released. These benchmarks are designed in such a way that (new) features specific to the program can be verified. The benchmarks are kept as simple as possible so that, per benchmark, only one specific feature is verified. As much as software developers would wish they could, it is impossible to prove the correctness of any non-trivial program. Re-calculating all the benchmarks in this report, and making sure the results are as they should be, will prove to some degree that the program works as it should. Nevertheless there will always be combinations of input values that will cause the program to crash or produce wrong results. Hopefully by using the verification procedure the number of times this occurs will be limited. The benchmarks will all be described to such detail that reproduction is possible at any time. In some cases, when the geometry is too complex to describe, the input file of the benchmark is needed. The results are presented in text format with each benchmark description. The input files belonging to the benchmarks can be found on CD-ROM or can be downloaded from our website Deltares 1 of 132

16 D-GEO STABILITY, Verification Report 2 of 132 Deltares

17 1 Group 1: Benchmarks from literature (exact solution) The different benchmarks with an exact solution (group1) are described in the next following paragraphs, from section 1.1 to section 1.5. This chapter describes a number of benchmarks for which an exact analytical solution can be found in the literature on slope stability calculations. As is often the case with analytical-solution benchmarks, the calculation result will usually be equal to 1.0. In the tables the results of a calculation with Bishop and Fellenius are shown. The difference between Bishop and Fellenius is that iteration is necessary in case of Bishop. In case of a safety factor of 1 both methods should give the same result. 1.1 Vertical slope in strictly cohesive material Fellenius shows that the safety factor has value 1.0 if the height of the vertical slope in a frictionless soil (ϕ = 0 ) is equal to: h = 3.83 c γ [Lit 12, page 42.3] (1.1) where: h is the height vertical slope, in m; c is the cohesion, in kn/m 2 ; γ is the volumetric mass, in kn/m 3. Figure 1.1: Geometry of benchmark 1-1 Table 1.2: Grid properties Left/top Right/bottom Number X co-ordinate [m] Y co-ordinate [m] Deltares 3 of 132

18 D-GEO STABILITY, Verification Report Benchmark results With γ = 1 kn/m 3 and c = 1 kn/m 2, the safety factor for a vertical slope of 3.83 m according to the above formula is equal to 1.0. D-GEO STABILITY results While trying to calculate this benchmark using D-GEO STABILITY, the required safety factor could not be found. This was caused by the way D-GEO STABILITY handles slip circles that intersect the geometry at more than 2 positions. With the vertical slope in this benchmark, many circles intersect both the vertical slope part of the geometry, and the horizontal surface at the low end of the slope. When this occurs, D-GEO STABILITY finds the largest of the two circle parts, and calculates the safety factor for that part. In many cases the part intersecting the horizontal lower surface is the largest, and therefore calculated giving a high safety factor. The above problem was solved by lowering the surface another 2 meters, and adding a fixed point 3.83 m below the top of the vertical slope. This prevents D-GEO STABILITY from finding circle parts through the horizontal surface, thereby only calculating slip circles including the vertical slop, as the benchmark was intended. Results are compared in Table 1.3. Table 1.3: Results of benchmark 1-1 Circle Benchmark D-GEO STABILITY Rel. error [%] X co-ordinate [m] Y co-ordinate [m] Radius [m] Safety factor Bishop (bm1-1a) [-] Safety factor Fellenius (bm1-1b) [-] Use D-GEO STABILITY input files bm1-1a.sti and bm1-1b.sti to run this benchmark. 1.2 Slope 1:1 with ϕ = 45 This benchmark is taken from problem 12 (page 431) of [Lit 7] and describes a slip circle in a dry soil without cohesion, a 1:1 slope and an angle of internal friction ϕ of 45 (Figure 1.2). Table 1.4: Grid and tangent line properties Left/top Right/bottom Number X co-ordinate [m] Y co-ordinate [m] Tangent line [m] Benchmark results With γ = 1 kn/m 3 and c = 0 kn/m 2, the safety factor for this slope is equal to 1.0. The circle with the minimum safety factor just intersects with the slope, so the safety factor found can be regarded as the safety factor for micro-stability. 4 of 132 Deltares

19 Group 1: Benchmarks from literature (exact solution) Figure 1.2: Geometry of benchmark 1-2, slope 1:1 with ϕ = 45 D-GEO STABILITY results The circle with the minimum safety factor has been found using the search algorithm in D-GEO STABILITY. In the final calculation a small, fine grid is placed around the centre point with the minimum safety factor so that the center point of the minimum circle is enclosed. Results are compared in Table 1.5. Table 1.5: Results of benchmark 1-2 Circle Benchmark D-GEO STABILITY Rel. error [%] X co-ordinate [m] Y co-ordinate [m] Radius [m] Safety factor Bishop (bm1-2a) [-] Safety factor Fellenius (bm1-2b) [-] Use D-GEO STABILITY input files bm1-2a.sti and bm1-2b.sti to run this benchmark. 1.3 Distributed load on horizontal surface This benchmark is taken from problem 13, page 432 of [Lit 7] and describes a distributed load on a horizontal surface in a strictly cohesive material. For a circular failure surface, the maximum load allowed is equal to: p = 5.52 c (1.2) where: p is a uniform distributed load in kn/m 2 ; c is the cohesion in kn/m 2. Table 1.7: Grid and tangent line properties Deltares Left/top Right/bottom Number X co-ordinate [m] of 132 Y co-ordinate [m] Tangent line [m]

20 D-GEO STABILITY, Verification Report Figure 1.3: Geometry of benchmark 1-3 D-GEO STABILITY results The circle with the minimum safety factor has been found using the search algorithm in D-GEO STABILITY. In the final calculation a small, fine grid is placed around the centre point with the minimum safety factor so that the centre point of the minimum circle is enclosed. Results are shown in Table 1.8. Table 1.8: Results of benchmark 1-3 Circle Benchmark D-GEO STABILITY Rel. error [%] X co-ordinate [m] Y co-ordinate [m] Radius [m] Safety factor Bishop (bm1-3a) [-] Safety factor Fellenius (bm1-3b) [-] Use D-GEO STABILITY input files bm1-3a.sti and bm1-3b.sti to run this benchmark. 1.4 Submerged slope 1:1 with ϕ = 45 6 of 132 Deltares

21 Group 1: Benchmarks from literature (exact solution) Figure 1.4: Geometry of benchmark 1-4 This benchmark is taken from problem 14 (page 433) of [Lit 7] and describes a slope 1:1 submerged under water with free water on the slope surface (Figure 1.4). The soil has no cohesion, and an angle of internal friction equal to 45. Table 1.9: Grid and tangent line properties Left/top Right/bottom Number X co-ordinate [m] Y co-ordinate [m] Tangent line [m] Benchmark results With γ = 20 kn/m 3, c = 0 kn/m 2 and the angle of internal friction ϕ = 45, the safety factor for this slope is equal to 1.0. The volumetric weight of the water is equal to 9.81 kn/m 3. D-GEO STABILITY results The circle with the minimum safety factor has been found using the search algorithm in D-GEO STABILITY. In the final calculation a small, fine grid is placed around the centre point with the minimum safety factor so that the centre point of the minimum circle is enclosed. The critical slip circle just touches the slope, so that the safety factor represents micro stability. The results of the D-GEO STABILITY calculation and the benchmark are given in Table Table 1.10: Results of benchmarks 1-4a and 1-4b Circle Benchmark D-GEO STABILITY Rel. error [%] X co-ordinate [m] Y co-ordinate [m] Radius [m] Safety factor Bishop (bm1-4a) [-] Safety factor Fellenius (bm1-4b) [-] Deltares 7 of 132

22 D-GEO STABILITY, Verification Report Use D-GEO STABILITY input files bm1-4a.sti and bm1-4b.sti to run this benchmark. 1.5 Phreatic water along slope 1:2 with ϕ = 45 This benchmark is taken from problem 15 (page 433) of [Lit 7] and describes a slope 1:2 with the phreatic water line along the slope surface (Figure 1.5). The soil has no cohesion, and an angle of internal friction ϕ equal to 45. The safety factor can be determined by: F s = γ γ w (1 + tan 2 α) tan ϕ γ tan α (1.3) where: F s is the safety factor; γ is the volumetric weight soil in kn/m 3 ; γ w is the volumetric weight of water in kn/m 3 ; ϕ is the angle of internal friction in ; α is the slope angle in. Figure 1.5: Geometry of benchmark 1-5 Table 1.12: Grid and tangent line properties Left-top Right-bottom Number X co-ordinate [m] Y co-ordinate [m] Tangent line [m] Benchmark results With γ = 24.5 kn/m 3, c = 0 kn/m 2 and the angle of internal friction ϕ = 45, the safety factor for this slope is equal to 1.0 if the slope angle is 1:2. The volumetric weight of the water is equal to 9.81 kn/m 3. 8 of 132 Deltares

23 Group 1: Benchmarks from literature (exact solution) D-GEO STABILITY results The circle with the minimum safety factor has been found using the search algorithm in D-GEO STABILITY. In the final calculation a small, fine grid is placed around the centre point with the minimum safety factor so that the centre point of the minimum circle is enclosed. The critical slip circle just touches the slope, so that the safety factor represents micro stability. Results are shown in Table Table 1.13: Results of benchmarks 1-5a and 1-5b Circle Benchmark D-GEO STABILITY Rel. error [%] X co-ordinate [m] Y co-ordinate [m] Radius [m] Safety factor Bishop (bm1-5a) [-] Safety factor Fellenius (bm1-5b) [-] Use D-GEO STABILITY input files bm1-5a.sti and bm1-5b.sti to run this benchmark. Deltares 9 of 132

24 D-GEO STABILITY, Verification Report 10 of 132 Deltares

25 2 Group 2: Benchmarks from literature (approximate solution) The benchmarks in this chapter have no exact analytical solution, but are documented in literature and therefore approximate solutions are available that are either calculated by hand, or by some other slope stability program. The D-GEO STABILITY results, when calculating these benchmarks, should approximate the results found in the literature. There may be a number of reasons why the results found by D-GEO STABILITY do not match the results found in literature exactly. Differences can be caused by: A difference in the number of slices. In D-GEO STABILITY the number of slices to use is determined automatically, so it is not possible to adjust this number to the number used in the literature examples. For most examples in literature the number of slices used is unknown. The position of the slice boundaries. D-GEO STABILITY places a slice boundary at every significant point (see the users manual for more details) while other programs use slices with constant width, divided evenly over the slip circle area. The results for these two strategies will almost never be equal. The spacing between the grid points. Between any two grid points there may always be another point which yields an even lower safety factor. In literature only the minimum safety factor is mentioned, not the grid size used to find it. The accuracy of the computer used to calculate the results. Although stability calculations are not very demanding with regard to numerical calculations made, there will still be small differences between the results calculated on different computers. However, within the accuracy used in these calculations (with about 3 digits behind the decimal point) it is not expected that the accuracy of the computer has much influence. Differences between the benchmark results and the D-GEO STABILITY result of 2% or less can be attributed to the above mentioned causes. If the difference becomes larger, it is important to check if the difference can be explained by inherent differences in the way both programs handle part of the calculation. The way free water on the surface is treated by different programs is, for example, often a reason why results differ. If no reasonable explanation can be found, there is a serious problem and research has to be done to find out what causes the difference. 2.1 Verification search algorithm This benchmark is taken from problem 23, page 447 of [Lit 7] and tests the search algorithm to see if the algorithm is able to find the position of the slip circle with the smallest safety factor. For this benchmark a complete geometry is used including multiple layers, a phreatic level line and a special piezometric level line for one of the soil layers (Figure 2.1). Table 2.1: Soil properties (bm2-1) Clay1 Clay2 Peat Sand Uns. total unit weight [kn/m 3 ] Sat. total unit weight [kn/m 3 ] Cohesion [kn/m 2 ] Friction angle ϕ [ ] Deltares 11 of 132

26 D-GEO STABILITY, Verification Report Figure 2.1: Geometry of benchmark 2-1 Table 2.2: Grid and tangent line properties (bm2-1) Left/top Right/bottom Number X co-ordinate [m] Y co-ordinate [m] Tangent line [m] Benchmark results The position of the critical slip circle and its radius is rounded to whole meters. In the above report, this problem has been calculated using three different slope stability programs. The results are shown in the table below. D-GEO STABILITY results The calculated safety factor differs less than 1% from the results of program (1) & (2), and less than 3% compared to program (3). The results are shown in Table 2.3. Table 2.3: Results of benchmark 2-1 Circle Benchmark D-GEO STABILITY Relative error [%] X co-ordinate [m] Y co-ordinate [m] Radius [m] Safety factor Bishop [-] (1) (2) (3) 2.89 Use D-GEO STABILITY input file bm2-1.sti to run this benchmark. 12 of 132 Deltares

27 Group 2: Benchmarks from literature (approximate solution) 2.2 Verification search algorithm with external load This benchmark is taken from problem 24, page 447 of [Lit 7] and tests the search algorithm to see if the algorithm is able to find the position of the slip circle with the smallest safety factor. The same geometry as benchmark 2-1 (section 2.1) is used. The vertical position of the grid is shifted (Table 2.4). A uniform distributed load on a part of the top of the dike is added (Figure 2.2). Table 2.4: Grid and tangent line properties (bm2-2) Left/top Right/bottom Number X co-ordinate [m] Y co-ordinate [m] Tangent line [m] Figure 2.2: Geometry of benchmark 2-2 Benchmark results The position of the critical slip circle and its radius is rounded to whole meters. In the above report, this problem has been calculated using two different slope stability programs. The results are shown in the table below. D-GEO STABILITY results The results of the other programs and the D-GEO STABILITY are compared in Table 2.5. Table 2.5: Results of benchmark 2-2 Circle Benchmark D-GEO STABILITY Relative error [%] X co-ordinate [m] Y co-ordinate [m] Radius [m] Safety factor Bishop [-] (1) (2) 0.99 Deltares 13 of 132

28 D-GEO STABILITY, Verification Report Use D-GEO STABILITY input file bm2-2.sti to run this benchmark. 2.3 Verification shear stress restriction 1 This benchmark is taken from problem 24, page 447 of [Lit 7] and tests the restriction of shear stress values for one particular slip circle in the geometry also used for benchmark 2-1 (section 2.1). The Grid and tangent line properties are changed (Table 2.6). The value for the shear stress should be restricted for values of α > 45 - ϕ/2, where α is the angle of the bottom of a slice and f is the angle of internal friction of the soil at that same position. For higher values of α the shear stress becomes extremely high. Therefore, for α > 45 - ϕ/2 the shear stress calculated at α = 45 - ϕ/2 should be used. Table 2.6: Grid and tangent line properties (bm2-3) X co-ordinate [m] 70 Y co-ordinate [m] 10 Tangent line [m] -25 Benchmark results In the above report there are three results available from three different slope stability programs. In the table below only two of the results are presented, because the third result was obtained from a program that does not support the above method for restricting shear stress values. The result obtained without restrictions on the shear stress values is 4.273, which gives an indication of the influence this restriction method for this particular problem. The difference with the other two programs is not so big because the part of the circle where restriction of shear stresses takes place is through a peat and clay layer with a relatively low ϕ value. D-GEO STABILITY results The results of the other programs and the D-GEO STABILITY are compared in Table 2.7. Table 2.7: Results of benchmark 2-3 Circle Benchmark D-GEO STABILITY Relative error [%] X co-ordinate [m] Y co-ordinate [m] Radius [m] Safety factor Bishop [-] (1) (2) 2.50 Use D-GEO STABILITY input file bm2-3.sti to run this benchmark. 2.4 Verification shear stress restriction 2 This benchmark is taken from problem 26, page 447 of [Lit 7] and is also intended to test the restriction of shear stress values for α > 45 - ϕ/2 as for benchmark 2-3 (section 2.3). 14 of 132 Deltares

29 Group 2: Benchmarks from literature (approximate solution) However, in this case a different geometry and circle is calculated so that the influence of this restriction method becomes clearer. In this benchmark the part of the circle where restriction of shear stress values takes place is through a sand layer with a high ϕ value. Figure 2.3: Geometry of benchmark 2-4 Table 2.8: Soil properties (bm2-4) Sand1 Sand2 Clay Uns. total unit weight [kn/m 3 ] Sat. total unit weight [kn/m 3 ] Cohesion [kn/m 2 ] Friction angle phi [ ] Table 2.9: Grid and tangent line properties (bm2-4) X co-ordinate [m] 10 Y co-ordinate [m] 11.5 Tangent line [m] 4 Benchmark results There are two results available, one of which by a program that does not support the restriction of shear stress method. The result with this program is F s = 2.003, which is a lot larger than the value found by the other program that does support the method. As was expected, the influence of the method for this particular calculation is significant. Deltares 15 of 132

30 D-GEO STABILITY, Verification Report D-GEO STABILITY results The D-GEO STABILITY result is exactly the same as the benchmark result. The results are shown in Table Table 2.10: Results of benchmark 2-4 Circle Benchmark D-GEO STABILITY Relative error [%] X co-ordinate [m] Y co-ordinate [m] Radius [m] Safety factor Bishop [-] Use D-GEO STABILITY input file bm2-4.sti to run this benchmark. 2.5 Verification deep circle This benchmark is taken from problem 27, page 447 of [Lit 7]. The geometry is the same as benchmark 2-1 (section 2.1). One circle is calculated which passes below the entire dike (Table 2.11). Table 2.11: Grid and tangent line properties (bm2-5) X co-ordinate [m] 40 Y co-ordinate [m] 30 Tangent line [m] -15 Benchmark results Three results are available, presented in the table below. D-GEO STABILITY results The difference between D-GEO STABILITY and programs (1) & (2) is less then 1%. Only the difference with program (3), which is approximately 5%, is rather large. The results of the other programs and the D-GEO STABILITY are compared in Table Table 2.12: Results of benchmark 2-5 Circle Benchmark D-GEO STABILITY Relative error [%] X co-ordinate [m] Y co-ordinate [m] Radius [m] Safety factor Bishop [-] (1) (2) (3) 5.51 Use D-GEO STABILITY input file bm2-5.sti to run this benchmark. 16 of 132 Deltares

31 Group 2: Benchmarks from literature (approximate solution) 2.6 Verification uniform distributed load This benchmark is taken from problem 28, page 447 of [Lit 7]. The same configuration as benchmark 2-3 is used (section 2.3). This benchmark tests the uniform distributed load on the top of a dike for one particular slip circle (Table 2.13). There is no distribution of the load with increasing depth. Table 2.13: Grid and tangent line properties (bm2-6) X co-ordinate [m] 70 Y co-ordinate [m] 25 Tangent line [m] -12 Benchmark results Two of the programs tested in the above report are able to process uniform distributed loads. The results are presented in the table below. D-GEO STABILITY results The results of the other programs and the D-GEO STABILITY are compared in Table Table 2.14: Results of benchmark 2-6 Circle Benchmark D-GEO STABILITY Relative error [%] X co-ordinate [m] Y co-ordinate [m] Radius [m] Safety factor Bishop [-] (1) (2) 0.36 Use D-GEO STABILITY input file bm2-6.sti to run this benchmark. 2.7 Verification water treatment This benchmark is taken from problem 29, page 447 of [Lit 7]. The same configuration as benchmark 2-1 is used (section 2.1). The free water on the surface of geometry can be treated in 2 ways: as an external load, taken into account in the moment equilibrium, as an extra soil layer, with its weight taken into account in the moment equilibrium. This benchmark checks if this free water is taken into account correctly. Therefore the Grid and tangent line properties are changed (Table 2.15). Table 2.15: Grid and tangent line properties (bm2-7) X co-ordinate [m] 70 Y co-ordinate [m] 15 Tangent line [m] -8 Deltares 17 of 132

32 D-GEO STABILITY, Verification Report Benchmark results Programs (1) & (2) use the first method for the free water: as external load. Program (3) uses the soil-layer method. The results are shown in the table below. D-GEO STABILITY results D-GEO STABILITY uses the external load method for taking the free water into account. For free water on a slope, the water force on that slope is decomposed into a horizontal and vertical component, and these forces are used in the moment equilibrium. The results of the other programs and the D-GEO STABILITY are compared in Table Table 2.16: Results of benchmark 2-7 Circle Benchmark D-GEO STABILITY Relative error [%] X co-ordinate [m] Y co-ordinate [m] Radius [m] Safety factor Bishop [-] (1) (2) (3) 2.13 Use D-GEO STABILITY input file bm2-7.sti to run this benchmark. 18 of 132 Deltares

33 3 Group 3: Benchmarks from spreadsheets These benchmarks are intended to verify specific features of D-GEO STABILITY. The results obtained from D-GEO STABILITY are compared to calculation made in a spreadsheet. The results that are compared will usually be intermediate result as opposed to the final safety factor used in the previous benchmarks. 3.1 Verification Bishop stress-dependent (linear) This benchmark is taken from problem 1 [ 4.3.1] page 26 of [Lit 13]. Using the stressdependent option in D-GEO STABILITY, it is possible to model a non-linear relation between the shear stress τ and the vertical stress σ. This non-linear relation can be modeled by specifying a number of (straight) segment of the σ-τ curve. If this curve only consists of one segment, thereby modeling a linear relation between τ and σ, the result should be the same as a calculation not using the stress dependent option. A simple geometry is used in this benchmark, in which just one circle is calculated (Figure 3.1). Using D-GEO STABILITY, the calculation is made using the regular Bishop method with c = 0 kn/m 2 and ϕ = 45, and using the Bishop stress dependent method with the σ-τ curve from point (σ = 0, τ = 0) to point (σ = 150, τ = 150). Figure 3.1: Geometry of benchmark 3-1 Table 3.1: Soil properties (bm3-1) Sandy clay Uns. total unit weight [kn/m 3 ] 15 Sat. total unit weight [kn/m 3 ] 15 Table 3.2: Grid and tangent line properties (bm3-1) X co-ordinate [m] 0 Y co-ordinate [m] 10 Tangent line [m] -5 Deltares 19 of 132

34 D-GEO STABILITY, Verification Report Benchmark results The problem is calculated in an Excel spreadsheet, using 31 slices, as in D-GEO STABILITY. The results are presented in Table 3.3. D-GEO STABILITY results The D-GEO STABILITY results for both methods is the same, while that result differs less than 1% from the handmade calculation. The results are shown in Table 3.3. Table 3.3: Results of benchmark 3-1 Circle Benchmark D-GEO STABILITY Relative error [%] X co-ordinate [m] Y co-ordinate [m] Radius [m] Safety factor Bishop [-] Use D-GEO STABILITY input file bm3-1.sti to run this benchmark. 3.2 Verification Bishop stress dependent (non-linear) This benchmark is taken from problem 3 [ 4.3.3] page 28 of [Lit 13]. Using the stressdependent option in D-GEO STABILITY, it is possible to model a non-linear relation between the shear stress τ and the vertical stress σ. This non-linear relation can be modeled by specifying a number of (straight) segment of the σ-τ curve. Using D-GEO STABILITY, the calculation is made using the Bishop stress dependent method. The same configuration as benchmark 3-1 (section 3.1) is used. For the soil, a stress dependent curve having two segments is used (Table 3.4). Table 3.4: Sigma-Tau curve (bm3-2) σ [kn/m 2 ] τ [kn/m 2 ] Benchmark results The problem is calculated in an Excel spreadsheet, using 31 slices, as in D-GEO STABILITY. The results are presented in the table below. 20 of 132 Deltares

35 Group 3: Benchmarks from spreadsheets D-GEO STABILITY results The results of D-GEO STABILITY and the benchmark are shown in Table 3.5. Table 3.5: Results of benchmark 3-2 Circle Benchmark D-GEO STABILITY Relative error [%] X co-ordinate [m] Y co-ordinate [m] Radius [m] Safety factor Bishop [-] Use D-GEO STABILITY input file bm3-2.sti to run this benchmark. 3.3 Verification geotextiles To check the results using geotextiles, three calculations are performed: First a calculation is made with no geotextile to check on the non iterated resisting moment(bm3-3a). Secondly a horizontal geotextile is used situated 2 m below the surface level with a tensile strength of S = 100 kn/m (bm3-3b). Thirdly a similar geotextile is used, but then under a slope of approximate 30 degrees (bm3-3c). Figure 3.2 shows a schematic outline for benchmarks 3-3b and 3-3c with a geotextile. Using a material with only cohesion (meaning no iteration is needed in the Bishop calculation), the result can be checked manually. A unique slip circle (with a unique tangent line at level -5 m) is defined with a centre situated at co-ordinates (0, 10 m) and a radius of R = 15 m, as shown in Figure 3.3. Figure 3.2: Geometry and geotextiles of benchmark 3-3 Deltares 21 of 132

36 D-GEO STABILITY, Verification Report Benchmark results When calculating the safety factor for Bishop, an extra resisting moment M G due to the geotextile is introduced and equal to: where: M G = R S cos α (3.1) S is the effective tensile strength in kn; R is the radius of the slip circle in m; α is the angle between the geotextile and the tangent line along the circle where the geotextile intersects the slip circle in. According to Figure 3.3 for benchmark 3-3c, the intersection point between the slip circle (centre (X = 0; Y = 10) and radius R = 15 m) and the inclined geotextile (with equation Y = 0.5 X 3.42) has the following co-ordinates: X = m and Y = m. Therefore, the angle α between the geotextile and the tangent line along the circle where the geotextile intersects the slip circle ) is: 180 for bm3-3b: α = arccos ( 12 [ 15 for bm3-3c: α = arctan ( = π ) ( )] arctan 4.08-(-5.95) 15-(-5.06) Then, the resisting moment from geotextile is: M G = cos ( ) = kn/m for bm 3-3b, M G = cos ( ) = kn/m for bm 3-3c. 180 π = Therefore, the total resisting moment is the sum of the resisting moment from soil (equals to knm/m according to benchmark 3-3a results) and the resisting moment from geotextile, which writes: R = M S + M G = = kn/m for bm3-3b, R = M S + M G = = kn/m for bm3-3c. D-GEO STABILITY results The results of the three calculations are presented in Table 3.7. Table 3.7: Results of benchmark 3-3 Case Resisting Benchmark D-GEO STABILITY Rel. error moment [knm/m] [knm/m] [%] No geotext. (bm3-3a) Total Horiz. geotext. (bm3-3b) Geotextile Total Inclined geotext. (bm3-3c) Geotextile Total Use D-GEO STABILITY input files bm3-3a.sti, bm3-3b.sti and bm3-3c.sti to run this benchmark. 3.4 Earthquake forces 22 of 132 Deltares

37 Group 3: Benchmarks from spreadsheets Figure 3.3: Calculation of angle α (bm3-3b and bm3-3c) The effect of earthquake is checked in this benchmark by calculating the driving and resisting moments for a simple problem, using the Fellenius model. The input values for the earthquake are: a h = 0.3 a v = 0.2 f w = 0.4 β = 50 % Horizontal earthquake factor; Vertical earthquake factor; Free water coefficient; Degree of consolidation. The unit weight of water is 9.81 kn/m 3. Deltares 23 of 132

38 D-GEO STABILITY, Verification Report Benchmark results The additional driving moment due to the horizontal earthquake coefficient, called M quake, is given by equation: M quake = a h n d i h i b i γ i (3.2) i=1 where: n is the number of slices; h i is the height of slice i in m; γ i is the volumetric mass of slice i in kn/m 3 ; b i is the width of slice i in m; d i is the vertical distance between circle center point and slice center in m. This leads to M quake = knm/m. The pore pressure at the bottom of slice i is given by equation: σ W ;q = σ W + σ hydro a v + σ W ;excess (3.3) The different components are equal to: σ W = (1 β) (γ sat γ water ) h i + (ϕ P L z i ) γ water σ W ;excess = a v (1 β) γ sat h i σ hydro = γ w (z phreatic z top,i ) where: β is the degree of consolidation of the soil (90%); ϕ P L is the piezometric level of the soil layer (30 m); z phreatic is the phreatic level (30 m); z top,i is the vertical co-ordinate top of slice i in m. The free water moment that simulates the temporary drawdown of the water is given by equation: M W ;q = M W (1 f w ) (3.4) where M W is the driving water moment. This leads to M W ;q = (1-0.4) = knm/m. The resisting moment from soil is given by equation: n M R = R τ i L i (3.5) where: i=1 24 of 132 Deltares

39 Group 3: Benchmarks from spreadsheets R is the radius of the slip circle in m; τ i is the shear stress along bottom of slice i in kn/m 2 ; L i is the length of arc at bottom of slice i in m. This leads to: M R = knm/m. The safety factor is: F s = M R (M soil + M W ;q + M quake ) (3.6) This leads to F s = The analytical calculations are worked out in an Excel spreadsheet by dividing the circle slip into n = 11 slices. D-GEO STABILITY results In D-GEO STABILITY, the driving moment of soil corresponds in fact to the total driving moment (soil + external load + free water + earthquake). Table 3.14: Results of benchmark 3-4 Benchmark D-GEO STABILITY Error [%] Safety factor [-] Driving moment soil [knm/m] Driving moment free water [knm/m] Quake reduced moment free water [knm/m] Driving moment external loads [knm/m] Driving moment quake horizontal [knm/m] Available resisting moment [knm/m] Use D-GEO STABILITY input file bm3-4.sti to run this benchmark. 3.5 Functioning of tree on slope The effect of trees on slope is checked using the values of Table A simple geometry is used (Figure 3.5). Table 3.15: Tree on slope properties Magnitude of the wind F wind 100 [kn/m] X co-ordinate application point of the wind X wind 15 [m] Y co-ordinate application point of the wind Y wind 12 [m] Vertical distance slope/wind h 7 [m] Horizontal width of the root zone w 5 [m] Angle of distribution δ 30 [ ] Deltares 25 of 132

40 D-GEO STABILITY, Verification Report Table 3.16: Soil properties Unsaturated total unit weight γ unsat 18 [kn/m 3 ] Saturated total unit weight γ sat 18 [kn/m 3 ] Cohesion c 10 [kn/m 2 ] Friction angle ϕ 28 [ ] Table 3.17: Grid and tangent line properties X co-ordinate [m] 20 Y co-ordinate [m] 10 Tangent line [m] 0 Figure 3.5: Geometry of benchmark 3-5 Benchmark results The effect of the wind in the trees is equivalent to the effect of two uniform loads at both sides of the application point, with a magnitude of: q = F wind h = = 112 kn/m 2 (w/2) 2 (5/2) 2 The driving moment due to external loads is: M L = F wind (Y wind Y circle ) = = -200 knm/m The driving moment due to the soil weight is given by equation: M S = n γ i h i x i r (3.7) i=1 where: n is the number of slices (8); 26 of 132 Deltares

41 Group 3: Benchmarks from spreadsheets γ i is the unit weight of soil in slice i in kn/m 3 ; h i is the height of slice i in m; x i is the width of slice i in m; r is the horizontal distance from the center point of the circle to the center of the upper boundary of slice i in m. This leads to M S = knm/m. The resisting moment is given by equation 3.4 and leads to M R = knm/m. The analytical solution is worked out in an Excel spreadsheet by dividing the circle slip into n = 11 slices. According to Fellenius method, the safety factor is: F s = M R M S + M L = = (3.8) D-GEO STABILITY results In D-GEO STABILITY, the driving moment of soil corresponds in fact to the total driving moment (soil + external load). Table 3.19: Results of benchmark 3-5 Benchmark D-GEO STABILITY Rel. error [%] Driving moment soil [knm/m] ( ) 0.00 = Driving moment load [knm/m] Resisting moment [knm/m] Safety factor [-] Use D-GEO STABILITY input file bm3-5.sti to run this benchmark. 3.6 Functioning of the reference level for ratio S = S u /σ y (Su-calculated model) Deltares 27 of 132

42 D-GEO STABILITY, Verification Report The functioning of the reference level for the calculation of the pre-consolidation stress is checked in this benchmark. A dike, with a unit weight of 20 kn/m 3 is constructed on a clay layer with a unit weight of 18 kn/m 3. A uniform load of q = 5 kn/m 2 is present at the top (Figure 3.6) and the phreatic line is at the top of the clay layer. Table 3.20: Properties of the soil materials (bm3-6) Parameter Unit Clay Dike Unsaturated total unit weight [kn/m 3 ] Saturated total unit weight [kn/m 3 ] Ratio [-] POP [kn/m 2 ] 12 0 Table 3.21: Grid and tangent line properties (bm3-6) X co-ordinate [m] 5 Y co-ordinate [m] 13 Tangent line [m] 8 Figure 3.6: Geometry of benchmark 3-6, dike and load at the top of a clay layer The reference level corresponds with the top of the clay layer. In this case, the pre-consolidation stress is defined as the highest value between: 1 the initial pre-consolidation stress before the construction of the dike: σ y = P OP + (γ clay γ w ) h clay (3.9) 2 the effective stress after the construction of the dike: σ y = σ v = γ dike h dike + q + (γ clay γ w ) h clay (3.10) 28 of 132 Deltares

43 Group 3: Benchmarks from spreadsheets where: P OP = 12 kn/m 2 γ w = 10 kn/m 3 is the pre-overburden pressure; is the unit weight of water. The calculation is performed for the Bishop / Cu-calculated model, by dividing the slip circle into 8 slices (see benchmark 3-8 in section 3.8 for the geometric characteristics of the slices). Benchmark results The pre-consolidation stress at the bottom of each slice is: σ y = (γ clay γ w ) h clay + max(p OP ; q + γ dike h dike ) (3.11) This leads to: P c1 = 0.5 (18-10) + max (12; ) = 24 kn/m 2 P c2 = (18-10) + max (12; ) = kn/m 2 P c3 = (18-10) + max (12; ) = kn/m 2 P c4 = (18-10) + max (12; ) = kn/m 2 P c5 = (18-10) + max (12; 10) = kn/m 2 P c6 = (18-10) + max (12; 10) = kn/m 2 P c7 = (18-10) + max (12; 10) = kn/m 2 P c8 = 0.5 (18-10) + max (12;10) = 16 kn/m 2 The safety factor is solution of equation: 8 i=1 F = R L i C u σ y P ci F = M = M load + M dike (3.12) R M load C u σ y 8 L i P ci (3.13) i=1 = (1.414 ( ) ( )+ (3.14) ( ) ( )) (3.15) = (3.16) where: C u /σ y = 0.1 M load = 0 knm/m because of symmetry M dike = 1 20 = 120 knm/m Deltares 29 of 132

44 D-GEO STABILITY, Verification Report D-GEO STABILITY results Table 3.23: Results of benchmark 3-6 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Und. cohesion slice 1 [kn/m 2 ] Und. cohesion slice 2 [kn/m 2 ] Und. cohesion slice 3 [kn/m 2 ] Und. cohesion slice 4 [kn/m 2 ] Und. cohesion slice 5 [kn/m 2 ] Und. cohesion slice 6 [kn/m 2 ] Und. cohesion slice 7 [kn/m 2 ] Und. cohesion slice 8 [kn/m 2 ] Use D-GEO STABILITY input file bm3-6.sti to run this benchmark. 3.7 Zone areas acc. to zone plot method A zone plot calculation is performed for a dike. The properties of the slip circle and the zone areas for safety are given in the tables below. The safety factor of such a geometry is equal to The entrance point of the slip circle is in zone 2B, but the slip circle passes through zone 1B. According to the zone plot method, this slip circle is defined as a slip circle in zone 2B only if the modified slip surface (deformed situation) leaves the rest profile intact. Figure 3.7: Geometry of benchmark of 132 Deltares

45 Group 3: Benchmarks from spreadsheets Table 3.24: Zone area for safety bm3-7a bm3-7b Dike table height [m] Start X co-ordinate rest profile [m] X co-ordinate design level influence [m] Y co-ordinate design level influence [m] Required safety in zone 1B [-] Required safety in zone 2B [-] Table 3.25: Slip circle definition (bm3-7) X co-ordinate [m] 10 Y co-ordinate [m] 11 Tangent line [m] 1 Benchmark results For benchmark 3-7a, the required safety factor in zone 1B is more than the calculated safety factor (2.00 > 1.10) therefore the slip circle is in zone 1B. For benchmark 3-7b, the required safety factor in zone 1B is less than the calculated safety factor (0.90 < 1.10) therefore the slip circle is in zone 2B. D-GEO STABILITY results Table 3.26: Results of benchmark 3-7 Benchmark D-GEO STABILITY Relative error [%] Zone of the slip circle (bm3-7a) 1B 1B - Zone of the slip circle (bm3-7b) 2B 2B - Use D-GEO STABILITY input files bm3-7a.sti and bm3-7b.sti to run this benchmark. 3.8 Deterministic calculation using design values c-phi model A deterministic calculation using design values is performed for a simplified configuration (Figure 3.8). Because of the symmetry of the problem, the soil weight of the active side (left side) is the same as the passive side (right side). Therefore the soil driving moment is nil. Only the driving moment due to load is considered. Calculations are performed be dividing the slip circle into 8 slices of equal width (X = 1 m). Therefore, an analytical solution can be found. Table 3.27: Unit weight of soil and water Unsatatured total unit weight γ unsat 18 [kn/m 3 ] Saturated total unit weight γ sat 18 [kn/m 3 ] Unit weight water γ water 9.81 [kn/m 3 ] Deltares 31 of 132

46 D-GEO STABILITY, Verification Report Figure 3.8: Division of the slip circle in 8 slices of 1 meter Table 3.28: Grid and tangent line properties X co-ordinate [m] 5 Y co-ordinate [m] 13 Tangent line [m] 8 Three types of statistical distribution are considered: none, normal and logarithmic normal. Slices 1 and 3 use probabilistic defaults whereas others slices use inputs for standard deviation σ, partial factor f partial and standard deviation factor k. Slices 1, 2, 7 and 8 use mean values whereas slices 3 to 6 use design values as input values as shown in Table 3.29 below. 32 of 132 Deltares

47 Group 3: Benchmarks from spreadsheets Table 3.29: Probabilistic soil properties (bm3-8) Slice Variable Input value V f partial k σ Distribution 1 c 1 [kn/m 2 ] Mean Normal ϕ 1 [ ] Mean LogNormal 2 c 2 [kn/m 2 ] Mean Normal ϕ 2 [ ] Mean LogNormal 3 c 3 [kn/m 2 ] Design Normal ϕ 3 [ ] Design LogNormal 4 c 4 [kn/m 2 ] Design Normal ϕ 4 [ ] Design LogNormal 5 c 5 [kn/m 2 ] Design LogNormal ϕ 5 [ ] Design Normal 6 c 6 [kn/m 2 ] Design None ϕ 6 [ ] Design None 7 c 7 [kn/m 2 ] Mean LogNormal ϕ 7 [ ] Mean Normal 8 c 8 [kn/m 2 ] Mean None ϕ 8 [ ] Mean None Benchmark results According to Figure 3-8, the X and Y co-ordinates, the length, the height and the angle at bottom of the different are given in Table Because of symmetry, only results for slices 1 to 4 are given. The degree of consolidation of all the layers is set equal to 0%. Then, the effective stress is equal to the load q = 30 kn/m 2 for slices in the active side (slices 1 to 4) and is nil for slices in the passive side (slices 5 to 8). Table 3.30: Geometrical data s of the different slices Slice X i top[m] Y i top[m] Y i bottom[m] h i [m] L i [m] α i [ ] σ i [kn/m 2 ] i According to Bishop method [ ], the safety factor F s is solution of equation: R 8 c i + σ i L i tan ϕ i = M L + M S (3.17) F s + tan α i tan ϕ i i=1 where: M L = q w L (w L /2) = 30 4 (4/2) = 240kNm/m M S = 0 (symmetry of the problem) α i ϕ i /2 45 For slices 3 to 6, the design values are the input values. For parameters c and ϕ of slice 8 with a none distribution, the design values is equal to: x design = x = µ[x] f (3.18) Deltares 33 of 132

48 D-GEO STABILITY, Verification Report This formula leads to the following numerical values: c 8 = kn/m2 and ϕ 8 = For parameter c of slice 1 with a normal distribution and a coefficient of variation V, the design value is equal to: x design = x = µ[x] (1 + k V [x]) f (3.19) This leads to c 1 = 4.7 kn/m2. For parameters c of slice 2 and ϕ of slice 7 with a normal distribution and a standard deviation σ, the design values are equal to: x design = x = µ[x] + k σ[x] f (3.20) This leads to the following numerical values: c 2 = kn/m2 and ϕ 7 = For parameter ϕ of slice 1 with a logarithmic normal distribution and a coefficient of variation V, the design value is equal to: x design = x = 1 f µ 2 [x] 1 + V 2 [x] exp[k ln(1 + V 2 [x])] (3.21) This formula leads to: ϕ 1 = For parameters c of slice 7 and ϕ of slice 2 with a logarithmic distribution and a standard deviation σ, the design values are equal to: x = 1 f µ 2 [x] 1 + (σ[x]/σ[x]µ[x]µ[x]) 2 exp[k ln(1 + (σ[x]/σ[x]µ[x]2 )] (3.22) This formula leads to the following numerical values: ϕ 2 = and c 7 = kn/m2. Equation (7) from which the safety factor is deduced is solved in an Excel spreadsheet starting with F s = 1. Finally: F s = The iterated resisting moment is equal to: R iter = M S + M L = 240kNm/m (3.23) The non-iterated resisting moment is equal to: R non iter = M R = R 8 c i + σ i L i tan ϕ i 1 + tan α i tan ϕ i i=1 (3.24) This formula leads to R non iter = knm/m. 34 of 132 Deltares

49 Group 3: Benchmarks from spreadsheets D-GEO STABILITY results Calculation of the design values of c and ϕ from input mean values is checked in the window Materials in the Soil menu. Table 3.31: Results for the design values (bm3-8) Design values Benchmark D-GEO STABILITY Relative error [%] c 1 [kn/m 2 ] ϕ 1 [ ] c 2 [kn/m 2 ] ϕ 2 [ ] c 7 [kn/m 2 ] ϕ 7 [ ] c 8 [kn/m 2 ] ϕ 8 [ ] The results of the deterministic calculation using design values are given in Table Table 3.32: Results of benchmark 3-8 Benchmark D-GEO STABILITY Rel. error [%] Safety factor [-] Driving moment external load [knm/m] Iterated resisting moment [knm/m] Non-iterated resisting moment [knm/m] Use D-GEO STABILITY input file bm3-8.sti to run this benchmark. 3.9 Deterministic calculation using mean values c-phi model The same input data s as benchmark bm3-8 (section 3.8) are used. A deterministic calculation is performed using the mean values of c and ϕ Benchmark results For slices 1, 2, 7 and 8, the mean values are the input values. For parameters c and ϕ of slice 6 with a none distribution, the mean values are: µ[x] = f x design (3.25) This formula leads to the following numerical values: µ[c 6 ] = 13.5 kn/m 2 and µ[ϕ 6 ] = For parameter c of slice 3 with a normal distribution and a coefficient of variation V, the mean value is: µ[x] = f x design 1 + k V [x] (3.26) This formula leads to: ï [c 3 ] = kn/m 2. Deltares 35 of 132

50 D-GEO STABILITY, Verification Report For parameters c of slice 4 and ϕ of slice 5 with a normal distribution and a standard deviation σ, the mean values are: µ[x] = f x design k σ[x] (3.27) This formula leads to the following numerical values: µ[c 4 ] = 18 kn/m 2 and µ[ϕ 5 ] = For parameter ϕ of slice 3 with a logarithmic distribution and a coefficient of variation V : µ[x] = 1 + V [x] 2 x design f exp[ k ln(1 + V [x]2 )] (3.28) This formula leads to ï [ϕ 3 ] = For parameters c of slice 5 and ϕ of slice 4 with a logarithmic distribution and a standard deviation σ, the following equation has to be solved: µ[x] = 1 + σ2 [x] µ 2 [x] x design f exp[ k ln(1 + σ2 [x] )] µ 2 (3.29) [x] This equation is solved in an Excel spreadsheet using an iterative process. The initial value is set equal to the mean value associated to normal distribution. This leads to µ[c 5 ] = kn/m 2 and µ[ϕ 4 ] = Equation (7) from which the safety factor is deduced is solved in an Excel spreadsheet starting with F s = 1. Finally: F s = The iterated and non-iterated resisting moments are the same as respectively Equation 3.23 and Equation 3.24 in section 3.8 and lead to R iter = 240 knm/m and R non iter = knm/m. D-GEO STABILITY results Calculation of the design values of c and ϕ from input mean values is checked in the window Materials in the Soil menu. Table 3.33: Results for the mean values (bm3-9) Mean values Benchmark D-GEO STABILITY Relative error [%] c 3 [kn/m 2 ] ϕ 3 [ ] c 4 [kn/m 2 ] ϕ 4 [ ] c 5 [kn/m 2 ] ϕ 5 [ ] c 6 [kn/m 2 ] ϕ 6 [ ] The results of the deterministic calculation using mean values are given in Table of 132 Deltares

51 Group 3: Benchmarks from spreadsheets Table 3.34: Results of benchmark 3-9 Benchmark D-GEO STABILITY Rel. error [%] Safety factor [-] Driving moment external load [knm/m] Iterated resisting moment [knm/m] Non-iterated resisting moment [knm/m] Use D-GEO STABILITY input file bm3-9.sti to run this benchmark Deterministic calculation using design values Stress table model The same configuration as benchmark 3-8 (section 3.8) is considered. A deterministic calculation using design values is performed with the stress table model. The probabilistic input values given in Table 3.35 apply to the different slices. Table 3.35: Probabilistic soil properties (bm3-10) Slice Input value V f partial k Use proba. Distribution Benchmark defaults 1 Mean Yes LogNormal bm3-10a Normal bm3-10b None bm3-10c 2 Mean No LogNormal all 3 Mean No Normal 4 Mean No None 5 Design Yes LogNormal bm3-10a Normal bm3-10b None bm3-10c 6 Design No LogNormal all 7 Design No Normal 8 Design No None Table 3.36: Sigma-Tau curve (bm3-10) σ [kn/m 2 ] τ [kn/m 2 ] When the option Use probabilistic defaults is selected, only the first column of the σ-τ curve is used considering that it corresponds with the specified input values (mean or design): for slice 1, it is the mean value whereas for slice 5 it is the design value. For other slices (without the option Use probabilistic defaults), the values of the σ-τ curve are directly use in the calculation, never mind what is the specified input value. Deltares 37 of 132

52 D-GEO STABILITY, Verification Report Benchmark results For slice 1, the stress table corresponds with the mean values, which writes: µ[c 1 ] = 5 kn/m 2 and µ[tan ϕ 1 ] = (105-5)/100 = 1. For benchmark bm3-10a, the design values of parameters c and ϕ of slice 1 with a logarithmic distribution are equal to: c 1 = 1 f µ 2 [c 1 ] 1+V1 2 exp[k ln(1 + V 2 1 )] = kN/m 2 (3.30) ϕ 1 = arctan{ 1 f µ 2 [tan ϕ 1 ] 1+V1 2 exp[k ln(1 + V 2 1 )]} = (3.31) For benchmark bm3-10b, the design values of parameters c and ϕ of slice 1 with a normal distribution are equal to: c 1 = µ[c 1] (1 + k V 1 ) f = kN/m 2 (3.32) ϕ 1 = arctan( µ[tan ϕ 1] (1 + k V 1 ) ) = (3.33) f For benchmark bm3-10c, the design values of parameters c and ϕ of slice 1 with a none distribution are equal to: c 1 = µ[c 1] f = = kn/m2 (3.34) ϕ 1 = arctan( µ[tan ϕ 1] ) = arctan( 1 f 1.15 ) = (3.35) For slices 2 to 8, the stress table corresponds with the design values which writes: c 2 8 = 5kN/m 2 and ϕ 2 8 = 45. Equation (7) in section 3.8 from which the safety factor is deduced is solved in an Excel spreadsheet starting with F s = 1. Finally: 1 F s = for benchmark bm3-10a, 2 F s = for benchmark bm3-10b, 3 F S = for benchmark bm3-10c. The iterated and non-iterated resisting moments are respectively given by Equation 3.23 and Equation 3.24 in section 3.8 and lead to R iter = 240 knm/m and: 1 R non iter = knm/m for benchmark 3-10a, 2 R non iter = knm/m for benchmark 3-10b, 3 R non iter = knm/m for benchmark 3-10c. 38 of 132 Deltares

53 Group 3: Benchmarks from spreadsheets D-GEO STABILITY results Table 3.37: Results of benchmark 3-10a Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Iterated resisting moment [knm/m] Non-iterated res. moment [knm/m] Design cohesion slice 1 [kn/m 2 ] Design cohesion slices [kn/m 2 ] Design friction angle slice 1 [ ] Design friction angle slice 2-8 [ ] Table 3.38: Results of benchmark 3-10b Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Iterated resisting moment [knm/m] Non-iterated res. moment [knm/m] Design cohesion slice 1 [kn/m 2 ] Design cohesion slices [kn/m 2 ] Design friction angle slice 1 [ ] Design friction angle slice 2-8 [ ] Table 3.39: Results of benchmark 3-10c Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Iterated resisting moment [knm/m] Non-iterated res. moment [knm/m] Design cohesion slice 1 [kn/m 2 ] Design cohesion slices [kn/m 2 ] Design friction angle slice 1 [ ] Design friction angle slice 2-8 [ ] Use D-GEO STABILITY input files bm3-10a.sti, bm3-10b.sti and bm3-10c.sti to run this benchmark Deterministic calculation using mean values Stress table model Deltares 39 of 132

54 D-GEO STABILITY, Verification Report The same input data s as benchmark 3-10 (section 3.10) are used. Only the calculation type is different as a deterministic calculation is performed using the mean values of the stress table. Benchmark results For slice 5, the stress table corresponds with the design values, which writes:c 5 = 5 kn/m2 and tan ϕ 5 = (105-5)/100 = 1. For benchmark bm3-11a, the mean values of parameters c and ϕ of slice 5 with a logarithmic distribution are equal to: µ[c 5 ] = 1 + V5 2 c 5 γ exp[ k ln(1 + V5 2 )] = kn/m 2 (3.36) µ[ϕ 5 ] = arctan{ 1 + V 2 5 tan ϕ 5 γ exp[ k ln(1 + V 2 5 )]} = (3.37) For benchmark bm3-11b, the mean values of parameters c and ϕ of slice 5 with a normal distribution are equal to: µ[c 5 ] = γ c k V 5 = kn/m 2 (3.38) ( ) γ tan ϕ µ[ϕ 5 ] = arctan 5 = (3.39) 1 + k V 5 For benchmark bm3-11c, the mean values of parameters c and ϕ of slice 5 with a none distribution are equal to: µ[c 5 ] = γ c 5 = = 5.75 kn/m 2 (3.40) µ[ϕ 5 ] = arctan(γ tan ϕ 5) = arctan(1.15 1) = (3.41) For the other slices, the stress table corresponds with the design values: µ[c] = 5 kn/m 2 and µ[ϕ] = 45. Equation (7) in section 3.8 from which the safety factor is deduced is solved in an Excel spreadsheet starting with F s = 1. Finally: 1 F s = for benchmark bm3-11a, 2 F s = for benchmark bm3-11b, 3 F s = for benchmark bm3-11c. The iterated and non-iterated resisting moments are respectively given by Equation 3.23 and Equation 3.24 in section 3.8 and lead to R iter = 240 knm/m and: 1 R non iter = knm/m for benchmark 3-11a, 2 R non iter = knm/m for benchmark 3-11b, 3 R non iter = knm/m for benchmark 3-11c. 40 of 132 Deltares

55 Group 3: Benchmarks from spreadsheets D-GEO STABILITY results Table 3.40: Results of benchmark 3-11a Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Iterated resisting moment [knm/m] Non-iterated res. moment [knm/m] Mean cohesion slice 5 [kn/m 2 ] Mean cohesion other slices [kn/m 2 ] Mean friction angle slice 5 [ ] Mean friction angle other slices[ ] Table 3.41: Results of benchmark 3-11b Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Iterated resisting moment [knm/m] Non-iterated res. moment [knm/m] Mean cohesion slice 5 [kn/m 2 ] Mean cohesion other slices [kn/m 2 ] Mean friction angle slice 5 [ ] Mean friction angle other slices[ ] Table 3.42: Results of benchmark 3-11c Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Iterated resisting moment [knm/m] Non-iterated res. moment [knm/m] Mean cohesion slice 5 [kn/m 2 ] Mean cohesion other slices [kn/m 2 ] Mean friction angle slice 5 [ ] Mean friction angle other slices[ ] Use D-GEO STABILITY input files bm3-11a.sti, bm3-11b.sti and bm3-11c.sti to run this benchmark Deterministic calculation using design values Cu-calculated model Deltares 41 of 132

56 D-GEO STABILITY, Verification Report The same configuration as benchmark 3-8 (section 3.8) is considered. A deterministic calculation using design values is performed with the Cu-calculated model. The probabilistic input values given in Table 3.43 apply to the different slices. Table 3.43: Probabilistic soil properties (bm3-12) POP [kpa] Slice Ratio C u /σ y Distrib σ k Input Avg Pas Act 1 10 Mean values Log Log Normal None Design values Log Log Normal None V = σ/µ fpartial Benchmark results For slices 5 to 8, the input values of S u /P c are design values. Therefore, they are directly use in the calculation. For slice 1 with a logarithmic normal distribution and a coefficient of variation V, the design values are calculated using Equation 3.21 which leads to: (C u1/c u1σ y ) avg = , (C u1/c u1σ y ) pas = and (C u1/c u1σ y ) act = For slice 2 with a logarithmic normal distribution and a standard deviation σ, the design values are calculated using Equation 3.22 which leads to: (C u2/c u2σ y ) avg = , (C u2/c u2σ y ) pas = and (C u2/c u2σ y ) act = For slice 3 with a normal distribution, the design values are calculated using Equation 3.20 which leads to: (C u3/c u3σ y ) avg = , (C u3/c u3σ y ) pas = and (C u3/c u3σ y ) act = For slice 4 with a none distribution, the design values are calculated using Equation 3.18 which leads to: (Cu4/C u4σ y ) avg = = , (C u4/c u4σ y ) pas = = and (C u4/c u4σ y ) act = = The calculated undrained cohesion is equal to: C ui = C ui [( σ y ) avg + (( C ui σ y ) act ( C ui σ y ) avg ) sin(2α i )] σ y for active side [( C ui σ y ) avg (( C ui σ y ) avg ( C ui σ y ) pas ) sin(2α i )] σ y for passive side. (3.42) 42 of 132 Deltares

57 Group 3: Benchmarks from spreadsheets where σ y = P OP + q and q = { 30 kn/m2 for active side 0 kn/m 2 for passive side.. Slices 1 to 4 and slices 5 to 8 are respectively in the active and passive sides of the slip circle. Therefore: C u1 = [ ( ) sin(2 45)] ( ) = kn/m 2 C u2 = [ ( ) sin( )] ( ) = kn/m 2 C u3 = [ ( ) sin( )] ( ) = kn/m 2 C u4 = [ ( ) sin(2 5.77)] ( ) = kn/m 2 C u5 = [0.42 ( ) sin(2 5.77)] (10 + 0) = 4.16 kn/m 2 C u6 = [0.42 ( ) sin( )] (20 + 0) = kn/m 2 C u7 = [0.42 ( ) sin( )] (30 + 0) = kn/m 2 C u8 = [0.42 ( ) sin(2 45)] (40 + 0) = 16 kn/m 2 The non-iterated resisting moment is equal to R non iter = knm/m as the Bishop method gives the following formula: R non iter = R 8 i=1 L i C ui 1 (3.43) The iterated moment is the same as Equation 3.24 which leads to R iter = 240 knm/m. The safety factor is: F s = R non iter R iter = = D-GEO STABILITY results Calculation of the design values of ratio C u /σ y from input mean values is checked in the window Soil in the Material menu and results are given in Table Table 3.44: Results for the design values (bm3-12) Layer Ratio C u /σ y Benchmark[-] D-GEO STABILITY[-] Relative error[%] 1 Average Passive Active Average Passive Active Average Passive Active Average Passive Active Results of the deterministic calculation using design values are given in Table Deltares 43 of 132

58 D-GEO STABILITY, Verification Report Table 3.45: Results of benchmark 3-12 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Driving moment load [knm/m] Iterated resisting moment [knm/m] Non-iterated res. moment [knm/m] Design S u slice 1 [kn/m 2 ] Design S u slice 2 [kn/m 2 ] Design S u slice 3 [kn/m 2 ] Design S u slice 4 [kn/m 2 ] Design S u slice 5 [kn/m 2 ] Design S u slice 6 [kn/m 2 ] Design S u slice 7 [kn/m 2 ] Design S u slice 8 [kn/m 2 ] Use D-GEO STABILITY input file bm3-12.sti to run this benchmark Deterministic calculation using mean values Cu-calculated model The same input data s as benchmark 3-12 (section 3.12) are used. Only the calculation type is different as a deterministic calculation is performed using the mean value of ratio C u /σ y. Benchmark results For slices 1 to 4, the input values of C u /σ y are design values. Therefore, they are directly used in the calculation. For slice 5 with a logarithmic normal distribution and a coefficient of variation V, the mean values are calculated using Equation 3.28 which leads to: µ[c u5 /σ y ] avg = , µ[c u5 /σ y ] pas = and µ[c u5 /σ y ] act = For slice 6 with a logarithmic normal distribution and a standard deviation σ, the mean values are deduced from Equation This equation is solved in an Excel spreadsheet using an iterative process. The initial value is set equal to the mean value associated to normal distribution. This leads to: µ[c u6 /σ y ] avg = , µ[c u6 /σ y ] pas = and µ[c u6 /σ y ] act = For slice 7 with a normal distribution and a standard deviation σ, the mean values are calculated using Equation 3.27 which leads to: µ[c u7 /σ y ] avg = = 1.442, µ[c u7 /σ y ] pas = = 1.415, µ[c u7 /σ y ] act = = For slice 8 with a none distribution, the mean values are calculated using Equation 3.25 which leads to: 44 of 132 Deltares

59 Group 3: Benchmarks from spreadsheets µ[c u8 /σ y ] avg = = 0.609, µ[c u8 /σ y ] pas = = 0.58, µ[c u8 /σ y ] act = = The calculated undrained cohesion is calculated using Equation 3.42 which leads to: C u1 = [0.42+( ) sin(2 45)] ( ) = 20 kn/m 2 C u2 = [0.42+( ) sin( )] ( ) = kn/m 2 C u3 = [0.42+( ) sin( )] ( ) = kn/m 2 C u4 = [0.42+( ) sin(2 5.77)] ( ) = kn/m 2 C u5 = [ ( ) sin(2 5.77)] (10 + 0) = kn/m 2 C u6 = [ ( ) sin( )] (20 + 0) = kn/m 2 C u7 = [1.442-( ) sin( )] (30 + 0) = kn/m 2 C u8 = [0.609-( ) sin(2 45)] (40 + 0) = 23.2 kn/m 2 The non-iterated resisting moment is the same as Equation 3.43 and leads to: R non iter = knm/m. The iterated moment is the same as Equation 3.24 which leads to: R iter = 240 knm/m. The safety factor is: F s = R non iter = = R iter 240 D-GEO STABILITY results Calculation of the design values of ratio S = S u /σ y from input mean values is checked in the window Soil in the Material menu and results are given in Table The passive and active values can be checked in the Shear Strength tab if advanced is selected. Table 3.46: Results for the mean values (bm3-13) Layer Ratio S Benchmark [-] D-GEO STABILITY [-] Relative error [%] 5 Average Passive Active Average Passive Active Average Passive Active Average Passive Active Results of the deterministic calculation using mean values are given in Table Deltares 45 of 132

60 D-GEO STABILITY, Verification Report Table 3.47: Results of benchmark 3-13 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Driving moment load [knm/m] Iterated resisting moment [knm/m] Non-iterated res. moment [knm/m] Mean S u slice 1 [kn/m 2 ] Mean S u slice 2 [kn/m 2 ] Mean S u slice 3 [kn/m 2 ] Mean S u slice 4 [kn/m 2 ] Mean S u slice 5 [kn/m 2 ] Mean S u slice 6 [kn/m 2 ] Mean S u slice 7 [kn/m 2 ] Mean S u slice 8 [kn/m 2 ] Use D-GEO STABILITY input file bm3-13.sti to run this benchmark Deterministic calculation using design values Cu-measured model The same configuration as benchmark 3-8 (section 3.8) is considered. A deterministic calculation using design values is performed with the Su-measured model. The probabilistic input values given in Table 3.48 apply to the different slices. Table 3.48: Probabilistic soil properties (bm3-14) Slice Input S u top S u bottom Distrib σ V k Avg Pas Act Avg Pas Act 1 Mean values Log Log Normal None Design values Log Log Normal None fpartial Benchmark results The undrained cohesion is not uniform along the layer: S u,i = S top u,i h i Stop u,i Sbottom u,i (3.44) h 0 where h 0 = 5 m is the thickness of the layer. For slices 5 to 8, the input values of Su top and Su bottom are design values. Therefore, they are directly used in Equation 3.44 which leads to: 46 of 132 Deltares

61 Group 3: Benchmarks from spreadsheets (Su5) avg = (Su5) pas = kn/m 2, (Su6) avg = kn/m 2, (Su6) pas = kn/m 2, (Su7) avg = kn/m 2, (Su7) pas = kn/m 2, (Su8) avg = 30.1 kn/m 2, (Su8) pas = 26 kn/m 2. = kn/m 2, For slice 1 with a logarithmic normal distribution and a coefficient of variation V, the design values are calculated using Equation 3.21 which leads to: (S top u1 ) avg = kn/m 2, (S bot u1 ) avg = kn/m 2, (S u1) avg = kn/m 2, (S top u1 ) pas = kn/m 2, (S bot u1 ) pas = kn/m 2, (S top (S bot u1 ) act = kn/m 2, u1 ) act = kn/m 2, (Su1) act = kn/m 2. For slice 2 with a logarithmic normal distribution and a standard deviation s, the design values are calculated using Equation 3.22 which leads to: (S top u2 ) avg = kn/m 2, (S bot u2 ) avg = kn/m 2, (S u2) avg = kn/m 2, (S top u2 ) pas = kn/m 2, (S bot u2 ) pas = kn/m 2, (S top (S bot u2 ) act = kn/m 2, u2 ) act = kn/m 2, (Su2) act = kn/m 2. For slice 3 with a normal distribution, the design values are calculated using Equation 3.20 which leads to: (S top u3 ) avg = kn/m 2, (S bot u3 ) avg = kn/m 2, (S u3) avg = kn/m 2, (S top u3 ) pas = kn/m 2, (S bot u3 ) pas = kn/m 2, (S top (S bot u3 ) act = kn/m 2, u3 ) act = kn/m 2, (Su3) act = kn/m 2. For slice 4 with a none distribution, the design values are calculated using Equation 3.18 which leads to: (S top u4 ) avg = kn/m 2, (S bot u4 ) avg = kn/m 2, (S u4) avg = kn/m 2, (S top u4 ) pas = kn/m 2, Deltares 47 of 132

62 D-GEO STABILITY, Verification Report (S bot u4 ) pas = kn/m 2, (S top (S bot u4 ) act = kn/m 2, u4 ) act = kn/m 2, (Su4) act = kn/m 2. The measured undrained cohesion is equal to: Sui = { (S ui) avg + ((Sui) act (Sui) avg ) sin(2α i ) for active side (Sui) avg ((Sui) avg (Sui) pas ) sin(2α i ) for passive side. (3.45) Slices 1 to 4 and slices 5 to 8 are respectively in the active and passive sides of the slip circle. Therefore: S u1 = ( ) sin (2 45) = kn/m2, S u2 = ( ) sin ( ) = kn/m2, S u3 = ( ) sin ( ) = kn/m2, S u4 = ( ) sin (2 5.77) = kn/m2, S u5 = ( ) sin (2 5.77) = kn/m2, S u6 = ( ) sin ( )= kn/m2, S u7 = ( ) sin ( ) = kn/m2. The non-iterated resisting moment is equal to R non iter = knm/m as the Bishop method gives the following formula: R non iter = R 8 i=1 L i Cui (3.46) The iterated moment is the same as Equation 3.24 which leads to R iter = 240 knm/m. The safety factor is: F s = R non iter = = R iter 240 D-GEO STABILITY results Calculation of the design values of S u from input mean values is checked in the window Soil in the Material menu and results are given in Table of 132 Deltares

63 Group 3: Benchmarks from spreadsheets Table 3.49: Results for the design values(bm3-14) Slice Design S u Benchmark D-GEO STABILITY Relative error [kn/m 2 ] [kn/m 2 ] [%] 1 Average Top Bottom Passive Top Bottom Active Top Bottom Average Top Bottom Passive Top Bottom Active Top Bottom Average Top Bottom Passive Top Bottom Active Top Bottom Average Top Bottom Passive Top Bottom Active Top Bottom Results of the deterministic calculation using design values are given in Table Table 3.50: Results of benchmark 3-14 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Driving moment load [knm/m] Iterated resisting moment [knm/m] Non-iterated res. moment [knm/m] Design S u slice 1 [kn/m 2 ] Design S u slice 2 [kn/m 2 ] Design S u slice 3 [kn/m 2 ] Design S u slice 4 [kn/m 2 ] Design S u slice 5 [kn/m 2 ] Design S u slice 6 [kn/m 2 ] Design S u slice 7 [kn/m 2 ] Design S u slice 8 [kn/m 2 ] Use D-GEO STABILITY input file bm3-14.sti to run this benchmark. Deltares 49 of 132

64 D-GEO STABILITY, Verification Report 3.15 Deterministic calculation using mean values Cu-measured model The same input data s as benchmark 3-14 (section 3.14) are used. Only the calculation type is different as a deterministic calculation is performed using the mean values of C u. Benchmark results For slices 1 to 4, the input values of Cu top and Cu bottom are mean values. Therefore, they are directly used in Equation 3.44 which leads to: µ[c u1 ] avg = (12 19) = 12.7 kn/m 2, µ[c u1 ] act = (11 21) = 12 kn/m 2, µ[c u2 ] avg = kn/m 2, µ[c u2 ] act = kn/m 2, µ[c u3 ] avg = kn/m 2, µ[c u3 ] act = kn/m 2, µ[c u4 ] avg = kn/m 2, µ[c u4 ] act = kn/m 2. For slice 5 with a logarithmic normal distribution and a coefficient of variation V, the mean values are calculated using Equation 3.28 which leads to: µ[c top u5 ] avg = kn/m 2, µ[c bot u5 ] avg = kn/m 2, µ[c u5 ] avg = kn/m 2 µ[c top u5 ] pas = kn/m 2, µ[c bot u5 ] pas = kn/m 2, µ[c u5 ] pas = kn/m 2 µ[c top µ[c bot u5 ] act = kn/m 2, u5 ] act = kn/m 2. For slice 6 with a logarithmic normal distribution and a standard deviation σ, the mean values are deduced from Equation This equation is solved in an Excel spreadsheet using an iterative process. The initial value is set equal to the mean value associated to normal distribution. This leads to: µ[c top u6 ] avg = kn/m 2, µ[c bot u6 ] avg = kn/m 2, µ[c u6 ] avg = kn/m 2, µ[c top u6 ] pas = kn/m 2, µ[c bot u6 ] pas = kn/m 2, µ[c u6 ] pas = kn/m 2, µ[c top µ[c bot u6 ] act = kn/m 2, u5 ] act = kn/m 2. For slice 7 with a normal distribution and a standard deviation σ, the mean values are calculated using Equation 3.27 which leads to: µ[c top u7 ] avg = kn/m 2, µ[c bot u7 ] avg = kn/m 2, µ[c u7 ] avg = kn/m 2, 50 of 132 Deltares

65 Group 3: Benchmarks from spreadsheets µ[c top u7 ] pas = kn/m 2, µ[cu7 bot ] pas = kn/m 2, µ[c u7 ] pas = kn/m 2, µ[c top µ[c bot u7 ] act = 33.6 kn/m 2, u7 ] act = 47.1 kn/m 2. For slice 8 with a none distribution, the mean values are calculated using Equation 3.25 which leads to: µ[c top u8 ] avg = 43.5 kn/m 2, µ[c bot u8 ] avg = kn/m 2, µ[c u8 ] avg = kn/m 2, µ[c top u8 ] pas = kn/m 2, µ[c bot u8 ] pas = kn/m 2, µ[c u8 ] pas = 37.7 kn/m 2, µ[c top µ[c bot u8 ] act = 37.7 kn/m 2, u8 ] act = 52.2 kn/m 2. The measured undrained cohesion is equal to: µ[c ui ] = { µ[c ui] avg + (µ[c ui ] act µ[c ui ] avg ) sin(2α i ) for active side µ[c ui ] avg (µ[c ui ] avg µ[c ui ] pas ) sin(2α i ) for passive side. (3.47) Slices 1 to 4 and slices 5 to 8 are respectively in the active and passive sides of the slip circle. Therefore: µ[c u1 ] = ( ) sin(2 45) = 12 kn/m 2, µ[c u2 ] = ( ) sin( ) = kn/m 2, µ[c u3 ] = ( ) sin( ) = kn/m 2, µ[c u4 ] = ( ) sin(2 5.77) = kn/m 2, µ[c u5 ] = ( ) sin(2 5.77) = kn/m 2, µ[c u6 ] = ( ) sin( ) = kn/m 2, µ[c u7 ] = ( ) sin( ) = kn/m 2, µ[c u8 ] = ( ) sin(2 45) = 37.7 kn/m 2. The non-iterated resisting moment is equal to R non iter = knm/m as the Bishop method gives the following formula: R non iter = R 8 i=1 L i µ(c ui ) (3.48) The iterated moment is the same as Equation 3.24 which leads to R iter = 240 knm/m. The safety factor is: F = R non iter R iter = = Deltares 51 of 132

66 D-GEO STABILITY, Verification Report D-GEO STABILITY results Calculation of the mean values of S u from input design values is checked in the Soil window from the Material menu and results are given in Table Table 3.51: Results for the mean values (bm3-15) Slice Mean C u Benchmark D-GEO STABILITY Relative error [kn/m 2 ] [kn/m 2 ] [%] 5 Average Top Bottom Passive Top Bottom Active Top Bottom Average Top Bottom Passive Top Bottom Active Top Bottom Average Top Bottom Passive Top Bottom Active Top Bottom Average Top Bottom Passive Top Bottom Active Top Bottom Table 3.52: Results of benchmark 3-15 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Driving moment load [knm/m] Iterated resisting moment [knm/m] Non-iterated res. moment [knm/m] Mean S u slice 1 [kn/m 2 ] Mean S u slice 2 [kn/m 2 ] Mean S u slice 3 [kn/m 2 ] Mean S u slice 4 [kn/m 2 ] Mean S u slice 5 [kn/m 2 ] Mean S u slice 6 [kn/m 2 ] Mean S u slice 7 [kn/m 2 ] Mean S u slice 8 [kn/m 2 ] of 132 Deltares

67 Group 3: Benchmarks from spreadsheets Use D-GEO STABILITY input file bm3-15.sti to run this benchmark Probabilistic calculation c-phi model 6 variables: cohesions, degree of cons. and hydraulic pressures with a normal distribution A probabilistic calculation in association with the c-ϕ model is performed using the same geometry as benchmark 3-8 (section 3.8) except for slice 2 which is divided into 2 layers (height of the top layer: h top = 1 m) as shown in Figure 3.9. Figure 3.9: Configuration of benchmark 3-16: slice 2 divided into 2 layers Six variables with a normal distribution are considered, called X 1 to X 6, corresponding to the cohesion, the degree of consolidation and the hydraulic pressure of slices 1 and 2. The properties given in Table 3.53 apply to the different slices. Deltares 53 of 132

68 D-GEO STABILITY, Verification Report Table 3.53: Probabilistic soil properties (bm3-16) Var. Slice µ σ V Distrib. X 1 1 c 1 Cohesion [kn/m 2 ] Normal 1 ϕ 1 Friction angle [ ] 45 Deterministic X 2 1 u 1 Hydraulic press. [kn/m 2 ] Normal X 3 1 U 1 D.O.C. in slice 1 caused by slice 1 [%] 40 σ 50 = 10% - Normal X 4 2 c 2 Cohesion [kn/m 2 ] Normal 2 ϕ 2 Friction angle [ ] 45 Deterministic X 5 2 u 2 Hydraulic press. [kn/m 2 ] Normal X 6 2 U 2 D.O.C. in bottom of slice 2 20 σ 50 = caused by top of slice 2 [%] 10% 3-8 c 3 Cohesion [kn/m 2 ] 0 Deterministic 3-8 ϕ 3 Friction angle [ ] 0 Deterministic 1-8 f Limit value model factor [-] 1 Deterministic - Normal The reliability function is linear and the geometry is simple, an exact analytical solution can be found. Benchmark results According to Bishop method, the safety factor F s is solution of equation: R 8 c i + σ i L i tan ϕ i = M = M load (3.49) F s + tan α i tan ϕ i i=1 This leads to: where: c 1 + σ 1 L 1 F s L c 2 + σ 2 2 M load F s + tan α 2 R = 0 (3.50) σ 1 = q + U 1 γ h 1 = = kn/m 2 σ 2 = q u 2 + γ (h 2 h top + U 2 h top ) σ 2 = ( ) = kn/m 2 γ = γ γ w = = 8.19 kn/m 3 tan α 1 = tan ϕ 1 = tan ϕ 2 = 1 This leads to a quadratic equation: a F 2 s + b F s + c = 0 where: a = M load /R = 48, b = L 1 (c 1 + σ 1) + L 2 (c 2 + σ 2) M load (1 + tan α 2 )/R = , c = L 1 (c 1 + σ 1) tan α 2 + L 2 (c 2 + σ 2) M load tan α 2 /R = The positive root is: F s = b b 2 4ac 2a = of 132 Deltares

69 Group 3: Benchmarks from spreadsheets A probabilistic analysis is performed using the First Order Reliability Method (FORM). The slope instability occurs if: F s f a f 2 + b f + c 0 c 1 + σ 1 L 1 f L c 2 + σ 2 2 M load f + tan α 2 R 0 g(x) = L 1 X 1 + q X 2 + X 3 γ h 1 f + 1 where function g defines the slope stability condition. X 4 + q X 5 + γ (h 2 h t + X 6 h t ) + L 2 f + tan α 2 (3.51) The reliability function Z is expressed as: n Z = (X i Xi g ) with n = 6 (3.52) X i i=1 M load R 0 Function g is linear, therefore: Z(X) = g(x) g(x ) (3.53) And the design point is defined by: g(x ) = 0 (3.54) Therefore: Z(X) = g(x) (3.55) The mean value of the reliability function is: µ[z] = g(µ[x]) (3.56) This formula leads to: µ 1 + q µ 2 + µ 3 γ h 1 µ 4 + q µ 5 + γ (h 2 h t + µ 6 h t ) µ[z] = L 1 +L 2 M load f + 1 f + tan α 2 R,µ[Z] = The standard deviation of the degrees of consolidation U 1 and U 2 is related to the standard deviation of the consolidation coefficient at 50% by the equations: σ(u 1 ) = σ(x 3 ) = 4 σ 50 U 1 (1 U 1 ) = (1 0.4) = σ(u 2 ) = σ(x 6 ) = 4 σ 50 U 2 (1 U 2 ) = (1 0.2) = The standard deviation of the reliability function Z is: σ[z] = n ( g σ[x i ]) X 2 (3.57) i with n = 6. i=1 This formula leads to: L 2 1 σ(z) = [σ2 1 + ( σ 2 ) 2 + (γ h 1 σ 3 ) 2 ] + L2 2 [σ2 3 + ( σ 4 ) 2 + (γ h t σ 5 ) 2 ] (f + tan α 1 ) 2 (f + tan α 2 ) 2 Deltares 55 of 132

70 D-GEO STABILITY, Verification Report (3.58) Therefore: σ(z) = The reliability index is equal to β = by applying the following formula: β = µ(z) σ(z) (3.59) And the probability of failure is equal to P f = by applying the following formula: P f (β > 0) = 1 2 [1 Erf( β 2 )] (3.60) with: Erf(x) = 2 x exp( t 2 ) dt (3.61) π 0 The factor of uncertainty of the different variables is: α i = g σ i X i σ(z) (3.62) and their relative contribution is α i 2. The formula above leads to: α 1 = L tan α 1 α 2 1 = % α 2 = L tan α 1 α2 2 = % α 3 = L 1 h 1 γ σ tan α 1 σ[z] α3 2 = % α 4 = L tan α 2 α4 2 = % L 2 α 5 = 1 + tan α 2 σ 1 σ[z] = tan 45 σ 2 σ[z] = tan = = = (1 + tan 45 ) = σ 4 σ[z] = tan σ 5 σ[z] = tan = , = α5 2 = % α 6 = L 2 h t γ σ tan α 2 σ[z] = (1 + tan ) = α6 2 = % The design value of variable X i is: X i = µ[x i ] α i β σ[x i ] (3.63) 56 of 132 Deltares

71 Group 3: Benchmarks from spreadsheets This formula leads to: X1 = c 1 = = kn/m2, X2 = u 1 = = kn/m2, X3 = U1 = = , X4 = c 2 = = kn/m2, X5 = u 2 = = kn/m2, X6 = U2 = = D-GEO STABILITY results Table 3.54: Results of benchmark 3-16 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design value c 1 [kn/m 2 ] Design value u 1 [kn/m 2 ] Design value U 1 [-] Design value c 2 [kn/m 2 ] Design value u 2 [kn/m 2 ] Design value U 2 [-] Contribution of c 1 [%] Contribution of u 1 [%] Contribution of U 1 [%] Contribution of c 2 [%] Contribution of u 2 [%] Contribution of U 2 [%] Use D-GEO STABILITY input file bm3-16.sti to run this benchmark Probabilistic calculation c-phi model 2 variables: cohesion and model factor with a normal distribution A probabilistic calculation in association with the c-ϕ model is performed using the same geometry as benchmark 3-8 (section 3.8). Two variables with a normal distribution are considered, called X 1 and X 2, corresponding respectively to the cohesion of slice 1 and the limit value of the model factor. The properties given in Table 3.55 apply to the different slices. Table 3.55: Probabilistic soil properties (bm3-17) Variable Slice µ i σ i Distribution X 1 1 c 1 Cohesion [kn/m 2 ] 40 4 Normal - ϕ 1 Friction angle [ ] 0 Deterministic Cohesion [kn/m 2 ] 0 Deterministic - Friction angle [ ] 0 Deterministic X f Limit value model factor [-] Normal Deltares 57 of 132

72 D-GEO STABILITY, Verification Report Benchmark results According to Bishop method, the safety factor F s is solution of Equation 3.49 which leads to: F s = R L 1 c 1 M load = = (3.64) A probabilistic calculation is performed using the First Order Reliability Method (FORM). The slope instability occurs if F s f which writes: R L 1 c 1 M load f 0 g(x) = X 1 M load R L 1 X 2 0 (3.65) where function g defines the slope stability condition. The reliability function Z is expressed as Equation 3.52 with n = 2. The same chain of reasoning as the one performed in section 3.16 (see Equation 3.53 to Equation 3.55) leads to: µ [Z] = µ [X 1 ] M load µ [X 2 ] = = R L The standard deviation of the reliability function is expressed as Equation 3.57 with n = 2, which leads to: ( ) 2 ( ) 2 Mload σ[z] = σ 2 (X 1 ) + σ (X 2 ) = 4 R L 2 + = The reliability index is: β = µ [Z] σ [Z] = = And the probability of failure is equal to: P f (β > 0) = 1 2 [ ( )] β 1 Erf = The factor of uncertainty of the different variables given by Equation 3.62 leads to:α 1 = σ 1 σ [Z] = = α 2 = M load σ 1 R L 1 σ[z] = = Therefore, their relative contribution are: α 2 1 = % and α 2 2 = %. The design values are: X 1 = c 1 = µ 1 α 1 β σ 1 = = kN/m 2 X 2 = f = µ 2 α 2 β σ 2 = = of 132 Deltares

73 Group 3: Benchmarks from spreadsheets D-GEO STABILITY results Table 3.56: Results of benchmark 3-17 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design cohesion [kn/m 2 ] Design model factor [-] Contribution of cohesion [%] Contribution of model factor [%] Use D-GEO STABILITY input file bm3-17.sti to run this benchmark Probabilistic calculation c-phi model 1 variable: friction angle with a normal distribution A probabilistic calculation in association with the c-ϕ model is performed using the same geometry as benchmark 3-8 (section 3.8). The uniform load is q = 2 kn/m 2 instead of 30 kn/m 2. The only variable, with a normal distribution, is the friction angle of slice 1. The properties given in Table 3.57 apply to the different slices. Table 3.57: Probabilistic soil properties (bm3-18) Variable Slice µ i σ i Distribution X 1 1 c 1 Cohesion [kn/m 2 ] 0 Deterministic ϕ 1 Friction angle [ ] 35 5 Normal 2-8 Cohesion [kn/m 2 ] 0 Deterministic Friction angle [ ] 0 Deterministic 1-8 f Limit value model factor [-] 1 Deterministic Benchmark results According to Bishop method, the safety factor F s is solution of Equation 3.49 which leads to: R L 1 σ 1 tan ϕ 1 F s + tan α 1 tan ϕ 1 = M L (3.66) where: σ 1 = q + γ h 1 = =6.095 kn/m 2 M L = 0.5 q w 2 = =16 knm/m tan α 1 = tan 45 = 1. Then: F s = tan ϕ 1(RL 1 σ 1 M L) M L = tan 35 ( ) 16 = A probabilistic calculation is performed using the First Order Reliability Method (FORM). The slope instability occurs if F s f which writes: tan ϕ 1 (R L 1 σ 1 M L ) M L f M L f 0 g(x) = X 1 arctan( σ 1 R L ) 0 1 M L Deltares 59 of 132

74 D-GEO STABILITY, Verification Report (3.67) where function g defines the slope stability condition. The reliability function Z is expressed as Equation 3.52 with n = 1. The same chain of reasoning as the one performed in section 3.16 (see Equation 3.53 to Equation 3.55) leads to: µ[z] f M L = µ[x 1 ] arctan( σ 1 R L ) 1 M L 1 16 = 35 arctan( ) 180 π = The standard deviation of the reliability function is expressed as Equation 3.57 with n = 1, which leads to: σ(z) = g X 1 σ(x 1 ) = σ(x 1 ) = 5. The reliability index β is: β = µ[z] σ[z] = = And the probability of failure is equal to: P f (β > 0) = 1 2 [1 Erf( β 2 )] = The factor of uncertainty of the variable ϕ 1 given by Equation 3.62 leads to: α 1 = g σ 1 X 1 σ(z) = 1, therefore its relative contribution is α1 2 = 100%. The design value of the friction angle is: X 1 = ϕ 1 = µ 1 α 1 β σ 1 = = D-GEO STABILITY results Table 3.58: Results of benchmark 3-18 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design friction angle [ ] Contribution of ϕ [%] Use D-GEO STABILITY input file bm3-18.sti to run this benchmark Probabilistic calculation c phi model 1 variable: degree of cons. with a logarithmic distribution A probabilistic calculation with the c phi model is performed using the same geometry as benchmark 3-8 (section 3.8), except that the uniform load is q = 2 kn/m 2. The only variable, with a logarithmic distribution, is the degree of consolidation of slice 1. The properties given in Table 3.59 apply to the different slices. 60 of 132 Deltares

75 Group 3: Benchmarks from spreadsheets Table 3.59: Probabilistic soil properties (bm3-19) Slice µ i σ i Distribution 1 c 1 Cohesion [kn/m 2 ] 0 Deterministic ϕ 1 Friction angle [ ] 45 Deterministic U 1 Degree of consolidation [-] 0.6 σ 50% = 0.2 LogNormal 2-8 Cohesion [kn/m 2 ] 0 Deterministic Friction angle [ ] 0 Deterministic 1-8 f Limit value model factor [-] 1 Deterministic Benchmark results The only variable X 1 is defined as: X 1 = ln U 1 and has therefore a normal distribution with a standard deviation and a mean value equal to: σ[x 1 ] = ln((σ[u 1 ]/µ[u 1 ]) 2 + 1) = ln((0.192/0.6) 2 + 1) = µ[x 1 ] = 1 2 ln( µ 2 [U 1 ] 1 + (σ[u 1 ]/µ[u 1 ]) 2 ) = 1 2 ln( (0.192/0.6) 2 ) = where: σ(u 1 ) is related to the standard deviation of the consolidation at 50% by the equation: σ[u 1 ] = 4 σ 50 U 1 (1 U 1 ) = (1 0.6) = According to Bishop method, the safety factor F s is solution of Equation 3.49 which leads to: F s = R L 1 σ 1 M L M L = where: σ 1 = q + U 1 γ h 1 = = kn/m 2 M L = 0.5 q w 2 = = 16kNm/m tan ϕ 1 = tan α 1 = tan 45 = 1. = (3.68) A probabilistic calculation is performed using the First Order Reliability Method (FORM). The slope instability occurs if F s f which writes: R L 1 (q + U 1 γ h 1 ) M L M L f 0 g(x) = X 1 ln( M L (1 + tan α 1 ) q R L 1 R L 1 γ h 1 ) 0 where function g defines the slope stability condition. (3.69) The reliability function Z is expressed as Equation 3.52 with n = 2. The same chain of reasoning as the one performed in section 3.16 (see Equation 3.53 to Equation 3.55) leads to: µ[z] = µ[x 1 ] ln( M L (1+tan α 1 ) q R L 1 R L 1 γ h 1 ) = ln( 16 (1+tan 45 ) ) = The standard deviation of the reliability function is expressed as Equation 3.57 with n = 1, which leads to: σ[z] = g X 1 σ[x 1 ] = σ[x 1 ] = The reliability index β is: β = µ[z] σ[z] = = Deltares 61 of 132

76 D-GEO STABILITY, Verification Report The probability of failure is: P f (β < 0) = 1 2 [1 + Erf( β 2 )] = The factor of uncertainty of variable X 1 given by Equation 3.62 leads to: α 1 = g X 1 σ(x 1) σ(z) = 1 therefore its relative contribution is α 2 1 = 100%. The design value of the degree of consolidation of slice 1 is: U 1 = exp X 1 = exp(µ(x 1 ) α 1 β σ(x 1 )) = exp( ( ) ) = D-GEO STABILITY results Table 3.60: Results of benchmark 3-19 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design value of U 1 [-] Contribution of U 1 [%] Use D-GEO STABILITY input file bm3-19.sti to run this benchmark Probabilistic calculation c-phi model 2 variables: cohesion and model factor with a logarithmic distribution This benchmark is identical to benchmark 3-17 (section 3.17) except that the cohesion of slice 1 and the model factor have a logarithmic distribution instead of a normal distribution. The properties given in Table 3.61 apply to the different slices. Table 3.61: Probabilistic soil properties (bm3-20) Variable Slice µ σ Distribution X 1 1 c 1 Cohesion [kn/m 2 ] 40 4 Logarithmic - ϕ 1 Friction angle [ ] 0 Deterministic Cohesion [kn/m 2 ] 0 Deterministic - Friction angle [ ] 0 Deterministic X f Limit value model factor [-] Logarithmic Benchmark results The same procedure as benchmark 3-17 (section 3.17) is used by defining variables X 1 and X 2 as: X 1 = ln c 1 and X 2 = ln f. Then, variables X 1 and X 2 have a normal distribution with a standard deviation and a mean value equal to: σ[x 1 ] = ln((σ[c 1 ]/µ[c 1 ]) 2 + 1) = ln((4/40) 2 + 1) = µ[x 1 ] = 1 ln( µ 2 [c 1 ] 2 1+(σ[c 1 ])/µ[c 1 ) = 1 ln( 40 2 ) ])) (4/40) 2 = σ[x 2 ] = ln((σ[f ]/µ[f ]) 2 + 1) = ln((0.1/0.9) 2 + 1) = µ[x 2 ] = 1 ln( µ 2 [F ] ) = 1 ln( ) 2 1+(σ[F ]/µ[f ]) (0.1/0.9) 2 = of 132 Deltares

77 Group 3: Benchmarks from spreadsheets The safety factor is the same as Equation 3.68: F s = and the expression of function g is identical to Equation The reliability function Z is expressed as Equation 3.52 with n = 2. The same chain of reasoning as the one performed in section 3.16 (see Equation 3.53 to Equation 3.55) leads to: µ[z] = µ[x 1 ] µ[x 2 ] ln( M L 240 ) = ln( R L ) = The standard deviation of the reliability function is expressed as Equation 3.57 with n = 2, which leads to: σ[z] = σ 2 [X 1 ] + σ 2 [X 2 ] = = Finally, the reliability index is equal to: β = µ[z] σ[z] = = And the probability of failure is equal to: P f (β > 0) = 1 2 [1 Erf( β 2 )] = The factor of uncertainty of the different variables given by Equation 3.62 leads to the following relative contributions: α 1 = g X 1 σ(x 1) σ(z) = σ(x 1) σ(z) = = , α 2 1 = % α 2 = g X 2 σ(x 2) σ(z) = σ(x 2) σ(z) = = , α 2 2 = % The design values are: c 1 = exp X 1 = exp(µ[x 1 ] α 1 β σ[x 1 ]) = exp( ) = kN/m 2 f = exp X 2 = exp(µ[x 2 ] α 2 β σ[x 2 ]) = exp( ) = D-GEO STABILITY results Table 3.62: Results of benchmark 3-20 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design cohesion slice 1 [kn/m 2 ] Design model factor [-] Contribution of cohesion [%] Contribution of model factor [%] Use D-GEO STABILITY input file bm3-20.sti to run this benchmark Probabilistic calculation c-phi model 1 variable: friction angle with a logarithmic distribution Deltares 63 of 132

78 D-GEO STABILITY, Verification Report This benchmark is identical to benchmark 3-18 (section 3.18) except that the friction angle of slice 1 has a logarithmic distribution instead of a normal distribution. The properties given in Table 3.63 apply to the different slices. Table 3.63: Probabilistic soil properties (bm3-21) Variable Slice µ i σ i Distribution X 1 1 c 1 Cohesion [kn/m 2 ] 0 Deterministic - ϕ 1 Friction angle [ ] 35 5 Logarithmic Cohesion [kn/m 2 ] 0 Deterministic - Friction angle [ ] 0 Deterministic f Limit value model factor [-] 1 Deterministic Benchmark results The same procedure as benchmark 3-18 (section 3.18) is used by defining variable X 1 as X 1 = ln ϕ 1. Then, X 1 has a normal distribution with a standard deviation and a mean value equal to: σ[x 1 ] = ln(σ 2 [ϕ 1 ]/µ 2 [ϕ 1 ] + 1) = ln(5 2 / ) = µ X 1 = 1 ln( µ 2 [ϕ 1 ] ) = 1 ln( σ 2 [ϕ 1 ]/.µ 2 [ϕ 1 ) = ] 2 1+(5/35) 2 The safety factor is the same as benchmark 3-18 (section 3.18): F s = And function g becomes: M load g(x) = X 1 ln[arctan( σ 1 R L )] 0 1 M load where:σ 1 = q+γ h 1 = = kn/m 2 M load = 0.5 q w 2 = = 16kN m/m The reliability function Z is expressed as Equation 3.52 with n = 1. The same chain of reasoning as the one performed in section 3.16 (see Equation 3.53 to Equation 3.55) leads to: M µ[z] = µ[x 1 ] ln[arctan( load σ 1 R L 1 M load )] 16 = ln[arctan( )] = The standard deviation of the reliability function is: σ[z] = g X 1 σ[x 1 ] = σ[x 1 ] = The reliability index is: β = µ[z] σ[z] = = The probability of failure is: P f (β > 0) = 1 2 [1 Erf( β 2 )] = = 1 and its relative contri- The factor of uncertainty of the friction angle is α 1 = g σ[x 1] X 1 σ[z] bution α1 2 = 100%. The design value of the friction angle is: X 1 = ln ϕ 1 = µ(x 1 ) α 1 β σ(x 1 ).Then: ϕ 1 = exp( ) = of 132 Deltares

79 Group 3: Benchmarks from spreadsheets D-GEO STABILITY results Table 3.64: Results of benchmark 3-21 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design value friction angle [ ] Contribution of ϕ [%] Use D-GEO STABILITY input file bm3-21.sti to run this benchmark Probabilistic calculation c-phi model 2 variables: cohesion with a normal and logarithmic distribution A probabilistic calculation in association with the c-ϕ model is performed using the same geometry as benchmark 3-8 (section 3.8). Two variables are considered, called X 1 and X 2, corresponding respectively to the cohesion of slice 1 and slice 2, with respectively a normal and lognormal distribution. The properties given in Table 3.65 apply to the different slices. Table 3.65: Probabilistic soil properties (bm3-22) Slice µ i σ i Distribution 1 c 1 Cohesion [kn/m 2 ] 20 4 Normal ϕ 1 Friction angle [ ] 0 Deterministic 2 c 2 Cohesion [kn/m 2 ] 20 4 LogNormal ϕ 2 Friction angle [ ] 0 Deterministic 3-8 Cohesion [kn/m 2 ] 0 Deterministic Friction angle [ ] 0 Deterministic 1-8 f Limit value model factor [-] 1 Deterministic In this case, the reliability function is not linear, therefore no exact analytical solution can be found. An Excel spreadsheet is used. Benchmark results Two variables X 1 and X 2 are defined as X 1 = c 1 and X 2 = ln c 2. Then, X 2 have a normal distribution with a standard deviation and a mean value equal to: σ[x 2 ] = ln((σ[c 2 ]/µ[c 2 ]) 2 + 1) = ln((4/20) 2 + 1) = µ[x 2 ] = 1 ln( µ 2 [c 2 ] 2 1+(σ[c 2 ]/µ[c 2 ) = 1 ln( 20 2 ) ]) (4/20) 2 = According to Bishop method, the safety factor F s is solution of Equation 3.49 which leads to: F s = R (L 1 c 1 + L 2 c 2 ) M load = 5 20 ( ) 240 = (3.70) A probabilistic calculation is performed using the First Order Reliability Method (FORM). The Deltares 65 of 132

80 D-GEO STABILITY, Verification Report slope instability occurs if F s f which writes: R (L 1 c 1 + L 2 c 2 ) M load f 0 (3.71) g(x) = L 1 X 1 + L 2 exp(x 2 ) f M load R 0 (3.72) where function g defines the slope stability condition. The reliability function Z is expressed as: Z(X) = 2 i=1 (X i X i ). g X i X = L 1 (X 1 X 1)+L 2 exp(x 2) (X 2 X 2) (3.73) where: X 1 = µ(x 1 ) α 1 β σ(x 1 ) X 2 = µ(x 2 ) α 2 β σ(x 2 ) α 1 = g σ(x 1) X 1 σ(z) = L 1 σ(x 1) σ(z) α 2 = g σ(x 2) X 2 σ(z) = L 2 exp(x2) σ(x 2) σ(z) The design point is defined by: g(x ) = 0 = L 1 X1 + L 2 exp(x2) M load R X2 = ln( M load X1 L 1 R ) R L 2 (3.74) Therefore, the mean value of the reliability function is: µ[z] = L 1 (µ[x 1 ] X 1) + ( M load R X 1 L 1) [µ[x 2 ] ln( M load X 1 L 1 R R L 2 )] The standard deviation of the reliability function is: ( ) 2 σ[z] = 2 g (L1 i=1 σ [X i ] = σ [X 1 ]) 2 + (L 2 exp(x X 2) σ (X 2 ) ) 2 i The reliability index is: β = µ(z) σ(z). As the design values X 1 and X 2 are function of reliability index β, a non-linear equation has to be solved. For this, an iterative process is used, starting with an assumed value of X 1 = µ[x 1 ] = 20 kn/m 2 and ending when the difference between the assumed and the calculated values of X 1 is less than Calculations are worked out in an Excel spreadsheet. Results are: µ[z] = σ[z] = β = p f = of 132 Deltares

81 Group 3: Benchmarks from spreadsheets α 1 = α 2 1 = % α 2 = α 2 2 = % c 1 = kn/m2 = kn/m2 c 2 D-GEO STABILITY results Table 3.66: Results of benchmark 3-22 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design cohesion slice 1 [kn/m 2 ] Design cohesion slice 2 [kn/m 2 ] Contribution of c 1 [%] Contribution of c 2 [%] Use D-GEO STABILITY input file bm3-22.sti to run this benchmark Probabilistic calculation Cu-measured model 2 variables: und. cohesion and model factor with a normal distribution A probabilistic calculation in association with the Cu-measured model is performed. This benchmark is identical to benchmark 3-17 (section 3.17) except that the cohesion is replaced by the undrained cohesion with a uniform distribution within the layer. Therefore, the same analytical solution as benchmark 3-17 is expected. The properties given in Table 3.67 apply to the different slices. Table 3.67: Probabilistic soil properties (bm3-23) Slices C u top [kn/m 2 ] C u bottom [kn/m 2 ] Distribution σ [kn/m 2 ] Normal 4 2 tot None 0 Benchmark results Same analytical solution as benchmark 3-17 (section 3.17). Deltares 67 of 132

82 D-GEO STABILITY, Verification Report D-GEO STABILITY results Table 3.68: Results of benchmark 3-23 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [%] Design undrained coh. [kn/m 2 ] Design model factor [-] Contribution of C u [%] Contribution of f [%] Use D-GEO STABILITY input file bm3-23.sti to run this benchmark Probabilistic calculation Cu-measured model 2 variables: und. cohesion and model factor with a logarithmic distribution A probabilistic calculation in association with the Cu-measured model is performed. This benchmark is identical to benchmark 3-20 (section 3.20) except that the cohesion is replaced by the undrained cohesion with a uniform distribution within the layer. Therefore, the same analytical solution as benchmark 3-20 is expected. The properties given in Table 3.69 apply to the different slices. Table 3.69: Probabilistic soil properties (bm3-24) Slices C u top [kn/m 2 ] C u bottom [kn/m 2 ] Distribution σ [kn/m 2 ] Log Normal 4 2 tot None 0 Benchmark results Same analytical solution as benchmark 3-20 (section 3.20). D-GEO STABILITY results Table 3.70: Results of benchmark 3-24 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design cohesion slice 1 [kn/m 2 ] Design model factor [-] Contribution of cohesion [%] Contribution of model factor [%] Use D-GEO STABILITY input file bm3-24.sti to run this benchmark. 68 of 132 Deltares

83 Group 3: Benchmarks from spreadsheets 3.25 Probabilistic calculation Cu-measured model 2 variables: und. cohesions with a normal and logarithmic distribution A probabilistic calculation in association with the Cu-measured model is performed. This benchmark is identical to benchmark 3-22 (section 3.22) except that the cohesion is replaced by the undrained cohesion with a uniform distribution within the layer. Therefore, the same analytical solution as benchmark 3-22 is expected. The properties given in Table 3.71 apply to the different slices. Table 3.71: Probabilistic soil properties (bm3-25) Slices C u top [kn/m 2 ] C u bottom [kn/m 2 ] Distribution σ [kn/m 2 ] Normal LogNormal 4 3 to None 0 Benchmark results Same analytical solution as benchmark 3-22 (section 3.22). D-GEO STABILITY results Table 3.72: Results of benchmark 3-25 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design cohesion slice 1 [kn/m 2 ] Design cohesion slice 2 [kn/m 2 ] Contribution of C u;1 [%] Contribution of C u;2 [%] Use D-GEO STABILITY input file bm3-25.sti to run this benchmark Probabilistic calculation Cu-measured model 2 variables: und. cohesions (non-uniform distribution) with a normal distribution Deltares 69 of 132

84 D-GEO STABILITY, Verification Report A probabilistic calculation in association with the Cu-measured model is performed using the same geometry as benchmark 3-8 (section 3.8). Two variables with a normal distribution are considered, called X 1 and X 2, corresponding respectively to the undrained cohesion of slices 1 and 2 (with a non-uniform distribution within the layer). The properties given in Table 3.73 apply to the different slices. Table 3.73: Probabilistic soil properties (bm3-26) Slice µ σ V Distrib. 1 C avg u1,top Average und. cohesion top [kn/m 2 ] Normal C avg u1,bot Average und. cohesion bottom [kn/m 2 ] 14 Deterministic C act u1,top Active und. cohesion top [kn/m 2 ] 20 Deterministic C act u1,bot Active und. cohesion bottom [kn/m 2 ] 23 Deterministic 5 C avg u5,top Average und. cohesion top [kn/m 2 ] Normal C avg u5,bot Average und. cohesion bottom [kn/m 2 ] 25 Deterministic C pas u5,top Passive und. cohesion top [kn/m 2 ] 10 Deterministic C pas u5,bot Passive und. cohesion bottom [kn/m 2 ] 15 Deterministic Average undrained cohesion [kn/m 2 ] 0 Deterministic Active undrained cohesion [kn/m 2 ] 0 Deterministic Passive undrained cohesion [kn/m 2 ] 0 Deterministic 1-8 f Limit value model factor [-] 1 Deterministic Benchmark results According to Bishop method, the safety factor F s is solution of equation: R 8 i=1 L i Cui F s = M = M load (3.75) where: R ( L 1 C u1 F s + L 5 C u5 ) = M load F s F s = R (L 1 C u1 +L 5 C u5 ) M load = C u1 = C avg u1 C avg u1 C act u1 + (C act u1 5 ( ) 240 = C avg u1 ) sin(2α 1 ) = ( ) sin(2 45 ) = 20.3 kn/m 2 = C avg u1,top + h 1 (C avg u1,bot h Cavg u1,top) = (14 10) = 10.4 kn/m2 layer 5 = Cu1,top act + h 1 (Cu1,bot act C h u1,top) act = (23 20) = 20.3 kn/m2 layer 5 C u5 = C avg u5 C avg u5 C pas u5 + (C avg u5 C pas u5 ) sin(2α 5 ) = ( ) sin( ) = kn/ = C avg u5,top + h 5 (C avg u5,bot h Cavg u5,top) = (25 16) = kn/m 2 layer 5 = C pas u5,top + h 5 (C pas u5,bot h Cpas u5,top) = (15 10) = kn/m 2 layer 5 h layer : thickness of the layer [m] 70 of 132 Deltares

85 Group 3: Benchmarks from spreadsheets A probabilistic analysis is performed using the First Order Reliability Method (FORM). The slope instability occurs if F s f which writes: R (L 1 C u1 + L 5 C u5 ) f 0 g(x) = L 1 X 1 +L 5 X 5 f M load M load R where function g defines the slope stability condition. 0 (3.76) The reliability function Z is expressed as Equation 3.52 with n = 2. The same chain of reasoning as the one performed in section 3.16 (see Equation 3.53 to Equation 3.55) leads to: µ[z] = g(µ[x]) = µ[x 1 ] L 1 + µ[x 2 ] L 2 f M load R = = The standard deviation of the reliability function is expressed as Equation 3.57 with n = 2. which leads to: where: σ[z] = (L 1 σ[x 1 ]) 2 + (L 2 σ[x 2 ]) 2 = ( ) 2 + ( ) 2 = σ[x 1 ] = σ[c u1 ] = σ[cavg u1,top] µ[cu1,top] avg µ[c u1] = = 6.09 kn/m2 10 σ[cu1,top] avg = V µ[cavg u1,top] + µ[c avg u1,bot ] 2 = = 3 kn/m 2 σ[x 5 ] = σ[c u5 ] = σ[cavg u5,top] µ[cu5,top] avg µ[c u1] = = kn/m2 16 The reliability index is: β = µ[z] σ[z] = = The probability of failure is: P f (β < 0) = 1 2 [ ( )] β 1 + Erf = The factor of uncertainty and the relative contribution of the different variables are: α 1 = L 1 σ[x 1] σ[z] α 2 = L 5 σ[x 2] σ[z] 6.09 = = , α2 1 = % = = , α2 2 = % The design values are: X1 = Cu1 = µ[x 1 ] α 1 β σ[x 1 ] = kn/m 2 C avg u1,top = µ[cu1,top] avg α 1 β σ[cu1,top] avg = kn/m 2 X2 = Cu5 = µ[x 2 ] α 2 β σ[x 2 ] = kn/m 2 C avg u5,top = µ[cu5,top] avg α 2 β σ[cu5,top] avg = kn/m 2 Deltares 71 of 132

86 D-GEO STABILITY, Verification Report D-GEO STABILITY results Table 3.74: Results of benchmark 3-26 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design value C avg u1,top [kn/m2 ] Design value C avg u5,top [kn/m2 ] Design value C u1 [kn/m 2 ] Design value C u5 [kn/m 2 ] Contribution of C u1 [%] Contribution of C u5 [%] Use D-GEO STABILITY input file bm3-26.sti to run this benchmark Probabilistic calculation Su-calculated model 5 variables: ratio S, hydraulic press., D.O.C. and model factor with a normal distribution The same problem as benchmark 3-16 (section 3.16) is considered. The five variables, called X 1 to X 5, have a normal distribution. The properties given in Table 3.75 apply to the different slices. Table 3.75: Probabilistic soil properties (bm3-27) Slice µ σ V Distrib X 1 1 (C u;1 /P c;1 ) avg Ratio (C u /σ y ) (average) [kn/m 2 ] Normal - (C u;1 /P c;1 ) act Ratio C u /σ y (active side) [kn/m 2 ] 0.16 Deterministic - 2 C u;2 /P c;2 Ratio C u /σ y [kn/m 2 ] 0.4 Deterministic X 2 u 2 Hydraulic pressure [kn/m 2 ] Normal - 3 C u;3 /P c;3 Ratio C u /σ y [kn/m 2 ] 0.4 Deterministic X 3 U 3 Degree of consol. of slice 3 [%] 40 σ 50 =20 - Normal - 5 (C u;5 /P c;5 ) avg Ratio S u /P c (average) [kn/m 2 ] Normal - (C u;5 /P c;5 ) pas Ratio C u /σ y (passive side) [kn/m 2 ] 0.11 Deterministic - others Ratio C u /σ y [kn/m 2 ] 0 Deterministic X f Limit value model factor [-] Normal Benchmark results Variables X 1 and X 5 correspond to the ratio C u /σ y of slices 1 and 5 which are respectively in the active and passive side. Their mean value and standard deviation are: µ[x 1 ] = (C u1 /P c1 ) avg + [(C u1 /P c1 ) act (C u1 /P c1 ) avg ] sin(2α 1 ) = 0.1 ( ) sin(2 45 ) = 0.16 µ[x 5 ] = (C u5 /P c5 ) avg [(C u5 /P c5 ) avg (C u5 /P c5 ) pas ] sin(2α 5 ) = 0.2 ( ) sin( ) = µ(x σ[x 1 ] = σ(c u1 /P c1 ) avg 1 ) µ(c u1 /P c1 ) avg = = µ(x σ[x 5 ] = σ(c u5 /P c5 ) avg 5 ) µ(c u5 /P c5 ) avg = = The Pre-Overburden Pressure is set equal to zero and the reference level for ratio S u /P c 72 of 132 Deltares

87 Group 3: Benchmarks from spreadsheets corresponds with slice 1. Therefore, the pre-consolidation pressure is equal to the effective stress: P c;1 = q + γ h 1 = = kn/m 2 P c;2 = q + γ h 2 = = kn/m 2 P c;3 = q + γ h 3 U 3 = = kn/m 2 P c;5 = γ h 5 = = kn/m 2 According to Bishop method, the safety factor F s is solution of equation: R 8 i=1 L i C u,i P c,i P c,i F s = M = M load (3.77) F s = R M load (L 1 C u1 P c1 P c1 + L 2 C u2 P c2 P c1 + L 3 C u3 P c3 P c1 + L 5 C u5 P c5 P c1 ) = ( ) = A probabilistic analysis is performed using the First Order Reliability Method (FORM). The slope instability occurs if F s f which writes: ( ) R C u1 C u2 C u3 L 1 P c1 + L 2 (P c2 u 2 ) + L 3 (P OP 3 + U 3 γ C u5 h 3 ) + L 5 P c5 P c1 P c2 P c3 P c5 M load (3.78) f 0 C u2 C u3 g (X) = X 1 L 1 P c1 + L 2 (P c2 X 2 ) + L 3 (P OP 3 + X 3 γ h 3 ) + X 5 L 5 P c5 MX 4 P c2 P c3 R where function g defines the slope instability condition. The reliability function Z is expressed as Equation 3.52 with n = 5. The same chain of reasoning as the one performed in section 3.16 (see Equation 3.53 to Equation 3.55) leads to: µ[z] C u2 C u3 = g (µ [X]) = L 1 µ 1 P c1 + L 2 (P c2 µ 2 ) + L 3 (P OP 3 + µ 3 γ h 3 ) + L 5 µ 5 P c5 M P c2 P c3 R = ( ) = (3.79) The standard deviation of the degree of consolidation U 3 is related to the standard deviation of the consolidation coefficient at 50% by the equation: σ(u 3 ) = σ(x 3 ) = 4 σ 50 U 3 (1 U 3 ) = (1 0.4) = (3.80) The standard deviation of the reliability function is expressed as Equation 3.57 with n = 1, which leads to: σ[z] = (L 1 P c1 σ 1 ) 2 + (L 2 σ 2 ) 2 + (L 3 γ h 3 σ 3 ) 2 + (M load σ 4 /R) 2 + (L 5 P c5 σ 5 ) 2 = The reliability index is: β = µ[z] σ[z] = = The probability of failure is: P f (β > 0) = 1 2 [1 Erf( β 2 )] = Deltares 73 of 132

88 D-GEO STABILITY, Verification Report The factor of uncertainty and the relative contribution are: α 1 = L 1P c1 σ = = and α 2 1 σ[z] = % α 2 = L 2σ 2 = = and σ[z] α 2 2 = % α 3 = L 3γ h 3 σ = = and σ[z] α 2 3 = % M load α 4 = R σ 4 σ[z] = 240 = and α 2 4 = % α 5 = L 5P c5 σ = = and σ[z] α 2 5 = % The design value of variable X i is Xi = µ i α i β σ i, which leads to: (C u1 /P c1 ) avg = = X2 = u 2 = 0 - ( ) = kn/m2 X3 = U3 = = X4 = f = ( ) = (C u5 /P c5 ) avg = = D-GEO STABILITY results Table 3.76: Results of benchmark 3-27 Benchmark D-GEO STABILITY Error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design value average ratio C u /σ y slice 1 [-] Design value hydraulic press. slice 2 [kn/m 2 ] Design value D.O.C. slice 3 [-] Design value model factor [-] Design value average ratio C u /σ y slice 5 [-] Contribution of average ratio C u /σ y slice 1 [%] Contribution of hydraulic pressure slice 2 [%] Contribution of D.O.C. slice 3 [%] Contribution of model factor [%] Contribution of average ratio C u /σ y slice 5 [%] Use D-GEO STABILITY input file bm3-27.sti to run this benchmark Probabilistic calculation Su-calculated model 3 variables: ratio S, degree of cons. and model factor with a logarithmic distribution 74 of 132 Deltares

89 Group 3: Benchmarks from spreadsheets The same problem as benchmark 3-16 (section 3.16) is considered with a lognormal distribution for three variables: the ratio C u /σ y and the degree of consolidation of slice 1 and the model factor. The properties given in Table 3.77 apply to the different slices. Table 3.77: Probabilistic soil properties (bm3-28) Slice ï σ V Distrib. 1 C u;1 /P c;1 Ratio und. coh. / precons. stress [-] Log P OP 1 Pre-overburden pressure [kn/m 2 ] 0 Deterministic U 1 Degree of consolidation [%] 0.4 σ 50 =0.05 Log U load Degree of cons. caused by load [%] 0 Deterministic 2-8 Ratio und. coh. / precons. stress [-] 0 Deterministic 1-8 f Limit value model factor [-] Log Benchmark results Variables X 1 to X 3 are defined as: X 1 = ln(c u1 /P c1 ), X 2 = ln(u 1 ) and X 3 = ln(f). Then, they have a normal distribution with a standard deviation and a mean value equal to: σ[x 1 ] = ln(σ 2 1/µ ) = ln(1.5 2 / ) = µ[x 1 ] = 1 2 ln( µ (σ 1 /µ 1 ) 2 ) = 1 2 ln( (1.5/15) 2 ) = σ[x 2 ] = ln(σ 2 2/µ ) = ln( / ) = µ[x 2 ] = 1 2 ln( µ (σ 2 /µ 2 ) 2 ) = 1 2 ln( (0.048/0.4) 2 ) = σ[x 3 ] = ln(σ 2 3/µ ) = ln( / ) = µ[x 3 ] = 1 2 ln( µ (σ 3 /µ 3 ) 2 ) = 1 2 ln( (0.05/0.9) 2 ) = According to Bishop method, the safety factor F s is solution of equation: R 8 i=1 L i Cu,i P c,i Pc,i F s = M = M load (3.81) R L 1 Cu1 P c1 Pc1 F s = F s = M load R M load L 1 Cu1 P c1 P c1 = where: P c;1 = P OP 1 + U 1 γ h 1 = = kn/m 2. A probabilistic analysis is performed using the First Order Reliability Method (FORM). The slope instability occurs if F s f which writes: R M load L 1 Cu1 P c1 U 1 γ h 1 f 0 (3.82) g(x) = X 1 + X 2 X 3 + ln( L 1 γ h 1 R ) M load Deltares 75 of 132

90 D-GEO STABILITY, Verification Report The reliability function Z is expressed as Equation 3.52 with n = 3. The same chain of reasoning as the one performed in section 3.16 (see Equation 3.53 to Equation 3.55) leads to: µ[z] = µ[x 1 ] + µ[x 2 ] µ[x 3 ] + ln( L 1 γ h 1 R M load ) = ln( ) = The standard deviation of the reliability function is expressed as Equation 3.57 with n = 3, which leads to: σ[z] = σ 2 [X 1 ] + σ 2 [X 2 ] + σ 2 [X 3 ] = The reliability index is: β = µ[z] σ[z] = = And the probability of failure is equal to: P f (β < 0) = 1 2 [1 + Erf ( β 2 )] = The factor of uncertainty and the relative contributions of the different variables are: α 1 = g σ[x 1] = σ[x 1] = X 1 σ[z] σ[z] = , α1 2 = % α 2 = g σ[x 2] = σ[x 2] = X 2 σ[z] σ[z] = , α2 2 = % α 3 = g σ[x 3] = σ[x 3] = X 3 σ[z] σ[z] = , α3 2 = % The design values are: C u1 P c1 = exp(µ[x 1 ] α 1 βσ[x 1 ]) = exp( ) = U 1 = exp(µ[x 2 ] α 2 βσ[x 2 ]) = exp( ) = F = exp(µ[x 3 ] α 3 β σ[x 3 ]) = exp( ) = D-GEO STABILITY results Table 3.78: Results of benchmark 3-28 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design value C u;1 /P c;1 [-] Design value U 1 [-] Design value f [-] Contribution of C u /σ y [%] Contribution of D.O.C. [%] Contribution of model factor [%] Use D-GEO STABILITY input file bm3-28.sti to run this benchmark Probabilistic calculation Su-calculated model 2 variables: ratio S with a normal and a logarithmic distribution 76 of 132 Deltares

91 Group 3: Benchmarks from spreadsheets The same problem as benchmark 3-22 (section 3.22) is considered with the soil properties given in Table Table 3.79: Probabilistic soil properties (bm3-29) Slice µ σ V Distrib. 1 C u1 /P c1 Ratio und. coh. / precons. stress [-] Normal POP 1 Pre-overburden pressure [kn/m 2 ] 1 Deterministic 2 C u2 /P c2 Ratio und. coh. / precons. stress [-] Log POP 2 Pre-overburden pressure [kn/m 2 ] 1 Deterministic 3-8 Ratio und. coh. / precons. stress [-] 0 Deterministic 1-8 f Limit value model factor [-] 1 Deterministic The degree of consolidation of slices 1 and 2 and the degree of consolidation due to uniform load are both set equal to zero. Then, the effective stress at the bottom of slices 1 and 2 is nil and the pre-consolidation stress is equal to the POP value, which means 1 kn/m 2. Therefore, this problem is identical to benchmark 3-22 (section 3.22) as the ratio C u /σ y is equal to cohesion c for slices 1 and 2 in benchmark Benchmark results Same analytical solution as benchmark 3-22 (section 3.22). D-GEO STABILITY results Table 3.80: Results of benchmark 3-29 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design value C u;1 /P c;1 [-] Design value C u;2 /P c;2 [-] Contribution of C u;1 /P c;1 [%] Contribution of C u;2 /P c;2 [%] Use D-GEO STABILITY input file bm3-29.sti to run this benchmark Probabilistic calculation Stress table model 2 variables: tangential stress and model factor with a normal distribution Deltares 77 of 132

92 D-GEO STABILITY, Verification Report A probabilistic calculation in association with the stress table model is performed using the same geometry as benchmark 3-8 (section 3.8) with a uniform load of q = 5 kn/m 2.Two variables are considered, called X 1 and X 2, corresponding respectively to the normalized tangential stress and to the model factor, with a normal distribution. The properties given in Table 3.81 apply to the different slices. Table 3.81: Probabilistic soil properties (bm3-30) Slice µ σ Distribution 1 c 1 Cohesion [kn/m 2 ] Normal ϕ 1 Friction angle [ ] 45 - Normal 2-8 Cohesion [kn/m 2 ] 0 Deterministic Friction angle [ ] 0 Deterministic 1-8 f Limit value model factor [-] Normal The σ-τ curve of slice 1 has the tangential stress mean values given in Table Table 3.82: Sigma-Tau curve of slice 1 (bm3-30b) σ [kn/m 2 ] τ charac [kn/m 2 ] τ mean [kn/m 2 ] The standard deviation is inputted using two methods: using a coefficient of variation equals to V = σ/µ = 0.1 (benchmark 3-30a) using the characteristic curve (benchmark 3-30b) Benchmark results For benchmark 3-30b, the normalized standard deviation σ 100 for a mean tangential stress of τ = 100 kn/m 2 varies as a function of the effective stress. For slice 1, the effective stress is: σ 1 = q + (γ γ w ) h 1 = 5 + ( ) 0.5 = kn/m 2. The mean and characteristic values of the tangential stress of slice 1 are: τ mean (σ 1) = τ mean (0) + σ 1 τmean(80) τmean(0) = kn/m τ charac (σ 1) = τ charac (0) + σ 1 τcharac(80) τ charac (0) = kn/m Therefore, the normalized standard deviation is: σ 100 = 100 τ mean(σ 1 ) τmean(σ 1 ) τ charac(σ 1 ) = 10 kn/m Therefore, both benchmarks (3-30a and 3-30b) are equivalent as: V = σ/µ = 10/100 = 0.1. According to Bishop method, the safety factor F s is solution of Equation 3.49 which leads to: F s = R L 1 (c 1 + σ 1 tan ϕ 1) M load tan ϕ 1 = (3.83) A probabilistic calculation is performed using the First Order Reliability Method (FORM). The 78 of 132 Deltares

93 Group 3: Benchmarks from spreadsheets slope instability occurs if F s f which writes: R L 1 (c 1 + σ 1 tan ϕ 1) M load tan ϕ 1 f 0 g(x) = τ 1 X1 100 X 2 M load R L 1 0 (3.84) where function g defines the slope stability condition and τ 1 is the tangential stress of slice 1: τ 1 = c 1 + tan ϕ 1 (σ 1 M load R L 1 ) = ( ) = kn/m M load = q w w/2 = 5 4 4/2 = 40kNm/m The reliability function Z is expressed as Equation 3.52 with n = 2. The same chain of reasoning as the one performed in section 3.16 (see Equation 3.53 to Equation 3.55) leads to: µ[z] = τ 1 µ[x 1] µ[x 2] M load 100 R L 1 = = The standard deviation of the reliability function is expressed as Equation 3.57 with n = 1, which leads to: σ[z] = ( τ σ[x 1]) 2 + ( M load R L 1 σ[x 2 ]) 2 = The reliability index is: β = µ[z] σ[z] = = The probability of failure is: P f (β > 0) = 1 2 [1 Erf ( β 2 )] = The factor of uncertainty and the relative contributions of the different variables are: α 1 = τ σ[x 1] = = and σ[z] α 2 1 = % α 2 = M load R σ[x 2] L 1 = = and σ[z] α 2 2 = % The design values are: X 1 = τ 100 = µ[x 1 ] α 1 β σ[x 1 ] = = kn/m 2 X 2 = f = µ[x 2 ] α 2 β σ[x 2 ] = ( ) = c 1 = µ[c 1 ] X 1/µ[X 1 ] = /100 = kn/m 2 ϕ 1 = arctan[µ[tan ϕ 1 ] X 1/µ[X 1 ]] = arctan [ /100] = D-GEO STABILITY results Table 3.83: Results of benchmark 3-30a (using a coefficient of variation) Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design normalized tau [kn/m 2 ] Design model factor [-] Design cohesion slice 1 [kn/m 2 ] Design friction angle slice 1 [ ] Contribution of stress table [%] Contribution of model factor [%] Deltares 79 of 132

94 D-GEO STABILITY, Verification Report Table 3.84: Results of benchmark 3-30b (using a characteristic curve) Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design normalized tau [kn/m 2 ] Design model factor [-] Design cohesion slice 1 [kn/m 2 ] Design friction angle slice 1 [ ] Contribution of stress table [%] Contribution of model factor [%] Use D-GEO STABILITY input files bm3-30a.sti and bm3-30b.sti to run this benchmark Probabilistic calculation Stress table model 4 variables: degree of cons. and hydraulic pressures with a normal distribution A probabilistic calculation in association with the stress table model is performed using the same geometry as benchmark 3-16 (section 3.16) (slice 2 divided into 2 layers) and the same properties, except that the cohesion of slices 1 and 2 has a deterministic value. Indeed, for the stress table model, it is not possible to separate the standard deviations of the cohesion and the friction angle (Figure 3.10). Therefore, the analytical resolution is identical to benchmark 3-16 (section 3.16) with a standard deviation of c 1 and c 2 equal to zero. Figure 3.10: Sigma-Tau curves for slice 1 and bottom of slice 2 Benchmark results The mean value of the reliability function is the same as benchmark 3-16 (section 3.16), µ[z] = But the standard deviation of the reliability function is modified: 6 i=1 ( g σ[z] = X i σ[x i ]) 2 L 2 = 1 [σ1 2+( σ 2) 2 +(γ h 1 σ 3 ) 2 ] [ ( ) 2 ] = = (f+tan α 1 + L2 ) 2 2 [σ2 3 +( σ 4) 2 +(γ h t σ 5 ) 2 ] (f+tan α 2 ) [ ( ) 2 ] (1+tan 45 ) 2 (1+tan ) 2 80 of 132 Deltares

95 Group 3: Benchmarks from spreadsheets The reliability index is: β = µ[z] σ[z] = = And the probability of failure is equal to: P f (β > 0) = 1 2 [1 Erf( β 2 )] = The factor of uncertainty of X 1 given by Equation 3.62 leads to: α 1 = α 4 = 0 α 2 = L 1 σ 2 = tan α 1 σ[z] 1+tan = , α2 2 = % α 3 = L 1 h 1 γ σ tan α 1 = σ[z] (1+tan 45 ) = , α3 2 = % α 5 = L 2 σ 5 = tan α 2 σ[z] 1+tan = , α5 2 = % α 6 = L 2 h t γ σ tan α 2 = σ[z] (1+tan ) = , α6 2 = % The design value of variable X i is Xi = µ i α i β σ i which leads to: X 2 = u 1 = = kn/m2 X 3 = U 1 = = X 5 = u 2 = = kn/m2 X 6 = U 2 = = D-GEO STABILITY results Table 3.85: Results of benchmark 3-31 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design value u 1 [kn/m 2 ] Design value U 1 [-] Design value u 2 [kn/m 2 ] Design value U 2 [-] Contribution of u 1 [%] Contribution of U 1 [%] Contribution of u 2 [%] Contribution of U 2 [%] Use D-GEO STABILITY input file bm3-31.sti to run this benchmark Probabilistic calculation Stress table model 2 variables: tangential stress and model factor with a log. distribution Deltares 81 of 132

96 D-GEO STABILITY, Verification Report The same problem as benchmark 3-30 (section 3.30) is considered except that the tangential stress and the model factor have a logarithmic distribution instead of a normal distribution. The σ-τ curve of slice 1 has the tangential stress mean values given in Table Table 3.86: Sigma-Tau curve of slice 1 (bm3-32a) σ [kn/m 2 ] τ mean [kn/m 2 ] The standard deviation is inputted: 1 using the coefficient of variation (benchmark 3-32a); 2 using the characteristic curve (benchmark 3-32b). Benchmark results For benchmark 3-32b, the characteristic values of the tangential stresses must be: τ charac = exp[µ[ln τ] 1.64 σ[ln τ]] (3.85) where: µ[ln τ] = 1 2 ln( µ2 [τ] 1+V 2 ) and σ[ln τ] = ln(1 + V 2 ) = ln( ) = Then: τ charac = µ[τ] exp[ 0.5 σ 2 [ln τ] 1.64 σ[ln τ]] (3.86) This formula leads to the following numerical values: τ charac (0) = 2.5 exp[ ] = kn/m 2 τ charac (80) = 82.5 exp[ ] = kn/m 2 Only two figures after the decimal point are inputted in D-GEO STABILITY as shown in Table Table 3.87: Sigma-Tau curve of slice 1 (bm3-32b) σ [kn/m 2 ] τ mean [kn/m 2 ] τ charac [kn/m 2 ] The same procedure as benchmark 3-30 (section 3.30) is used by defining variables X 1 and X 2 as: X 1 = ln τ 100 and X 2 = ln f. Then, X 1 and X 2 have a normal distribution with a standard deviation and a mean value equal to: σ[x 1 ] = ln(v 2 + 1) = ln( ) = µ[x 1 ] = 1 ln( µ2 (τ 100 ) 2 ln( ) = ) = 1 1+V 2 2 The safety factor is the same as benchmark 3-30 (section 3.30): F s = and function g becomes: g(x) = ln( τ ) + X 1 X 2 ln( M load R L 1 ) 0 (3.87) 82 of 132 Deltares

97 Group 3: Benchmarks from spreadsheets where: τ 1 = c 1 + tan ϕ 1 (σ 1 M load R L 1 ) = kn/m 2 σ 1 = q + γ h 1 = = kn/m 2 M load = q w w/2 = 5 4 4/2 = 40 knm/m The reliability function Z is expressed as Equation 3.52 with n = 2. The same chain of reasoning as the one performed in section 3.16 (see Equation 3.53 to Equation 3.55) leads to: µ[z] = ln( τ 1 ) + µ[x 100 1] µ[x 1 ] ln( M load R L 1 ) = The standard deviation of the reliability function is: σ[z] = σ 2 [X 1 ] + σ 2 [X 2 ] = = The reliability index is: β = µ[z] σ[z] = = The probability of failure is: P f (β > 0) = 1 2 [1 Erf( β 2 )] = The factor of uncertainty and the relative contributions of the different variables are: α 1 = σ[x 1] σ[z] = = and α2 1 = % α 2 = σ[x 2] σ[z] = = and α2 2 = % The design values are: τ 100 = exp X 1 = exp[µ[x 1 ] α 1 β σ[x 1 ]] = exp[ ] = kn/m 2 f = exp X 2 = exp[µ[x 2 ] α 1 β σ[x 2 ]] = exp[ ] = c 1 = µ[c 1 ] τ 100/µ[τ 100 ] = /100 = kn/m 2 ϕ 1 = arctan[µ[tan ϕ 1 ] τ 100/µ[τ 100 ]] = arctan[ /100] = D-GEO STABILITY results Table 3.88: Results of benchmark 3-32a (using a coefficient of variation) Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design normalized tau [kn/m 2 ] Design model factor [-] Design cohesion slice 1 [kn/m 2 ] Design friction angle slice 1 [ ] Contribution of stress table [%] Contribution of model factor [%] Deltares 83 of 132

98 D-GEO STABILITY, Verification Report Table 3.89: Results of benchmark 3-32b (using a characteristic curve) Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design normalized tau [kn/m 2 ] Design model factor [-] Design cohesion slice 1 [kn/m 2 ] Design friction angle slice 1 [ ] Contribution of stress table [%] Contribution of model factor [%] Use D-GEO STABILITY input files bm3-32a.sti and bm3-32b.sti to run this benchmark Probabilistic calculation Stress table model 1 variable: degree of cons. with a logarithmic distribution The same problem as benchmark 3-19 (section 3.19) is considered except that the friction angle of 45 degrees of slice 1 is replaced by a σ-τ curve. The only variable is the degree of consolidation of slice 1 with a logarithmic distribution. Table 3.90: Sigma-Tau curve (bm 3-19) σ [kn/m 2 ] τ mean [kn/m 2 ] Benchmark results Same analytical solution as benchmark 3-19 (section 3.19). D-GEO STABILITY results Table 3.91: Results of benchmark 3-33 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design value D.O.C. slice 1 [-] Contribution of D.O.C. [%] Use D-GEO STABILITY input file bm3-33.sti to run this benchmark Probabilistic calculation Stress table model 2 variables: tangential stresses with a normal and a logarithmic distribution 84 of 132 Deltares

99 Group 3: Benchmarks from spreadsheets The same problem as benchmark 3-22 (section 3.22) is considered except that the cohesion of 20 kn/m 2 and the friction angle of 0 degrees of slices 1 and 2 are replaced by the σ-τ curve given in Table Table 3.92: Sigma-Tau curve (bm3-34) σ [kn/m 2 ] τ charac [kn/m 2 ] τ mean [kn/m 2 ] For slice 1, with a normal distribution, the normalized standard deviation of the tangential stress is determined from the characteristic curve: σ 100 = 100 τmean τ charac τ mean 1.64 = = 20 kn/m 2 For slice 2, with a logarithmic distribution, the normalized standard deviation of the tangential stress is determined from the coefficient of variation of V = σ/µ = 0.2. Therefore, both slices have the same standard deviation for the cohesion as for benchmark 3-22: σ(c) = σ 100 µ[c 1] 100 = V µ[c 2] = 4 kn/m 2 Benchmark results Same analytical solution as benchmark 3-22 (section 3.22). D-GEO STABILITY results Table 3.93: Results of benchmark 3-34 Benchmark D-GEO STABILITY Rel. error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design cohesion slice 1 [kn/m 2 ] Design cohesion slice 2 [kn/m 2 ] Contribution stress table slice 1 [%] Contribution stress table slice 2 [%] Use D-GEO STABILITY input file bm3-34.sti to run this benchmark Probabilistic calculation using all shear strength models A probabilistic calculation in association with the four shear strength models is performed using the same geometry as benchmark 3-8 (section 3.8). Five variables with a normal distribution are considered, called X 1 to X 5, corresponding to the cohesion of slices 1 to 4 and Deltares 85 of 132

100 D-GEO STABILITY, Verification Report to the limit value of the model factor. The properties given in Table 3.94 apply to the different slices. Table 3.94: Probabilistic soil properties (bm3-35) Var. Model Slice µ σ Distrib. X 1 Cu-calculated 1 C u;1 /σ y Ratio S u /P c [-] Normal - P OP 1 P OP [kn/m 2 ] 3 Deterministic X 2 c, ϕ 2 c 2 Cohesion [kn/m 2 ] Normal - ϕ 2 Friction angle [ ] 0 Deterministic X 3 Stress table 3 c 3 Cohesion [kn/m 2 ] Normal - ϕ 3 Friction angle [ ] 0 X 4 Cu-measured 4 C u;4 Und. cohesion [kn/m 2 ] 17 2 Normal X 5 all 1-8 f Model factor [-] Normal As the reliability function is linear and the geometry is simple, an exact analytical solution can be found. Benchmark results According to Bishop method, the safety factor F s is solution of equation: R (L 1 Cu1 P c1 + L 2 c 2 + L 3 c 3 + L 4 C u4 ) = M load (3.88) P c1 F s F s F s F s F s = R M load (L 1 Cu1 P c1 P c1 + L 2 c 2 + L 3 c 3 + L 4 C u4 ) = 5 ( ) 240 = where: P c1 = P OP 1 + q + γ h 1 = = kn/m 2 A probabilistic analysis is performed using the First Order Reliability Method (FORM). The slope instability occurs if F s f which writes: R M load (L 1 Cu1 P c1 P c1 +L 2 c 2 +L 3 c 3 +L 4 C u4 ) < f g(x) = L 1 X 1 P c1 +L 2 X 2 +L 3 X 3 +L 4 X 4 X 5 where function g defines the slope stability condition. (3.89) The reliability function Z is expressed as Equation 3.52 with n = 5. The same chain of reasoning as the one performed in section 3.16 (see Equation 3.53 to Equation 3.55) leads to: µ[z] = L 1 µ[x 1 ] P c1 + L 2 µ[x 2 ] + L 3 µ[x 3 ] + L 4 µ[x 4 ] µ[x 5 ] Mload R = /5 = The standard deviation of the reliability function is expressed as Equation 3.57 with n = 5, which leads to: σ[z] = (L 1 P c1 σ[x 1 ]) 2 + (L 2 σ[x 2 ]) 2 + (L 3 σ[x 3 ]) 2 + (L 4 σ[x 4 ]) 2 + (σ[x 5 ] M load R )2 = ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 + ( )2 = of 132 Deltares

101 Group 3: Benchmarks from spreadsheets The reliability index is: β = µ[z] σ[z] = = And the probability of failure is equal to: P f (β > 0) = 1 2 [1 Erf( β 2 )] = The factor of uncertainty of the different variables is: α i = g X i σ i σ[z] and their contribution is αi 2, which lead to: α 1 = L 1 P c1 σ 1 /σ[z] = and α1 2 = % α 2 = L 2 σ 2 /σ[z] = and α 2 = % α 3 = L 3 σ 3 /σ[z] = and α3 2 = % α 4 = L 4 σ 4 /σ[z] = and α4 2 = % α 5 = M load /R σ 5 /σ[z]= and α5 2 = % The design value of variable X i is Xi = µ i α i β σ i, which leads to: X1 = (C u1 /σ y ) = = X2 = c 2 = = kn/m 2 X3 = c 3 = = kn/m 2 X4 = Cu4 = = kn/m 2 X5 = f = 1.1 ( ) = D-GEO STABILITY results Table 3.95: Results of benchmark 3-35 Benchmark D-GEO STABILITY Rel. error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design value ratio C u /σ y slice 1 [-] Design value cohesion slice 2 [kn/m 2 ] Design value cohesion slice 3 [kn/m 2 ] Design value und. coh. slice 4 [kn/m 2 ] Design value model factor [-] Contribution of ratio C u /σ y slice 1[%] Contribution of cohesion slice 2 [%] Contribution of stress table slice 3 [%] Contribution of und. coh. slice 4 [%] Contribution of model factor [%] Use D-GEO STABILITY input file bm3-35.sti to run this benchmark Probabilistic calculation of the design water level from three external water cases (D-GEO STABILITY and MProStab) A probabilistic calculation is performed for a complex configuration with several layers and PLlines. The reference case (MHW) corresponds to a water level at 3.9 m. Two others external water cases are considered: Case MHW-1: Water level at 2.9 m (1 m below the reference water level); Deltares 87 of 132

102 D-GEO STABILITY, Verification Report Case GHW: Water level at 1 m. Each case has is own PL-line per layer distribution. For each case, a probabilistic calculation is performed and gives the following reliability index (see results in *.pcr file): Table 3.96: Reliability index (bm3-36) Case i Water level h i Reliability index β i D-GEO STABILITY (bm3-36a) GHW MHW MHW (reference) MProStab (bm3-36b) GHW MHW MHW (reference) The input data are: B = 0.31 m ef = 1/2000 = MHW = 3.9 m decimate height; exceeding frequency; design water level. Benchmark results The design value of the water level is: where: h = u + B 2.3 ln( ln(φ( β H α H ))) (3.90) µ H is the mean value of the water level: µ H = h Φ 1 (F H (h )) σ H ; α H is the uncertainty on the water level: α H =. β h h σ H σ(z) ; β H is the reliability index: β H = E(Z) σ(z) ; is the standard deviation of the water level: σ H σ H = exp[ 1 2 {Φ 1 (F H (h ))} 2 ] ; 2π fh (h ) Φ 1 F H (h) f H (h) u is the inverse of the standard normal cumulative distribution; is the probability density function (Gumbel): F H (h) = exp[ exp( 2.3 (h u))]; B is the probability density function from Gumbel: f H (h) = 2.3 B exp[ 2.3 (h u) exp( 2.3 (h u))] ; B B is the expected value of the reliability function: ln[ ln(1 ef)] u = + MHW = m; 2.3/B 88 of 132 Deltares

103 Group 3: Benchmarks from spreadsheets σ[z] is the standard deviation of the reliability function: σ[z] = 1 + (. β h h σ H) 2 ; β(h ) is the linear interpolation of the curve h β;. β h h is the slope of curve h β: β h h = β i+1 β i for h i < h < h i+1 h i+1 h i To solve this problem, an iterative process is used starting with h = MHW = 3.9 m. All the calculations are worked out in an Excel spreadsheet. Five iterations are needed to find a relative difference between h assumed and h calculated inferior to m. Analytical results are given in Table 3.99and Table below. D-GEO STABILITY results Table 3.99: Results of benchmark 3-36a Benchmark D-GEO STABILITY Rel. error [%] Reliability index [-] Design value high water [m] Uncertainty water level [-] Table 3.100: Results of benchmark 3-36b Benchmark MProStab Rel. error [%] Reliability index [-] Design value high water [m] Uncertainty water level [-] Use D-GEO STABILITY input files bm3-36a.sti and bm3-36b.sti to run this benchmark MProStab 2 variables: cohesion and model factor with a log. distribution This benchmark is identical to benchmark 3-20 (section 3.20) except that the calculation is performed with MProStab, which means based on the Bishop / c-ϕ model with a logarithmic distribution for the cohesion and the model factor. In order to compare those results with the analytical solution of benchmark 3-20, the parameters given in Table must be adapted in MProStab in order to minimize the autocorrelation effect. Table 3.101: Correlation parameters (bm3-37) D h horizontal autocorrelation parameters 1000 m D v vertical autocorrelation parameters 100 m Number of tests Ratio between local and total variance 1 Deltares 89 of 132

104 D-GEO STABILITY, Verification Report Table 3.102: Unit weight of soil (bm3-37) Slice 1 Slice 2 to 8 Load top Uns. total unit weight [kn/m 2 ] Sat. total unit weight [kn/m 2 ] Table 3.103: Grid and tangent line properties (bm3-37) X co-ordinate [m] 5 Y co-ordinate [m] 13 Tangent line [m] 8 Table 3.104: Cohesion and friction angle MProStab, slice 1 (bm3-37) Cohesion - mean value [kn/m 2 ] 0 Cohesion - standard deviation [kn/m 2 ] 0 Friction angle - mean value [ ] 40 Friction angle - standard deviation [ ] 4 Benchmark results The equations of the analytical solution are identical to those from benchmark 3-20 (section 3.20). However, the numerical solution is different as the length of each slice is calculated in MProStab differently than in D-GEO STABILITY: L i = R w i Y i Y 0 (3.91) where: R is the radius of the slip circle in m; Y i is the Y co-ordinate at the middle of slice i in m; Y 0 is the Y co-ordinate of the slip circle in m; w i is the width of slice i in m. This leads to: L 1 = m. Finally: F = µ[z] = σ[z] = β = P f = α1 2 = % α2 2 = % c 1 = kn/m2 f = of 132 Deltares

105 Group 3: Benchmarks from spreadsheets MProStab results Table 3.107: Results of benchmark 3-37 Benchmark MProStab Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design cohesion [kn/m 2 ] Design model factor [-] Contribution of c 1 [%] Contribution of f [%] Use D-GEO STABILITY input file bm3-37.sti to run this benchmark MProStab 1 variable: friction angle with a log. distribution This benchmark is identical to benchmark 3-21 (section 3.21) except that the calculation is performed with MProStab, which means based on the Bishop / c-ϕ model with a logarithmic distribution of the friction angle. To compare those results with the analytical solution of benchmark 3-21, the following parameters must be adapted in MProStab in order to minimize the autocorrelation effect: Table 3.108: Correlation parameters (bm3-38) D h Horizontal autocorrelation parameters 1000 m D v Vertical autocorrelation parameters 100 m Number of tests Ratio between local and total variance 1 Table 3.109: Unit weight of soil (bm3-38) Slice 1 Slice 2 to 8 Load top Uns. total unit weight [kn/m 2 ] Sat. total unit weight [kn/m 2 ] Table 3.110: Grid and tangent line properties (bm3-38) X co-ordinate [m] 5 Y co-ordinate [m] 13 Tangent line [m] 8 Table 3.111: Cohesion and friction angle MProStab, slice 1 (bm3-38) Cohesion - mean value [kn/m 2 ] 0 Cohesion - standard deviation [kn/m 2 ] 0 Friction angle - mean value [ ] 35 Friction angle - standard deviation [ ] 3 Deltares 91 of 132

106 D-GEO STABILITY, Verification Report Benchmark results The equations of the analytical solution are identical to those from benchmark 3-21 (section 3.21). However, the numerical solution is different as the length of each slice is calculated in MProStab differently than in D-GEO STABILITY, as shown by Equation 3.91, which leads to: L 1 = m. Finally: F = µ[z] = σ[z] = β = P f = tan ϕ 1 = tan( ) = MProStab results Table 3.112: Results of benchmark 3-38 Benchmark MProStab Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design value tan ϕ [-] Contribution of ϕ [%] Use D-GEO STABILITY input file bm3-38.sti to run this benchmark MProStab 1 variable: hydraulic pressure with a normal distribution A probabilistic calculation with MProStab is performed, using the same geometry as benchmark 3-8 (section 3.8). The uniform load is q = 5 kn/m 2 instead of 30 kn/m 2. The only variable, with a normal distribution, is the hydraulic pressure of slice 1. The properties given in Table apply to the different slices. Table 3.113: Probabilistic soil properties (bm3-39) Var. Slice µ σ Distrib. - 1 c 1 Cohesion [kn/m 2 ] 5 Deterministic - ϕ 1 Friction angle [ ] 45 Deterministic X 1 u 1 Hydraulic pressure [kn/m 2 ] 0 3 Normal Cohesion [kn/m 2 ] 0 Deterministic - Friction angle [ ] 0 Deterministic f Model factor [-] 1 Deterministic In order to compare MProStab results with the analytical solution, the parameters given in Table must be adapted in order to minimize the autocorrelation effect. 92 of 132 Deltares

107 Group 3: Benchmarks from spreadsheets Table 3.114: Correlation parameters (bm3-39) D h horizontal autocorrelation parameters 1000 m D v vertical autocorrelation parameters 100 m Number of tests Ratio between local and total variance 1 As the reliability function is linear and the geometry is simple, an exact analytical solution can be found. Benchmark results According to Bishop method, the safety factor F s is solution of Equation 3.49 which leads to: F s = R M load L 1 (c 1 + σ 1) 1 = (3.92) where: L 1 = m (see Equation 3.91) σ 1 = q + (γ γ w ) h 1 = 5 + ( ) 0.5 = kn/m 2 M load = q w (w/2) = 5 4 (4/2) = 40kNm/m A probabilistic analysis is performed using the First Order Reliability Method (FORM). The slope instability corresponds with exceeding the limit value of the model factor, which writes F s f and leads to: R L 1 (c 1 +σ M 1) 1 f 0 g(x) = c 1 + q + γ h 1 X 1 M load 0 (3.93) load f + 1 R L 1 where function g defines the slope stability condition. The reliability function Z is expressed as Equation 3.52 with n = 1. The same chain of reasoning as the one performed in section 3.16 (see Equation 3.53 to Equation 3.55) leads to: µ[z] = c 1+q+γ h 1 µ[x 1 ] M load f+1 R L 1 = = The standard deviation of the reliability function is expressed as Equation 3.57 with n = 1, which leads to: σ[z] = g X 1 σ[x 1 ] = 1 2 σ[x 1] = 5 2 = 2.5 The reliability index is: β = µ[z] = = σ[z] 2.5 And the probability of failure is equal to: P f (β > 0) = 1 2 [1 Erf( β 2 )] = The factor of uncertainty of the different variables is: α 1 = g σ 1 X 1 σ[z] = -1. The design value of the hydraulic pressure is: X 1 = u 1 = µ[x 1 ] α 1 β σ[x 1 ] = 0 ( 1) = kn/m 2 Therefore, the design effective stress at bottom of slice 1 is: σ 1 = q + γ h 1 u 1 = = 6.2 kn/m 2 Deltares 93 of 132

108 D-GEO STABILITY, Verification Report MProStab results In MProStab, a standard deviation on the cohesion of slice 1 must be introduced to make calculation possible. By choosing a standard deviation of 0.01, the effect is negligible and the analytical solution is still correct. Table 3.115: Results of benchmark 3-39 Benchmark MProStab Error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design value effective stress of slice 1 [kn/m 2 ] Contribution of hydraulic pressure [%] Use D-GEO STABILITY input file bm3-39.sti to run this benchmark MProStab 1 variable: D.O.C. with a normal distribution The same problem as benchmark 3.39 is used except that the hydraulic pressure in slice 1 is replaced by a degree of consolidation with a mean of µ[u 1 ] = 70% and a reference standard deviation of σ 50% = 30%. In order to compare MProStab results with the analytical solution, the parameters given in Table must be adapted in order to minimize the autocorrelation effect. Table 3.116: Correlation parameters (bm3-40) D h horizontal autocorrelation parameters 1000 m D v vertical autocorrelation parameters 100 m Number of tests Ratio between local and total variance 1 Table 3.117: Unit weight of soil (bm3-40) Slice 1 Slice 2 to 8 Load top Uns. total unit weight [kn/m 2 ] Sat. total unit weight [kn/m 2 ] Table 3.118: Grid and tangent line properties (bm3-40) X co-ordinate [m] 5 Y co-ordinate [m] 13 Tangent line [m] 8 Table 3.119: Probabilistic soil properties for slice 1 (bm3-40) Variable µ σ c Cohesion [kn/m 2 ] ϕ Friction angle [ ] of 132 Deltares

109 Group 3: Benchmarks from spreadsheets As the reliability function is linear and the geometry is simple, an exact analytical solution can be found. Benchmark results According to Bishop method, the safety factor F s is solution of Equation 3.49 which leads to: F s = R L 1(c 1 + σ 1) M load 1 = ( ) 240 where: L 1 = m (see Equation 3.91) σ 1 = q + U 1 (γ γ w ) h 1 = = kn/m 2 M load = q w (w/2) = 5 4 (4/2) = 40 knm/m 1 = (3.94) A probabilistic analysis is performed using the First Order Reliability Method (FORM). The slope instability corresponds with exceeding the limit value of the model factor, which writes F s f and leads to: R L 1 (c 1 +σ M 1) 1 f 0 g(x) = c 1 + q + γ h 1 X 1 M load 0 (3.95) load f + 1 R L 1 where function g defines the slope stability condition. The reliability function Z is expressed as Equation 3.52 with n = 1. The same chain of reasoning as the one performed in section 3.16 (see Equation 3.53 to Equation 3.55) leads to: µ[z] = c 1+q+γ h 1 µ[x 1 ] M load f+1 R L = = The standard deviation of the degree of consolidation is: σ[u 1 ] = 4 σ 50% U 1 (1 U 1 ) = (1 0.7) = The standard deviation of the reliability function is expressed as Equation 3.57 with n = 1, which leads to: σ[z] = g X 1 σ[x 1 ] = γ h 1 f+1 σ[x ] = = The reliability index is: β = µ[z] σ[z] = = And the probability of failure is equal to: P f (β > 0) = 1 2 [1 Erf( β 2 )] = The factor of uncertainty of the different variables is: α 1 = g σ 1 = X 1 σ[z] 1. The design value of the hydraulic pressure is: X 1 = U 1 = µ[x 1 ] α 1 β σ[x 1 ] = = Therefore, the design effective stress at bottom of slice 1 is: σ 1 = q + γ h 1 U 1 = = 6.2 kn/m 2. MProStab results In MProStab, a standard deviation on the cohesion of slice 1 must be introduced to make calculation possible. By choosing a standard deviation of 0.01, the effect is negligible and the analytical solution is still correct. Deltares 95 of 132

110 D-GEO STABILITY, Verification Report Table 3.120: Results of benchmark 3-40 Benchmark MProStab Relative error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design value effective stress slice [kn/m 2 ] Contribution of phreatic line [%] Use D-GEO STABILITY input file bm3-40.sti to run this benchmark Horizontal balance method A horizontal balance calculation is performed for a dike body retaining water with a high water level at 5 m and a low water level of 0 m, as shown in Figure Table 3.121: Soil properties (bm3-41) Unsaturated total unit weight [kn/m 2 ] 12.5 Saturated total unit weight [kn/m 2 ] 15 Cohesion c [kn/m 2 ] 9 Friction angle ϕ [ ] 22 Table 3.122: Calculation area (bm3-41) X co-ordinate left side [m] 30 X co-ordinate right side [m] 54 Y co-ordinate highest slip plane [m] -1.5 Y co-ordinate lowest slip plane [m] -1.5 Number of slip planes 1 Benchmark results The calculation area is divided into 4 slices. Calculations are worked out using the equations given in the Background Section of the user manual. According to Figure 3.11, the maximal horizontal force is:f max = (5 (-1.5)) 10 / 2 = 32.5 kn/m and the minimal horizontal force is: F min = (0 (-1.5)) 10 / 2 = 7.5 kn/m.the total resisting force is kn/m 2 (see table below). Therefore, the stability factor is equal to:f s = R / (F max - F min ) = /( ) = Table 3.123: Calculation of the resisting force (bm3-41) Slice i X[m] Y top [m] Y bot [m] Width l i [m] σ [kpa] τ [kpa] τ l i [kn/m] [-] sum of 132 Deltares

111 Group 3: Benchmarks from spreadsheets Figure 3.11: Geometry of benchmark 3-41, check of horizontal balance of a dike retaining water D-GEO STABILITY results Table 3.124: Results of benchmark 3-41 Benchmark D-GEO STABILITY Relative error [%] Stability factor [-] Maximal horizontal force [kn/m] Minimal horizontal force [kn/m] Total resisting force [kn/m] Use D-GEO STABILITY input file bm3-41.sti to run this benchmark Pseudo characteristic shear strength model (local measurements) A calculation using the Fellenius model in combination with the pseudo characteristic shear strength model with local measurements is performed for a simplified configuration (Figure 3.12). Because of the symmetry of the problem, the driving moment due to the soil weight is nil. A uniform load of q = 100 kn/m2 is applied at the left side of the geometry. The degree of consolidation of all layers is 0%, therefore the effective stress corresponds with the load at the top of slice. Calcualtions are performed be dividing the slip circle into 8 slices of equal width (w = 1 m). Therefore, an analytical can be found. Table shows that each layer is described by one of two σ-τ curves given in Table Table 3.125: Soil properties (bm3-42) Layer 1 Layer 2 Layer 3 Unit weight [kn/m 2 ] Slices 1 and 2 3 and 4 5 to 8 Sigma-Tau curve Nr. 1 Nr. 2 Nr. 1 Deltares 97 of 132

112 D-GEO STABILITY, Verification Report Table 3.126: Sigma-Tau curves (bm3-42) σ τ τ char τ mean [kn/m 2 ] [kn/m 2 ] [kn/m 2 ] [kn/m 2 ] Curve nr. 1 Point Point Curve nr. 2 Point Point Figure 3.12: Geometry of benchmark 3-42 Benchmark results The geometrical data s of the different slices are the same as for benchmark 3-8 (section 3.8). Equations given in the Background section of the User Manual are used and lead to the analytical solution of Table of 132 Deltares

113 Group 3: Benchmarks from spreadsheets Table 3.127: Analytical results of benchmark 3-42 Slice i Layer j σi τi(σ i) τi;mean(σ i) τi;char(σ i) Li δτi Tj Pf;j c 1 j;pseudo τ (1) pt2;j;pseudo ϕ (2) j;pseudo αi τ (3) i;pseudo ti;pseudo Li [kpa] [kpa] [kpa] [kpa] [kn/m] [kn/m] [kpa] [kpa] [kpa] [ ] [ ] [kpa] [kn/m] sum (1) Pseudo characteristic shear stress of point 1 (τ pt1;j;pseudo = cj;pseudo) and point 2 (τpt2;j;pseudo) of the σ-τ curve, see Equation 3.52 in the Background Section. (2) Pseudo characteristic friction angle: ϕ j;pseudo = arctan( τ pt2;j;pseudo τpt1;j;pseudo σpt2;j σpt1;j (3) Shear stress according to Fellenius, see Equation 3.18 in the Background Section. ). Deltares 99 of 132

114 D-GEO STABILITY, Verification Report The driving moment due to load is: M L = q 4 w 2w = = 800 knm/m. The stability factor is equal to: F s = R n i=1 τ i L i = M L 800 = = D-GEO STABILITY results Table 3.128: Results of benchmark 3-42 Benchmark D-GEO STABILITY Relative error [%] Cohesion of layer 1 [kn/m 2 ] Cohesion of layer 2 [kn/m 2 ] Cohesion of layer 3 [kn/m 2 ] Friction angle of layer 1 [ ] Friction angle of layer 2 [ ] Friction angle of layer 3 [ ] Stability factor [-] Total resisting moment [knm] Total driving moment [knm] Use D-GEO STABILITY input file bm3-42.sti to run this benchmark Pseudo characteristic shear strength model (global measurements) The same configuration as benchmark 3-42 is used (section 3.42) except that global measurements are used instead of local measurements. The σ-τ curves nr. 1 and 2 have a number of tests of 18 and 57 respectively. Layers 1 and 2 compose soil group 1 and layer 3 compose soil group 2. Benchmark results The geometrical data s of the different slices are the same as for benchmark 3-8 (section 3.8). Equations given in the Background section of the user manual are used and lead to the analytical solution of Table of 132 Deltares

115 Group 3: Benchmarks from spreadsheets Table 3.129: Analytical result of benchmark 3-43 Slice Layer j σ i τ i (σ i ) τ i;mean (σ i ) τ i;char (σ i ) L i δ τ i T j i [kpa] [kpa] [kpa] [kpa] [kn/m] [kn/m] Slice i P f P f;j c (1) j τ (1) pt2;j ϕ (2) j P f;group τ (3) i t i L i [-] [-] [kn/m 2 ] [ ] [ ] [kpa] [kn/m] sum (1) Pseudo characteristic shear stress of point 1 (τ pt1;j;pseudo = c j;pseudo ) and point 2 (τ pt2;j;pseudo ) of the σ-τ curve, see Equation 3.53 in the Background Section. (2) Pseudo characteristic friction angle: ϕ j;pseudo = arctan( τ pt2;j;pseudo τ pt1;j;pseudo ). (3) Shear stress according to Fellenius, σ pt2;j σ pt1;j see Equation 3.18 in the Background Section. The driving moment due to load is: M L = q 4w 2w = = 800 knm/m. The stability factor is equal to: F s = R n i=1 τ i L i M L = = = D-GEO STABILITY results Table 3.130: Results of benchmark 3-43 Benchmark D-GEO STABILITY Relative error [%] Cohesion of layer 1 [kn/m 2 ] Cohesion of layer 2 [kn/m 2 ] Cohesion of layer 3 [kn/m 2 ] Friction angle of layer 1 [ ] Friction angle of layer 2 [ ] Friction angle of layer 3 [ ] Stability factor [-] Total resisting moment [knm] Total driving moment [knm] Use D-GEO STABILITY input file bm3-43.sti to run this benchmark. Deltares 101 of 132

116 D-GEO STABILITY, Verification Report 3.44 Deterministic calculation using design values Uplift Van model The same benchmark as benchmark 3-6 is used, except that the Uplift Van model is used instead of the Bishop model. In order to make the comparison with the analytical solution possible, the centers of the two slip circles (right and left) of the Spencer method coincide with the centre of the slip circle of the Bishop method. Benchmarks results Same analytical solution as benchmark 3-8 (section 3.8). D-GEO STABILITY results Calculation of the design values of c and ϕ from input mean values is checked in the window Soil in the Material menu and results are given in Table Table 3.131: Results for the design values (bm3-44) Design values Unit Benchmark D-GEO STABILITY Relative error [%] c 1 [kn/m 2 ] ϕ 1 [ ] c 2 [kn/m 2 ] ϕ 2 [ ] c 7 [kn/m 2 ] ϕ 7 [ ] c 8 [kn/m 2 ] ϕ 8 [ ] The results of the deterministic calculation using design values are given in Table Table 3.132: Results of benchmark 3-44 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Driving moment = [knm/m] Resisting moment = [knm/m] Use D-GEO STABILITY input file bm3-44.sti to run this benchmark Deterministic calculation using mean values Uplift Van model The same benchmark as benchmark bm3-9 (section 3.9) is used, except that the Uplift Van model is used instead of the Bishop model. In order to make the comparison with the analytical solution possible, the centers of the two slip circles (right and left) of the Spencer method coincide with the centre of the slip circle of the Bishop method. 102 of 132 Deltares

117 Group 3: Benchmarks from spreadsheets Benchmarks results Same analytical solution as benchmark bm3-9 (section 3.9). D-GEO STABILITY results Calculation of the mean values of c and ϕ from input design values is checked in the Soil window of the Material menu. Table 3.133: Results for the mean values (bm3-45) Mean values Unit Benchmark D-GEO STABILITY Relative error [%] c 3 [kn/m 2 ] ϕ 3 [ ] c 4 [kn/m 2 ] ϕ 4 [ ] c 5 [kn/m 2 ] ϕ 5 [ ] c 6 [kn/m 2 ] ϕ 6 [ ] The results of the deterministic calculation using mean values are given in Table Table 3.134: Results of benchmark 3-45 Benchmark D-GEO STABILITY Relative error [%] Safety factor [-] Driving moment = [knm/m] Resisting moment = [knm/m] Use D-GEO STABILITY input file bm3-45.sti to run this benchmark Probabilistic calculation All shear strength models Uplift Van model The same benchmark as benchmark 3-35 (section 3.35) is used, except that the Uplift Van model is used instead of the Bishop model. In order to make the comparison with the analytical solution possible, the centers of the two slip circles (right and left) of the Spencer method coincide with the center of the slip circle of the Bishop method. Benchmarks results Same analytical solution as benchmark 3-35 (section 3.35). Deltares 103 of 132

118 D-GEO STABILITY, Verification Report D-GEO STABILITY results Table 3.135: Results of benchmark 3-46 Benchmark D-GEO STABILITY Rel. error [%] Safety factor [-] Reliability index [-] Probability of failure [-] Design value ratio C u /σ y slice 1 [-] Design value cohesion slice 2 [kn/m 2 ] Design value cohesion slice 3 [kn/m 2 ] Design value und. coh. slice 4 [kn/m 2 ] Design value model factor [-] Contribution of ratio C u /σ y slice 1 [%] Contribution of cohesion slice 2 [%] Contribution of cohesion slice 3 [%] Contribution of und. coh. slice 4 [%] Contribution of model factor [%] Use D-GEO STABILITY input file bm3-46.sti to run this benchmark Verification of the functioning of nails Two cases are tested: Case A : Input of ultimate lateral and shear stress along nail Case B : Use soil parameters c, phi and input of bond stress diagram Benchmarks results The analytical solution can be found in the Excel spreadsheet associated to benchmark D-GEO STABILITY results 104 of 132 Deltares

119 Group 3: Benchmarks from spreadsheets Table 3.136: Results of benchmark 3-47a Benchmark D-GEO STABILITY Error [%] Driving moment soil [knm/m] Available resisting moment [knm/m] Total resis. mom. from soil nails [knm/m] Safety factor [-] Nail 1: Intersection point X-coord. [m] Intersection point Y-coord. [m] Normal force from nails [kn] Shear force from nails [kn] Resisting moment [knm/m] Nail 2: Intersection point X-coord. [m] Intersection point Y-coord. [m] Normal force from nails [kn] Shear force from nails [kn] Resisting moment [knm/m] Nail 3: Intersection point X-coord. [m] Intersection point Y-coord. [m] Normal force from nails [kn] Shear force from nails [kn] Resisting moment [knm/m] Deltares 105 of 132

120 D-GEO STABILITY, Verification Report Table 3.137: Results of benchmark 3-47b Benchmark D-GEO STABILITY Error [%] Driving moment soil [knm/m] Available resisting moment [knm/m] Total resis. mom. from soil nails [knm/m] Safety factor [-] Nail 1: Intersection point X-coord. [m] Intersection point Y-coord. [m] Normal force from nails [kn] Shear force from nails [kn] Resisting moment [knm/m] Nail 2: Intersection point X-coord. [m] Intersection point Y-coord. [m] Normal force from nails [kn] Shear force from nails [kn] Resisting moment [knm/m] Nail 3: Intersection point X-coord. [m] Intersection point Y-coord. [m] Normal force from nails [kn] Shear force from nails [kn] Resisting moment [knm/m] Use D-GEO STABILITY input files bm3-47a.sti and bm3-47b.sti to run this benchmark Verification of the shear strength model Su-calculated with yield stress A calculation with the Bishop model is performed for materials using the Su-calculated with yield stress model. The 10 yield stress measurements are randomly positioned (Table 3.139). The values of the soil parameters for both models are given in Table Table 3.138: Material properties (benchmark 3-48) Clay Sand Peat Pleistocene Unsaturated unit weight of soil [kn/m 3 ] Saturated unit weight of soil [kn/m 3 ] Ratio S [-] Exponent m [-] of 132 Deltares

121 Group 3: Benchmarks from spreadsheets Table 3.139: Pre-consolidation stress at different positions (benchmark 3-48) σ y X Z [kn/m 2 ] [m] [m NAP] Figure 3.13: Geometry of benchmark 3-48 Benchmarks results The analytical solution can be found in the Excel spreadsheet associated to benchmark D-GEO STABILITY results Table 3.140: Results of benchmark bm Safety factor and moments Spreadsheet D-GEO STABILITY Error [%] Safety factor [-] Driving moment [knm/m] Soil moment [knm/m] Water moment [knm/m] Load moment [knm/m] Resisting moment [knm/m] Resisting moment uniterated [knm/m] Deltares 107 of 132

122 D-GEO STABILITY, Verification Report Use D-GEO STABILITY input file bm3-48.sti to run this benchmark. 108 of 132 Deltares

123 4 Group 4: Benchmarks generated by the program The benchmarks in this chapter are generated by the D-GEO STABILITY. 4.1 Verification Cu-measured method Using the Cu-measured method a result is obtained without iterating. This should give the same result as an iterated Bishop calculation where the angle of internal friction ϕis zero and the cohesion is equal to the S u values in the Cu-measured calculation. Figure 4.1: Geometry of benchmark 4-1 Table 4.1: Soil properties, Cu-measured model (bm4-1) Layer1 Layer2 Layer3 Layer4 Layer5 Layer6 Uns. total unit weight [kn/m 3 ] Sat. total unit weight [kn/m 3 ] C u top [kn/m 2 ] C u bottom [kn/m 2 ] Table 4.2: Soil properties, c-phi model (bm4-1) Layer1 Layer2 Layer3 Layer4 Layer5 Layer6 Uns. total unit weight [kn/m 3 ] Sat. total unit weight [kn/m 3 ] Cohesion c [kn/m 2 ] Friction angle ϕ [ ] D-GEO STABILITY results Deltares 109 of 132

124 D-GEO STABILITY, Verification Report Cu-measured model A so called Cu-measured calculation with 6 layers having an undrained cohesion C u of 50, 30 and 10 kn/m 2 on top and bottom, a line load and a degree of consolidation were added to the problem (bm4-1a). c-ϕ model A so called c-ϕ model calculation (bm4-1b) with the same geometry, loads and degree of consolidation as the Cu calculation is executed. The friction angle in this calculation was zero while the cohesion had the same value as the C u values in the calculation mentioned above. The results of the Cu-model and c-ϕ model are compared in Table 4.3. Table 4.3: Results of benchmark 4-1 Circle D-GEO STABILITY Relative error [%] (bm4-1a) (bm4-1b) Shear strength model Cu-measured c-ϕ X co-ordinate [m] Y co-ordinate [m] Radius [m] Available resisting moment [knm/m] Safety factor [-] Use D-GEO STABILITY input files bm4-1a.sti and bm4-1b.sti to run this benchmark. 4.2 Verification MSeep results with piezometric lines It is possible to use pore pressures from a MSeep calculation in D-GEO STABILITY. To check if interpolation from those values in a D-GEO STABILITY calculation is correct, two calculations are done. One with pore pressures from a MSeep calculation and one with pore pressures from a piezometric level line. The MSeep calculation is added too. When calculating with PL-lines the pore pressure in a point is calculated by the vertical distance between the point and the PL-line. When using MSeep results the pore pressure is got from interpolation in the flow net. So there will be differences between the two methods when the streamlines are curved. In order to get a possible comparison it was necessary to get little curvature in the stream lines so a special MSeep calculation with almost only horizontal flow was used. 110 of 132 Deltares

125 Group 4: Benchmarks generated by the program D-GEO STABILITY results The results of the two calculations are presented in Table 4.4. Table 4.4: Results of benchmark 4-2 Circle D-GEO STABILITY Relative error (bm4-2a) (bm4-2b) [%] PL-line From MSeep User defined X co-ordinate [m] Y co-ordinate [m] Radius [m] Safety factor [-] Use D-GEO STABILITY input files bm4-2a.sti and bm4-2b.sti to run this benchmark. 4.3 Comparison between D-Geo Stability and MProStab (without autocorrelation) A probabilistic calculation is performed with D-GEO STABILITY and MProStab for a complex problem, with several layers and PL-lines. Results are compared. MProStab results Calculations with MProStab are performed in benchmark 4-3a. In order to compare MProStab results with D-GEO STABILITY results, the parameters of Table 4.5 must be adapted in order to minimize the autocorrelation effect. Table 4.5: Correlation parameters (bm4-3) D h Horizontal autocorrelation parameters 1000 m D v Vertical autocorrelation parameters 100 m Number of tests Ratio between local and total variance 1 D-GEO STABILITY results Table 4.6: Results of benchmark 4-3 MProStab (bm4-3a) D-GEO STABILITY (bm4-3b) Error [%] X co-ordinate center point [m] Y co-ordinate center point [m] Radius critical circle [m] Safety factor [-] Reliability index [-] Probability of failure [-] Contribution of cohesion [%] Contribution of friction angle [%] Contrib. of consoled. (excess pore press.) [%] Contrib. of hydraulic press. (phreatic line) [%] Contribution of model factor [%] Deltares 111 of 132

126 D-GEO STABILITY, Verification Report Use D-GEO STABILITY input files bm4-3a.sti and bm4-3b.sti to run this benchmark. 4.4 Comparison between D-Geo Stability and MProStab (with autocorrelation) D-GEO STABILITY results The results of a probabilistic calculation performed with D-GEO STABILITY are compared to those of MProStab, for a simple problem: only one layer loaded with a uniform load of 500 kn/m 2 at the left side. Table 4.7: Unit weight of soil (bm4-4) Soil Load top Unsaturated total unit weight [kn/m 3 ] Saturated total unit weight [kn/m 3 ] Table 4.8: Grid and tangent line properties (bm4-4) X co-ordinate [m] 25 Y co-ordinate [m] 30 Tangent line [m] -2 Table 4.9: Probabilistic soil properties for MProStab (bm4-4a) Variable Unit σ c Cohesion [kn/m 2 ] 20 4 ϕ Friction angle [ ] Table 4.10: Correlation parameters (bm4-4a) D h Horizontal autocorrelation parameters 50 m D v Vertical autocorrelation parameters 0.25 m Number of tests 40 Ratio between local and total variance 0.75 Figure 4.2: Geometry of benchmark 4-4, load on left side To make the comparison with D-GEO STABILITY possible, the standard deviation of the cohesion and the friction angle in D-GEO STABILITY must be adapted in order to take into account 112 of 132 Deltares

127 Group 4: Benchmarks generated by the program the autocorrelation effect: σ local = σ spatial 1/n + Γ 2 (4.1) where Γ = 0.5 for the spatial variables considered above. Then, the following standard deviations in D-GEO STABILITY must be inputted: σ[c] = 4 1/ = 2.10 kn/m 2 σ[ϕ] = 0.5 1/ = 0.26 Table 4.11: Probabilistic soil properties for D Geo Stability (bm4-4b) Variable Unit σ Distribution c 1 Cohesion [kn/m 2 ] Log normal ϕ 1 Friction angle [ ] Log normal D-GEO STABILITY results A calculation with D-GEO STABILITY without adapting the standard deviations of the cohesion and the friction angle leads to a reliability index of 1.388, which corresponds to a relative error of 85.45%. Table 4.12: Results of benchmark 4-4 MProStab (bm4-4a) D-GEO STABILITY (bm4-4b) Rel. error [%] Safety factor [-] Reliability index [-] Contribution Spatial variability of cohesion Est. mean value [%] Total Contribution Spatial variability of friction Est. mean value angle [%] Total Use D-GEO STABILITY input files bm4-4a.sti and bm4-4b.sti to run this benchmark. 4.5 Comparison between Uplift Van and Uplift Spencer A dike with a height of 5 m and a slope of 1:2 on 7 m peat. The peat is placed on the Pleistocene. At the top layers a piezometric level of -1 m is present. At the bottom, a piezometric level of 0.98 m is present (Figure 4.3). Table 4.13: Soil properties (bm4-6) Parameter Unit Dike Peat Sand Unsaturated total unit weight [kn/m 2 ] Saturated total unit weight [kn/m 2 ] Cohesion c [kn/m 2 ] Friction angle ϕ [ ] Deltares 113 of 132

128 D-GEO STABILITY, Verification Report Figure 4.3: Geometry of benchmark 4-5 The calculation is a comparison of two methods: 1 the Uplift Van method using simplifications (bm4-5a), 2 the Uplift Spencer method (bm4-5b). D-GEO STABILITY results The circle with the minimum safety factor has been found using the search algorithm in both grids. In the final calculation a smaller grid is placed around the centre point with the minimum safety factor. Table 4.14: Results of benchmarks 4-5a and 4-5b Circle Van Uplift method (bm4-5a) Spencer Uplift method (bm4-5b) Rel. error [%] Left X co-ordinate [m] grid Y co-ordinate [m] Radius [m] Right X co-ordinate [m] grid Y co-ordinate [m] Radius [m] Safety factor [-] Use D-GEO STABILITY input files bm4-5a.sti and bm4-5b.sti to run this benchmark. 4.6 Comparison between Genetic Algorithm and Grid search methods (Spencer) Simple case 114 of 132 Deltares

129 Group 4: Benchmarks generated by the program Figure 4.4: Geometry of benchmark 4-6 The geometry is simple with a uniform load of 10 kn/m 2. Table 4.15: Soil properties (bm4-6) Parameter Unit Soft clay Unsaturated total unit weight kn/m 2 ] 0.01 Saturated total unit weight [kn/m 2 ] 0.01 Cohesion c [kn/m 2 ] 10 Friction angle ϕ [ ] 0 The calculation is a comparison of two search methods: 1 the Grid search method (bm4-6a). 2 the Genetic Algorithm search method (bm4-6b), It is expected that the Genetic Algorithm search method will lead to a more accurate safety factor than the classical Grid search method. D-GEO STABILITY results As expected, the safety factor found using the Genetic Algorithm method is smaller than using the grid method. Table 4.16: Results of benchmarks 4-6a and 4-6b Circle Grid search method GA search method Relative (bm4-6a) (bm4-6b) [%] Safety factor [-] error Use D-GEO STABILITY input files bm4-6a.sti and bm4-6b.sti to run this benchmark. 4.7 Comparison between Genetic Algorithm and Grid search methods (Spencer) Complex case Deltares 115 of 132

130 D-GEO STABILITY, Verification Report The geometry is complex as shown in Figure 4.5. Figure 4.5: Geometry of benchmark 4-7 The calculation is a comparison of two search methods: 1 the Grid search method (bm4-7a). 2 the Genetic Algorithm search method (bm4-7b), It is expected that the Genetic Algorithm search method will lead to a more accurate safety factor than the classical Grid search method. D-GEO STABILITY results As expected, the safety factor found using the GA method is smaller than using the grid method. Table 4.17: Results of benchmarks 4-7a and 4-7b Circle Grid search method GA search method Relative error (bm4-7a) (bm4-7b) [%] Safety factor [-] Use D-GEO STABILITY input files bm4-7a.sti and bm4-7b.sti to run this benchmark. 116 of 132 Deltares

131 5 Group 5: Benchmarks compared with other programs The benchmarks in this chapter have no exact solution, but are compared with other programs (using the same method). In D-GEO STABILITY the Spencer method is used. There can easily be some differences due to most points as described in [chapter 3]. But if the results are close this should give confidence in the calculation method. Most of the data in this chapter was found on the internet. 5.1 Simple homogeneous slope without pore water A slope 2:3 without water; the cohesion of the soil is 10 kn/m 2 and the angle of internal friction is equal to 20. Figure 5.1: Geometry of benchmark 5-1 with slip plane 1 Benchmark results The benchmark calculations are performed with the geotechnical program UTEXAS3 Web. The UTEXAS3 slope stability computer program uses the method of slices and Spencer s limit equilibrium procedure to determine a factor of safety for a given potential failure geometry. The program can take an initial estimate of the failure surface and search for the critical surface (i.e., the surface that has the lowest factor of safety). D-GEO STABILITY results D-GEO STABILITY calculations are performed using the Spencer method which gives an inclination angle of the interslice forces as well as a safety factor. The safety factor satisfies both moment and forces equilibrium. Results are presented in Table 5.1. Table 5.1: Results of benchmark 5-1 Circle UTEXAS3 D-GEO STABILITY Relative error [%] Safety factor Spencer [-] Interslice force angle [ ] Deltares 117 of 132

132 D-GEO STABILITY, Verification Report Use D-GEO STABILITY input file bm5-1.sti to run this benchmark. 5.2 Simple homogeneous slope with pore water The geometry is the same as benchmark 5-1 (section 5.1) but now there is water in- and outside the slope. Figure 5.2: Geometry of benchmark 5-2 with slip plane and water Benchmark results The benchmark calculations are performed with the geotechnical program UTEXAS3 [Lit 15]. D-GEO STABILITY results D-GEO STABILITY calculations are performed using the Spencer method which gives an inclination angle of the interslice forces as well as a safety factor. The safety factor satisfies both moment and forces equilibrium. Results are presented in Table 5.2. Table 5.2: Results of benchmark 5-2 Circle UTEXAS3 D-GEO STABILITY Relative error [%] Safety factor Spencer [-] Interslice force angle [ ] Use D-GEO STABILITY input file bm5-2.sti to run this benchmark. 5.3 A simple slope from Fredlund and Krahn 118 of 132 Deltares

133 Group 5: Benchmarks compared with other programs Figure 5.3: Geometry of benchmark 5-3 A slope 1:4 with three layers. Table 5.3: Soil properties (bm5-3) Parameter Unit Silty sand Peat Bedrock Cohesion c [kn/m 2 ] Friction angle ϕ [ ] Benchmark results The benchmark calculations are performed with the geotechnical program UTEXAS3 [Lit 15]. D-GEO STABILITY results D-GEO STABILITY calculations are performed using the Spencer method which gives an inclination angle of the interslice forces as well as a safety factor. The safety factor satisfies both moment and forces equilibrium. Results are presented in Table 5.4. Table 5.4: Results of benchmark 5-3 Circle UTEXAS3 D-GEO STABILITY Relative error [%] Safety factor Spencer [-] Interslice force angle [ ] Use D-GEO STABILITY input file bm5-3.sti to run this benchmark. 5.4 A simple slope from Fredlund and Krahn, with both pore water and weak layer Deltares 119 of 132

134 D-GEO STABILITY, Verification Report This benchmark is identical to benchmark 5-3 (section 5.3) with three layers; a bead rock layer, a weak layer and a soil layer. The soil properties are given in Table 5.5. Figure 5.4: Geometry of benchmark 5-4 Table 5.5: Soil properties (bm5-4) Silty sand Peat Bedrock Cohesion c [kn/m 2 ] Friction angle ϕ [ ] Benchmark results The benchmark calculations are performed with the geotechnical program UTEXAS3 [Lit 15]. D-GEO STABILITY results D-GEO STABILITY calculations are performed using the Spencer method which gives an inclination angle of the interslice forces as well as a safety factor. The safety factor satisfies both moment and forces equilibrium. Results are presented in Table 5.6. Table 5.6: Results of benchmark 5-4 Circle UTEXAS3 D-GEO STABILITY Relative error [%] Safety factor Spencer [-] Interslice force angle [ ] Use D-GEO STABILITY input file bm5-4.sti to run this benchmark. 5.5 A complex slope with various layers 120 of 132 Deltares

135 Group 5: Benchmarks compared with other programs This benchmark has a complex slope under different angles. The ground profile is also made of various layers. The soil properties are given in Table 5.7. Loam with little sand Table 5.7: Soil properties (bm5-5) Clay with little sand Sandy clay Organic clay c [kn/m 2 ] ϕ [ ] Stiff clay with sand Benchmark results The benchmark calculations are performed with the geotechnical program UTEXAS3 [Lit 15]. D-GEO STABILITY results D-GEO STABILITY calculations are performed using the Spencer method which gives an inclination angle of the interslice forces as well as a safety factor. The safety factor satisfies both moment and forces equilibrium. Results are presented in Table 5.8. Table 5.8: Results of benchmark 5-5 Circle Benchmark D-GEO STABILITY Relative error [%] Safety factor Spencer [-] Interslice force angle [ ] unknown Use D-GEO STABILITY input file bm5-5.sti to run this benchmark. 5.6 Su-calculated model with yield stress measurements A comparison with the results given by the WTI kernel Macro-Stability has been performed as the Su-calculated with yield stress model is also implemented in this kernel. Five test cases have been used, for the three methods Bishop, Spencer and Uplift-Van and for a single slip plane: Case A: Lekdijk dp 183 Case B: Lekdijk dp 190 Case C: Markermeerdijk Case D: Wolpherensedijk Case E: Lekdijk west Deltares 121 of 132

136 D-GEO STABILITY, Verification Report Figure 5.5: Geometry of benchmark 5-6a Figure 5.6: Geometry of benchmark 5-6b 122 of 132 Deltares

137 Group 5: Benchmarks compared with other programs Figure 5.7: Geometry of benchmark 5-6c Figure 5.8: Geometry of benchmark 5-6d Deltares 123 of 132

138 D-GEO STABILITY, Verification Report Figure 5.9: Geometry of benchmark 5-6e D-GEO STABILITY results The results of the comparison are given in Figure The calculated safety factors are very close (and sometimes exactly the same) for the first test case, but not for the other ones because in those test cases the same material is used in different layers and/or contains several yield stress measurements: D-GEO STABILITY uses an average value of all the POP values available for the material whereas WTI MacroStability uses also an average value of the POP values but per column (of the pre-process calculation) not for the complete section. 124 of 132 Deltares

139 Group 5: Benchmarks compared with other programs Figure 5.10: Results of benchmark 5-6 Deltares 125 of 132

140 D-GEO STABILITY, Verification Report 5.7 Verification of the reference line for the undrained shear strength models To analyse the stability of a slope based on the undrained shear strength of the soil the yield stress of the soil plays a role in the calculation of the undrained shear strength. Building a new embankment influences the yield stress of the subsoil. Excavations influence the yield stress of the subsoil too. In D-GEO STABILITY the reference level is used to calculate the yield stress correctly. Calculations with the Su calculated with POP model and the Su calculated with yield stress model are executed. For the Sand layers the C-phi model is used. The Spencer slip plane model is used. Figure 5.11: Geometry of benchmark 5-7 Table 5.9: Soil properties for benchmarks 5-7 Material name Sand Soft clay Soft clay Shear strength model C-phi Su calculated with POP Su calculated with yield stress γ unsat [kn/m 3 ] γ sat [kn/m 3 ] Cohesion [kn/m 2 ] 0 Phi [ ] 35 Dilatancy [ ] 0 Ratio S [-] POP [kn/m 2 ] 25 Yield stress [kn/m 2 ] (1) Exponent m [-] 1 (1) Yield stress of kn/m 2 at coordinate X = 30 m and Y = -3 m. D-GEO STABILITY results The results of the calculations are compared in Table Table 5.10: Results of benchmark 5-7 Case Location reference line Safety factor Su calc. with POP Su calc. with yield stress 1 Original horizontal surface (bm5-7a) (bm5-7b) 2 Along embankment surface and ditch (bm5-7d) (bm5-7d) bottom 3 Original horizontal surface and along (bm5-7e) (bm5-7f) ditch bottom 4 Original horizontal surface and along embankment surface (bm5-7g) (bm5-7h) 126 of 132 Deltares

141 Group 5: Benchmarks compared with other programs Use D-GEO STABILITY input files bm5-07a.sti to bm5-07h.sti to run this benchmark. 5.8 Verification of the dilatancy Using the C-phi shear strength model also the dilatancy angle plays a role. Increasing the dilatancy angle will give higher shear strength and decreasing the dilatancy angle result in lower shear strength. Calculations with the C-phi shear strength model with different values of the dilatancy angle are executed. The slip plane models Bishop, Uplift-Van and Spencer are used. Figure 5.12: Geometry of benchmark 5-8 Table 5.11: Soil properties for benchmarks 5-8 Medium clay Soft clay Sand γ unsat [kn/m 3 ] γ sat [kn/m 3 ] Cohesion [kn/m 2 ] Phi [ ] Table 5.12: Variations of the dilatancy angle in the calculations Material name Dilatancy [ ] Case 1 Case 2 case 3 Medium clay Soft clay Sand D-GEO STABILITY results The results of the calculations are compared in Table Table 5.13: Results of benchmark 5-8 Case Safety factor Bishop Uplift-Van Spencer (bm5-08a) (bm5-08b) (bm5-08c) (bm5-08d) (bm5-08e) (bm5-08f) (bm5-08g) (bm5-08h) (bm5-08i) Use D-GEO STABILITY input files bm5-08a.sti to bm5-08i.sti to run this benchmark. Deltares 127 of 132

142 D-GEO STABILITY, Verification Report 128 of 132 Deltares

143 References Program UTEXAS3. URL engcomp.htm. Alonso, E., Risk Analysis of Slopes and its Application to Canadian Sensitive Clays. Geotechnique vol. 26, no 3. American Petroleum Institute Washington, D., Recommended practice for planning, designing, and constructing fixed offshore platforms.. Beacher, G. B. and J. T. Christian, Reliability and Statistics in Geotechnical Engineering.. Calle, E. O. F., Probabilistic Approach of Stability of Earth Slopes. Proc. XI-th ICSMFE, San Francisco. Calle, E. O. F., PROSTAB: A Computer Model for Probabilistic Analysis of Stability of Slopes. GeoDelft report CO /32 (in Dutch). Calle, E. O. F., Aaanpassing MPROSTAB. GeoDelft report CO /04 (in Dutch). CUR, Publicatie 162: Construeren met grond Grondconstructies op en in weinig draagkrachtige en sterk samendrukbare ondergrond. Civieltechnischcentrum Uitvoering en Regelgeving. Dutch) report SE-52029/2 (in, February Development of Uplift Stability Theory. Tech. rep., GeoDelft. Geodelft. Dutch) report SE /02 (in, January Tech. rep., GeoDelft. Spanningsafhankelijk rekenen in MStab. NEN, NEN 6740:2006. Geotechniek - TGB Basiseisen en belastingen (Geotechnics - TGB Basic requirements and loads), in Dutch. Phoon, K. K. and F. H. Kulhawy, 1999a. Characterization of geotechnical variability. Can. Geotech. Journal no 36: Phoon, K. K. and F. H. Kulhawy, 1999b. Evaluation of geotechnical property variability. Can. Geotech. Journal no 36: TAW, Leidraad voor het ontwerpen van rivierdijken, Deel 1 Bovenrivierengebied. Tech. rep., Technische Adviescommissie voor de Waterkeringen. TAW, Leidraad voor het ontwerpen van rivierdijken, Deel 2 Benedenrivierengebied. Tech. rep., Technische Adviescommissie voor de Waterkeringen. Uitgeverij Waltman. Delft. TAW, Handreiking Constructief ontwerpen. Tech. rep., Technische Adviescommissie voor de Waterkeringen. Van Marcke, E., Random Fields, Analysis and Synthesis.. Verruijt, Collegedictaat b22: Grondmechanica CUR Civiele. Technische Hogeschool Delft, afdeling Civiele Techniek, vakgroep Geotechniek. Deltares 129 of 132

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