RESULTS FROM NUMERICAL BENCHMARK EXERCISES IN GEOTECHNICS

Transcription

1 RESULTS FROM NUMERICAL BENCHMARK EXERCISES IN GEOTECHNICS H.F. Schweiger Institute for Soil Mechanics and Foundation Engineering, Computational Geotechnics Group, Graz University of Technology, Austria ABSTRACT. Results of two geotechnical benchmarking exercises are presented. A deep excavation problem in Berlin sand has been specified by the working AK 1.6 of the German Society for Geotechnics and a simple tunnel excavation was used by working group A of Cost Action C7. The results of these exercises clearly emphasize the need for establishing recommendations and guidelines for numerical analysis in practice. RÉSUMÉ. 1. Introduction The significant progress made in the understanding of the behaviour of geomaterials would not have been possible without the use of numerical methods. In particular, developments in constitutive modelling are closely related to advances made in the field of numerical analysis and therefore finite element (and other) methods have had a significant impact on geotechnical research since the 197s. Advances in computer hardware and, more importantly, in geotechnical software over the past ten years have resulted in a widespread application also in practical geotechnical engineering. These developments enable the geotechnical engineer to perform very advanced numerical analyses at low cost and with relatively little computational effort. Commercial codes, fully integrated into the PC-environment, have become so user-friendly that little training is required for operating the programme. They offer sophisticated types of analysis, such as fully coupled consolidation analysis with elasto-plastic material models. However, for performing such complex calculations and obtaining sensible results a strong background in numerical methods, mechanics and, last but not least, theoretical soil mechanics is essential. The potential problems arising from the situation that geotechnical engineers, not sufficiently trained for that purpose, perform complex numerical analyses and may produce unreliable results have been recognized within the profession and some national and international committees have begun to address this problem, amongst them the working group AK 1.6 "Numerical Methods in Geotechnics" of the German Society for Geotechnics (DGGT) and working group A "Numerical Methods" of the COST Action C7 (Co-Operation in Science and Technology of the European Union). One of the main goals of AK 1.6 of the DGGT is to provide recommendations for numerical analyses in geotechnical engineering. In addition benchmark examples are specified and the results obtained by various users employing different software are compared. Relatively little attention has been paid in the literature on validation and reliability of numerical models in general and on specific software in particular, although some attempts have been made (e.g. Schweiger 1998, Schweiger 2). More recently the problem has also been addressed by Potts and Zdravkovic (21) and Carter et al. (2). In this paper solutions for two benchmark problems, namely a deep excavation in Berlin sand, specified by the AK 1.6 of the DGGT, and a simple tunnel excavation, specified by the working group A of the COST Action C7, will be discussed.

2 2. Undrained analysis of a shield tunnel excavation 2.1. Specification of problem This example has been specified by the Working Group A of COST Action C7 and has been deliberately chosen very simple (e.g. constant undrained shear strength instead of increasing with depth). Undrained conditions are considered and 3 analyses should be performed in terms of total stresses in plane strain conditions: Analysis A: elastic, no lining, uniform initial stress state Analysis B: elastic-perfectly plastic, no lining, K o = 1. Analysis C: elastic-perfectly plastic, segmental lining, K o = 1., given ground loss The tunnel diameter is given as 1 m and the overburden (measured from crown to surface) is assumed to be 15 m. At a depth of 45 m below surface bedrock can be assumed (see Figure 1). The material parameters for all analysis are given in Table 1. Table 1. Material parameters for analyses A, B and C Analysis γ G ν σ v = σ h (K o =1.) c u E lining ν lining γ lining kn/m 3 kpa - kpa kpa kpa - kn/m 3 A B (z C (z x Computational step to be performed: Analyses A and B: full excavation Analysis C: full excavation with assumed ground loss of 2%. m surface A 15 m z tunnel diameter = 1 m B thickness of lining =.3 m D C -45. m bedrock Figure 1. Geometry for benchmark shield tunnel 2.2 Results Some selected results are presented in the following. 12 solutions (termed ST1 to ST12 respectively) have been submitted. Table 2 summarizes calculated displacements at various

3 locations which are indicated in Figure 1. It follows that there is a 2% difference of maximum settlement of point A, which is by no means acceptable for an elastic solution. As will be seen later this is entirely due to the different assumptions for the lateral boundary condition. Table 2. Analysis A - calculated displacements of points A, B, C and D [mm] A B C D vert. D horiz. ST ST ST ST ST ST ST ST ST ST ST ST Figures 2 and 3 show settlements and horizontal displacements at the surface for the plastic solution with constant undrained shear strength (Analysis B). In Figure 2 a similar scatter as in Analysis A is observed with the exception of ST4, ST9 and ST1 which show an even larger deviation from the "mean" of all analyses submitted. ST5 restrained vertical displacements at the lateral boundary and thus the settlement is zero here. ST9 used a Von-Mises and not a Tresca failure criterion which accounts for the difference. The strong influence of employing a Von-Mises criterion as follows from Figure 2 has been verified by separate studies. It is emphasized therefore that a careful choice of the failure criterion is essential in a non-linear analysis even for a simple problem as considered here. The significant variation in predicted horizontal displacements, mainly governed by the placement of the lateral boundary condition, is evident from Figure 3. Taking the settlement at the surface above the tunnel axis (point A) the minimum and maximum value calculated is 76 mm and 159 mm respectively. Thus differences are - as expected - significantly larger than in the elastic case and again not acceptable. distance from tunnel axis [m] vertical displacements [mm] ST1 ST3 ST4 ST5 ST6 ST7 ST8 ST9 ST1 ST11 ST12-18 Figure 2. Calculated surface settlements - analysis B

4 distance from tunnel axis [m] horizontal displacements [mm] ST1 ST3 ST4 ST5 ST6 ST7 ST8 ST9 ST1 ST11 ST12-6 Figure 3. Calculated horizontal displacements at surface - analysis B Figure 4 plots surface settlements for the elastic-perfectly plastic analysis with a specified volume loss of 2% and the even wide scatter in results is indeed not very encouraging. The significant effect of the vertically and horizontally restrained boundary condition used in ST5 is apparent. However in the other solutions no obvious cause for the differences could be found except that the lateral boundary has been placed at different distances from the symmetry axes and that the specified volume loss is modelled in different ways. The range of calculated values for the surface settlement above the tunnel axis is between 1 and 25 mm and for the crown settlement between 17 and 45 mm respectively. The normal forces in the lining and the contact pressure between soil and lining do not differ that much (variation is within 15 and 2% respectively), with the exception of ST9 who calculated significantly lower values. distance from tunnel axis [m] vertical displacements [mm] ST1 ST2 ST3 ST4 ST5 ST6 ST8 ST9 ST1 ST11 ST12-4 Figure 4. Calculated surface settlements - analysis C After comparing all results submitted, a second round of calculations has been performed. All authors were asked to redo their analysis with the lateral boundary placed at a distance of 1 m from the tunnel axis with horizontal displacements restrained. By doing so all solutions for analyses A and B were within acceptable limits, for analysis C however, still significant differences in results were obtained, although the range of scatter was reduced (Figure 5). These differences are most likely due to the way different software handles the specified volume loss. Again this is a strong case for developing guidelines and reference examples how to model this (and other) excavation problems.

5 distance from tunnel axis [m] vertical displacements [mm] ST1 ST3 ST5 ST7 ST8 ST9 ST1-25 Figure 5. Calculated surface settlements - analysis C with lateral boundary at 1 m 3. Deep excavation 3.1. Geometry, basic assumptions and computational steps The general layout of the problem follows from Figure 6 and the following additional specifications have been given: - plane strain - influence of diaphragm wall construction is neglected, i.e. initial stresses are calculated without the wall, then wall is "wished-in-place" - diaphragm wall modelling: beam elements or continuum elements (E b = 3.e6 kpa, ν =.15, d =.8 m) - interface elements between wall and soil - horizontal hydraulic cut off at -3. m is not considered as structural support, the same mechanical properties as for the surrounding soil are assumed - hydrostatic water pressures correspond to water levels inside and outside excavation (groundwater lowering is performed in one step before excavation starts) - anchors are modelled as rods, the grouted body as membrane element which guarantee a continuous load transfer to the soil - given anchor forces in Figure 1 are design loads Computational steps to be performed: stage : initial stress state (given by σ' v = γz, σ' h = K o γz, K o =.43) stage 1: activation of diaphragm wall and groundwater lowering to m stage 2: excavation step 1 (to level -4.8 m) stage 3: activation of anchor 1 at level -4.3 m and prestressing stage 4: excavation step 2 (to level -9.3 m) stage 5: activation of anchor 2 at level -8.8 m and prestressing stage 6: excavation step 3 (to level m) stage 7: activation of anchor 3 at level m and prestressing stage 8: excavation step 4 (to level m) Distance and prestressing loads for anchors follow from Figure 6.

6 z x 3 m excavation step 1 = - 4.8m excavation step 2 = - 9.3m 2-3 x width of excavation.m GW = -3.m below surface m 8.m excavation step 3 = m excavation step 4 = -16.8m -17.9m m 23.8m 8.m 8.m top of hydraulic barrier = -3.m -32.m = base of diaphragm wall 2-3 x width of excavation γ'=γ' sand Specification for anchors: prestressed anchor force: 1. row: 768KN 2. row: 945KN 3. row: 98KN distance of anchors: 1. row: 2.3m 2. row: 1.35m 3. row: 1.35m cross section area: 15 cm 2 Young's modulus E = 2.1 e8 kn/m 2 sand.8m Figure 6. Geometry and excavation stages 3.2. Material parameters Some reference values for stiffness and strength parameters from the literature, frequently used in the design of excavations in Berlin sand, were given (z = depth below surface): E s 2 z kpa E s 6 z kpa ϕ = 35 γ = 19 kn/m 3 γ' = 1 kn/m 3 K o = 1 sin ϕ for < z < 2 m for z > 2 m (medium dense) In addition to these values from literature, results from oedometer tests (on loose and dense samples) and triaxial tests (confining pressures σ 3 = 1, 2 and 3 kpa) have been provided. It was not possible to include a significantly large number of test results and thus the question arose whether the stiffness values obtained from the oedometer test have been representative. If, for example, the constitutive model requires a tangential oedometric stiffness at a reference pressure of 1 kpa as an input parameter, a value of only Es 12 kpa was found based on these experiments. If a secant modulus for a pressure range beyond 2 kpa is determined a value of about 4 kpa is obtained. This was considered as too low by many authors and indeed other test results from Berlin sand in the literature indicate higher values. For example from Ohde (1951) values of about 35 to 45 kpa could be estimated as reference loading modulus of a medium dense sand at a reference pressure of 1 kpa. Properties for the diaphragm wall (linear elastic): E = 3 x 1 3 kpa ν =.15 γ = 24 kn/m 3

7 3.3. Comments on solutions submitted A wide variety of programmes and constitutive models has been employed to solve this problem. Simple elastic-perfectly plastic material models such as the Mohr-Coulomb or Drucker-Prager failure criterion (B1, B4, B5, B6, B7, B9, B12 and B16), still widely used in practice have been chosen by a number of authors. Several entries utilized the computer code PLAXIS (Brinkgreve & Vermeer 1998) with the so-called Hardening Soil model. One submission used a similar plasticity model with a simplified small strain stiffness formulation for the elastic range (B14). Three entries employed a hypoplastic formulation (B3, B3a and B13), B3 without and B3a and B13 with considering intergranular strains (Niemunis & Herle 1997). Only marginal differences exist in the assumptions of strength parameters (everybody trusted the experiments in this respect), the angle of internal friction ϕ was taken as 36 or 37 and a small cohesion was assumed to increase numerical stability by some authors. A significant variation was observed however in the assumption of the dilatancy angle ψ, ranging from to 15. For reasons mentioned earlier only a limited number of analysts used the provided laboratory test results to calibrate their material model. Most of the analysts used data from the literature from Berlin sand or their own experience to arrive at input parameters for their analysis assuming an increase with depth either by introducing some sort of power law or by defining different layers with different (constant) Young's moduli. However the choice of the reference moduli for primary loading and unloading/reloading varied significantly. Additional variation was introduced through different formulations for interface elements (zero thickness, finite thickness), element types (linear, quadratic), domains analysed (the width of meshes varied from 8 to 16 m, the depth from 5 to 16 m), modelling of the prestressed anchors, implementation details of constitutive models and the solution procedure with respective convergence criteria. The latter aspect is commonly ignored in practice but it can be easily shown that it may have a significant influence not only for stress levels near failure but also for working load conditions (Potts and Zdravkovic, 1999) Selected results Because some of the analyses made extremely unrealistic assumptions for the material parameters (B2, B3, B7, B9 and B17), they have been excluded for the comparison presented in the following. Figure 7a depicts lateral displacements of the diaphragm wall due to lowering of the groundwater level inside the excavation pit to m below surface. No clear trend e.g. with respect to the constitutive model could be identified, B6 is an elastic-perfectly plastic model but so is B16, both on the opposite sides of the range of results. Observing this variety of results already in the first construction stage, it is of course not surprising that the scatter increases with further calculation steps which will be shown later. It should be emphasized at this stage that not only the assumption of the constitutive model and the parameters have a significant influence on the result of this construction stage but also the way the groundwater lowering is simulated in the numerical analysis. Programme specific implementation details, the commercial user of a particular software may not be aware of, will contribute to the differences shown in Figure 7a. Because of these possible differences in modelling the groundwater lowering depending on the software used, it was investigated whether a more clear picture would evolve if a construction stage without the influence of the groundwater lowering is considered. For that purpose the wall deflection for excavation step 1 (to -4.8 m below surface) was plotted setting displacements to zero before this construction stage. The result follows from Figure 7b and the significant scatter already at this stage is obvious. Although most of the differences can be attributed to the stiffness parameters chosen as input, a few additional conclusions can be drawn. The largest horizontal displacement is obtained from the hypoplastic analysis, which was not the case in the previous construction stage (groundwater lowering). This indicates the strong response of these models on the stress paths, which are obviously quite different for these two construction steps. This effect of different stress paths is also observed in the other models but by far not to the same extent. The elasticplastic models with stress dependent stiffness (B2a, B8, B1 and B14) tend to give smaller

8 displacements compared to the elastic-perfectly plastic models. Exceptions are B5 and B16, which show a distinctly different deflection curve although the Young's modulus chosen is similar to other entries. Most probably is due to the fact that they did not use an interface element for modelling the soil/wall interaction B1 B2a B4 B5 B6 B8 B9 B1 B11 B12 B13 B14 B depth below surface [m] B1 B2a B4 B5 B6 B8 B9 B1 B11 B12 B13 B14 B depth below surface [m] horizontal displacement [mm] horizontal displacement [mm] Figure 7. Wall deflection a) after groundwater lowering, b) for first excavation step Limited in situ measurements are available for this project and although some simplifications compared to the actual construction have been introduced for this benchmark exercise in order to facilitate the calculations, the order of magnitude of displacements can be assumed to be known. Figure 8 shows the measured wall deflection for the final construction stage together with calculated values. It should be mentioned that measurements have been taken by inclinometer readings, fixed at the base of the wall, but unfortunately no geodetic survey of the wall head is available. It is very likely that the wall base moves horizontally and a parallel shift of the measurement is thought to reflect the in situ behaviour more closely, and therefore the measurement readings have been shifted by 1 mm in Figure 8. This is confirmed by other measurements under similar conditions. The calculated maximum horizontal wall displacement for all results considered varies between approximately 1 to 65 mm (exception B6). The shape of the deflection curves is also quite different. Some results indicate the maximum displacement slightly above the final excavation level, others show the maximum value at the top of the wall.

9 measurement (corrected) B1 B2a B4 B5 B6 B8 B9 B1 B11 B12 B13 B14 B15 B16 measurement (corrected) depth below surface [m] horizontal displacement [mm] Figure 8. Calculated wall deflections after final excavation step distance from wall [m] vertical displacement of surface [mm] B1 B2a B4 B5 B6 B8 B9 B1 B11 B12 B13 B14 B15 B16 Figure 9. Calculated surface settlements after final excavation step

10 When comparing the results of the calculations with the measurements it has to be pointed out that the simplification introduced in modelling the groundwater lowering (one step lowering instead of step-wise lowering according to the excavation progress) leads to higher horizontal displacements. Further studies revealed that the difference in calculated horizontal displacements due to the difference in modelling the groundwater lowering is strongly dependent on the constitutive law employed and ranges in the order of 5 to 15 mm. This may be one of the reasons why B15, which is an elastic-plastic analysis with stepwise groundwater lowering, is close to the measurement, but it also means that all solutions predicting less than 3 mm of horizontal displacement are far off reality. Figure 9 depicts the calculated surface settlements. Settlements of 45 mm (B11) have to be compared with a heave of about 15 mm (B4). Considering the fact that calculation of surface settlements is one of the main goals of such an analysis these results are not very encouraging. 4. Conclusion Results from a geotechnical benchmark exercise have been presented. A typical problem of a deep excavation in Berlin sand, formulated by the working group 1.6 of the German Society for Geotechnics, has been solved by a number of geotechnical engineers from universities and consulting companies utilizing different finite element codes and constitutive models. The comparison of the solutions submitted showed a wide scatter in results and only the most extreme solutions on the far end of the range could be explained with respect to assumptions of input parameters made in the analysis. Some of the results showed obvious errors such as incorrect prestress forces of anchors but most analyses made reasonable assumptions for parameters, discretisation and other modelling details. This benchmark exercise demonstrates the strong need for guidelines and recommendations how to model typical geotechnical problems in practice. Pitfalls and unrealistic modelling assumptions, the commercial user may not be aware of, have to be pointed out and procedures have to be developed to identify these. 5. References Brinkgreve R.B.J., Vermeer P.A. (1998), PLAXIS: Finite element code for soil and rock analyses, version 7. Balkema. Carter, J.P, Desai, C.S., Potts, D.M., Schweiger, H.F. & S.W. Sloan 2. Computing and Computer Modelling in Geotechnical Engineering. Proc. GeoEng2, Melbourne, Technomic Publishing, Lancaster. (Vol. 1: invited papers), Niemunis, A. & I. Herle Hypoplastic model for cohesionless soils with elastic strain range. Mechanics of cohesive-frictional materials, 2, Ohde, J Grundbaumechanik (in German), Huette, BD. III, 27. Auflage. Potts, D.M & L. Zdravkovic (1999). Finite element analysis in geotechnical engineering Theory. Thomas Telford Potts, D.M & L. Zdravkovic (21). Finite element analysis in geotechnical engineering Application. Thomas Telford Schweiger, H. F Results from two geotechnical benchmark problems. Proc. 4th European Conf. Numerical Methods in Geotechnical Engineering, Cividini, A. (ed.), Springer, Schweiger, H. F. 2. Ergebnisse des Berechnungsbeispieles Nr. 3 "3-fach verankerte Baugrube". Tagungsband Workshop "Verformungsprognose für tiefe Baugruben", Stuttgart, (in German) REFERENCE: Results from numerical benchmark exercises in geotechnics Proc. 5th European Conf. Numerical Methods in Geotechnical Engineering, Presses Ponts et chaussees, Paris, 22,