Using numerical analysis in geotechnical engineering practice 1 … · 2016-09-19 · Using numerical analysis in geotechnical engineering practice 1! Professor David M. Potts First

Using numerical analysis in geotechnical engineering practice 1 Professor David M. Potts First Leonardo Zeevaert Lecture

SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA, A.C.

“Classical soil mechanics has evolved around a few simplified models which do not fit the

properties of most real soils sufficiently for useful and safe predictions to be made ... Since we

cannot change the soil to fit the soil mechanics, perhaps we should change the soil mechanics to

fit the soil. The theory which fails to fit their behaviour is problematic, not the soil.”

Peter R. Vaughan

2 XXVI RNMSeIG, Cancún, Quintana Roo, 14-16 noviembre 2012

SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA, A.C

Using numerical analysis in geotechnical engineering practice

David M. Potts

Imperial College London, UK ABSTRACT Numerical analysis, in the form of the finite element, finite difference or boundary element method, has become a preferred calculation tool in the design of complex geotechnical structures. Even new codes of practice, such as the European EUROCODE 7, encourage the use of numerical analysis in design. However, as there is no general guidance and agreement in the international community on appropriate numerical solvers, constitutive models and boundary conditions, there are often examples of bad practice in the numerical analysis of geotechnical structures. Through examples of the analyses of real problems, this paper investigates the use of appropriate constitutive models and boundary conditions, to highlight both the successful use and the potential pitfalls of modern numerical analysis.

1. INTRODUCTION The increasing complexity of geotechnical construction has increased the application of advanced numerical analysis in the design of geotechnical structures. Classical methods of analysis, such as limit equilibrium or limit analysis, cannot provide an answer to all design requirements, in particular those related to the effects of new construction on existing structures and services.

However, for an analysis to be successful, appropriate soil constitutive models, boundary conditions and solution algorithms have to be applied (Potts, 2003; Potts & Zdravkovic, 2010). These are, however, the causes of many problems with numerical analysis, as there is currently no internationally agreed best practice for their implementation and application in geotechnical software packages. Consequently, the same problem could have different answers from different software. Moreover, users are often not familiar with the software and use it as a ‘black box’, which further increases the potential for erroneous results.

This paper uses examples of some common geotechnical problems to demonstrate both pitfalls with, and the correct use of, constitutive models and boundary conditions. All analyses have been



performed using the Imperial College finite element program ICFEP (Potts & Zdravković, 1999, 2001), which employs a modified Newton-Raphson nonlinear solver with an error-controlled sub-stepping stress-point algorithm.

2. SOIL MECHANICAL BEHAVIOUR – CONSTITUTIVE MODELS 2.1 MOHR-COULOMB MODEL 2.1.1 ULTIMATE LIMIT STATES The Mohr-Coulomb constitutive model is the simplest representation of soil behaviour and forms part of most geotechnical software packages. This is a linear elastic – perfectly plastic model, requiring only a few input parameters that can be obtained from standard laboratory testing: Young’s modulus E and Poisson ratio ν, to describe the elastic part of the model, and cohesion c' and angle of shearing resistance φ', to describe the plastic (failure) part of the model. If no other input parameter is required, this implies associated plasticity for the model and negative (dilative) plastic volumetric strains. Problems with such a formulation of the model are two-fold: (i) soil dilation is usually smaller than that implied by associated plasticity (i.e. angle of dilation ψ equal to φ'); and (ii) once the soil starts to dilate, it will dilate for ever, without reaching a limit load for volumetrically constrained problems.

Figure 1. Load-displacement curves for vertically loaded pile



An example of such model performance is given in Figure 1, which shows load-displacement curves for a vertically loaded pile, 1.0m diameter and 20m long, in a drained soil with c' =0 and φ' =25o (Potts & Zdravković, 2001). If the angle of dilation, ψ, is equal to φ' (i.e. associated plasticity), the pile never reaches a limit load, no matter how far it is pushed into the ground. Even if the model has the flexibility to input a value of ψ smaller than φ', for any value of ψ that is greater than zero the model still does not predict a limit load. The magnitude of ψ makes a difference only in the magnitude of the dilative volumetric plastic strains, which reduce with a reduction of ψ and hence reduce the magnitude of the pile load for a particular displacement of the pile head. Consequently, a practical approach in determining a limit load from such analysis has been to arbitrarily adopt the load value that corresponds to the displacement equal to 10% of the pile diameter (0.1m in this case). This can result in a significant over-estimate in the load capacity of the pile in the case of associated plasticity, as soil dilation is generally over-predicted. Only if ψ=0 can the pile reach a limit load. This is a conservative prediction as many soils dilate to some extent, but it is at least a theoretically correct value obtained without any arbitrary decisions from the user.

2.1.2 SERVICEABILITY LIMIT STATES Apart from failure conditions it is also necessary to design geotechnical structures for working conditions, where ground deformations are limited. What becomes important for design is the determination of ground movements imposed by new construction and whether they can cause any damage to existing structures and services.

Figure 2 shows the layout of two London Underground (LU) tunnels as they pass into St. James’s park in London, UK. The 30m deep westbound and the 20m deep eastbound tunnels, of 3km total length, are part of the extension of the LU Jubilee Line, and were constructed between 1994 and 1996. The figure shows that at this location they pass directly underneath the Treasury building, with several other buildings (shaded areas) in direct vicinity. To the left of the Treasury building the tunnels pass underneath the greenfield area of St. James’s park, which was heavily instrumented for monitoring of both ground movements and stress changes in the soil due to the tunnels advancement (Standing et al., 1996). Both tunnels were excavated rapidly (i.e. under undrained conditions) in the London Clay formation.

The measured settlement trough above the westbound tunnel at St. James’s park, which was the first of the two tunnels to be constructed, is used here to demonstrate the necessity for advancing the capabilities of a constitutive model in order to obtain reasonable predictions of tunnel induced ground movements.



Figure 2. Layout of Jubilee Line tunnels at St. James’s park Since in this type of a boundary value problem the strains are small, the pre-failure characteristics of the model dominate the predicted behaviour. If a simple linear elastic-perfectly plastic Mohr-Coulomb model is applied, of the type used in the case of pile loading, the predicted surface settlement trough is shown in Figure 3. In this case, the pre-failure behaviour of the Mohr-Coulomb model is characterised by an isotropic linear elastic Young’s modulus, which increases linearly with depth, producing a surface settlement trough which does not resemble the measurements.

Figure 3. Predicted and measured surface settlements above west bound tunnel



If an anisotropic linear elastic stiffness for London Clay, still increasing with depth, is introduced in the pre-failure behaviour of the Mohr-Coulomb model, where the horizontal Young’s modulus is larger than the vertical due to the overconsolidated nature of the London Clay, the predicted settlement trough in Figure 3 is slightly better than that of the isotropic linear elastic pre-failure model, but is still too shallow and wide compared to the measurements.

Figure 4. Shear stiffness degradation of London Clay from triaxial tests It is only with the advancement in laboratory testing in the past 20 years, where local instrumentation has been introduced on soil samples, that soil stiffness has been shown to be highly nonlinear, varying with both stress and strain levels, as shown in Figure 4 for the shear stiffness, G, of London Clay (note that p’ is the mean effective stress). When an isotropic nonlinear elastic model, in this case of the Jardine et al. (1986) type, is introduced to describe the pre-failure behaviour of the Mohr-Coulomb model, the predicted surface settlement trough in Figure 3 above the west bound tunnel plots much closer to the measurements (Addenbrooke et al., 1997).

2.2 ADVANCED KINEMATIC SURFACE MODEL Figure 3 shows that although the modelling of nonlinearity of soil stiffness at small strains significantly improves predictions of ground deformations above tunnels, these do not completely agree with measurements, as the settlement trough is still shallower and wider than that observed. This result is characteristic of soils with a high earth pressure coefficient Ko, such as London



Clay. Therefore further attempts have been taken to improve constitutive models for stiff overconsolidated clays.

The framework that has been applied mostly in recent years is that of multi-surface kinematic plasticity (Mroz et al. 1978), resulting in models with one or two kinematic surfaces inside the bounding surface (e.g. Al Tabbaa & Wood, 1989; Stallebrass & Taylor, 1997; Kavvadas & Amorosi, 2000; Grammatikopoulou et al., 2006). Compared to the nonlinear elastic Mohr-Coulomb model, this class of model has a sound theoretical framework based on critical state soil mechanics and, apart from nonlinearity, it also introduces plasticity in soil behaviour from early stages of loading. In terms of other important facets of real soil behaviour, effects of recent stress history and stress path direction are automatically incorporated in this framework. The models also need a larger number of parameters, but these can still be obtained from standard laboratory testing: compression, λ, and swelling, κ, indices, specific volume, v1, at unit pressure, Poisson’s ratio, µ, and angle of shearing resistance, φ'. Two or three additional parameters, depending on the number of kinematic surfaces, are calibrated parametrically from stiffness degradation curves. The models are suited for simulating both the ultimate states, always predicting a limit load when critical state conditions in the soil are mobilised, and serviceability limit states, as they account for nonlinearity, plasticity, recent stress history and stress path direction.

Figure 5. Configuration of kinematic surfaces at the end of geological history Analysing again the case of the St. James’s park tunnel construction, using a model with two kinematic surfaces inside the bounding surface (M3-SKH model of Grammatikopoulou et al. 2006), the latter effects on predictions of the tunnel settlement trough have been investigated (Grammatikopoulou et al. 2008). Figure 5 shows the configurations of all surfaces following the geological history of the London Clay formation before tunnel construction, and whether, in option A, there is an effect of recent stress history, or, in option B, there is only the effect of stress path direction. The paths in Figure 5 show the initial deposition of London Clay to point A,

p'

q

(a)

(b) Configuration A

Configuration B

A

B

C

A

C

B



about 200m of subsequent erosion to point B and re-loading with Terrace Gravels to point C, which represents the current stress state before tunnel construction.

Figure 6. Surface settlement troughs predicted with the M3-SKH model and for two configurations of kinematic surfaces Figure 6 shows, as before, the predicted surface settlement profiles for the west bound tunnel construction, for both configurations of kinematic surfaces. It is disappointing that such an advanced model is still incapable of improving the predictions of the surface settlement trough in high Ko soils, compared to the simple nonlinear Mohr-Coulomb model. It therefore needs further development to be able to simulate the behaviour of stiff overconsolidated clays.

2.3 SOFT CLAYS Another basic constitutive model of soil behaviour is the modified Cam clay model (MCC) of Roscoe & Burland (1968). This is a critical state model that has formed the basis for the development of the kinematic surface models mentioned before and also exists in some form in most geotechnical software. The model requires 5 input parameters that can be estimated from standard laboratory experiments: compression, λ, and swelling, κ, coefficients, specific volume v1 at unit mean effective stress, angle of shearing resistance φ' and shear stiffness G. It is particularly suited for modelling problems in soft, normally consolidated clays. The examples in this section demonstrate the necessity for further upgrading of this model in order to deal with real problems.



2.3.1 SHORT TERM STABILITY OF EMBANKMENTS ON SOFT CLAY The case study of the Saint-Alban embankment construction on soft Champlain clays in Canada is an example of the necessity to account for strength anisotropy in constitutive models for soft clays (Zdravković et al. 2002). The embankment was built rapidly to failure on highly anisotropic clay foundations (La Rochelle et al. 1974), with the undrained strength profile from triaxial compression (CIU) and shear vane experiments in Figure 7 showing clearly different strengths.

Figure 7. Measured and predicted undrained strength profiles of Saint-Alban clays; test data from Trak et al. 1980 The embankment was observed to fail when reaching 3.9m height (La Rochelle et al., 1974). The finite element analysis of the embankment was first attempted with the MCC model, which can only simulate isotropic strength. In the first instance the undrained strength, Su, in the model was matched to that corresponding to the triaxial compression profile in Figure 11. This analysis significantly over-predicted the embankment failure height, ato 4.9m. The following analysis, adopting the vane shear strength profile, which was similar to the direct simple shear (DSS) strength, significantly under-predicted the failure height, at 3.3m. After several variations of the Su profile, the 3.9m failure height using the MCC model was achieved with the Su profile 25% larger than the DSS profile. Figure 8 shows the development of embankment toe movement to failure from the two MCC analyses, as well as the sketch of the embankment cross-section.



Figure 8. Embankment heights predicted with two different models

Figure 9. Predicted embankment heights of the altered embankment In addition to this back-analysis of the embankment failure height with an isotropic constitutive model, a more advanced constitutive model was applied, which can simulate the strength anisotropy observed in Figure 7. This was the MIT-E3 model (Whittle, 1993), from the class of bounding surface plasticity models, which adopts the inclined MCC surface as the model’s bounding surface. Similar to the kinematic surface models, the MIT-E3 also introduces nonlinearity and plasticity in soil behaviour beneath the bounding surface and requires a large number of input parameters. Figure 7 shows the full range of strength anisotropy of the Saint-Alban clays, from plane strain compression (PSC) to plane strain extension (PSE), that was



simulated with the MIT-E3 model. As a result, with such an Su variation the model predicted the correct failure height of 3.9m, without any adjustment to the input parameters.

Both the back-analysed isotropic MCC and the above anisotropic MIT-E3 strengths were applied further in the analysis of the altered embankment geometry, shown in the sketch in Figure 9. Although it might be expected that both analyses would predict the same failure height of the altered embankment, as they have both accurately predicted it in the case of the original geometry, the differences are significant. Whereas the MIT-E3 model predicts a 4.4m failure height, the back-analysed MCC model produces a greater height of 4.9m. To achieve the same 4.4m height, the MCC strength now needs to be only 15% higher than the DSS strength.

Clearly, for highly anisotropic soils, the back-calculated strength in one problem does not necessarily apply if the problem changes. Therefore, either a better estimate of an average undrained strength from available experiments has to be performed, or a more advanced constitutive model has to be applied in the analysis.

2.3.2 LONG TERM STABILITY OF EMBANKMENTS ON SOFT CLAY A new challenge in geotechnical engineering arises from the necessity to account for the effects of climate change, which are likely to, for example, increase current sea levels. Consequently, existing flood defences, in terms of embankments, will have to be increased in height to sustain the rise in sea levels. By how much they can be raised becomes an important design question, as the foundation soil, usually a soft clay, would have increased its strength due to the dissipation of excess pore water pressures during the time elapsed since the construction of the existing embankment.

A field study of embankments on soft clays was conducted in the 1980’s in the Thames estuary in the UK, on the site known as Mucking Flats (Pugh, 1978). Two embankments of different cross-sections were constructed on the site and instrumented with settlement gauges and piezometers. The monitoring was carried out both during the construction and for a period of time after the construction of the embankments. Therefore, this case history was used in recent numerical studies (Zdravković et al. 2012), to first calibrate the numerical model against the measurements, and then raise the existing embankments to failure.

The foundation soil was initially represented with the MIT-E3 model, which was shown above to work well for the short-term behaviour of anisotropic soft clays. However, for longer term conditions, the model only simulates the dissipation of excess pore water pressures and consequently was unable to match accurately the field observations at Mucking Flats. Namely, Figure 10 shows a comparison of field measurements and predictions of the settlement profile



under the embankment 2, which had 2.8m original height, at different times. It is clear that after 7 months the measured settlements are much larger than those predicted due to the pore pressure dissipation for the same period. Equally, to achieve the measured settlement profile at 21 months, the analysis needed to apply 32 years of pore pressure dissipation. Clearly, there is an additional mechanism in the soil that generates these large settlements, which is not captured by pure pore water pressure dissipation.

Figure 10. Measured and predicted settlement profiles under the BANK2; predictions account only for excess pore water pressure dissipation

Figure 11. Measured and predicted settlement profiles under the BANK2; predictions account for both excess pore water pressure dissipation and creep



Creep, or secondary compression, is a process of soil deformation under constant loading, after all excess pore water pressures have dissipated. Soft clays can have appreciable creep and not modelling creep was deemed the reason for not being able to recover the measured deformations at Mucking Flats. Therefore, in subsequent analyses, in addition to pore water pressure dissipation, a creep model (Bodas Freitas et al., 2007), which is also an extension of the basic MCC model, was applied to model the foundation soil. The predicted settlement profiles presented in Figure 11 are now much closer to those observed and therefore give confidence in the subsequent prediction of a raised embankment height of 5.5m.

3. SOIL HYDRAULIC BEHAVIOUR – PERMEABILITY MODELS Apart from the soil’s mechanical behaviour, represented through constitutive models, it is also important to have appropriate models for its hydraulic behaviour. Permeability is one of the dominant parameters when performing a fully coupled numerical analysis, which is also a difficult parameter to accurately determine from any laboratory or field experiments.

Figure 12. Under-drained pore water pressure profile for London Clay



For example, when setting up initial stresses in a finite element analysis, it is important to specify the correct pore water pressure profile for the site, as this will affect effective stresses in the soil and thus its strength and stiffness. In the case of many urban areas, such as London or Bangkok, this profile is often under-drained (i.e. less than hydrostatic), as shown in Figure 12 for a typical situation in London, due to the historic extraction of water from an underlying aquifer. To account for such a profile in a coupled analysis, it is incorrect to apply a simple constant permeability to the soil (with a coefficient of permeability k), as such a permeability model will attempt to modify the initial profile, even if no other perturbation is made in the analysis. For example, in the situation presented in Figure 12, the pore water pressure profile consistent with a constant permeability (isotropic or anisotropic) for the London Clay is linear, varying from the pore water pressure in the sands immediately above the London Clay to the pore water pressure in the aquifer immediately below it (shown as a dashed line in Figure 12). Instead, a non-linear permeability model has to be applied, that allows a variation of permeability either with depth, or stress level, or void ratio. From a fundamental perspective the latter two scenarios are more realistic, since they will model permeability changes as the initial stress state is modified due to any construction process (e.g. tunnel excavation). The under-drained profile in Figure 12 has been obtained from the following logarithmic relationship (Vaughan, 1989):

Eq. (1) )(e paokk ʹ′⋅−⋅=

where ko is the coefficient of permeability at zero mean effective stress p’, and a is a constant.

The following example demonstrates the effect of the permeability model on predictions of stability of temporary slopes in London Clay, which were excavated in the recent development of the new Terminal 5 at Heathrow airport in London (Kovačević et al., 2007). The mechanical behaviour of London Clay was simulated with the nonlinear elastic Mohr-Coulomb model used in the tunnelling study. In terms of its hydraulic behaviour, Figure 13 shows a range of permeability measurements, as well as two models adopted to simulate this variation. One model considers the permeability to be constant but anisotropic, with different horizontal and vertical permeability coefficients in different geological units of the London Clay. The second model considers permeability to be nonlinear and isotropic, represented with the above logarithmic relationship.



Figure 13. Adopted permeability profiles in London Clay Figure 14 shows vectors of displacements at failure for a 25m slope excavated in several benches in the London Clay, from the two analyses with different permeability models, but the same ground conditions otherwise, simulated with the same constitutive models and input parameters. Excavation was performed quickly and then the excess pore water pressure were allowed to dissipate with time in a coupled analysis. Whereas the anisotropic, but constant, permeability model predicts a deep seated failure after 6.7 years of slope excavation, the isotropic nonlinear model predicts a shallow failure after only 0.2 year of slope excavation.

The permeability model becomes even more complicated when analysing unsaturated soils exposed to climate effects of rainfall and evapotranspiration. For example, as the soil loses moisture during the summer months due to evaporation from the ground surface and transpiration from vegetation, the negative pore water pressure (suction) in the soil increases, hence reducing its saturated permeability ksat. This relationship can be represented with a model sketched in Figure 15a, where s1 is the value of suction at which the permeability coefficient ksat starts to reduce, s2 is the value of suction at which this reduction ceases, reaching the kmin value, and RS is the overall reduction factor between s1 and s2. For any suction, s, between these limits the permeability is found using a linear interpolation.



(a) (b)

Figure 14. a) Deep seated failure predicted with the linear anisotropic model and b) superficial failure predicted with the nonlinear isotropic model

Figure 15. Model of soil permeability in relation with a) soil suction and b) minimum total principal stress However, as permeability due to drying can become several orders of magnitude smaller than the saturated permeability, it becomes difficult in any subsequent rainfall period during the winter months for the water to enter into the soil. This arises because the infiltration is controlled by the permeability which is low and consequently most of the rainfall cannot enter. The analysis will therefore indicate ponding or run-off of water at the ground surface, which is not necessarily realistic. In fact, drying of the soil in the summer months is likely to cause cracking of the superficial soil, thus increasing its overall permeability and allowing subsequent rainfall infiltration. To be able to simulate this successfully, the above suction permeability model could be coupled with a model that will allow the permeability to increase to a maximum value kmax when the minor total principal stress, σ3, reaches the soil’s tensile capacity (Nyambayo & Potts, 2009; Zdravković et al. 2009).



4. BOUNDARY CONDITIONS Boundary conditions represent any constraints on or within the finite element mesh boundaries that are applied to simulate the real problem that is being analysed. Standard boundary conditions for coupled static finite element analysis are normally related to prescribed displacements, loads/stresses acting in the problem and pore water pressures. However, when dealing with simulations of climatic conditions or soil dynamic problems, these are not always sufficient and therefore need to be further developed. This section presents examples that require the development of new boundary conditions for realistic geotechnical analysis.

4.1 PRECIPITATION BOUNDARY CONDITION The precipitation boundary condition controls the seepage in and out of the soil, for problems that have a transitional change from a pore pressure to a flow boundary condition and vice versa. An example of its application is that of tunnel excavation in a coupled analysis, where the tunnel boundary in the long term is supposed to be permeable.

For the long term conditions, this is achieved simply by prescribing a zero pore water pressure around the tunnel boundary, pb=0. However, immediately after excavation, in the soil around the tunnel the pore pressures, ps, are depressed and can be tensile (i.e. suction, ps<0), as shown in the sketch in Figure 16. If a zero pore water pressure is prescribed on the boundary (pb=0) immediately after excavation, this would imply the flow of water from the tunnel into the soil across the tunnel boundary (i.e. water always flows from the high potential to the low potential, and in this case pb>ps). This is clearly unrealistic and what should be applied on such a tunnel boundary in the short term is a zero flow boundary condition (qb=0).

However, with reference to Figure 17, in the transient state, between the end of tunnel construction and the long term, the pore water pressures in the soil will dissipate and suctions around the tunnel will gradually change to positive pore water pressures (i.e. ps>0). This will not happen simultaneously at all nodes around the tunnel boundary and the precipitation boundary condition is therefore needed to automatically determine at each transient increment which node changes from a zero flow to a zero pore pressure boundary condition. If in a transient increment pb at a node remains greater than ps, the boundary condition at that node remains that of qb=0. If, on the other hand, pb becomes smaller than ps, the boundary condition at that node changes to pb=0. In the long term all nodes on the tunnel boundary will have a pb=0 boundary condition.



Figure 16. Example of wrong application of pore pressure boundary condition

Figure 17. Precipitation boundary condition on tunnel boundary The precipitation boundary condition is also necessary for problems that involve simulation of rainfall infiltration through a soil boundary. Whether the soil will absorb the rain, or whether the rain will run off or pond on the surface depends again on the trade off between the intensity of the flow and the soil permeability and pore pressures. The precipitation boundary condition determines this automatically for each increment of the analysis.



4.2 DASHPOT BOUNDARY CONDITION AND THE DOMAIN REDUCTION METHOD (DRM)

Dynamic analysis of geotechnical problems involves wave propagation through the soil domain, induced either form earthquake action or by the dynamic loading of structures, such as vibrating machinery or pile driving. The energy created due to wave propagation towards and from the structure dissipates into the soil mass. It is therefore important to have boundary conditions that can simulate this dissipation, but also reproduce the free-field response of the soil in the vicinity of the boundaries. The standard boundary conditions that normally apply in static analysis, such as zero displacements or forces, cannot simulate energy dissipation through such a boundary as they cause reflection of the waves from the boundary and thus lead to trapping of the wave inside the mesh.

The simplest type of boundary condition that can absorb energy is the viscous boundary of Lysmer & Kuhlemeyer (1969). This consists of a series of dashpots placed normally and tangentially at boundary nodes. The optimal absorption for such a boundary condition is achieved only for perpendicularly impinging waves. Reasonable absorption in two- and three-dimensional problems is achieved for angles of incidence of 30o or more. It is therefore important that viscous boundaries are placed at a distance from the source of excitation in order to achieve a reasonable solution. This implies that they are applicable for problems where the source of excitation is concentrated, like pile driving, in which case the viscous boundaries in a reasonably large mesh are far from this source. However, this also implies that this boundary condition does not work well for earthquake problems, where excitation is normally applied along the whole of the bottom boundary of the mesh, in which case some of the waves propagate almost parallel to the vertical viscous boundaries.

A more appropriate way of analysing earthquake problems is to use the recently developed sub-structuring approach of Bielak et al. (2003), known as the Domain Reduction Method (DRM). This is a two-step procedure which aims to reduce the domain that has to be modelled numerically by changing the governing variables. The seismic excitation is introduced directly into the computational domain, while the local boundaries (e.g. viscous dashpots) are needed to absorb only the scattered energy of the system. The method has been further developed by Kontoe et al. (2008, 2009) to enable dynamic coupled consolidation analyses to be performed.

The example of excavation in front of a retaining wall, in the ground conditions of the Montenegro Adriatic coast, is used here to demonstrate how erroneous viscous boundary conditions can be in an earthquake finite element analysis (Zdravković & Kontoe, 2008). The excitation applied to the problem in Figure 18 corresponds to the Hercegnovi recording of the 1979 Montenegro earthquake (Ambraseys & Douglas, 2004), which devastated the area. Two conventional analyses were performed first with the mesh in Figure 18, one having vertical



displacement along the vertical mesh boundaries prescribed to zero, and the other having both normal and tangential dashpots applied to the vertical mesh boundaries. The mesh for the DRM analysis is presented in Figure 19 and the input for this mesh was obtained from the column analysis of the free-field response for the whole domain.

Figure 18. Finite element mesh for the conventional earthquake analysis

Figure 19. Finite element mesh for the DRM analysis



As an example, Figure 20 compares the development of shear strain with time at point A, chosen in the middle of the clayey sand layer, from the three analyses, with each other and with the shear strain developed at the same elevation in the column analysis. As noted above, the column analysis represents the free-field conditions, which should be established some distance away from the structure. It can be seen in the figure that the prescribed displacement boundary conditions significantly amplify the response, due to the wave trapping inside the mesh, while the dashpot boundary conditions over-damp the response, compared to the result from the column analysis. The response at point A from the DRM analysis agrees very well with that from the column analysis, confirming that this is the correct boundary condition to be applied.

Figure 20. Comparison of shear strain evolution with time at point A

5. NUMERICAL ANALYSIS AND CODES OF PRACTICE The concept of a safety factor in the design of geotechnical structures has been traditionally developed within the framework of classical soil mechanics, where the analysis methods for its calculation involve simple limit equilibrium or limit analysis approaches. Therefore, the inclusion of a safety factor within an advanced analysis method, such as finite elements or finite differences, is a more complex issue. In particular, the problem arises with design codes, such as Eurocode 7, in which partial factors on soil strength (or partial material factors) must be accounted for. Namely, Eurocode 7 implies that a numerical analysis should be performed accounting for a characteristic strength which is reduced by partial factors.



For geotechnical problems which are dominated by undrained behaviour and where a total stress constitutive model using the undrained strength Su is employed, the application of partial factors implies that the design (factored) strength used in an analysis, Su,d, is estimated as:

Eq. (2) m

chu,du, γ

SS =

whereas for problems based on drained strength – the angle of shearing resistance φ’ and cohesion c’ – the design (factored) strength used in an analysis, φ’d and cΝd, is estimated as:

Eq. (3) ( )

m

chd

m

chd and

tanarctan

γγφ

φc

cʹ′

=ʹ′⎟⎟⎠

⎞⎜⎜⎝

⎛ ʹ′=ʹ′

In the above equations γm is a partial material factor and Su,ch, φ' ch and c' ch are the characteristic (unfactored) values of soil strength, estimated from the site investigation data. The strength reduction given by these equations can be achieved in two ways in numerical analysis, which both have advantages and disadvantages in their application.

The first approach, SR1, is to start the analysis with the characteristic strength (Su,ch or φ' ch and c'ch as applicable) directly, without modification, and then at relevant stages of this analysis to gradually increase the partial material factor (i.e. to reduce the strength), until failure in the soil is fully mobilised. The advantage of this strength reduction approach is that a single analysis could be used for assessing both the serviceability and ultimate limit states for the problem being analysed. It is also possible to obtain, from this single analysis, the magnitude of the factor of safety at the ultimate limit state (i.e. collapse). However, the disadvantage of this approach is that it requires modification of the numerical software. As there is no agreed unique way of how this strength reduction should be numerically implemented, different software accounts for this in different ways, which are not always clearly explained. Also, most software can only perform such reductions if simple constitutive models are used in the analysis. A theoretically correct approach, which is applicable to any constitutive model, has been proposed by Potts and Zdravkovic (2012) and this procedure has been used for the examples presented below.

The second approach, SR2, is to start the analysis with the factored strength (Su,d or φ' d and c'd), as given by Eqs. (2) or (3), and continue until the analysis is completed. The advantage of this approach is that no modification to the analysis software is needed, which makes it an easier option to use. The disadvantage is that such a reduced strength may require initial stresses which are not consistent with those in-situ, resulting, for example, in the wrong structural forces being calculated in retaining walls or tunnel linings that are present in the analysis. In addition, in an analysis with the SR2 approach all stages of the analysis may be completed without reaching failure, which ensures the stability of the problem, but it does not produce information on the real



magnitude of the safety factor. Another disadvantage of this approach is that it may not be easy to use in combination with advanced constitutive models in which strength is stress and/or strain dependent.

An additional issue with the two strength reduction approaches is whether they produce the same result for a given problem (i.e. the same limit state). This will be investigated by considering the results of bearing capacity analyses using three different constitutive models. See Potts & Zdravkovic (2012) for further details.

The geometry of the problem being analysed is shown by the finite element mesh in Figure 21. The strip footing is 2 m wide and due to symmetry in both its geometry and loading conditions (i.e. vertical load) only half of the problem is discretised in Figure 21. The footing itself is not discretised in the mesh. Its rough interface is simulated by prescribing zero horizontal displacements at the nodes at the soil-footing interface, whereas the rigid conditions are simulated by the uniform incremental vertical displacements applied at the same nodes. The vertical load on the footing is then calculated from the reactions to the prescribed vertical displacements.

Figure 21. Finite element mesh for the bearing capacity problem



5.1 ANALYSES WITH TRESCA MODEL The Tresca model is one of the simplest constitutive models, formulated in terms of total stresses, where the soil’s strength is characterised by the undrained strength Su. The first analysis using this model adopted the characteristic strength in the soil Su,ch = 100 kPa. The resulting load-settlement curve for the footing is shown in Figure 22 as a solid line. The predicted ultimate load on the footing is Qf = 519 kN, which results in the bearing capacity factor Nc = 5.19 (= Qf/(A.Su), where A is the base area of the footing, 1 m2 in this case). This result is within 1% of the theoretical value of the bearing capacity factor Nc = 5.14, which confirms sufficient accuracy of the analysis procedure for this problem.

Figure 22. Load-displacement curves for undrained fooring capacity using Tresca model In the second analysis the strength reduction approach SR1 is applied, in that the characteristic strength in the soil Su,ch = 100 kPa is adopted at the beginning of the analysis. The footing is initially loaded to a working load of 273 kN, which represents a load factor, Lf, of 1.9 with respect to the ultimate load (i.e. Lf = =519/273). This load was then maintained in the analysis while the partial material factor was incrementally increased (i.e. the undrained strength was reduced) until failure occurred. The load-displacement curve (solid line with symbols) in Figure 22 shows that the footing deforms further while the load is maintained, due to the reduction in the



soil’s strength. The resulting partial material factor, γm, was 1.9, which is identical to the load factor for the applied working load, as would be expected.

The final analysis then applied the strength reduction approach SR2, such that the undrained strength at the beginning of this analysis was reduced by the partial material factor 1.9 obtained in the second analysis, i.e. Su,d = 100/1.9 = 52.6 kPa. The footing was then loaded to failure with this undrained strength in the soil, resulting in the ultimate load of 273 kN, as shown by the dashed line in Figure 22.

These analyses show that with the Tresca constitutive model both options for incorporating partial material factors in numerical analysis (i.e. strength reduction approaches SR1 and SR2) result in the same failure loads for the same partial factor. Additional analyses, not presented here, also indicate that the results are not dependent on the value of Ko, as would be expected.

5.2 ANALYSES WITH MCC MODEL The Modified Cam Clay (MCC) is a critical state based, strain-hardening/softening, effective stress model, with the soil’s strength expressed in terms of the angle of shearing resistance φ'. The analyses presented here assume the soil to be normally consolidated (i.e. OCR = 1), with the ground water table (GWT) 2 m below the ground surface. A hydrostatic pore water pressure profile is adopted, which gives suction above the GWT. The form of the MCC model used in these analyses adopts the Mohr-Coulomb hexagon for the shape of the yield surface and a circle for the shape of the plastic potential surface in the deviatoric plane. The relevant strength parameter φ' ch = 30o is assumed. The remaining parameters for this model can be found in Potts& Zdravkovic (2012).

Undrained bearing capacity analyses adopting Ko = 1 – sinφ' are considered. The first analysis adopts φ' ch = 30o which results in Ko = 0.5 and the load-displacement curve in Figure 23 shown by the solid line, indicating the maximum footing load of 29 kN.

In the second analysis φ' ch = 30o and Ko = 0.5 are adopted at the beginning of the analysis and the footing is loaded to a working load of 14.5 kN (i.e. Lf = 2). The load is then maintained at this level and the soil strength is reduced until the failure of the footing (SR1 approach), leading to further footing settlement. The resulting partial material factor is 2.25.

The third analysis now starts with a reduced (i.e. design) strength φ' d = 14.4o (= arc tan(tan30o/2.25)), but there is a dilemma as to the choice of the value of Ko. If Ko = 0.751 is adopted, which is consistent with the modified value of φ' (i.e. = 1 – sin14.4o), then the dashed line in Figure 23 is obtained for the resulting load-displacement curve, indicating the ultimate



footing load of 17.1 kN. This is different from 14.5 kN in the above SR1 approach. If this analysis is then repeated with the same reduced strength of φ' d = 14.4o, but adopting the original Ko = 0.5 value in the initial stresses, then the dot-dashed line in Figure 23 is obtained for the load-settlement curve and the maximum footing load is 27.6 kN. This result is also different from 14.5 kN.

Figure 23. Load-displacement curves for undrained footing capacity using the MCC model

5.3 ANALYSES WITH LADE’S SINGLE HARDENING MODEL The Lade’s single hardening model is also a strain-hardening/softening effective stress model, with both the φ' and c' (cohesion) strength parameters. More importantly, the φ' is not a constant value (like in the MCC model), but varies with stress and strain level and is not a direct input parameter to the model.

The soil is assumed to be normally consolidated (i.e. OCR = 1) and the model parameters are discussed in Potts & Zdravkovic (2012). Dry sand is considered in the analyses, with a surcharge load of 10 kPa on the ground surface and a Ko = 0.4 throughout. Drained bearing capacity analyses are considered.



Lade’s model is formulated in terms of effective stress but does not include the angle of shearing resistance φ' as an input parameter. Instead, the peak strength is controlled by several of the input parameters. As a result, the φ' varies with stress level. This implies that in the footing analyses the angle of shearing resistance varies spatially. Consequently, when applying the SR1 strength reduction approach, it is necessary to account for this variation and adopt a particular Lode’s angle at which the partial factor is to be applied. Here it has been assumed that it is the angle of shearing resistance in triaxial compression that is factored. For further details see Potts & Zdravkovic (2012).

Following the same methodology as described so far, the first analysis of the bearing capacity problem with the Lade’s model adopts the characteristic (unfactored) soil strength and the footing is loaded to failure. The resulting initial variation with depth of the angle of shearing resistance, which is available in the ground before application of the footing loading, is shown in Figure 24. The relationship is highly non-linear, with φ' varying from about 38o on the ground surface (i.e. lower stress levels) to about 33.5o at 20 m depth (i.e. higher stress levels). Because of this variation, Jaky’s formula is not applied for calculating the Ko, instead a constant value of Ko = 0.4 is adopted in all analyses. The resulting load-displacement curve from this first analysis is shown as the solid line in Figure 25, which indicates the maximum footing load at failure of 1614 kN.

Figure 24. Distributions of the angle of shearing resistance in the soil for analyses with Lade’s single hardening model



The second analysis starts with the same characteristic strength and other ground conditions and the footing is loaded to 806 kPa (load factor Lf = 2). This load is then maintained while the soil’s strength is reduced (strength reduction SR1), resulting in a failure of the footing when γm = 1.15 is achieved.

In the third analysis it is now not possible to factor a single value of φ' (strength reduction SR2) as φ' varies with depth. Therefore, the γm = 1.15 from the second analysis is applied on the whole distribution of the characteristic angle of shearing resistance in Figure 24, which results in the factored initial strength of the soil, at the beginning of the third analysis, as shown by the dashed line in Figure 24. Since φ' is not an input parameter to the model, the model parameters are adjusted iteratively until the same distribution of φ' is obtained. This distribution is shown by symbols in Figure 24 (which plot on top of the dashed line). The ultimate load on the footing from this analysis (load-displacement curve shown as a dashed line in Figure 25) is 923 kN, which is different from 806 kN in the SR1 strength reduction approach.

Figure 25. Load-displacement curves for drained bearing capacity using Lade’s single hardening model As with the MCC model the two strength reduction approaches applied in the drained bearing capacity calculations with the Lade’s model do not result in the same footing capacity for the same partial material factor. The reason for this is the non-linear variation of the angle of shearing resistance with stress level in the formulation of this model, which has resulted in



different failure mechanisms mobilised in the soil for the two approaches (see Potts & Zdravkovic (2012)).

6. CONCLUDING REMARKS This paper considers the application of the finite element method to the analyses of geotechnical engineering problems. Although the method is commonly used in current engineering practice, there are far too often examples of bad practice and misuse of numerical tools. Through examples of some common problems, the author has presented his own experience of some of the potential pitfalls associated with advanced numerical analysis and advantages that can be achieved with numerical analysis if these pitfalls are properly understood.

ACKNOWLEDGEMENTS The content of this lecture is based on work that I have performed in collaboration with my research students and colleagues at Imperial College and with my colleagues at the Geotechnical Consulting Group, London. I am indebted to them for their numerous and important contributions.

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