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Examples of successful numerical modelling of complex geotechnical problems

Examples of successful numerical modelling of complex geotechnical problems Over the last decades, numerical methods have gained increasing importance in practical geotechnical engineering and numerical methods have become a standard tool in geotechnical design, widely accepted by the geotechnical profession. The advantages of numerical analyses for solving practical problems have been recognised, and developments in software and hardware allow their application in practice with reasonable effort. However, there is still a gap between practice and research and, often unnecessary, simplifications are made in practice and therefore the full power of numerical analyses is not always utilised. One reason for this discrepancy is a lack of transfer of knowledge from research into practice but also a lack of theoretical background of numerical methods, constitutive modelling and modern soil mechanics in practice. In this paper, the application of advanced numerical models for solving practical geotechnical problems is shown, whereas the examples have been chosen in such a way that different aspects are highlighted in each case. Results from fibre-optic measurements for a pull-out test of a ground anchor in soft soil could be reproduced by employing advanced constitutive models, in particular for the grout, in the bonded length of the anchor. For this test, a class-A prediction has been made and numerical results have then been compared with in situ measurements. The back-analysis of a slow-moving landslide is presented next, where the rate of deformation is influenced by water level changes in a reservoir for a pumping power plant, creep of lacustrine sediments and environmental effects such as rainfall infiltration. Finally, some results of modelling cone penetration testing in silts are presented highlighting the effects of anisotropic permeability. Keywords Finite element analysis · Anchor load test · Slope stability · CPT Introduction Although the advantages of numerical modelling are obvious, it has to be mentioned that the role of numerical Numerical methods have proven to be an important and analyses in geotechnical engineering is different as com - powerful tool for solving practical geotechnical problems. pared to other engineering disciplines such as, for example, This has been possible on the one hand because finite ele - mechanical or structural engineering. The reasons for this ment/finite difference codes have been developed to a stage are specific aspects of geotechnical engineering, such as that they can be easily operated by geotechnical engineers. On the other hand, constitutive models which are able to 1. In geotechnics, the “construction material” is natural describe important features of soil behaviour have been ground (soil and rock) and not man-made such as con- implemented in a robust manner in these codes, although it crete and steel, fabricated to predefined specifications. has to be emphasised that open questions in soil modelling This inevitably means that the material is inhomoge- remain and there is still no generally accepted constitutive neous, its mechanical and hydraulical behaviour is not model for soils available. easily formulated in mathematical terms and material parameters are difficult to determine. 2. Even with a perfect site investigation scheme, significant * H. F. Schweiger uncertainties remain with respect to the soil profile and helmut.schweiger@tugraz.at thus with the geotechnical model which forms the basis Institute of Soil Mechanics, Foundation Engineering for the numerical model. and Computational Geotechnics, Graz University of Technology, Graz, Austria Vol.:(0123456789) 1 3 2 Page 2 of 10 Innovative Infrastructure Solutions (2019) 4:2 3. Installation processes, such as construction of piles, dia- in the bonded length could be identified. In order to take phragm walls, stone columns, mixed-in-place columns, into account cracking in the numerical model, an advanced jet grout panels, have an influence on the stress regime constitutive model for the grout has been employed. The test in the soil, which is still extremely difficult, if not impos- was performed on a construction site in St. Kanzian, Aus- sible, to quantify numerically. tria. The anchor was vertically installed, was post-grouted 4. Geometric simplification has to be introduced (2D vs and had a free length of 12 m and a fixed length of 8 m, 3D), and the domain of the model to be analysed may respectively. not always be easily identified. Laboratory testing of soil samples extracted close to the test area classified the soil as clayey, sandy silt of low In the following, an attempt is made to show the benefits plasticity and the grain size distribution was approximately of using numerical methods in geotechnical engineering by 70% silt, 15% clay and 15% sand. Direct shear tests and means of practical examples, addressing an in situ anchor oedometer tests were performed in samples located at 17 m load test, a complex slope stability problem and cone pen- depth and 24 m depth. In addition, a seismic dilatometer test etration testing. (sDMT) was conducted to determine the shear wave velocity. The soil is referred herein as “seeton”. Thin sand layers are also present. Based on this information, the parameters for Example: anchor pull‑out test the employed constitutive model (the Hardening Soil Small model as implemented in the finite element code Plaxis 2D) Soil conditions and test arrangement were determined (see Table 1). The tendon was modelled as a linear elastic material and the grout with the so-called Analysing an anchor pull-out test by means of numerical Plaxis Shotcrete model, which is a nonlinear constitutive modelling provides a very useful tool not only to predict the model allowing for post-peak softening in compression and ultimate load of the anchor but also to have a better insight tension and is therefore able to capture the development of into the interactions between the tendon, the grout and the cracking in the grout, at least in an approximate manner. soil. In this particular case, a class-A prediction of an in situ Parameters are summarised in Table 2. For more details on test was performed and these results were subsequently com- this model, the reader is referred to Schädlich and Schweiger pared with the in situ performance of the tested anchor. The [9]. monitoring system not only involved the standard set-up for an anchor load test to obtain the load–displacement curve but included fibre-optic measurements in tendon and grout of the anchor. In this way, for example, cracking of the grout Table 1 Parameters for “seeton” Parameter Description Unit Seeton Sand E Primary loading stiffness at ref. pressure kPa 6625 24,000 50,ref E Oedometric stiffness at ref. pressure kPa 5300 24,000 oed,ref E Un/reloading stiffness at ref. pressure kPa 48,000 72,000 ur,ref G Small strain shear modulus kPa 120,000 120,000 0ref γ Shear strain at 70% G – 0.15E−3 0.15E−3 0.7 0ref c′ Effective cohesion kPa 10 5 φ′ Effective friction angle ° 29 35 Table 2 Tendon and grout Parameter Description Unit Tendon Grout properties E Young’s modulus kPa 195,000,000 16,260,000 f Uniaxial compressive strength kPa – 32,120 c,28 f Uniaxial tensile strength kPa – 2000 t,28 G Compressive fracture energy kN/m – 50 c,28 G Tensile fracture energy kN/m – 0.15 t,28 f Ratio residual/peak tensile strength – – 0.05 tun φ′ Maximum friction angle ° – 40 1 3 Innovative Infrastructure Solutions (2019) 4:2 Page 3 of 10 2 increased to 280 mm (borehole diameter was 178 mm) based on the amount of grout pumped into the soil. It is acknowl- edged that this approach of taking into account the ee ff cts of pressure grouting is highly simplified; nevertheless, it can be justified from a practical point of view and can be considered to be sufficient for the purpose of this study. Figure 2a shows the load–displacement curves obtained numerically (class-A prediction) and measured results. The behaviour in the numerical model is slightly stiffer, but it has to be taken into account that in the test the load has been kept constant at each loading stage and the “creep” has been measured. Up to about 1100 kN, the “creep” was small; however, in the numerical model, it is completely neglected. At 1100 kN, the creep increased significantly, and accord- ing to testing standards, this would be considered as failure. Considering these aspects, the comparison between predic- tion and test results can be considered very reasonable. The shear stress distribution along the interface grout–soil is Fig. 1 Model geometry a overview and b anchor detail presented in Fig. 2b and, as expected, stresses are higher within the stiffer sand layers. The ultimate pull-out load of about 1100 kN is achieved when the maximum shear stress Numerical model and simulation results is mobilised along the entire fixed length. The contribution of the grout on the strain distribution The numerical simulation was performed using the finite element software Plaxis 2D 2016 [4]. Because the anchor along the tendon can be evaluated by the tension softening parameter H , an output of the Shotcrete model. If H exceeds was installed vertically, the model is axisymmetric and only t t half of the anchor geometry was modelled. The model geom- zero, softening in tension starts and cracks start to develop, leading to an increase in the strains along the tendon. If H etry is presented in Fig. 1. The tendon was considered only in the fixed length, and the grout was applied in the free and is larger than 1, the tensile stress decreases to its residual value (practically zero) and the strains oscillate along the fixed length, whereas pressure grouting was only assumed in the latter. The load was applied by means of vertical pre- tendon. The strain distribution in the tendon and the vari- ation of H in the grout are shown in Fig. 3a, b. Although scribed displacements on top of the tendon. The pressure grouting was considered by increasing the radial stresses an exact comparison with respect to crack location is not possible, a qualitative assessment can be made as shown in along the entire section comprising the fixed length and thus the earth pressure coefficient at rest (K ) was set equal to Fig. 4a, b where strains from the fibre-optic measurements in the grout and the H -parameter are compared for different one. In addition, the diameter of the bonded section was Fig. 2 Numerical results a load–displacement curve and b shear stress distribution 490 kN 700 kN 900 kN 1100 kN In-situ Numerical 0 8 020406080 100 120 140 160 180 200 Displacement (mm) (kPa) (a) (b) 1 3 Load (kN) Depth (m) 2 Page 4 of 10 Innovative Infrastructure Solutions (2019) 4:2 Fig. 3 Numerical results a 0 strains along the tendon and b H distribution in the grout 2 2 3 3 5 5 6 6 490 kN - Numerical 490 kN - Numerical 7 7 700 kN - Numerical 700 kN - Numerical 010002000300040005000 0,00,2 0,40,6 0,81,0 Strain ( m/m) H (a) (b) Fig. 4 Crack development in 0 grout of fixed length—numeri- -1 cal results versus measurements -1 -2 -2 -3 -3 -4 -4 -5 -5 -6 -6 Grout 900 kN Grout 485 kN -7 -7 Num. grout - 485 kN Num. grout - 900 kN -8 -8 -1000 0 1000 2000 3000 4000 -10000 1000 2000 3000 4000 Strain (m/m) Strain (m/m) (a) (b) load levels. Again, the comparison can be considered satis- The horizontal length of the landslide is roughly 270 m. factory. At present, a more detailed analysis of the results Movements were detected over a large part of the storage (numerical and measurements) is in progress to gain a clear basin length. The inclination of the slope is 30° on aver- picture of the load transfer between tendon/grout and grout/ age. Based on inclinometer measurements, the sliding sur- soil. Furthermore, two more tests, with similar instrumenta- faces could be identified between 20 and 40 m depth below tion, are underway in different ground conditions in order to ground surface. The subsurface explorations showed a identify differences in load transfer mechanisms depending sliding mass consisting mainly of weathered and sheared on ground conditions. rock. Below the sliding mass and below the water storage basin, lacustrine fine sediments, mainly silt, are present. A plan view and a layout of the slow-moving landslide Example: slow‑moving landslide and the water storage basin are shown in Fig. 5. Due to the operation of the pump storage power plant, the water Problem description level in the basin changes up to three times a day, whereas the maximum level change is 7.0 m. In the course of extension works for a water storage Due to the increased risk of a destructive flood wave, basin, a slow-moving landslide was identified next to a comprehensive monitoring system was installed includ- the storage basin. The dimensions of the water storage ing inclinometers and pore water pressure gauges. The basin are roughly 400 m in length and 100 m in width. pore pressure measurements showed changes in pore water 1 3 Depth (m) Depth (m) Depth (m) Depth (m) Innovative Infrastructure Solutions (2019) 4:2 Page 5 of 10 2 Fig. 5 Layout and plan view of water storage basin and slow- moving landslide Fig. 6 a Excess pore water pressures due to water level changes; b correlation between deformation rates and water level changes pressures in the subsoil at the toe of the landslide linked to subsoil. A typical comparison of water level changes and the water table changes in the water storage basin, but it was rotation of the inclinometer probe in the major sliding zone observed that this was not a one-to-one relationship because is shown in Fig. 6b. positive and negative excess pore pressures have been meas- ured, i.e. the pressure level of the pore pressure measurement Numerical model and results is above or below the corresponding water level in the water reservoir. Typical measurement results of the pore pressure Preliminary numerical analyses and careful examination of gauges are shown in Fig. 6a. For a better understanding of the measurements revealed that water level changes alone the reasons for these measured excess pore water pressures, could not explain measured displacements. Therefore, a a numerical study was carried out. These analyses reveal, finite element model (Fig.  7), using the code Plaxis 2D as expected, that the magnitude of these excess pore water [4], was set up incorporating the water level changes from pressures depends on the ratio between drawdown velocity the storage basin but also environmental influences such and soil permeability and furthermore on the ratio between as rainfall events assuming site-specific precipitation and pore water compressibility and soil skeleton compressibil- evaporation. The precipitation was measured on site. The ity [3]. Also, the presence of air bubbles in the water may potential evaporation was estimated according to Thorn- contribute to this effect [ 2]. In Fig. 6a, the excess pore water thwaite [10], whereas the influences of the soil suction on pressure p_excess is defined by the difference of measured the evaporation and the transpiration were neglected due and hydrostatic pore water pressure. Deformation measure- to the lack of measurement data. Furthermore, the lacus- ments using in-place inclinometers revealed a correlation trine fine sediments at the slope toe were modelled with between deformation rates and excess pore water pressures the Soft Soil Creep model [11] to model the creep behav- at the slope toe. This shows that the slope deformations are iour of these soil layers (see Fig. 7). To achieve appropri- influenced by the water level changes in the water storage ate initial conditions concerning the stress state and the basin, as these control the excess pore water pressures in the hydraulic conditions, the geological history was modelled 1 3 2 Page 6 of 10 Innovative Infrastructure Solutions (2019) 4:2 Fig. 7 FE model for numerical back-calculations [1] Fig. 8 Back-calculated excess pore water pressures [1] in a simplified way. Afterwards, 1 year with characteristic level changes. Figure 8 shows a comparison of the meas- precipitation and water level changes was simulated [1]. ured and calculated excess pore water pressures for the The back-calculation of the pore water pressures was per- two installed pore water pressure gauges PPG 1 (21 m) and formed for several periods with different types of water PPG 2 (33 m). Furthermore, the water level in the storage 1 3 Innovative Infrastructure Solutions (2019) 4:2 Page 7 of 10 2 Fig. 9 Displacements at node E (in the middle of the slope) for different influencing factors [1] basin is plotted in the diagram. From the comparison in with increasing distance from the slope toe. The influence of Fig. 8, it can be seen that a good agreement between cal- precipitation and evaporation is increasing in the middle and culations and measurements could be achieved. upper part of the slope, and the influence of the water level In order to identify the quantitative contribution of the dif- changes on the slope deformations is almost constant over the ferent influencing factors on the total displacements, separate entire slope. Based on the back-calculations of the slope defor- calculation phases were performed. In each phase, a new influ- mations, it can be argued that the water level changes are the encing factor was considered. As an example, the results for a main reason for the slope movements. However, slope move- point in the middle of the slope are shown in Fig. 9. The dif- ments would also occur without the storage operation but the ference between two time–displacement curves is the influence magnitude of the deformations would be smaller. of each additionally considered factor as indicated by the label of the curves. A comparison between the in situ measurements and the total displacements from the calculation indicates a Example: numerical simulation of cone good agreement. According to Fig. 9, the displacements are penetration test mainly due to the water level changes in the storage basin but creep behaviour of the lacustrine sediments at the base of the Particle finite element method slope and precipitation add to the displacements. The results were evaluated for several points along the slope. It could be In situ investigation methods, such as cone penetration clearly shown that the influence of the creep behaviour of testing (CPT), are frequently used to derive hydraulic and the lacustrine fine sediments on the deformations decreases mechanical soil properties from measured tip resistance, 1 3 2 Page 8 of 10 Innovative Infrastructure Solutions (2019) 4:2 sleeve friction and pore water pressure via empirical cor- added and old ones removed, in order to deal with large relations. Experience has shown that CPT provides reason- deformations and avoid excessive mesh distortion. This able results for applications in sand or clay where either strategy results in an increased computational cost and drained or undrained behaviour governs the penetration therefore low-order elements in combination with a mixed process. However, correlations for partial drainage, as it formulation of the problem are used. In order to avoid occurs during penetration in intermediate soils such as locking effects, an additional degree of freedom, namely silts, are still an ongoing research topic. Recent advances the determinant J of the deformation gradient, is intro- in the numerical simulation of large deformation prob- duced on top of the displacement and water pressure fields lems based on a Particle Finite Element Method (PFEM, u and pw [7]. Furthermore, the problem is stabilised using see [8]) allow to model this kind of penetration problems the Polynomial Pressure Projection. The interested reader where a rigid cone penetrates a fully water-saturated soil is referred to Monforte et al. [6] and Monforte et al. [7] for body. The simulations here are carried out using the plat- a more detailed outline of the PFEM. form G-PFEM, short for Geotechnical-PFEM [6, 7], which has been developed within the Kratos framework [5] at the Polytechnic University of Catalonia (UPC) and the Numerical model for cone penetration Center for Numerical Methods in Engineering (CIMNE). In G-PFEM, the quasi-static linear momentum and mass Modelling a CPT involves an ideally rigid cone that pene- balance equations are formulated for a solid and fluid trates a deformable two-phase medium at a constant velocity. phase adopting an updated Lagrangian description. The This leads to an axisymmetric model consisting of a rectan- basic idea behind the PFEM is a continuous remeshing of gular box with a height of 1.1 m and a width of 0.5 m. The critical regions of the domain, where new nodes can be cone radius R measures 1.78 cm with a tip angle of 60° cor- responding to the standard geometry (base area of 10 cm ). The penetration starts from an initial position where the cone is located at a depth of 10 cm. The lateral and lower boundaries are fixed in normal direction while an overbur - den pressure can be applied at the top of the domain. Moreo- ver, free drainage is allowed at the boundaries except along the symmetry axis. Figure 10 shows the basic model. From a mathematical point of view, the contact between cone and soil body is described by a set of constraints and a penalty method is adopted. The contact algorithm is explained in detail in Monforte et al. [6]. The constitutive behaviour of the soil is described by means of the Modified Cam Clay (MCC) model, and the parameters for this particular analysis are given in Table 3. Influence of anisotropic permeability In order to study the effect of anisotropic permeability, the set-up described in the previous section was analysed with a standard penetration velocity of 2 cm/s, a smooth interface between cone and soil and anisotropic perme- Fig. 10 Axisymmetric model (left) and refined mesh during penetra- ability, whereas two cases have been investigated. Test tion process (right) Table 3 Input parameters for MCC model 3 3 * * ρ (kg/m ) ρ (kg/m ) λ (–) κ (–) φ′ (°) M (–) G (kPa) α (–) s w 0 1700 1000 0.015 0.005 22.5 0.88 2900 0 OCR (kPa) p (kPa) K (m/s) k (m/s) k (–) K (°) φ (–) e (–) c0 w v h 0 int 0 8 −8 −7 1 100 1 × 10 2 × 10 2 × 10 0.7 7 0.5 1 3 Innovative Infrastructure Solutions (2019) 4:2 Page 9 of 10 2 is vertical. Consequently, although the overall behaviour is rather undrained, differences can be observed locally which may become important when deriving coefficients of consolidation from dissipation tests. The effect of anisotropic permeability on the pore pressure field is shown in Fig.  12. It is observed that the order of magnitude of the calculated water pressures is the same, again as expected for undrained conditions, but the distribution differs. For case (b), the pressure bulb has an increased vertical extension along the preferred drain- age direction. Further studies are currently undertaken to assess in more detail the effect of partial drainage in inter - mediate soils on tip resistance, sleeve friction and pore water pressures. Conclusion The numerical simulation of an anchor pull-out test showed that the predicted load–displacement curve (“class-A” pre- diction) and the one obtained in  situ showed very good agreement. The ultimate pull-out load is achieved when the shear strength is mobilised along the entire interface grout–soil. The numerical simulation indicated that the Fig. 11 Comparison of q and u for inverted anisotropic permeabili- strains in the tendon are highly influenced by the develop- c 2 −8 −7 ties; case (a) with k = 2 × 10   m/s, k = 2 × 10   m/s and case (b) v h ment of cracks in the grout of the fixed length which could −7 −8 with k = 2 × 10  m/s, k = 2 × 10  m/s v h be captured by applying an advanced constitutive model for the grout. case (a) considers an increased horizontal permeability The second example, namely the back-analysis of a slow- −7 −8 with k = 2 × 10  m/s and k = 2 × 10  m/s, while case moving landslide at a water reservoir, proved that com- h v −8 (b) simulates the opposite case (k = 2 × 10   m/s and plex mechanisms contributing to slope movement, such as −7 k = 2 × 10  m/s). The anisotropy is crucial for the hydrau- water level fluctuations in the reservoir, creep phenomena lic behaviour and prescribes the preferred flow direction in soft lacustrine deposits present at the toe of the slope which is horizontal for case (a) and vertical for case (b). and environmental effects such as rainfall infiltration can The profiles of q and u are compared in Fig. 11 and in be accounted for in numerical analyses. It is acknowledged, c 2 both cases (a) and (b) overall undrained behaviour with u however, that due to significant uncertainties in input param- (u = most common position in CPT for measuring pore eters the solutions presented are by no means rigorous and pressures) being around 320  kPa and q of 600  kPa is unique but still provide a better insight into the mechani- observed. However, a closer look reveals that test case (a) cal behaviour of such slopes and will help the experienced gives a slightly lower pore pressures (315 kPa vs 330 kPa) geotechnical engineer in defining appropriate mitigating and a higher tip resistance (610 kPa vs 585 kPa) compared measures. to case (b). These observations are consistent with the In the last example, it is shown that G-PFEM is an assumption of a preferred flow direction due to anisotropy. appropriate tool for simulating CPT. The application pre- As the cone penetrates the soil, radial (or rather horizon- sented looked into the effect of considering anisotropic tal) flow is dominantly caused by the geometric boundary permeability of the soil layer. It could be shown that ani- conditions of the test. Thus, for case (a), the preferred flow sotropy influences the local pore pressure field under glob- directions due to anisotropy and problem geometry coin- ally undrained conditions which may have a consequence cide which leads to increased drainage of the system. For when deriving coefficients of consolidation from dissipa- case (b), the opposite occurs and the main flow direction tion tests. 1 3 2 Page 10 of 10 Innovative Infrastructure Solutions (2019) 4:2 Fig. 12 Pore pressure -100 fields for inverted aniso- tropic permeabilities; case −8 (a) with k = 2 × 10  m/s, −7 k = 2 × 10  m/s and case −7 (b) with k = 2 × 10  m/s, −8 -190 k = 2 × 10  m/s -190 -130 -130 -250 -250 -85 -85 -100 -100 (a) (b) Acknowledgements Open access funding provided by Graz University 4. Brinkgreve RBJ, Kumarswamy S, Swolfs WM (2016) PLAXIS of Technology. 2016. Finite element code for soil and rock analyses, user manual. Plaxis bv, Delft 5. Dadvand P, Rossi R, Oñate E (2010) An object-oriented environ- Open Access This article is distributed under the terms of the Crea- ment for developing finite element codes for multidisciplinary tive Commons Attribution 4.0 International License (http://creat iveco applications. Arch Comput Methods Eng 17(3):253–297 mmons.or g/licenses/b y/4.0/), which permits unrestricted use, distribu- 6. Monforte L, Arroyo M, Carbonell JM, Gens A (2017) Numerical tion, and reproduction in any medium, provided you give appropriate simulation of undrained insertion problems in geotechnical engi- credit to the original author(s) and the source, provide a link to the neering with the particle finite element method (PFEM). Comput Creative Commons license, and indicate if changes were made. Geotech 82:144–156 7. Monforte L, Carbonell JM, Arroyo M, Gens A (2017) Performance of mixed formulations for the particle finite element method in soil mechanics problems. Comput Part Mech 4(3):269–284 8. Oñate E, Idelsohn SR, Celigueta MA, Rossi R, Marti J, Carbonell References JM, Ryzakov P, Suárez B (2011) Advances in the particle finite element method (PFEM) for solving coupled problems in engi- 1. Ausweger GM (2017) Influences of water level changes on the neering. Part Based Methods 25:1–49 behaviour of a slow moving landslide—in situ measurements, 9. Schädlich B, Schweiger HF (2014) A new constitutive model for model test and numerical analyses. PhD thesis, Graz University shotcrete. In: Hicks MA, Brinkgreve RBJ, Rohe A (eds) Proceed- of Technology ings of the numerical methods in geotechnical engineering. Taylor 2. Ausweger GM, Schweiger HF (2016) Numerical study on the & Francis Group, London, pp 103–108 influence of entrapped air bubbles on the time-dependent pore 10. Thornthwaite CW (1948) An approach toward a rational classifica- pressure distribution in soils due to external changes in water tion of climate. Geogr Rev 38(1):55–94 level. In: Proceedings of the 3rd European conference on unsatu- 11. Vermeer PA, Neher HP (1999) A soft soil model that accounts rated soils, paper #16010, Paris, 12–14 Sept 2016 for creep. In: Proceedings of the international symposium beyond 3. Ausweger GM, Schweiger HF (2017) Numerical investigation 2000 in computational geotechnics of excess pore water pressures due to external fluctuating water tables. In: Proceedings 15th international conference computer methods and recent advances in geomechanics, Wuhan, China 1 3 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Innovative Infrastructure Solutions Springer Journals

Examples of successful numerical modelling of complex geotechnical problems

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Springer Journals
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Copyright © 2018 by The Author(s)
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Earth Sciences; Geotechnical Engineering & Applied Earth Sciences; Environmental Science and Engineering; Geoengineering, Foundations, Hydraulics
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Abstract

Over the last decades, numerical methods have gained increasing importance in practical geotechnical engineering and numerical methods have become a standard tool in geotechnical design, widely accepted by the geotechnical profession. The advantages of numerical analyses for solving practical problems have been recognised, and developments in software and hardware allow their application in practice with reasonable effort. However, there is still a gap between practice and research and, often unnecessary, simplifications are made in practice and therefore the full power of numerical analyses is not always utilised. One reason for this discrepancy is a lack of transfer of knowledge from research into practice but also a lack of theoretical background of numerical methods, constitutive modelling and modern soil mechanics in practice. In this paper, the application of advanced numerical models for solving practical geotechnical problems is shown, whereas the examples have been chosen in such a way that different aspects are highlighted in each case. Results from fibre-optic measurements for a pull-out test of a ground anchor in soft soil could be reproduced by employing advanced constitutive models, in particular for the grout, in the bonded length of the anchor. For this test, a class-A prediction has been made and numerical results have then been compared with in situ measurements. The back-analysis of a slow-moving landslide is presented next, where the rate of deformation is influenced by water level changes in a reservoir for a pumping power plant, creep of lacustrine sediments and environmental effects such as rainfall infiltration. Finally, some results of modelling cone penetration testing in silts are presented highlighting the effects of anisotropic permeability. Keywords Finite element analysis · Anchor load test · Slope stability · CPT Introduction Although the advantages of numerical modelling are obvious, it has to be mentioned that the role of numerical Numerical methods have proven to be an important and analyses in geotechnical engineering is different as com - powerful tool for solving practical geotechnical problems. pared to other engineering disciplines such as, for example, This has been possible on the one hand because finite ele - mechanical or structural engineering. The reasons for this ment/finite difference codes have been developed to a stage are specific aspects of geotechnical engineering, such as that they can be easily operated by geotechnical engineers. On the other hand, constitutive models which are able to 1. In geotechnics, the “construction material” is natural describe important features of soil behaviour have been ground (soil and rock) and not man-made such as con- implemented in a robust manner in these codes, although it crete and steel, fabricated to predefined specifications. has to be emphasised that open questions in soil modelling This inevitably means that the material is inhomoge- remain and there is still no generally accepted constitutive neous, its mechanical and hydraulical behaviour is not model for soils available. easily formulated in mathematical terms and material parameters are difficult to determine. 2. Even with a perfect site investigation scheme, significant * H. F. Schweiger uncertainties remain with respect to the soil profile and helmut.schweiger@tugraz.at thus with the geotechnical model which forms the basis Institute of Soil Mechanics, Foundation Engineering for the numerical model. and Computational Geotechnics, Graz University of Technology, Graz, Austria Vol.:(0123456789) 1 3 2 Page 2 of 10 Innovative Infrastructure Solutions (2019) 4:2 3. Installation processes, such as construction of piles, dia- in the bonded length could be identified. In order to take phragm walls, stone columns, mixed-in-place columns, into account cracking in the numerical model, an advanced jet grout panels, have an influence on the stress regime constitutive model for the grout has been employed. The test in the soil, which is still extremely difficult, if not impos- was performed on a construction site in St. Kanzian, Aus- sible, to quantify numerically. tria. The anchor was vertically installed, was post-grouted 4. Geometric simplification has to be introduced (2D vs and had a free length of 12 m and a fixed length of 8 m, 3D), and the domain of the model to be analysed may respectively. not always be easily identified. Laboratory testing of soil samples extracted close to the test area classified the soil as clayey, sandy silt of low In the following, an attempt is made to show the benefits plasticity and the grain size distribution was approximately of using numerical methods in geotechnical engineering by 70% silt, 15% clay and 15% sand. Direct shear tests and means of practical examples, addressing an in situ anchor oedometer tests were performed in samples located at 17 m load test, a complex slope stability problem and cone pen- depth and 24 m depth. In addition, a seismic dilatometer test etration testing. (sDMT) was conducted to determine the shear wave velocity. The soil is referred herein as “seeton”. Thin sand layers are also present. Based on this information, the parameters for Example: anchor pull‑out test the employed constitutive model (the Hardening Soil Small model as implemented in the finite element code Plaxis 2D) Soil conditions and test arrangement were determined (see Table 1). The tendon was modelled as a linear elastic material and the grout with the so-called Analysing an anchor pull-out test by means of numerical Plaxis Shotcrete model, which is a nonlinear constitutive modelling provides a very useful tool not only to predict the model allowing for post-peak softening in compression and ultimate load of the anchor but also to have a better insight tension and is therefore able to capture the development of into the interactions between the tendon, the grout and the cracking in the grout, at least in an approximate manner. soil. In this particular case, a class-A prediction of an in situ Parameters are summarised in Table 2. For more details on test was performed and these results were subsequently com- this model, the reader is referred to Schädlich and Schweiger pared with the in situ performance of the tested anchor. The [9]. monitoring system not only involved the standard set-up for an anchor load test to obtain the load–displacement curve but included fibre-optic measurements in tendon and grout of the anchor. In this way, for example, cracking of the grout Table 1 Parameters for “seeton” Parameter Description Unit Seeton Sand E Primary loading stiffness at ref. pressure kPa 6625 24,000 50,ref E Oedometric stiffness at ref. pressure kPa 5300 24,000 oed,ref E Un/reloading stiffness at ref. pressure kPa 48,000 72,000 ur,ref G Small strain shear modulus kPa 120,000 120,000 0ref γ Shear strain at 70% G – 0.15E−3 0.15E−3 0.7 0ref c′ Effective cohesion kPa 10 5 φ′ Effective friction angle ° 29 35 Table 2 Tendon and grout Parameter Description Unit Tendon Grout properties E Young’s modulus kPa 195,000,000 16,260,000 f Uniaxial compressive strength kPa – 32,120 c,28 f Uniaxial tensile strength kPa – 2000 t,28 G Compressive fracture energy kN/m – 50 c,28 G Tensile fracture energy kN/m – 0.15 t,28 f Ratio residual/peak tensile strength – – 0.05 tun φ′ Maximum friction angle ° – 40 1 3 Innovative Infrastructure Solutions (2019) 4:2 Page 3 of 10 2 increased to 280 mm (borehole diameter was 178 mm) based on the amount of grout pumped into the soil. It is acknowl- edged that this approach of taking into account the ee ff cts of pressure grouting is highly simplified; nevertheless, it can be justified from a practical point of view and can be considered to be sufficient for the purpose of this study. Figure 2a shows the load–displacement curves obtained numerically (class-A prediction) and measured results. The behaviour in the numerical model is slightly stiffer, but it has to be taken into account that in the test the load has been kept constant at each loading stage and the “creep” has been measured. Up to about 1100 kN, the “creep” was small; however, in the numerical model, it is completely neglected. At 1100 kN, the creep increased significantly, and accord- ing to testing standards, this would be considered as failure. Considering these aspects, the comparison between predic- tion and test results can be considered very reasonable. The shear stress distribution along the interface grout–soil is Fig. 1 Model geometry a overview and b anchor detail presented in Fig. 2b and, as expected, stresses are higher within the stiffer sand layers. The ultimate pull-out load of about 1100 kN is achieved when the maximum shear stress Numerical model and simulation results is mobilised along the entire fixed length. The contribution of the grout on the strain distribution The numerical simulation was performed using the finite element software Plaxis 2D 2016 [4]. Because the anchor along the tendon can be evaluated by the tension softening parameter H , an output of the Shotcrete model. If H exceeds was installed vertically, the model is axisymmetric and only t t half of the anchor geometry was modelled. The model geom- zero, softening in tension starts and cracks start to develop, leading to an increase in the strains along the tendon. If H etry is presented in Fig. 1. The tendon was considered only in the fixed length, and the grout was applied in the free and is larger than 1, the tensile stress decreases to its residual value (practically zero) and the strains oscillate along the fixed length, whereas pressure grouting was only assumed in the latter. The load was applied by means of vertical pre- tendon. The strain distribution in the tendon and the vari- ation of H in the grout are shown in Fig. 3a, b. Although scribed displacements on top of the tendon. The pressure grouting was considered by increasing the radial stresses an exact comparison with respect to crack location is not possible, a qualitative assessment can be made as shown in along the entire section comprising the fixed length and thus the earth pressure coefficient at rest (K ) was set equal to Fig. 4a, b where strains from the fibre-optic measurements in the grout and the H -parameter are compared for different one. In addition, the diameter of the bonded section was Fig. 2 Numerical results a load–displacement curve and b shear stress distribution 490 kN 700 kN 900 kN 1100 kN In-situ Numerical 0 8 020406080 100 120 140 160 180 200 Displacement (mm) (kPa) (a) (b) 1 3 Load (kN) Depth (m) 2 Page 4 of 10 Innovative Infrastructure Solutions (2019) 4:2 Fig. 3 Numerical results a 0 strains along the tendon and b H distribution in the grout 2 2 3 3 5 5 6 6 490 kN - Numerical 490 kN - Numerical 7 7 700 kN - Numerical 700 kN - Numerical 010002000300040005000 0,00,2 0,40,6 0,81,0 Strain ( m/m) H (a) (b) Fig. 4 Crack development in 0 grout of fixed length—numeri- -1 cal results versus measurements -1 -2 -2 -3 -3 -4 -4 -5 -5 -6 -6 Grout 900 kN Grout 485 kN -7 -7 Num. grout - 485 kN Num. grout - 900 kN -8 -8 -1000 0 1000 2000 3000 4000 -10000 1000 2000 3000 4000 Strain (m/m) Strain (m/m) (a) (b) load levels. Again, the comparison can be considered satis- The horizontal length of the landslide is roughly 270 m. factory. At present, a more detailed analysis of the results Movements were detected over a large part of the storage (numerical and measurements) is in progress to gain a clear basin length. The inclination of the slope is 30° on aver- picture of the load transfer between tendon/grout and grout/ age. Based on inclinometer measurements, the sliding sur- soil. Furthermore, two more tests, with similar instrumenta- faces could be identified between 20 and 40 m depth below tion, are underway in different ground conditions in order to ground surface. The subsurface explorations showed a identify differences in load transfer mechanisms depending sliding mass consisting mainly of weathered and sheared on ground conditions. rock. Below the sliding mass and below the water storage basin, lacustrine fine sediments, mainly silt, are present. A plan view and a layout of the slow-moving landslide Example: slow‑moving landslide and the water storage basin are shown in Fig. 5. Due to the operation of the pump storage power plant, the water Problem description level in the basin changes up to three times a day, whereas the maximum level change is 7.0 m. In the course of extension works for a water storage Due to the increased risk of a destructive flood wave, basin, a slow-moving landslide was identified next to a comprehensive monitoring system was installed includ- the storage basin. The dimensions of the water storage ing inclinometers and pore water pressure gauges. The basin are roughly 400 m in length and 100 m in width. pore pressure measurements showed changes in pore water 1 3 Depth (m) Depth (m) Depth (m) Depth (m) Innovative Infrastructure Solutions (2019) 4:2 Page 5 of 10 2 Fig. 5 Layout and plan view of water storage basin and slow- moving landslide Fig. 6 a Excess pore water pressures due to water level changes; b correlation between deformation rates and water level changes pressures in the subsoil at the toe of the landslide linked to subsoil. A typical comparison of water level changes and the water table changes in the water storage basin, but it was rotation of the inclinometer probe in the major sliding zone observed that this was not a one-to-one relationship because is shown in Fig. 6b. positive and negative excess pore pressures have been meas- ured, i.e. the pressure level of the pore pressure measurement Numerical model and results is above or below the corresponding water level in the water reservoir. Typical measurement results of the pore pressure Preliminary numerical analyses and careful examination of gauges are shown in Fig. 6a. For a better understanding of the measurements revealed that water level changes alone the reasons for these measured excess pore water pressures, could not explain measured displacements. Therefore, a a numerical study was carried out. These analyses reveal, finite element model (Fig.  7), using the code Plaxis 2D as expected, that the magnitude of these excess pore water [4], was set up incorporating the water level changes from pressures depends on the ratio between drawdown velocity the storage basin but also environmental influences such and soil permeability and furthermore on the ratio between as rainfall events assuming site-specific precipitation and pore water compressibility and soil skeleton compressibil- evaporation. The precipitation was measured on site. The ity [3]. Also, the presence of air bubbles in the water may potential evaporation was estimated according to Thorn- contribute to this effect [ 2]. In Fig. 6a, the excess pore water thwaite [10], whereas the influences of the soil suction on pressure p_excess is defined by the difference of measured the evaporation and the transpiration were neglected due and hydrostatic pore water pressure. Deformation measure- to the lack of measurement data. Furthermore, the lacus- ments using in-place inclinometers revealed a correlation trine fine sediments at the slope toe were modelled with between deformation rates and excess pore water pressures the Soft Soil Creep model [11] to model the creep behav- at the slope toe. This shows that the slope deformations are iour of these soil layers (see Fig. 7). To achieve appropri- influenced by the water level changes in the water storage ate initial conditions concerning the stress state and the basin, as these control the excess pore water pressures in the hydraulic conditions, the geological history was modelled 1 3 2 Page 6 of 10 Innovative Infrastructure Solutions (2019) 4:2 Fig. 7 FE model for numerical back-calculations [1] Fig. 8 Back-calculated excess pore water pressures [1] in a simplified way. Afterwards, 1 year with characteristic level changes. Figure 8 shows a comparison of the meas- precipitation and water level changes was simulated [1]. ured and calculated excess pore water pressures for the The back-calculation of the pore water pressures was per- two installed pore water pressure gauges PPG 1 (21 m) and formed for several periods with different types of water PPG 2 (33 m). Furthermore, the water level in the storage 1 3 Innovative Infrastructure Solutions (2019) 4:2 Page 7 of 10 2 Fig. 9 Displacements at node E (in the middle of the slope) for different influencing factors [1] basin is plotted in the diagram. From the comparison in with increasing distance from the slope toe. The influence of Fig. 8, it can be seen that a good agreement between cal- precipitation and evaporation is increasing in the middle and culations and measurements could be achieved. upper part of the slope, and the influence of the water level In order to identify the quantitative contribution of the dif- changes on the slope deformations is almost constant over the ferent influencing factors on the total displacements, separate entire slope. Based on the back-calculations of the slope defor- calculation phases were performed. In each phase, a new influ- mations, it can be argued that the water level changes are the encing factor was considered. As an example, the results for a main reason for the slope movements. However, slope move- point in the middle of the slope are shown in Fig. 9. The dif- ments would also occur without the storage operation but the ference between two time–displacement curves is the influence magnitude of the deformations would be smaller. of each additionally considered factor as indicated by the label of the curves. A comparison between the in situ measurements and the total displacements from the calculation indicates a Example: numerical simulation of cone good agreement. According to Fig. 9, the displacements are penetration test mainly due to the water level changes in the storage basin but creep behaviour of the lacustrine sediments at the base of the Particle finite element method slope and precipitation add to the displacements. The results were evaluated for several points along the slope. It could be In situ investigation methods, such as cone penetration clearly shown that the influence of the creep behaviour of testing (CPT), are frequently used to derive hydraulic and the lacustrine fine sediments on the deformations decreases mechanical soil properties from measured tip resistance, 1 3 2 Page 8 of 10 Innovative Infrastructure Solutions (2019) 4:2 sleeve friction and pore water pressure via empirical cor- added and old ones removed, in order to deal with large relations. Experience has shown that CPT provides reason- deformations and avoid excessive mesh distortion. This able results for applications in sand or clay where either strategy results in an increased computational cost and drained or undrained behaviour governs the penetration therefore low-order elements in combination with a mixed process. However, correlations for partial drainage, as it formulation of the problem are used. In order to avoid occurs during penetration in intermediate soils such as locking effects, an additional degree of freedom, namely silts, are still an ongoing research topic. Recent advances the determinant J of the deformation gradient, is intro- in the numerical simulation of large deformation prob- duced on top of the displacement and water pressure fields lems based on a Particle Finite Element Method (PFEM, u and pw [7]. Furthermore, the problem is stabilised using see [8]) allow to model this kind of penetration problems the Polynomial Pressure Projection. The interested reader where a rigid cone penetrates a fully water-saturated soil is referred to Monforte et al. [6] and Monforte et al. [7] for body. The simulations here are carried out using the plat- a more detailed outline of the PFEM. form G-PFEM, short for Geotechnical-PFEM [6, 7], which has been developed within the Kratos framework [5] at the Polytechnic University of Catalonia (UPC) and the Numerical model for cone penetration Center for Numerical Methods in Engineering (CIMNE). In G-PFEM, the quasi-static linear momentum and mass Modelling a CPT involves an ideally rigid cone that pene- balance equations are formulated for a solid and fluid trates a deformable two-phase medium at a constant velocity. phase adopting an updated Lagrangian description. The This leads to an axisymmetric model consisting of a rectan- basic idea behind the PFEM is a continuous remeshing of gular box with a height of 1.1 m and a width of 0.5 m. The critical regions of the domain, where new nodes can be cone radius R measures 1.78 cm with a tip angle of 60° cor- responding to the standard geometry (base area of 10 cm ). The penetration starts from an initial position where the cone is located at a depth of 10 cm. The lateral and lower boundaries are fixed in normal direction while an overbur - den pressure can be applied at the top of the domain. Moreo- ver, free drainage is allowed at the boundaries except along the symmetry axis. Figure 10 shows the basic model. From a mathematical point of view, the contact between cone and soil body is described by a set of constraints and a penalty method is adopted. The contact algorithm is explained in detail in Monforte et al. [6]. The constitutive behaviour of the soil is described by means of the Modified Cam Clay (MCC) model, and the parameters for this particular analysis are given in Table 3. Influence of anisotropic permeability In order to study the effect of anisotropic permeability, the set-up described in the previous section was analysed with a standard penetration velocity of 2 cm/s, a smooth interface between cone and soil and anisotropic perme- Fig. 10 Axisymmetric model (left) and refined mesh during penetra- ability, whereas two cases have been investigated. Test tion process (right) Table 3 Input parameters for MCC model 3 3 * * ρ (kg/m ) ρ (kg/m ) λ (–) κ (–) φ′ (°) M (–) G (kPa) α (–) s w 0 1700 1000 0.015 0.005 22.5 0.88 2900 0 OCR (kPa) p (kPa) K (m/s) k (m/s) k (–) K (°) φ (–) e (–) c0 w v h 0 int 0 8 −8 −7 1 100 1 × 10 2 × 10 2 × 10 0.7 7 0.5 1 3 Innovative Infrastructure Solutions (2019) 4:2 Page 9 of 10 2 is vertical. Consequently, although the overall behaviour is rather undrained, differences can be observed locally which may become important when deriving coefficients of consolidation from dissipation tests. The effect of anisotropic permeability on the pore pressure field is shown in Fig.  12. It is observed that the order of magnitude of the calculated water pressures is the same, again as expected for undrained conditions, but the distribution differs. For case (b), the pressure bulb has an increased vertical extension along the preferred drain- age direction. Further studies are currently undertaken to assess in more detail the effect of partial drainage in inter - mediate soils on tip resistance, sleeve friction and pore water pressures. Conclusion The numerical simulation of an anchor pull-out test showed that the predicted load–displacement curve (“class-A” pre- diction) and the one obtained in  situ showed very good agreement. The ultimate pull-out load is achieved when the shear strength is mobilised along the entire interface grout–soil. The numerical simulation indicated that the Fig. 11 Comparison of q and u for inverted anisotropic permeabili- strains in the tendon are highly influenced by the develop- c 2 −8 −7 ties; case (a) with k = 2 × 10   m/s, k = 2 × 10   m/s and case (b) v h ment of cracks in the grout of the fixed length which could −7 −8 with k = 2 × 10  m/s, k = 2 × 10  m/s v h be captured by applying an advanced constitutive model for the grout. case (a) considers an increased horizontal permeability The second example, namely the back-analysis of a slow- −7 −8 with k = 2 × 10  m/s and k = 2 × 10  m/s, while case moving landslide at a water reservoir, proved that com- h v −8 (b) simulates the opposite case (k = 2 × 10   m/s and plex mechanisms contributing to slope movement, such as −7 k = 2 × 10  m/s). The anisotropy is crucial for the hydrau- water level fluctuations in the reservoir, creep phenomena lic behaviour and prescribes the preferred flow direction in soft lacustrine deposits present at the toe of the slope which is horizontal for case (a) and vertical for case (b). and environmental effects such as rainfall infiltration can The profiles of q and u are compared in Fig. 11 and in be accounted for in numerical analyses. It is acknowledged, c 2 both cases (a) and (b) overall undrained behaviour with u however, that due to significant uncertainties in input param- (u = most common position in CPT for measuring pore eters the solutions presented are by no means rigorous and pressures) being around 320  kPa and q of 600  kPa is unique but still provide a better insight into the mechani- observed. However, a closer look reveals that test case (a) cal behaviour of such slopes and will help the experienced gives a slightly lower pore pressures (315 kPa vs 330 kPa) geotechnical engineer in defining appropriate mitigating and a higher tip resistance (610 kPa vs 585 kPa) compared measures. to case (b). These observations are consistent with the In the last example, it is shown that G-PFEM is an assumption of a preferred flow direction due to anisotropy. appropriate tool for simulating CPT. The application pre- As the cone penetrates the soil, radial (or rather horizon- sented looked into the effect of considering anisotropic tal) flow is dominantly caused by the geometric boundary permeability of the soil layer. It could be shown that ani- conditions of the test. Thus, for case (a), the preferred flow sotropy influences the local pore pressure field under glob- directions due to anisotropy and problem geometry coin- ally undrained conditions which may have a consequence cide which leads to increased drainage of the system. For when deriving coefficients of consolidation from dissipa- case (b), the opposite occurs and the main flow direction tion tests. 1 3 2 Page 10 of 10 Innovative Infrastructure Solutions (2019) 4:2 Fig. 12 Pore pressure -100 fields for inverted aniso- tropic permeabilities; case −8 (a) with k = 2 × 10  m/s, −7 k = 2 × 10  m/s and case −7 (b) with k = 2 × 10  m/s, −8 -190 k = 2 × 10  m/s -190 -130 -130 -250 -250 -85 -85 -100 -100 (a) (b) Acknowledgements Open access funding provided by Graz University 4. Brinkgreve RBJ, Kumarswamy S, Swolfs WM (2016) PLAXIS of Technology. 2016. Finite element code for soil and rock analyses, user manual. Plaxis bv, Delft 5. Dadvand P, Rossi R, Oñate E (2010) An object-oriented environ- Open Access This article is distributed under the terms of the Crea- ment for developing finite element codes for multidisciplinary tive Commons Attribution 4.0 International License (http://creat iveco applications. Arch Comput Methods Eng 17(3):253–297 mmons.or g/licenses/b y/4.0/), which permits unrestricted use, distribu- 6. Monforte L, Arroyo M, Carbonell JM, Gens A (2017) Numerical tion, and reproduction in any medium, provided you give appropriate simulation of undrained insertion problems in geotechnical engi- credit to the original author(s) and the source, provide a link to the neering with the particle finite element method (PFEM). Comput Creative Commons license, and indicate if changes were made. Geotech 82:144–156 7. Monforte L, Carbonell JM, Arroyo M, Gens A (2017) Performance of mixed formulations for the particle finite element method in soil mechanics problems. Comput Part Mech 4(3):269–284 8. Oñate E, Idelsohn SR, Celigueta MA, Rossi R, Marti J, Carbonell References JM, Ryzakov P, Suárez B (2011) Advances in the particle finite element method (PFEM) for solving coupled problems in engi- 1. Ausweger GM (2017) Influences of water level changes on the neering. Part Based Methods 25:1–49 behaviour of a slow moving landslide—in situ measurements, 9. Schädlich B, Schweiger HF (2014) A new constitutive model for model test and numerical analyses. PhD thesis, Graz University shotcrete. In: Hicks MA, Brinkgreve RBJ, Rohe A (eds) Proceed- of Technology ings of the numerical methods in geotechnical engineering. Taylor 2. Ausweger GM, Schweiger HF (2016) Numerical study on the & Francis Group, London, pp 103–108 influence of entrapped air bubbles on the time-dependent pore 10. Thornthwaite CW (1948) An approach toward a rational classifica- pressure distribution in soils due to external changes in water tion of climate. Geogr Rev 38(1):55–94 level. In: Proceedings of the 3rd European conference on unsatu- 11. Vermeer PA, Neher HP (1999) A soft soil model that accounts rated soils, paper #16010, Paris, 12–14 Sept 2016 for creep. In: Proceedings of the international symposium beyond 3. Ausweger GM, Schweiger HF (2017) Numerical investigation 2000 in computational geotechnics of excess pore water pressures due to external fluctuating water tables. In: Proceedings 15th international conference computer methods and recent advances in geomechanics, Wuhan, China 1 3

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