Workpackage 3 Innovative design methods in geotechnical ...

GeoTechNet – European Geotechnical Thematic Network

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Workpackage 3 Innovative design methods in geotechnical engineering Background document to part 2 of the final WP3 report on the use of finite element and finite difference methods in geotechnical engineering Author: - Monika De Vos & Valerie Whenham, Belgian Building Research Inst., Belgium List of Members which contributed to the input of this report: - Noël Huybrechts (WP3-leader), Belgian Building Research Institute, Belgium - Prof. Jan Maertens, Jan Maertens bvba & Cath. University of Leuven, Belgium - Dinesh Patel & Hoe-Chian Yeow, Arup Geotechnics, London, UK


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TABLE OF CONTENT

1. Overview of different methods of analysis of a geotechnical problem ......................................................4

2. Design of a geotechnical construction by means of

FEM: formulation of sources of inaccuracies and difficulties encountered...................................................6

2.1 Introduction..................................................................................................6 2.2 Problem idealisation ....................................................................................7 2.3 Specific FEM aspects ...................................................................................7

3 Some guidelines for the proper use of FEM..................8

3.1 Defining the geometry of the project..........................................................8 3.1.1 2D/3D ANALYSIS & SYMMETRY SIMPLIFICATIONS .................8 3.1.2 EXTENT OF THE GEOMETRY..........................................................9 3.1.3 STRATIGRAPHY ...............................................................................10

3.2 Meshing.......................................................................................................10

3.2.1. COARSENESS OF THE MESH.........................................................11 3.2.2. ELEMENT SHAPE .............................................................................11 3.2.3. ELEMENT ORDER ............................................................................11 3.2.4. ELEMENT COMPATIBILITY...........................................................12 3.2.5. BOUNDARY CONDITIONS .............................................................13

3.3 Numerical aspects ......................................................................................13

3.4 Soil model....................................................................................................14

3.4.1. DESCRIPTION OF THE MOST COMMONLY USED MODELS...15 3.4.1.1. Elastic material models ......................................................................15 3.4.1.2. Elastic-plastic material models ...........................................................16

3.4.1.2.1. Simple elastic-plastic constitutive models (elastic perfectly plastic models)..........................................................................18

3.4.1.2.2. An elastic strain hardening/softening Mohr Coulomb model..22 3.4.1.2.3. Cam Clay and Modified Cam Clay models (critical state

models) .....................................................................................23 3.4.2. RECOMMENDATIONS FOR THE SELECTION OF THE SOIL

MODEL FOR SPECIFIC GEOTECHNICAL PROBLEMS...............29 3.4.2.1. Shallow foundations.....................................................................29 3.4.2.2. Embankments...............................................................................29 3.4.2.3. Excavations and retaining walls...................................................30 3.4.2.4. Tunnels.........................................................................................31 3.4.2.5. Deep foundations .........................................................................31

3.4.3. RECOMMENDATIONS FOR THE SELECTION OF THE SOIL MODEL AS A FUNCTION OF THE SOIL TYPE.............................33


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3.4.3.1. Soft soils (clays slightly overconsolidated, silts, peats)...............33 3.4.3.2. Stiff overconsolidated clays.........................................................33 3.4.3.3. Sands ............................................................................................33

3.5 Input soil parameters.................................................................................34

3.5.1. STIFFNESS PARAMETERS (E,υ, G, K, m)......................................34 3.5.2. CONSOLIDATION & CREEP PARAMETERS: SWELLING,

COMPRESSION AND CREEP INDICES ..........................................48 3.5.3. SHEAR STRENGTH PARAMETERS ...............................................53 3.5.4. STATE PARAMETERS (K0 AND OCR)...........................................66 3.5.5. PERMEABILITY ................................................................................68

3.6 Models and parameters for structural elements .....................................71

3.6.1. RETAINING WALLS.........................................................................71 3.6.2. SHALLOW FOUNDATIONS ............................................................74

3.7 Analysis aspects................................................................................................75

3.7.1. THE INITIAL CONDITIONS.............................................................75 3.7.2. THE SEQUENCE OF CONSTRUCTION..........................................75 3.7.3. THE CONSOLIDATION (DRAINAGE) CONDITIONS ..................76

3.8 Supplementary guidelines specific for different types of construction.78

4 Pitfalls .............................................................................81 5. Inventory of resources ..................................................83

5.1 Benchmarks ................................................................................................83 5.2 Relevant committees:.................................................................................83 5.3 Existing (validated) softwares...................................................................83 5.4 Internet resources ......................................................................................83

6. List of references ...........................................................86


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1. Overview of different methods of analysis of a geotechnical problem Current methods of analysis for geotechnical engineers can be grouped into the following categories:

- Theoretical (closed form) analysis; - Simple analysis such as lower or upper bound solutions; - Numerical analysis.

For a particular geotechnical structure, if it is possible to satisfy the four requirements for a general solution:

- Equilibrium (i.e. well known equation of Timoshenko, 1951); - Compatibility (i.e. no overlapping of material and no generation of holes); - Material constitutive behaviour (i.e. description of material behaviour in terms of stress-strain relationship); - Boundary conditions.

Then an ‘exact theoretical’ solution can be obtained. But as soil is a highly complex non linear material, full analytical solutions to geotechnical problems are usually not available.

Therefore approximations must be introduced. Geotechnical engineers normally resort to three main design analysis methods to solve their geotechnical problems. These are the "simple analyses" approach, simplified numerical approach (beam-spring) and full numerical analysis approach. In a ‘simple analysis’, the constraints on satisfying the basic solution requirements are not fully met, but mathematics is still used to obtain an approximate analytical solution. Examples of such 'simple analyses' are limit equilibrium method, upper bound theory etc.. All these methods essentially assume the soil is at failure and/or a specific failure mechanism is assumed, but differ in the manner in which they arrive at a solution.

Typical shortcomings of simple analyses: - for ULS calculations, the soil is assumed to be everywhere at failure; - soil stress histories are not taken into account; - information is provided on local stability, but no information on soil or structural

movements is given and separate calculations are required to investigate overall stability; - these methods are less reliable for cases with complex soil-structure interaction (because

structure stiffness is often not considered).

A more comprehensive way of introducing approximations in solving geotechnical problems is through numerical approximations. Using this approach, all requirements of a theoretical solution are considered, but may only be satisfied in an approximate manner, hence the ’simplified numerical approach’.

The simplified numerical approach used to model soil-structure interaction behaviour approximates the soil as a set of unconnected vertical and horizontal springs, or as a set of linear elastic interaction factors, and represents structural supports (e.g. props, anchors…) by simple springs. This is commonly known as the "beam-spring" approach.


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Some limitations of numerical analyses based on ‘beam-spring’ approaches: - the response of the ground as a whole to loads from the structure is not properly

represented; - soil continuum effects are ignored (i.e. interaction between neighbouring springs and

redistribution of stresses in the soil); - Tension cut offs usually required for spring forces and/or interaction factors (representing

soil behaviour) are not a direct result of the beam-spring calculation; - it is often difficult to select appropriate spring stiffness and strength and to simulate support

features; - information is provided on local stability, but no information is given concerning global

stability or movements in the adjacent soil, nor are considered adjacent structures. In the full numerical analysis approach, attempts are made to satisfy all theoretical requirements, include realistic soil constitutive models and boundary conditions that realistically simulate field conditions. Approaches based on finite difference, boundary element and finite element methods are those most widely used. These methods essentially involve computer simulation of the history of the boundary value problem from green field conditions, through construction and in the long term. Their ability to accurately reflect field conditions essentially depends on the ability of the constitutive model to represent real soil behaviour and correctness of the boundary conditions imposed. Some relevant advantages of the FEM as compared to the previously mentioned methods:

- more appropriate soil constitutive models and boundary conditions can be included to simulate field conditions;

- the complex interaction between the structural elements and the soil can be accounted for, and the influence of the construction process on the environment (adjacent soil) can be assessed;

- no postulated failure mechanism or mode of failure of the problem is required ; - the effect of time on the development and variation of pore water pressures can be

simulated; - it is relatively simple to vary the material parameters and loading conditions to address

questions concerning the reliability of predictions; - No separate analysis of ULS and SLS.

Besides, with the FEM, the complete history of the problem can be simulated by a single analysis, and information can be provided on all design stages. In that way the full numerical approach can contribute to efficient application of the observational method: - The analysis results can be used to identify both representative and critical locations to

install instrumentation that will be used to monitor performance during and after construction;

- Field measurements obtained from the instrumentation during the early stages of construction can be used to calibrate the finite element model;

- The calibrated finite element model can then be used to make more reliable predictions of final displacements and stresses and to evaluate whether specific contingency plans should be implemented or not.

Notwithstanding the above advantages, full numerical analyses, in particular finite element analyses are complex and can be very confusing to inexperienced engineers. Therefore the overall objective of this report is to give the reader pragmatic recommendations when using such analysis tool.


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2. Design of a geotechnical construction by means of FEM: formulation of sources of inaccuracies and difficulties encountered 2.1 Introduction Several difficulties may be encountered when using FEM software. The problems can range from geometrical and numerical issues, selection and determination of input parameters, to the interpretation of the calculation results. An enquiry was organised within the framework of Geotechnet Workpackage 3 to identify the frequency of various difficulties that FE users had in the past. Some results are given in figures 1 and 2.

Figure 1 : Encountered difficulties in software use

Figure 2 : Encountered difficulties in software use, depending on type of user Out of the results of this enquiry, the following major problems of using FE software were identified: - parameter determination ; - definition of initial conditions ; - choice of the soil model ; - interpretation of the results.

These problems are highlighted below. For more details reference is made to the literature.

Sources of inaccuracies can be classified in those generally encountered when solving geotechnical problems and those more specifically related to the use of the FEM.

engineering/consultancy office DiscretisationType of elementsBoundary conditionsInitial conditionsChoice of soil modelChoice of type of analysis Parameters determinationResults interpretationOthers (calculations for ULS...)

contractorresearch institute universityend user

Encountered difficulties in software use

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ULS...)


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2.2 Problem idealisation Real geotechnical problems are complex by nature, involving complex 3D geometries (e.g. spatially heterogeneous soil strata), complex behaviours (for the soil, the soil-water coupling, the soil-structure interactions…) and complex process (stage constructions…) that would require infinite number of parameters. Continuous advances in computational geotechnics extend the modelling possibilities through the use of more realistic geometries or soil behaviour models, but they will never allow taking into account the exact conditions that constitute a real geotechnical problem. Therefore the phase of idealisation of the problem will always be a first important stage of analysis and will always remain the main source of inaccuracies and difficulties when solving real geotechnical problems. Generally speaking, the problem idealisation includes: - Simplification of the geometry (2D or 3D, extent of the model, soil layering etc.); - Characterisation of the soil and structural behaviour, and characterisation of their interactions; - Definition of the relevant stages of constructions and/or stress histories (from green field

conditions). 2.3 Specific FEM aspects Specific FEM difficulties are mainly related to discretisation and numerical aspects. Discretisation is related to the meshing process: selection of starting coordinates, element definition (shape, size, order, specific properties) and generation. Numerical difficulties are specific to the way the FEM has been implemented, whereas some difficulties are common to all softwares. For example, in all non-linear analyses, iterative techniques are needed to obtain acceptable solutions, and it is the user’s responsibility to decide what constitutes convergence. Discretisation aspects and numerical difficulties are general, but because part or details of it are also software specific, the use of benchmarks (specific to the software) is encouraged.


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3 Some guidelines for the proper use of FEM 3.1 Defining the geometry of the project 3.1.1 2D/3D ANALYSIS & SYMMETRY SIMPLIFICATIONS Real problems are always three-dimensional, so any 2D FE analysis will be an approximation of the reality. It is essential to be conscious of the approximations made when interpreting the results. For example the consideration of a symmetric axis is rigorously valid only if everything is symmetric (geometry, loads, boundaries…). Potts & Zdravkovic (2001) illustrate some limitations of 2D analyses (Figure 3). The figure shows a cross section of a road tunnel. The road alignment at the location of the tunnel is on a bend and consequently the tunnel roof and base slab are inclined to the horizontal (~4%). Because the fall across the structure does not appear to be significant, it is tempting to simplify the analysis by ignoring the fall across the structure and assuming symmetry about its centre. However, Figure 3 shows that wall displacements are not the same on either side of the excavation. An analysis of a ‘half section’ would not have predicted this asymmetrical behaviour. Since the displaced shape of one, or both, walls would not be correct, the predicted bending moments and forces in the walls would also have been incorrectly predicted.

Figure 3 (Potts and Zdravkovic, 1999) Cross section of a road tunnel : asymmetric effects

- The limitations of 3D FE analysis are mainly related to computer resources (time and memory

consummation) and software developments. - The necessity to take account of 3D effects depends on geometries of the problem and the

construction process. Shallow foundations can have different shapes in plane. If they are long in one dimension (strip foundation) they can be analysed assuming plane strain conditions. If they are circular and if the loading is vertical, they can be analysed assuming axi-symmetric conditions. For other shapes or loading conditions a full 3D analysis is ideally required. The influence of the shape of the foundation on its bearing capacity is illustrated below (Potts and Zdravkovic, 2001) for foundations on undrained clay (Figure 4).


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Figure 4 Influence of the shape of the foundation (load expressed by a mobilised bearing capacity factor Ncmob=Q/(A.Su), and the displacement is normalised by the width of the foundation, the halve-width of the square foundation or the halve-diameter of the circular foundation (Potts & Zdravkovic, 2001)

Piled raft design implies most often a three-dimensional modelling. The assumption of plane strain state can lead to large over-predictions of the settlements and loads carried by the piles (Van Impe, 2001).

Tunnel construction is a 3D process. If restricted to 2D analysis, often plane strain analysis is normally adopted but axi-symmetric analysis has been used, depending on what the analysis aims to achieve. Methods of simulating tunnel construction in plane strain require at least one assumption: the volume loss to be expected; the percentage of stress relief prior to lining construction; or the actual displacement of the tunnel boundary.

Anchors have 3D geometries and are therefore difficult to model in plane strain analyses (see CUR-178). 3.1.2 EXTENT OF THE GEOMETRY A volume of soil (surface and/or depth) or model boundaries that are not sufficiently large can lead to erroneous results. Various rules of thumb have been proposed in the past (Mestat, 2001, Mestat 2002, Potts et al.2002, Mestat et al. 2004…) and they depend on numerous factors such as type of problem (eg. larger model boundaries are required under dynamic loading…), type of analysis (eg. larger model boundaries are required under undrained conditions…), degree of nonlinearity of the soil model, level of stresses and strains etc. Of course the extent of the model boundaries is also closely related to the coarseness of the mesh. It is always advised to check for: - the independence of the results with the extent of the model boundaries; - the output given for the elements placed close to the model boundaries. An example is given by Potts and Zdravkovic (2001) on the necessary extent for modelling an excavation with two constitutive ground models, a linear elastic perfectly plastic model based on the Mohr Coulomb criterion taking into account the stiffness increase with depth, and a small strain soil. The soil parameters are typical for stiff clay, the excavation is modelled in undrained conditions up to 9.3m, and a stiff strut is placed on the top of the wall. Figure 5 shows that when the model taking into account the elastic small strain soil behaviour is used, a mesh with a side extent of 180 units is enough. When the elastic behaviour of the soil is modelled by a linear law, a more extended mesh must be used. According to the results of a benchmark organised by Plaxis, the influence of the geometry extent is still more important for undrained analyses.


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Figure 5 (Pötts and Zdravkovic, 2001) Influence de l’extension du maillage latéralement 3.1.3 STRATIGRAPHY The modelling of soil heterogeneity (stratigraphy) is purely a geotechnical problem. Attention should be paid when introducing thin layers or layers with sharp changes of properties (eg. stiffness) in the model due to risk of numerical oscillations and divergence in the calculations and/or inaccuracies in the results. Often, models which avoid these numerical problems are also closer to reality. 3.2 Meshing The amount of computer memory and processing time required to solve the finite element equations is a function of:

o the number of nodes in the model; o the difference between node numbers in each element (or bandwidth); o the integration order.

Furthermore, the accuracy of the results is affected to some extent by the shape of the elements and by the mixing of the different element types. The nodal number difference can be minimized by generating the elements in horizontal rows and vertical columns. It is good practice to choose a balance between ease of mesh generation and efficiency of processing. Directing a large amount of effort at mesh generation to gain marginal processing efficiency is not warranted. However, total disregard for computer memory and processing efficiency may result in a large amount of unnecessary computing time. As a general guideline, some thought should be given to processing efficiency but not at the expense of complicating the mesh generation. Another factor that should be considered in the design of a mesh is the selection of starting x and y-coordinates (datum) of a problem. Using a large starting x-or y-coordinate may affect the precision of the computed results due to round-off error. Round-off error occurs when a small number is added to a large number. The easiest way to minimize round-off error is ensure that both the starting x-and y coordinates are as close to zero as possible.


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3.2.1. COARSENESS OF THE MESH As a general rule, the element’s size should be related to the element type (with higher order elements a coarser mesh may be used) and to the gradient of the field variables (change of stress with distance,…, see Figure 6) : smaller elements in the region where the gradients are smaller. It is recommended to avoid increasing the size of an adjoining element by a factor of more than two. The coarseness of the mesh should be checked by checking the influence of refining (locally) the mesh on the output.

Figure 6. Coarseness of the mesh – illustration 3.2.2. ELEMENT SHAPE The performance of elements deteriorates rapidly if any interior angle approaches zero or 180°, or if the aspect ratio (length to height) increases (as a general rule, it is recommended to use an aspect ratio smaller than 5). Therefore it is recommended to keep triangular elements as equilateral and quadrilateral elements as square as possible (see Figure 7). Also, the best performance of long, thin elements is achieved by quadrilateral elements with eight nodes and nine point integration, while the poorest performance comes from three-noded long, thin elements with one point integration (generally, the higher order elements should be used when the aspect ratio is high). When elements with questionable interior angle or with very high aspect ratio are used, it should be verified that the results are realistic by trying different element shapes. This is generally of less importance in remote areas of the mesh where the rate of change of stress with distance is small.

Figure 7. Element shape - illustration 3.2.3. ELEMENT ORDER When an element has secondary nodes at the mid-points between the corner nodes, the element is known as a "higher order" element, since the equations describing the deformation


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within the element are of a higher order than when there are no secondary nodes. Higher order elements exhibit better behaviour than the ordinary 4 noded quadrilateral and 3 noded triangular elements. However, the higher order elements also greatly increase the processing time and memory requirements due to the additional nodes in the elements.

As a general rule, low-order elements (sometimes called simple elements) are adequate for the linear-elastic model, or if linear-elastic deformation is the primary objective of the analysis. While the ordinary elements result in a poor stress distribution within a single element, the stress distribution is reasonable when averaged to the nodes for contouring. Also, for a linear-elastic model, the material property is not a function of the computed stress; consequently, the poor stress distribution has little effect on the computed deformations. If linear-elastic deformation is the primary objective of the analysis, then ordinary 4-noded or 3-noded elements are usually adequate.

For the nonlinear constitutive soil models, the material properties are a function of the computed stresses. Therefore, a reasonable stress distribution within the element is essential and higher order elements should be used (as a minimum quadratic elements for elastic-plastic analysis are suggested); alternatively, the number of simple elements used must be relatively large. This is particularly true for elements which are subject to bending, such as elements located at the edge of a footing or in a retaining wall and in regions where the stress and strain changes are high. Alternatively a larger number of smaller elements are needed to improve the accuracy.

For axi-symmetric problems it is recommended to use higher order elements in order to correctly describe the non compressible materials (for example when the water must carry the load or when the soil fails and the plastic strains are zero). 3.2.4. ELEMENT COMPATIBILITY Note: Some software does not cater different element types and the following comment may be irrelevant. The memory and disk space requirements and the processing time can be significantly reduced by selectively using different element types and sizes in various regions of the mesh. In the transition zones, care must be taken to ensure that compatibility is maintained between elements. This is done by ensuring that the interpolating function along an edge common to the elements is of the same order. The function must either be linear in both elements or non linear in both elements. This is especially important at an interface between finite and infinite elements.

Figure 8 illustrates the requirements for compatibility. In Figure 8a, the interpolating function between Nodes A and C for Element 3 is non linear, while the interpolating function between Nodes A and B in Element 1 and between Nodes B and C in Element 2 is linear. It is therefore an unacceptable transition arrangement. In Figure 8b, the interpolating function is linear between Nodes A and B and linear between Nodes B and C in Elements 1, 2, 4, and 5. As a result, compatibility is maintained between the elements. The potential for creating unacceptable transition elements can be reduced by using triangular elements. Using quadrilateral transition elements can lead to unacceptable arrangements, such as illustrated in Figure 8a.


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Figure 8 Element Transitions and Compatibility 3.2.5. BOUNDARY CONDITIONS Inappropriate boundary conditions such as node fixities can lead to great inaccuracies in the results. Some examples are given in Potts & Zdravkovic (2001). Attention should be paid to the definition of the boundary conditions. 3.3 Numerical aspects In all non-linear finite element analyses, it is necessary to use iterative techniques to compute acceptable solutions. When the analysis has produced an acceptable match, the solution is deemed to have converged and the user must decide what constitutes convergence. In this regard it is important to be aware of the software limitations. Therefore two aspects are important: - Use of clear convergence criteria for the users; - Need of benchmarks (specific to the software used – this aspect is not considered below).

To decide what constitute convergence, different convergence criteria can be used, related to displacement and/or the unbalanced load criterion. The displacement criterion checks the ratio of the vector norm of incremental displacements in a iteration to the vector norm of the total displacements in the load step. The unbalanced load criterion compares the unbalanced load in an iteration to the applied load in the load step as a percentage ratio. Viewing the convergence parameters graphically makes it easier to judge whether the convergence criteria are being met. Generally, the convergence parameters should tend toward reasonable, stable, constant values which are below a user predefined acceptable values/limits. A trick that is sometimes useful in judging convergence is to set the maximum number of iterations to a very high number and also set the convergence tolerance to a very low number. This will force the solution far past an acceptable convergence, but it can give confidence that an acceptable, stable solution has been reached. However, this approach sometimes causes a long computation time. Graphical representation of the unbalanced load norm parameter can be particularly useful in judging convergence, since this parameter should ideally always diminish and trend towards zero. The solution is unacceptable if the unbalanced load norm is higher after several iterations than at the start of the analysis and it never diminishes below the starting position. Figure 9 illustrates the influence of the error tolerance when using the Plaxis software for the resolution of a deep excavation (see Schweiger 1998). It is worth noting that such variation is a function of the software package used.


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02468101214161820222426283032

-50 -40 -30 -20 -10 0Wall displacements [mm] depth [m]

Tol. error 0,03Tol.error 0,05

-30 -25 -20 -15 -10

-5 0

0 10 20 30 40 50 60 70 80 90 100 110 120Distance from the wall [m]Settlements

[mm]

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02468101214161820222426283032

-1000 -500 0 500 1000Bending moments [kN.m/m] Depth [m]

Tol.error 0,03Tol.error 0,05

Figure 9. Numerical aspects – illustration - Influence of the ‘tolerated error criteria’ with Plaxis software

3.4 Soil model Natural soils are complex and variable. None of the currently available soil constitutive models can reproduce all aspects of real soil behaviour; therefore the choice of soil model will depend on several factors: - which soil features govern the behaviour of a particular geotechnical problem; - the type of soil to model; - the analysis conditions; - the availability of soil data from which the parameters should be derived.


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3.4.1. DESCRIPTION OF THE MOST COMMONLY USED MODELS In the following, the soil models that are most commonly used in finite element analyses are presented. The first group consists of the elastic material models, of both linear and non linear type. The second group is the elastic-plastic models and include the models of Tresca, von Mises, Mohr Coulomb (with and without stress hardening), Drucker-Prager and Cam clay. Although there are many other advanced soil models available for use in finite element analyses, they are not covered under the context of this report. 3.4.1.1. Elastic material models (Figure 10) The basic assumption of elastic behaviour is that the directions of principal incremental stress and incremental strain coincide. Elastic constitutive models can take many forms: isotropic or anisotropic, linear or non linear.

The isotropic linear elastic models involve two elastic stiffness parameters: Young's modulus and Poisson's ratio, or shear modulus and effective bulk modulus. For geotechnical engineering it is often convenient to use bulk modulus K and shear modulus G. The reason for this is that the behaviour of soil under changing mean (bulk) stress is very different to that under changing deviatoric (shear) stress. For instance, under increasing mean stress the bulk stiffness of the soil will usually increase, whereas under increasing deviatoric stress the shear stiffness will reduce. Furthermore, in the formulation of the isotropic elastic model, the two modes of deformation are decoupled: changes in mean stress Δp’ do not cause distortion (shear strain) and changes in deviatoric stress do not cause volume change (Potts and Zdravkovic, 1999).

The linear elastic model is very limited for the simulation of soil behaviour. It is primarily used for stiff massive structures in the soil (structural elements e.g. retaining walls, slabs etc.).

(a) (b) Figure 10. Elastic model. a.linear elastic model, b. Non-linear elastic model (bilinear and hyperbolic) (Potts and Zdravkovic, 1999) In reality, the stress-strain behaviour of soil becomes nonlinear, particularly as failure conditions are approached. Therefore, nonlinear elastic models, in which the material parameters vary with stress and/or strain level are a substantial improvement over the linear models, whereas they still fail to model some of the important facets of real soil behaviour. In particular they cannot reproduce the tendency to change volume when sheared. Also, because of the assumption of coincidence of principal incremental stress and strain directions, they cannot accurately reproduce failure mechanisms. (Potts and Zdravkovic, 1999)

σ, τ σ, τ

ε ε


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Duncan and Chang model

One example of nonlinear elastic model was presented by Duncan and Chang (1970). In this formulation, the stress-strain curve is hyperbolic in the shear stress, (σ1-σ3), versus axial strain space and the soil modulus is a function of the confining stress and the shear stress that a soil is experiencing. This nonlinear material model is attractive since the soil properties it requires can be obtained quite readily from triaxial tests or the literature (for example, Duncan et al., 1980). Depending on the stress state and stress path, three soil moduli are required; namely, the initial modulus, Ei the tangential modulus, Et, and the unloading-reloading modulus, Eur (see figure 11).

Figure 11. Non-Linear Stress-Strain Behaviour 3.4.1.2. Elastic-plastic material models Elastic plastic theory provides probably the best framework available in which to formulate constitutive models that can realistically simulate real soil behaviour (Potts and Zdravkovic, 1999). It assumes elastic behaviour prior to yield and can therefore utilise the benefits of both elastic and plastic behaviour. Elastic-plastic models are based on the assumption that the principal directions of accumulated stress and incremental plastic strain coincide. They require the following piece of information for their definition: a yield function which separates purely elastic from elastic-plastic behaviour; a plastic potential (or flow rule) which prescribes the direction of plastic straining, and (optional) a set of hardening/softening rules which describe how the state parameters (for example strength) vary with plastic strain (or plastic work).

In uniaxial situations, the yield stress indicates the onset of plastic straining. In a multi-axial situation a yield function is defined instead, which is a scalar function of stress and state parameters {k}: F({σ},{k})=0.

Figure 12 gives examples of yield function with the Mohr Coulomb and Von Mises failure conditions (common plasticity rules used respectively to describe drained and undrained situations).


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(a) (b)

Figure 12. (a) Mohr Coulomb and (b) Von Mises failure criteria (and related yield functions) (EPFL, 1997) The yield function defines the state of stress at which material response changes from elastic to plastic. In general, the surface is a function of the stress state {σ} and its size also changes as a function of the state parameters {k}, which can be related to hardening/softening parameters.

In real soil plastic strain occurs even before reaching "failure". Strain hardening and softening models introduce plastic strains before reaching the ultimate yield surface. The plastic strains modify the yield surface during hardening or softening. The Figure 13 illustrates the three main forms of elastic-plastic behaviour. They can be compared with some of the more common features of soil behaviour. For example if the curve corresponding to the elastic plastic strain hardening model is replotted with the stress axis horizontal and the strain axis vertical, it bares some resemblance to the behaviour observed in an oedometer test. Soil behaviour on a swelling line is often assumed to be reversible and therefore is akin to behaviour on an elastic unload-reload loop. The behaviour on the virgin consolidation line is irreversible and results in permanent strains. It is therefore similar to behaviour along the strain hardening path. Similarity also exists between the strain softening behaviour and the shear stress-shear strain behaviour observed in a direct or simple shear test on dense sand. To simulate the behaviour of real soil it is necessary to have a model that involves both strain hardening and softening.

Elastic-perfectly plastic model El-pl. Strain hardening model El-pl. Strain softening model

ε

σ, τ εp εe

ε

σ, τ


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Mohr Coulomb with cap hardening Von Mises with resp.orthotropic and

kinematic hardening Figure 13. Elasto-plastic models - illustration

In the multi-axial case it is necessary to have some means of specifying the direction of plastic straining at every stress state. This is done by means of a flow rule that relates strain increments to stress increments after the onset of initial yielding. The flow rule can be expressed as follows:

σσε∂

∂Λ=Δ

}){},({ mPp where dεp is the incremental plastic strain, P is

the plastic potential function and Λ is a scalar multiplier. The plastic potential function is of the form P({σ},{m})=0 where {m} is essentially a vector of state parameters the values of which are immaterial, because only the differentials of P with respect to the stress components are needed in the flow rule.

Due to the complex nature of soil it has not been possible, to date, to develop an elastic-plastic model that can capture all the facets of real soil behaviour and be defined by a limited set of input parameters that can be readily obtained from simple laboratory tests. There are therefore many such models currently in the literature. These range from simple to extremely complicated models. 3.4.1.2.1. Simple elastic-plastic constitutive models (elastic perfectly plastic models) Tresca and van Mises elastic perfectly plastic models are expressed in terms of total stresses and apply to undrained soil behaviour, while the Mohr Coulomb and Drucker-Prager models are expressed in terms of effective stresses, and are therefore more suited for solving most geotechnical problems.

The implicit assumption of the Tresca and Mohr Coulomb models is that yield and strength are independent of the intermediate principal stress σ2.There is little experimental data available to accurately quantify the effect of the intermediate principal stress on soil behaviour. The limited data that does exist suggests that both yield and failure functions plot as smoothed surfaces (no corners) in the deviatoric plane, with a shape somewhere between that of the hexagons and circles (assumptions of Tresca / Mohr-Coulomb and von Mises /

σ’1=σ’2=σ’3 σ’1

1ε&

σ’1 1ε&

σ’1 1ε&

22

32

22

'2'2

εε

σσ

&& =

=


19

Drucker Prager). As conventional soil mechanics is based on the Tresca and Mohr Coulomb models, it seems sensible to use these models in preference to the von Mises and Drucker Prager models. This has the advantage that the finite element analysis is then compatible with conventional soil mechanics, but has the disadvantage that the software has to deal with the corners of the yield and plastic potential surfaces.

To increase the flexibility of the models, it is also possible to replace the linear elastic with nonlinear elastic behaviour, by allowing the elastic constants to vary with stress and/or strain level.

Tresca yield surface von Mises yield surface

Mohr Coulomb yield surf. Drucker Prager yield surf.

Figure 14. Elastic perfectly plastic models


20

Tresca model

If a conventional triaxial test is performed, it is common to plot the results in terms of the vertical and horizontal stresses σv

(‘) and σh(‘). If testing saturated clay, the Mohr’s circle of stress at failure is often

plotted in terms of total stress and may look like that given in figure 15.

Figure 15. Mohr’s circles of total stress (Potts and Zdravkovic, 1999) If two similar samples are tested at different cell pressures, without allowing any consolidation, conventional soil mechanics theory suggests that the Mohr’s circles of stress at failure for the two samples have the same diameter but plot at different positions on the σ axis. A failure criterion is then adopted which relates the undrained strength Su to the diameter of the Mohr’s circle at failure. Noting that in a conventional triaxial test σ1 = σv and σ3 = σh, this can be expressed as: σ1-σ3 = 2 Su. In the Tresca model this failure criterion is adopted as a yield surface and the yield function becomes: F({σ},{k}) = σ1 - σ3 – 2 Su = 0. For finite element analysis, it is more convenient to rewrite this equation in terms of the stress invariants p (mean effective stress), J (deviatoric stress), θ (Lode’s angle)

F({σ},{k})=J cosθ - Su = 0 where

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−

=

−+−+−=

++=

− 1)''()''(

23

1tan

)²''()²''()²''(61

)'''(31'

31

321

313221

321

σσσσ

ϑ

σσσσσσ

σσσ

J

p

In principal total stress space this yield function plots as a regular hexagonal cylinder, which has the space diagonal as its line of symmetry. This model is perfectly plastic, therefore there is no hardening/softening law required and the state parameter {k}=Su is assumed constant (independent of plastic strain). As the model is intended to simulate the undrained behaviour of saturated clay, it should predict zero volumetric strains. Since the soil is purely elastic (below the yield surface) or purely plastic (on the yield surface), both the elastic and plastic components of the volumetric strain use be zero. It can be shown that an associated plastic flow i.e. P({σ},{m}) = F({σ},{k}) satisfies the no plastic volumetric strain condition. Besides, as there should be no elastic volumetric strains, μ ~ 0.5. The model can therefore be defined by specifying the undrained strength Su and the undrained Young’s modulus Eu.


21

Von Mises model

When plotted in 3D principal total stress space, the Tresca yield surface has corners. In particular, the intersection of the surface with a deviatoric plane (i.e. a plane normal to the space diagonal) produces a regular hexagon which has its corners at triaxial compression and extension points. These corners imply singularities in the yield function which can cause difficulties in numerical analysis. Therefore applied mathematicians have often simplified the yield function expression so that it plots as a circular cylinder in principal stress space, instead of a hexagonal cylinder. However, there are ways that the numerical difficulties due to the singularities in the Tresca yield function can be overcome, and the Tresca model is thought to be more appropriate than von Mises because it is based on the same assumptions as conventional soil mechanics (Potts & Zdravkovic).

Figure 16. von Mises and Tresca yield surfaces in principal stress space

Mohr Coulomb model

If the results of laboratory tests are plotted in terms of effective stresses, the Mohr’s circles of stress at failure are often idealised as shown in figure17.

Figure 17. Morh’s circles of effective stress (Potts and Zdravkovic, 1999)

It is usual to assume that the tangent of the failure circles from several tests, performed with different initial effective stresses is straight. This line is called the Coulomb failure criterion and can be expressed as τf = c’ + σ’nf tan ϕ’ where τf and σ’nf are the shear and normal effective stresses on the failure plane, and the cohesion c’ and the angle of shearing resistance ϕ’ are material parameters. Using the Mohr’s circle of stress and noting that σ’1=σ’v and σ’3=σ’h the equation can be rewritten as σ’1-σ’3 = 2c’ cos ϕ’ + (σ’1+σ’3) sin ϕ’. This is often called the Mohr Coulomb failure criterion and in the present model is adopted as the yield function F({σ’},{k}) = σ’1-σ’3 - 2c’ cos ϕ’ - (σ’1+σ’3) sin ϕ’ or in terms of stress invariants

0)(''tan

'})}{'({ =⎟⎟⎠

⎞⎜⎜⎝

⎛+−= ϑ

ϕσ gpcJkF where

3'sinsincos

'sin)(ϕϑϑ

ϕϑ+

=g

In principal effective stress space the yield function plots as an irregular hexagonal cone. If Su is substituted for c’ and ϕ’ is set to zero, the Tresca yield function is obtained. As the Mohr Coulomb model is assumed to be perfectly plastic, there is no hardening/softening law required; the state parameter {k}={c’,ϕ’} is assumed constant, independent of plastic strain.


22

Similar to the Tresca model, an associated flow rule with P({σ},{m}) = F({σ},{k}) could be adopted for the plastic potential function, but there is two drawbacks to this approach. Firstly the magnitude of the plastic volumetric strains (the dilation) is much larger than that observed in real soils, and secondly, once the soil yields it will dilate for ever. Real soil, which may dilate initially on meeting the failure surface, will often reach a constant volume condition (zero incremental plastic volumetric strains) at large strains. The first drawbacks can be partly rectified by adopting a non-associated flow rule, where the plastic potential function is assumed to take a similar form to that of the yield surface, but with ϕ’ replaced by dilation υ (more often noted ψ).

( ) 0)('})}{'({ =+−= ϑσ pppp gpaJmP where

3sinsincos

sin)(νϑϑ

υϑ+

=ppg

and app is the distance of the apex of the plastic potential cone from the origin of principal effective stress space. It is akin c’/tan ϕ’ in the yield function.

Figure 18. Relationship between the yield and plastic potential functions (Potts & Zdravkovic, 1999) While the yield surface is fixed in p’-J-θ space, the plastic potential surface moves so as to pass through the current stress state. If υ=ϕ’, associated conditions arises. However υ<ϕ’ results in non-associated conditions, and υ reduces less dilation is generated. If υ=0°, zero plastic dilation (no plastic volume strain) occurs. Consequently, by prescribing the angle of dilation υ, the predicted plastic volumetric strains can be controlled.

Drucker-Prager model

As the Tresca model, the Mohr Coulomb yield function has corners when plotted in principal effective stress space. In order to simplify numerical difficulties, earlier pioneers sought simplifications. One way to overcome the corner problem is to modify the yield function so that it plots as a cylindrical cone. This form of the yield function is often called the Drucker-Prager yield function. 3.4.1.2.2. An elastic strain hardening/softening Mohr Coulomb model To improve the Mohr Coulomb model, the strength parameters c’ and ϕ’ and the angle of dilation υ can all be allowed to vary with the accumulated plastic strains. One example of a model assuming the variation of c’ and ϕ’ with accumulated deviatoric plastic strain Ep

d is given in figure 19. Strain hardening occurs in zone 1, behaviour is perfectly plastic in zone 2 and strain softening occurs in zone 3. The angle of dilation is assumed to be proportional to the angle of shearing resistance ϕ’ in zones 1 and 2 whereas in zone 3 it is assumed to reduce from the peak value assumed in zone 2 to a residual value υr in the same manner as ϕ’ reduces (i.e. either linearly or exponentially).


23

Figure 19. Hardening rules (Potts and Zdravkovic, 1999) 3.4.1.2.3.Cam Clay and Modified Cam Clay models (critical state models)

Cam Clay and Modified Cam Clay models were originally developed for triaxial loading conditions. The models are essentially based on the following assumptions:

- A piece of clay, which is subjected to slow, perfectly drained isotropic (σ1’=σ2’=σ’3) compression, moves along a trajectory in the v-lnp’ plane (v =specific volume = 1+e, p’ =(σ1’+σ2’+σ’3)/3 ), which consists of a virgin consolidation line and a set of swelling lines. Initially, on first loading, the soil moves down the virgin consolidation line.

Figure 20. Behaviour under isotropic compression Figue 21. Yield surface The virgin consolidation line and the swelling lines are assumed to be straight in v-lnp’ space and are given by the following equations: v+λ(ln p’) = v1 v+ κ(ln p’) = vs

The values of κ, λ and v1 are characteristics of the particular type of clay, whereas the value of

vs is different for each swelling line. Volume change along the virgin consolidation line is mainly irreversible or plastic, while volume change along a swelling line is reversible or elastic.

- The behaviour under increasing triaxial shear stress, q = σv’-σh’= √3J, (where J=[(σ1’-σ2’)²+(σ2’-σ3’)²+(σ1’-σ3’)²]0.5*(1/6)) is assumed to be elastic until a yield value of q is reached, which can be obtained from the yield function F({σ’},{k}) = 0. Behaviour is elastic along

Virgin consolidation line

Swelling line

Specific volume

Swelling line

Virgin consolidation line

Yield surface


24

swelling lines and therefore the yield function plots above each swelling line. For respectively Cam clay and modified Cam clay the yield surface are assumed to take the form:

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛=

0''7183,2ln

'.}){},'{(

pp

MpJkF

j

σ (Original Cam clay)

Figure 22. Projection of yield surface onto J-p’ plane Figure 23. State boundary surfaces where p’ is the mean effective stress, J is the deviatoric stress, MJ is another clay parameter, and po’ is the value of p’ at the intersection of the current swelling line with the virgin consolidation line. The projection of these curves onto the J-p’ plane is shown in Figure 22 where it can be seen that the modified Cam clay yield surface plots as an ellipse. The parameter po’ essentially controls the size of the yield surface and has a particular value for each swelling line.

As there is a yield surface for each swelling line, the yield function defines a surface in v-J-p’ space, called the Stable State Boundary Surface. If the v-J-p’ state of the clay plots inside this surface, its behaviour is elastic, whereas if its state lies on the surface it is elastic plastic. It is not possible for the clay to have a v-J-p’ state that lies outside this surface.

- Hardening/softening is isotropic and is controlled by the parameter po’ which is related to the plastic volumetric strain, εv

p by

When the soil is plastic (i.e. on the Stable State Boundary Surface), the plastic strain increment vector is taken normal to the yield curve. Consequently, the model is associated, with the plastic potential P({σ’},{m}) being equal to the yield function.

- Behaviour along a swelling line is elastic.

In the original formulation, no elastic shear strains are considered. To avoid numerical problems and to achieve a better modelling inside the state boundary surface, elastic shear strains are usually computed from an elastic shear modulus, G, which is an additional model parameter. In the above form, both the Cam Clay and modified Cam Clay models require 5

(hardening rule)

(Modified Cam clay)


25

material parameters: v1 , κ, λ, MJ and G. Sometimes an elastic Poisson’s ratio, µ, is specified instead of G. Remarks on the Cam clay model The material parameters used to define the modified Cam Clay model include the consolidation parameters (v1, κ and λ), the drained strength parameter (Φcs’ or MJ) and its variation in the deviatoric plane, and the elastic parameter (µ or G). They do not involve the undrained shear strength, Su . As this model is often used to represent the undrained behaviour of soft clays, whose strength is conventionally expressed in terms of Su , this can be inconvenient. The undrained shear strength, Su, can be derived from the input parameters and the initial state of stress as shown by Potts and Zdravkovic (Potts and Zdravkovic, 1999). The resulting equation is:

where

By using this equation it is possible (but not recommended!) to select input parameters (κ , λ , and Φcs’ or MJ) and initial stress conditions (OCR and coefficient of earth pressure at rest, Ko), so that the desired undrained strength distribution can be obtained. In this respect care must be exercised because the undrained strength is always zero when the initial vertical effective stress is zero. Consequently, if the undrained strength profile is known, it is possible to use Equation (1) to back calculate one of either OCR, K0

OC or K0NC (it will be necessary for the finite element software to be

flexible enough to allow the user to input such a variation of OCR). For modified Cam-clay the undrained strength, Su, is linearly related to the vertical effective stress, σ’vi. Consequently, if σ’vi=0, then so will the undrained strength. Therefore it is necessary for the OCR to increase rapidly near to the ground surface. However, even if σ’vi=0 at the ground surface (i.e. no pore water suctions present) it is still possible to perform finite element analysis which simulate a finite undrained strength at the surface. This is possible because the constitutive model is only evaluated at the integration points which lie a finite distance below the ground surface (Potts and Zdravkovic, 2000).

In their paper, Potts and Zdravkovic consider the influence of the shape of the yield and plastic potential surfaces in the deviatoric plane. Many different options have been described in the literature (and implemented in softwares) and this is one of the most uncertain areas of the model. They show that this shape can have a dominant effect on both the predicted drained and undrained strengths of the soil. In the strip footing example described in their paper an increase in the failure load of some 58% can be attributed to just changing the shape of the yield surface from a Mohr-Coulomb hexagon to a circle. Table 1 gives an overview of available soil models along with the aspects that are taken into account, their limitations and the input parameters needed.

(1)


26

Soil model : Stress-strain behaviour : Aspects taken into account : Limitations : Input parameters : Examples : Linear elastic

ε

σ'

Ε

1

- linear stress-strain behaviour

- no plasticity, - stiffness is not dependent of

the stress or strain level - stiffness is the same for

loading, unloading and reloading

E (or G) , ν

- Hooke

Non linear elastic

ε

σ'

Ε0

Ε5011

- stiffness is dependent of the stress or strain level e.g. hyperbolic law

- No plasticity, - Stiffness is the same for

loading, unloading and reloading

E (or G) as a function of the stress or strain level (e.g. E0 and E50) , ν

- Duncan-Chang

Elastic perfectly plastic

ε

σ'

- plasticity

- stiffness is the same for loading, unloading and reloading

Elastic behaviour : E (or G) , ν Plastic behaviour : ϕ’ , c’ , ψ

- Mohr-Coulomb - Von Mises - Drucker-Prager - Tresca

Soil hardening

εa

σa-σr

- Soil hardening Model specific Model specific - Hardening Soil model (Plaxis software)

- BRICK model (Oasys software)

- ICFEP model - Nova model

(CESAR-LCPC) - Many others…

1


27

Soil softening

ε

cu

Ε1 1

R

Peak cu

- Soil softening

Model specific Model specific - Multi-linear model (Oasys software)

Critical state model

P'0

J

CMJ

CSL

- Model specific Model specific Model specific - Cam Clay

With creep

- Creep Model specific Model specific - Soft Soil Creep (Plaxis software)

Cyclic Hardening Plasticity

- Model specific Model specific Model specific - Wang - …


28

3D models

- Model specific Model specific Model specific - Lade-Duncan - Matsuoka-Nakai - Willam-Warnke - Argyris-Gudehus - …

With small strain stiffness

- Small strain stiffness Model specific Model specific - BRICK model (Oasys software)

- ICFEP model - Von

Wolffersdorff Table 1 : Soil models most usually adopted for finite element calculations

GeoTechNet – European Geotechnical Thematic Network 29

3.4.2. RECOMMENDATIONS FOR THE SELECTION OF THE SOIL MODEL FOR SPECIFIC GEOTECHNICAL PROBLEMS 3.4.2.1. Shallow foundations - If only the bearing capacity of the foundation has to be considered, the soil model primarily has to

simulate correctly the strength properties of the ground. Besides it may be important for foundations constructed on clay soil to use a coupled numerical approach to simulate the variation of bearing capacity with time due to the consolidation process.

- If settlement is the main design criterion, and the settlement must be fairly small, the load-displacement response is likely linear elastic along the initial portion of the stress-strain curve and a simple linear elastic analysis is adequate. However this approach will not provide accurate predictions of differential settlements, deformations under combined loading, or movements in the soil adjacent to the foundation. If deformations are of concern and in particular those adjacent to the foundation, then it is advisable to use a constitutive model that can more accurately represent the nonlinear behaviour of the soil under small strains.

- If, as a consequence of the loading, a state of normal consolidation may be reached and significant creep may follow, it may be important to select a soil model capable of simulating the time dependent behaviour of the soil.

Results of finite element analyses using the sophisticated MIT-E3 model are presented by Potts & Zdravkovic (2001) to show how the effects of observed anisotropic soil behaviour can be reproduced in finite element analyses. Figure 23 shows an example of the results of ultimate vertical and horizontal load for a strip foundation and a circular foundation on a silt soil. In both cases the negative effect of anisotropy is clearly demonstrated.

Figure 23. Influence of the anisotropy on the bearing capacity of a direct foundation (Potts and Zdravkovic, 2001) In another example of strip footing described by Potts and Zdravkovic (2001), an ultimate load increase of about 58% has been attributed to differences in the shape of the plasticity surface (hexagon or circle), that shows the influence of the intermediate principal stresses, generally ignored in classical models. 3.4.2.2. Embankments - If considerable yielding and deformation can be tolerated without affecting the serviceability of the structure, a nonlinear analysis is required to obtain a realistic estimate of the potential displacements; a simple linear-elastic analysis could considerably underestimate the displacements. Indeed, a great variety of stress paths, accompanied by a rotation of principal stresses occurs in embankment dams during construction. Non-linear elastic constitutive models cannot account for stress path dependency which is caused by non elastic components of soil behaviour. - If excess pore pressures are generated (eg: by placing fill for an embankment on soft soil) to the

point where the stability is affected, one should use a more sophisticated effective stress model together with a consolidation analysis.


- When excavations are formed in clayey soils, the selection of the constitutive model mainly depends on the type of clay:

- Stiff plastic clays exhibit brittle properties and are susceptible for progressive failure. To model such complex failure mechanisms, it is necessary to use elastic-plastic models that account for softening by allowing the angle of shearing resistance ϕ’ and/or the cohesion intercept c’ to vary with strains.

- Slightly plastic clayey loam can be reasonably modelled by simple elastic perfectly plastic models.

- Soft clays normally consolidated or slightly overconsolidated are often subjected to plastic deformations during excavation, and it may be necessary to select models taking account of the pre-failure plastic behaviour of the soil (hardening models).

- Creep is important for problems involving large primary compression, especially if embankments are founded on initially overconsolidated soil layers that yield relatively small primary settlements, and depending on the type of soil it may be important to select a soil model able to simulate the time dependent behaviour of the soil.

- Filling material is generally much stiffer than the overlying soft clay, and is rarely critical for the analysis (simple elastic perfectly plastic models can be used). If the embankment stability has to be considered, the constitutive model must represent the plastic and ultimate soil resistances, and a model such as Mohr Coulomb can be selected (sometimes with nonlinear elasticity).

3.4.2.3. Excavations and retaining walls - In situ soils should be modelled using the most appropriate constitutive model that is available and

that can be justified from the site investigation data. Ideally, this model should account for both nonlinearity at small strains and soil plasticity. The former being required to enable realistic predictions of displacements and the latter to limit the magnitudes of the active and passive earth pressures.

- In overwhelming majority of cases simple, linear elastic analyses are entirely inappropriate and can be misleading. In such analyses there is no restriction on the tensile stresses, compressive or shear stresses that can develop within the soil, nor the magnitude of the active and passive earth pressures. For example considering a simple embedded cantilever wall with an excavation in front of it, tension in the soil behind the top of the wall will tend to hold the wall back. The displacement profile of the wall will be affected and consequently bending moments and forces induced in the structure.

- Linear-elastic perfectly plastic constitutive models do limit the tensile stresses in the soil and the active and passive pressures that can develop. However, they generally give poor predictions of both the extent and the distribution of ground movements adjacent to the structure under construction because the same stiffness is used for loading and unloading and because in the case of soil unloading, the deformation shape is mainly governed by the (medium) stress level decrease due to the excavation.

- A coupled consolidation/swelling analysis is useful for medium and long term situations. This requires good computation of the pore pressures generated during deformation, for which an advanced effective stress model is often needed. For the modelling of the long term behaviour and of the post-construction effects which can be significant in stiff overconsolidated clays (swelling of clay), more advanced models taking into account the non-linear small strain behaviour must be selected. For short term analyses, section of the soil model depends on the type of soil.

Comparison between simple linear elastic and linear elastic perfectly plastic models is illustrated below (Figure 24).


Figure 24 (Potts and Zdravkovic, 1999) Comparison between elastic linear models and elastic perfectly plastic models Hence models that account for the evolution of the soil elasticity modulus with stress and strain are necessary to model the displacement of the soil adjacent to the excavation. Grande (1998) showed that the deformations obtained from the elastic perfectly plastic model (Mohr Coulomb, Plaxis) are unrealistic because the same stiffness is used for loading and unloading and because in the case of soil unloading, the deformation shape is mainly governed by the medium stress level decrease due to the excavation. Swelling is predicted in surface, contrary to the predictions of the ‘Hardening soil’ model that accounts for soil hardening. These observations are illustrated below by Figure 25, showing the soil surface deformations just after excavation.

Figure 25 (Grande, 1998) Surface settlements due to sheet pile excavation 3.4.2.4. Tunnels Tunneling is a 3D problem and the effects of anisotropy is essential in any soil models used to analyse this construction problem. To model tunnel construction, it is important to select constitutive models capable of reproducing field behaviour. For example, in a situation where pre-yield behaviour dominates the ground response, it is essential to model the nonlinear elasticity at small strains.

3.4.2.5. Deep foundations - Under vertical loading, simple elastic perfectly plastic models are generally used. However, it can

be useful for clay soils to model the time dependent soil behaviour (creep…). For granular soils dilation can have an important effect on the pile behaviour.


- When analysing a single pile subject to axial loading either thin solid elements or special interface elements should be placed adjacent to the pile shaft. If solid elements are used and they are not sufficiently thin, the analysis will over estimate the pile shaft capacity. Most often an elastic perfectly plastic model like the Coulomb friction model is selected for these interface elements, with soil parameters possibly modified as compared with the adjacent soil in function of the real pile installation process. The Coulomb friction model allows distinguishing the elastic behaviour of the interface, with reduced displacements, as the plastic behaviour of the soil at the interface corresponding to sliding of the pile. It does not allow to model soil softening or lateral friction decrease between pile and soil when the Mohr Coulomb criterion is reached.

- If interface elements are positioned adjacent to the pile shaft care should be taken when selecting their normal and shear stiffness. These values, if not sufficiently large, can dominate pile behaviour. However, if the values are too large numerical ill conditioning may occur.

- When analysing a single pile subject to lateral loading the possibility of a crack forming down the back of the pile (i.e. gapping) should be considered. If this is likely to occur interface elements should be installed along the pile soil interface, which cannot sustain tensile normal stresses. Whether or not gapping is likely to occur depends on the soil strength and in particular its distribution with depth.

- In order to model tension pile, an interface layer with very low axial stiffness can be used allowing forces and water pressures to act on the boundary, while limiting the resistance to the uplift pile movement. Obviously the axial stiffness should reflect the stiffness of the pile under tensile load, when the concrete exceeds the tensile capacity. These elements do not allow to take into account the decrease in surface contact between pile and soil when pile uplift. Important shear deformations then appear at the base of the pile, and the modelling become not valuable.

- Most of the constitutive models do not consider excess water pressure generated by shearing respectively in loose and dense soils. Generation of these water pressures modifies the effective stress state in the soil and can have a great influence on the pile friction.

FE analyses using MIT-E3 model have shown how that anisotropic soil strength observed in the laboratory could be modelled in such constitutive soil model. Anisotropy in undrained shear strength will for example reduce the pull-out capacity of a pile as compared to equivalent analyses assuming isotropic behaviour.

The effect of dilation on the load-settlement curve of piles is illustrated in Figure 26. A finite element modelling of a static pile load test executed at the site of BBRI in Limelette, which soil consists in quaternary silty layers overlaying the tertiary Bruxellian sand, has been performed. Figure 26 shows the load-settlement curve obtained by finite element modelling, assuming different values of the dilation angle and the Mohr Coulomb soil model. When analysing axially loaded piles using an effective stress constitutive model it is unwise to use a model which predicts finite plastic dilation indefinitely, without reaching a critical state condition (e.g. Mohr-Coulomb model with ψ> 0°). Such analyses will not predict an ultimate pile capacity.

-35

-30

-25

-20

-15

-10

-5

00 500 1000 1500 2000 2500 3000

Q [kN]

s [m

m]

ref

psi 1-1

psi 1-2

psi 2-1

psi 2-2

Figure 26. influence of the dilation angle: ref : layer 1:ψ=4°, layer 2:ψ=4° ; psi 1-1: layer 1:ψ=0°, layer 2:ψ=4°; psi 1-2: layer 1:ψ=2°, layer 2:ψ=4° ;psi 2-1: layer 1:ψ=4°, layer 2:ψ=0° ;psi c2-2: layer 1:ψ=4°, layer


2:ψ=2°. 3.4.3. RECOMMENDATIONS FOR THE SELECTION OF THE SOIL MODEL AS A FUNCTION OF THE SOIL TYPE 3.4.3.1. Soft soils (clays slightly overconsolidated, silts, peats)

- The (modified) Cam Clay model, with the correct selection of input parameters, represents (at least

qualitatively) the strength and deformation properties of the soft soil realistically. Common observed properties such as an increasing stiffness as the material undergoes compression, hardening/softening and compaction/dilatancy behaviour, and tendency to eventually reach a state in which the strength and volume become constant (critical state) are all captured by the modified Cam Clay model. The model is most suitable for lightly overconsolidated clays, clayey silts and peat (soft soils), for which the compaction and hardening behaviour are well predicted, and is less suitable for hardly overconsolidated soils.

- When selecting a model for soft soil problems one has to consider the type of engineering problem being dealt with. When one is confronted with loading and consolidation (settlement of foundations or embankments eg), stresses will often increase beyond the existing level of preconsolidation and a critical state** model that models the plastic compaction beyond the preconsolidation stress is most suitable. In such cases, it can also be useful to take into account the viscous effects (creep and stress relaxation), most dominant in soft soils, i.e. normally consolidated clays, silts and peat. On the other hand, when considering unloading problems, normal stresses will not increase beyond the preconsolidation and most appropriated models are elastic plastic models that account for the hardening behaviour of the soil.

- Constitutive models are generally based on the assumption of isotropic soil, what is rarely the case for soft clays, whose strength and stiffness is usually function of the principal stresses orientation. Ideally, this would involve selection of more complex soil models which account for anisotropy.

** The “critical state” in a soil is defined as the state in which shear deformation can continue to

occur at constant effective stress and without change of volume. A critical state model is one which exhibits this fracture.

3.4.3.2. Stiff overconsolidated clays

- For stiff overconsolidated clays most appropriated models are elastic plastic models that account for the hardening behaviour of the soil. Non-linearity inside the yield surface is also an important soil behaviour that needs to be modelled.

- For overconsolidated London Clay, two of the most used soil models are the Jardine's small strain model (Jardine 1986) and the BRICK model (Simpson 1992). The Jardine model is based on a non-linear elasticity model while the BRICK model is based on the concept of a multi-surface kinematic hardening model

3.4.3.3. Sands

- Elastic plastic models taking into account dilatancy and soil shear hardening are suitable for describing sand behaviour. Dense sands often behave in a very stiff manner until close to their failure stress; they may reasonably be modelled by simple elastic perfectly plastic models.

Remark: At present, some of the (required) features (anisotropy, softening, small strain behaviour…) are not yet available in most commercial codes, but will be in near future.


3.5 Input soil parameters Results are greatly influenced by the parameters selection, and all these parameters do not have the same influence on the results. Several factors have to be considered in the parameters selection; these include the constitutive model adopted to describe the soil behaviour, the soil conditions and the available soil investigation data.

Soil properties are not dependent on the soil model but soil model parameters are dependent on the soil model selected. Different soil models will not only require different parameters but also for different types of tests and for different interpretations of these tests to derive the input parameters. For example in the case of an excavation, the important deformation parameters are the unloading modulus (which influence the swelling in the excavation and the support provided) and the shearing modulus which influence the bending moments in the retaining structure and the efforts in the anchorages. Therefore for trench excavation triaxial tests will be more appropriated than oedometer test to deduce the deformation parameters. In the cases of shallow foundations, the essential role of the soil elasticity modulus on the settlement and bearing capacity predictions of shallow foundations has been shown whitin the framework of the Labenne experiments (Mestat and Berthelon, 1998). Such a modelling using elastic perfectly plastic models leed to largely underestimated vertical displacements as compared to the observed displacements, because of erroneous estimate of the elasticity modulus. Laboratory values traditionally used correspond to the initial modulus, while a lower value such as obtained in situ is more realistic for shallow foundations.

Important note: In the following sections, correlations for various engineering parameters of soils from in-situ and laboratory tests are incorporated to show the approaches one could take to derive some of the input parameters needed to undertake FE analyses. However, some of these parameters, as explained above, are not constants for different soil-structure interaction problems and therefore their use must be treated with great care. It is usual to undertake some back analyses of monitored case histories to establish the appropriate correlation for the specific engineering problem in order to build up the confidence in the input parameters used. 3.5.1. STIFFNESS PARAMETERS (E,υ, G, K, m) The soil stiffness depends on the magnitude of the deformations (Figure 27), and any correlation shown below should be treated with care because of the importance of stiffness degradation with strain (Mair, 1993). The deformations under serviceability conditions for retaining walls range between 0.01-0.1%, which shows that the conventional laboratory tests are not suitable for the evaluation of the stiffness in situ. When such laboratory tests are used to evaluate such parameter, they underestimate the stiffness values significantly. The values of the stiffness parameters also depend on other factors, like the models selected to represent the soil and the structure behaviour (assumed constant or stress/strain dependent values…).


Figure 27. variations of the soil stiffness for service deformation levels corresponding to different geostructural systems – Mair (1993) Depending on the type of soil and precision required, elastic parameters can be determined from: - Pressuremeter tests (undrained situations); - Dilatometer test; - Cone penetration test (lower reliability, especially for clays); - Standard penetration test (lower reliability); - In-situ elastic wave velocities measurements (very small strain) (undrained situations); - Oedometer test; - Triaxial tests (complete history of the degradation of soil stiffness with increase in strain level); - Simple shear test (lower reliability); - Correlations with index parameters (plasticity and liquidity of the soils…) (lower reliability). Definition of the parameters a. Tangent module: A soil is said to be following a loading path when it is subjected to a shear stress higher than it has previously experienced. Along this loading path, its behaviour is governed by the tangent modulus Et. This modulus is, for example, defined in the Duncan and Chang model as a function of soil properties, triaxial deviatoric stress, (σ1-σ3), and

confining stress σ3, using the following equation: if

t Ec

RE

2

3

31

sin2)(cos2)sin1)((

1 ⎥⎦

⎤⎢⎣

⎡+

−−−=

φσφφσσ

with: Ei = initial tangent modulus Et = tangent modulus Φ= friction angle of soil c = cohesion strength of soil Rf = ratio between the asymptote to the hyperbolic curve and the maximum shear strength. This value is usually between 0.75 and 1.0. σ1 = major principal stress σ3 = minor principal stress

b. Oedometer module Eoed or M :vm

M 1= = inverse of the coeff. of volume compressibility.

c. E50 module : elasticity modulus corresponding to 50% of the ultimate load. d. Unloading-reloading modules: When the soil is unloaded from a higher shear stress state, the unloading-reloading modulus, Eur is used in the non linear model. The unloading-reloading modulus is computed


n

aaurur P

PKE ⎟⎟⎠

⎞⎜⎜⎝

⎛= 3σ . Unlike the tangent modulus, this unloading-reloading modulus is

unaffected by the shear stress level. e.Poisson’s ratio: The Poisson’s defines how much strain occurs in the lateral directions when strained in the vertical direction or vice versus. For the non linear elastic model, the Poisson’s ratio can either be specified as a constant which is independent of stress state, or it can be computed from the soil bulk modulus, which depends on the confining stress. For the latter case, the

bulk modulus is given by: )1(3 υ−

=EBm where

m

aabm P

PKB ⎟⎟⎠

⎞⎜⎜⎝

⎛= 3σ .

with: Bm = bulk module Kb = modulus parameter pa = atmospheric pressure m = exponent of the bulk modulus

For loading of normally consolidated materials, Poisson’s ratio plays a minor role, but it becomes important in unloading problems. If Poisson’s ratio is small, the horizontal stress diminishes a little with unloading and a characteristic OCR stress state starts with relative high horizontal stress

For example, for unloading in a one-dimensional compression test (oedometer), the relatively small Poisson’s ratio will result in a small decrease of the lateral stress compared with the decrease in vertical stress. As a result, the ratio of horizontal and vertical stress increases, which is a well-known phenomenon for overconsolidated materials. Hence Poisson’s ratio should not be based on the normally consolidated K0,NC –value, but on the ratio of difference in horizontal stress to difference in vertical stress in oedometer unloading and reloading. f. m : indicate how the stiffness depends on the stress level. The values of the stiffness parameters depend on several factors, in particular on the models selected to represent the soil and the structure behaviour. For example the Poisson’s ratio when using a critical state model (such as the Modified Cam Clay model) must be selected lower than when using elastic perfectly plastic model such as the Mohr Coulomb model. Some general recommendations are given below concerning the determination of the stiffness parameters in function of the selected soil model. Selection of the parameters – soil model a. ‘Cam clay’ models Undrained behaviour involves no volume change, the undrained stresses are therefore deviatoric. Clay behaviour under deviatoric stress development is stiffer than under compression. Compression is coupled in primary compression with plastic volume changes and the effective stiffness is limited. The behaviour by unloading and reloading is on the contrary stiffer than the behaviour in shear. In conclusion, we have the relation: compression stiffness < shear stiffness < unloading stiffness. Undrained loading from a normal consolidated stress condition involve compression and the stiffness modulus must be based on 1D compression. By loading in undrained condition or in condition with relatively larger horizontal deformation, a stiffness determination based on shear is adequate (undrained triaxial test). By unloading the determination of stiffness modulus can be based on 1D unloading.


b. Elastic perfectly plastic model

For soils, both Eur and E50 tend to increase with the confining pressure. Hence deeper soil layers tend to have a larger stiffness than shallow layers. The stiffness also depends on the stress path that is followed. The stiffness is much higher for unloading and reloading than for primary loading. The observed soil stiffness in terms of Young’s modulus may be lower for (drained) compression than for shearing. Hence, when using a constant stiffness to represent soil behaviour one should choose a value that is consistent with the stress level and the stress path development (this is not usually easy to do).

When using elastic perfectly plastic models in primary (deviatoric) loading cases, it is suggested to use E50 rather than E0. E50 is the secant stiffness at 50% of the peak strength. E50 better represents the average stiffness of the soil and is easier to determine than E0. When the elastic model or Mohr Coulomb model is used for gravity loading (one dimensional compression) it is possible to evaluate υ by matching K0. In many cases we will obtain υ values in the range between 0.3 and 0.4. In general, such values can also be used for loading conditions other than 1D compression. For unloading conditions, it is more common to use values in the range between 0.15 and 0.25 for the Poisson’s ratio.

Standard drained triaxial tests may yield a significant rate of volume decrease at the very beginning of axial loading and, consequently, a low initial value of Poisson’s ratio (υ0). For some cases, such as particular unloading problems, it may be realistic to use such a low initial value, but in general when using the Mohr Coulomb model the use of a higher value is recommended. c. Non linear (Duncan Chang) models: When a soil is subjected to zero shear stress, its stress-strain behaviour is modelled using the initial modulus Ei. This initial tangent modulus is controlled by the confining stress σ3 and is

calculated as follows:n

aaLi P

PKE ⎟⎟⎠

⎞⎜⎜⎝

⎛= 3σ .

with: Ei = initial tangent modulus as a function of the confining stress σ3 KL =loading modulus number pa = atmospheric pressure (used as a normalizing parameter) σ3 = confining stress n = exponent for defining the influence of the confining pressure on the initial modulus

Selection of the parameters – soil tests Determination from cone penetration test In Mitchell and Gardner (1975) a table is presented that gives an indication of the applicability of CPT tests in estimating various parameters according to five different classes of reliability:

1. High reliability 2. High to moderate reliability 3. Moderate reliability 4. Moderate to low reliability 5. Low reliability

For stiffness moduli this table indicates as class of reliability: - For clays: 4-5


- For sands: 2-3 Important note: In the context of FE analysis, the definition of reliability used in this in-situ test should only be used as a guideline and one should not use the correlations provided below without some calibration or back analyses. Most of the relations based on CPT results refer to the constrained modulus, M, which is similar to the oedometer modulus Eoed (1D compression). Considering first order models, one stiffness parameter can be determined from another by means of Hooke’s law of isotropic elasticity and by estimating proper value of Poisson’s ratio. However Hooke’s law does not accurately represent soil behaviour in general and care must be taken by using these correlations to calculate other stiffness parameters (E, G) as used in other situations than compression (e.g. deviatoric loading, shearing).

Hooke’s law: )1).(21(

).1(υυ

υ+−

−==

EEM oed and )1.(2 υ+= GE

a. Clays and silts: Most of the relations between Eoed and qc are expressed M or Eoed = α.qc. Table 2 gives some indication for the value of the cone factor α as function of the type of soil.

Table 2. Empirical cone factors after Sanglerat (1972) qc < 0.7 MPa 0.7 < qc < 2.0 MPa qc > 2.0 MPa

3 < α < 8 2 < α < 5 1 < α < 2.5

Clay of low plasticity

qc > 2.0 MPa qc < 2.0 MPa

3 < α < 6 1 < α < 3

Silts or low plasticity clays

qc < 2.0 MPa 2 < α < 6 Highly plastic silts and clays qc < 1.2 MPa 2 < α < 8 Organic silts qc < 1.2 MPa 50 < w < 100 100 < w < 200 w > 200 w = water content

1.5 < α < 4 1 < α < 1.5 0.4 < α < 1

Peat and organic clays

From other sources: (clays)

- Source - Correlation - Range of validity - Lunne and

Christoffersen - Eoed = 4.qc - -

- Trofimenkov - Eoed = 7.qc - - - Sanglerat - Eoed = (2,5 à

6,3).qc - 0,7≤qc≤2 MPa

- Meigh and Corbet - Eoed = (5 à 8).qc - - b. Sands: The correlations between CPT and deformation parameters are mainly used for granular soils because of the difficulty to perform laboratory tests on such soils (remoulded samples). For sands, it appeared from calibration tests that the Young modulus under drained conditions mainly depends on its relative density, on the overconsolidation ratio and on the stress level. Several correlations are given below.

- Eoed = M : most of the correlations between CPT results and M modulus refer to the tangent modulus as determined in the oedometer. The reference value for M is normally based on the effective vertical stress σ’v0, before execution of the in situ test; this value is identified as M0. Based on calibration tests, Lunne and Christophersen (1983) recommended following relations to estimate M0 of uncemented normally consolidated silica sands:


M0 = 4.qc for qc < 10 MPa M0 = 2.qc + 20 (MPa) for 10 MPa < qc<50 MPa M0 = 120 MPa for qc > 50 MPa

For overconsolidated sands, Lunne and Christophersen recommended the following in first approximation

M0 = 5.qc for qc < 50 MPa M0 = 250.qc for qc > 50 MPa

For supplementary stresses Δσ’v, Lunne and Christophersen suggested the Janbu’s

formulation (1963) to calculate M : 0

00 '

2/'

v

vvMMσ

σσ Δ+= .

Recently Elsaamizaad and Robertson (1996) proposed an alternative method to estimate M0 from CPT results, on the basis of calibration tests on quartz sands. The proposed correlation

can be written n

a

vaM p

pkM ⎟⎟⎠

⎞⎜⎜⎝

⎛= 0

0

'.. σwith

n = 0.2 for normally consolidated sands, 0.128 for overconsolidated sands pa = atmospheric pressure, in the same units as σ’v0, M0 and qc kM = adimensional number (see Figure 28)

Figure 28.Relation between qc and M0 (Elsaamizaad and Robertson,1996)

When applying this relation, the predicted M0 value is between 75 and 125% of the corresponding value as measured in a calibration test.

Other relations have been proposed by Vermeer (2000): Normally consolidated sands: Eoed ~ 3.qc

0'

.3v

refc

refoed

pqE

σ≈ 3.qc.

Overconsolidated sands: Eoed ~ 5.qc - E50 : Baldi (1989) proposed a chart to estimate the secant Young’s modulus (E’s) of silica sands for an average axial strain of 0,1%, for a range of stress histories and ageing (Figure 29). This level of strain is reasonably representative for most well designed foundations, but


is in most cases lower than the strain at 50% of the mobilised of deviator stress (ε50) in a standard triaxial test. For triaxial test, this strain values are between 0,2 and 0,6%, the denser the sand the stiffer the response and the lower the ε50 However, the results from triaxial test are often derived from disturbed and reconstituted samples. Under in situ conditions a stiff response for all sands is expected during the first stage of shearing and this is not the case in a triaxial test. Besides, recent improvements in small strain measurements indicate that stiffness derived from a standard triaxial test can be significantly underestimated by sources of error in external axial deformation measurements. Given that the strain level (ε50) is probably more than 0,1% and the E’50 might be overestimated. On the other hand most sands are slightly aged and overconsolidated while being considered as normally consolidated so the E’50 (ε50) might be underestimated. Therefore it seems reasonable to assume that the E’s of recent NC sands presented by Baldi can be compared to the E’50 modulus. It is also assumed that the E’s of overconsolidated sands can be compared to the unloading-reloading Eur modulus. For many geotechnical constructions the level of strain in the subsoil due to unloading is less than 0,1%. The estimated value can be seen as an upper boundary.

Figure 29. Correlations between Young module and cone résistance (normalised) (Baldi, 1989) Robertson and Campanella presented relationships between qc and Young’s and shear moduli at increasing vertical effective stress levels (fig 30). (Potts and Zdravkovic, 1999)

Figure 30. C one resistance vs. Young’s and shear moduli for sand (after Robertson and Campanella, 1983)


- G0 and Eur : the small strain shear modulus G0 is related to the density and shear wave velocity and it can be determined from empirical correlations. The graph from Rix and Stokoe (Figure 31) allows determining the modulus for uncemented normally consolidated sands. Soil compressibility can affect these correlations. More compressible sands give lower normalised cone resistance values and thus greater G0/qc ratios. From G0, the E0 modulus can be determined on the basis of Hooke’s law (assumption of elastic behaviour). E0 is the very small strain modulus or dynamic modulus and can be considered as an upper limit value for Eur.

Figure 31. Relation between qc and G0 for uncemented normally consolidated sands (Rix and Stokoe, 1992)

Figure 32. Effect of the cementation and of the aging of the ground on the relation between G0 and qt. (Robertson, 1990)

Figure 32 shows the effects of cementation and ageing on the relation between G0 and qt.

From cone pressiometer, Menard test The cone pressiometer is a CPTU cone combined with a pressiometer device. This apparatus is a very powerful tool in estimating the stratification and in-situ small strain stiffness. The tests are mainly used for determining the elastic stiffness of the soil. The shear modulus, G0, can be determined directly from the test. The elastic Young’s modulus can only be calculated when Poisson’s ratio is estimated.

The following formula can be used to determine the shear modulus from a pressiometer test:

RRP

GΔ

Δ=

.2. 0

In which: G: shear modulus ΔP: Increment of pressure R0: Initial radius ΔR: Increment of radius

With the following equations based on Hooke’s law of isotropic elasticity and an estimated value of Poisson’s ratio one can calculate Young’s moduli:

E0 = 2 (1+νur).Gur E100 = 2 (1+ν).Gmin

Gur is the shear modulus for unloading / reloading and Gmin is the shear modulus at the point of failure.


From Standard penetration test The applicability of the test varies from medium stiff clays up to very dense sands and gravel. Most experience using this test method is gained in sands and stiff clays. a. Sands. From granular soils, Menzenbach (1967) established a rough relationship between the deformation modulus Ed (approximately E50) and N-values from SPT (figure 33a). As first approximation for sands: - E/pa = 5 N60 sands with fines - E/pa = 10 N60 clean NC sands - E/pa = 15 N60 clean OC sands

(a) (b)

Figure 33. Relationships between the SPT N-value and stiffness moduli (CUR 195) Vesic (1973) performed tests with a loaded foundation of a constant width in sands of varying density. He found a unique relationship between settlement and the degree of loading q/qult. This suggested that q/qult could be used as a direct measure of shear strain and therefore linked to the average stiffness E’ (expressed in terms of Young modulus). For this purpose data from a wide range of strip footings, raft foundations and plate tests on both NC and OC sands have been collected (Stroud, 1989). They have been taken from case histories where SPT tests have been carried out and a correlation (fig 34) has been derived between the E’/N60 and q/qult values. Knowing the design loading and estimating the ultimate bearing capacity, it is possible to use this graph to estimate the average ground stiffness at working load on the basis of SPT blow count. Clay: For clays, when plotting the SPT N values with EPMT (approximately Eundrained), a large scatter is observed (see Figure 35). The drained stiffness E’ expressed in terms of Young’s modulus of OC clays in vertical loading can be estimated from case histories, in a similar manner as for sands. Fig.35 shows such data plotted as E’/N60 against q/qult.


Figure 34. Stiffness vs. degree of loading for sand Figure 35. Stiffness vs. degree of loading for (after Stroud 1989) clay (after Stroud 1989)

From Oedometer test

The oedometer test is mainly used to determine the: - constrained modulus M (=Eoed) - compressibility (λ, λ*, κ, κ*, μ*, Cc, Cr, Cs, Cα, C’p, C’s, Cp, Cs)

- preconsolidation pressure σp - coefficient of consolidation cv

The constrained modulus is determined from a fully consolidated sample in the following way: M = Eoed = ∂σ/∂ε = 1/mv with dε = vertical strain (Δεa)* dσ = differential vertical pressure for a particular stage (Δσ’a) mv = coefficient of volume compressibility * The normal stiffness is defined as: εv = Δh/h0 corresponding to a normal plastic analysis

The natural strain is defined as: εv = Δh/h corresponding to the updated mesh analysis Alternatively: M = Eoed ~ [2.3 (1+e0) σy] / Cc.

The above equation becomes inaccurate when large variations of the void ratio are considered. If the encountered stress path is such that the preconsolidation pressure is not exceeded, the Cc value should be replaced by the Cr-value (recompression).

If the soil is assumed to be isotropic elastic: Young’s modulus: E = (1+ν).(1-2ν)/(1-ν) Eoed ~2/3 Eoed From Triaxial tests

The following basic rule can be used to determine the drained E’50 from an undrained triaxial test (CU) Eu

50: E’50 = Eu

50.(1+ν’)/(1+νu) (theory, based on Hooke’s law) Or E’50 = 0.67 Eu

50 (practice)


The following equation can be used in determining the E50 from triaxial data at a given reference pressure of 100kPa. E0, u ~3,5.E50,u Eref

ur ~ 3.5.Eref50

Eref50 ~ 2 Eref

oed (qc < 5MPa): clay NC Eref

50 ~ Erefoed ( 10< qc < 25MPa): clay NC

Eref50 ~ Eref

oed : sand NC

The unloading-reloading modulus should be determined as the inclination of a line in the q-ε plot from the top to the bottom of an unload reload loop. Besides this, the Eur can be determined from unloading tests such as consolidated drained/undrained unloading tests (CDU/CUU tests). Due to the development of plasticity in this type of triaxial tests, the unloading stiffness will be different from the stiffness calculated from the unloading-reloading loop from the consolidated drained/undrained shear tests (CDS/CUS tests).

From the εv-ε1 diagram of a drained test, the Poisson’s ratio can also be calculated. Poisson’s ratio is defined in an analogous form for the triaxial tests in which both axial and volumetric strain is measured. From these data, the axial and radial strains can be obtained. Poisson’s ratio is the ratio of the radial strain εr to the axial strain εa: ν = -∂εr/∂εa where the radial strain is calculated from the volumetric strain by εr=(εν-εa)/2. Poisson’s ratio is measured from the secant line that intersects the εv-ε1 diagram at the same ε1 value as the corresponding Young’s modulus.

From triaxial tests both drained E’ and undrained Eu Young’s moduli can be estimated in both triaxial compression and extension. The triaxial test provides a complete history of the degradation of soil stiffness with increase in strain level (fig.36). The small strain behaviour is an essential part of constitutive modelling if soil deformations are of interest in the analysis. Ignoring this can result in the prediction of patterns of movement considerably different to those observed in the field. If the small strain modulus E’max can be obtained, the other elastic parameter that can be calculated from a drained triaxial compression test is Poisson’s ratio μar’ for straining in the radial direction due to changes in axial stress: μar’ = -Δεr/Δεa. The ability to measure the radial strain to a very high resolution is essential for this parameter to be estimated (Potts and Zdravkovic, 1999).

Figure 36. Soil stiffness curve (Potts and Zdravkovic, 1999) From simple shear test

These tests are mainly used to determine the (undrained) strength properties of the soil. It is also possible to determine the shear modulus from these tests.

G=τzx/γzx with τzx = applies shear stress in the zx-plane γzx = rotation in the zx plane of the sample

Excess pore water pressures do not affect the shear modulus (no difference between drained and undrained G).


From Geophysical techniques The bender element technique has been developed for very small strain (on the elastic plateau) shear stiffness measurements in the laboratory. From Correlations with index parameters Most correlations with stiffness or compressibility parameters are based on the plasticity and liquidity of the soils.

M = Eoed ~500.σ1/(Ip[%]); for a reference pressure of 100 kPa: Mref = Erefoed ~50MPa/(Ip[%])

Regarding the elastic deformability of clay soils, Vermeer, Termaat and Vergeer presented the following equation liking the plasticity index to the Young’s modulus at 25% of the strength (E25) for normally consolidated clays (OCR<2) Eu

25 ~ 15000.cu/Ip with cu ~ 0.5 σ’3: Eu25 ~7500.σ’3/Ip

As a first approximation for the shear modulus, Hardin and Black (1968) find the following equation for sands with angular particles and for several clays. G ~ [3230 (2.973-e)²(σ’c)1/2]/(1+e) where σ’c is the mean effective stress to which the soil has been consolidated

Vermeer suggests using the following equation for NC clays: G50 ~ (5.5+18.5*Ip)*σ1/Ip where σ1 = major principal stress From pressuremeter test (Potts and Zdravkovic, 1999) Interpretation of pressuremeter test results is based on cavity expansion theory. It consists of (i) an elastic phase, during which soil deformation is considered to be isotropic elastic (ii) a plastic phase, during which soil behaviour is assumed to be perfectly plastic From measurements taken during the elastic phase estimates of the shear modulus can be made, while during the plastic phase the measurements can be used to evaluate the shear strength.

Jardine (1992) compared triaxial Gs-εs curves, where εs is the shear strain (=2/3(εa-εr)), with the pressuremeter Gp-εc curves and derived an empirical relationship between εs and εc by comparing the strain at identical shear moduli levels: (εs/εc) = 1.2 + 0.8 log10(εc/10-5). With this expression pressuremeter Gp-εc curves can be transformed into triaxial Gs-εs curves (for small strain models).

When using the pressuremeter in sands it should be noted that the shear modulus is likely to be stress level dependent. As the stress level in the sand adjacent to the pressuremeter decreases with distance, the value determined from the pressuremeter test represents an average value.

It is not possible to obtain estimates of the bulk modulus K or Poisson’s ratio υ from pressuremeter tests. From dilatometer test (DMT) In most cases the DMT is used to determine "commonly used" geotechnical design parameters, notably the undrained shear strength cu and the constrained modulus M. The DMT can test from extremely soft to very stiff soils (clays with cu from 2 - 4 to 1000 kPa, moduli M from 0.5 to 400 MPa) (G. Totani, S. Marchetti, P. Monaco & M. Calabrese). However, there


are just a few correlations with other parameters. Correlations found in the literature are within a specific geological setting and therefore should not be used in general conditions (CUR 195). Table 3 – DMT formulae (Totani et al.)

Young's modulus E can be derived from M via theory of elasticity (E ≈ 0.8 M for ν = 0.20 - 0.30). The modulus obtained by expanding the DMT membrane (a "mini" load test) is physically more related to deformability than is the penetration resistance. Besides, the availability of a second independent parameter KD, reflecting σh stress history, leads to more realistic values of M (G. Totani, S. Marchetti, P. Monaco & M. Calabrese). Typical values & relations Some typical values and relations are given below (only for a first estimation!). As average values for various soil types, we have Eur ~3 E50 and Eoed ~E50, but both very soft and stiff soils tend to give other ratios of Eoed/E50. Table 4 Typical values for elastic moduli & Poisson ratios Elastic moduli (kN/m²) Poisson ratio Clay very soft 500-5000 clay saturated 0.5 Soft 5000-20000 clay with sand + silt 0,3 to 0,42 Medium 20000-50000 unsaturated clay 0,35 to 0,4 stiff clay, silty clay 50000-100000 Loess 0.44 sandy clay 25000-200000 Silt 0,3 to 0,35 clay shale 100000-200000 sandy soil 0,15 to 0,25 Sand sandy soil 0,3 to 0,35 loose sand 10000-25000 Rock 0,1 to 0,4 dense sand 25000-100000 Dense sand 0.3 to 0.4 dense sand and gravel 100000-200000 Loose sand 0.1 to 0.3 silty sand 25000-200000 Clay 0.2 to 0.4


Table 5 – Typical relations – elastic moduli (Plaxis bulletin) qc

[MPa] σ’v

[kPa] Dr [-]

E50 Eoed Eur E0 Mohr Coulomb Model [MPa]

E50,ref Eoed,ref Eur,ref E0,ref HS Model [Mpa]

5 5 5

20 50 100

0.70 0.53 0.41

12 15 18

20 20 20

48 60 72

99 140 182

38 30 25

45 28 20

151 120 101

222 198 182

15 15 15

50 100 200

0.90 0.76 0.63

26 31 37

50 50 50

104 124 147

184 239 310

52 44 37

71 50 35

208 175 147

261 239 219

25 25 25

100 200 400

0.94 0.80 0.67

40 48 57

70 70 70

160 190 226

272 352 457

57 48 40

70 49 35

226 190 160

272 249 228

Table 6 – Typical relations – elastic moduli (Plaxis bulletin)

σ’v [kPa]

Dr [-]

E50 Eoed Eur E0 [MPa]

Very loose Loose

Middle dense Dense

Very dense

0.00 0.15 0.35 0.65 0.85

0.15 0.35 0.65 0.85

1

16 23 35 50 62

8 17 37 58 73

66 93

138 199 246

146 174 212 255 283

For Poisson’s ratio’s the following values can be used: For primary loading: Sandy soils: ν = 0.27 up to 0.33 Clays: ν = 0.30 up to 0.38 For unloading and reloading: Sandy soils: ν = 0.12 up to 0.17 Clays: ν = 0.15 up to 0.20 Typical value of the parameter m for the hardening soil model from plaxis – Between 0.3 and 1. Value 1 corresponds to the behaviour modelled by the Cam Clay

model (for clay), and involves a logarithmic relation between the volumetric deformation and the middle stress ;

– Higher for low compacted sand – Often: 0.5 for coarse grained soils, 1 for soft soils The following table illustrates the relationships between elastic parameters.


Table 7 – Relationships between elastic parameters (EPFL website)

3.5.2. CONSOLIDATION & CREEP PARAMETERS: SWELLING, COMPRESSION AND CREEP INDICES When a soil deposit is loaded, one or a combination of the following will occur: deformation of the soil particles (negligible unless large loading conditions such as pile tip), compression of air and water (negligible) in the voids, re-arrangement of the soil skeleton or swelling depending on whether the load is increased or decreased. Deformation may be divided into three broad categories in soil mechanics:

- Immediate settlements (si): Elastic deformations, in dry, partially saturated and saturated soils without any change in the moisture content. This process is time independent and occurs instantaneously following a stress changes – in saturated soils it is assumed to be fully undrained loading. Immediate settlements are only possible in saturated soils provided the soil is free to strain laterally under the Poisson ratio effect, i.e. assuming incompressible grains and water. Thus immediate settlements are zero for one-dimensional (laterally confined) load conditions. Immediate settlements are usually calculated from simple elasticity theory.

- Primary Consolidation Settlements (sc): These result from volume changes in saturated soils that go hand-in-hand with the expulsion of pore water as excess pore pressures are dissipated. Excess pore pressures are set up immediately following a total stress change (Undrained) and will gradually transfer the total stresses increase to the effective stresses at a rate proportional to the soil permeability (Drained). In sandy, free draining soils, primary consolidation occurs almost instantaneously with relatively small settlements, but in low permeability clays the process can take several years to complete and lead to significant settlements.

- Secondary compression (or secondary consolidation or creep) (ss): Settlements observed in saturated clayey soils that follow primary consolidation and occur under constant effective


stress and no pore pressure changes. This process is also time dependent. This process can also occur simultaneously with primary consolidation. Secondary compression or creep is though to be due to the gradual readjustment of clay particles into a more stable arrangement, following disturbance of the skeletal structure by compression. The rate of secondary compression is though to be controlled by the highly viscous film of adsorbed water (double layer water) surrounding clay minerals. For certain highly plastic and especially organic clays secondary compression may dominate over primary consolidation. Determination of consolidation and creep parameters is possible from : - Oedometer (primary consolidation and secondary compression parameters); - Correlations with index parameters and empirical relations.

Definition of the parameters a. Primary consolidation parameters :

Figure 37. Consolidation parameters - illustration Coefficient of volume compressibility (mv) : Change in volumetric strain per unit volume per unit change in effective stress in one dimensional compression (m2/MN):

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+=

01

10

0 ''11

vvv

eee

mσσ

or ⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

=01

10

0 ''1

vvv

HHH

mσσ

with eo; Ho; σ'vo initial void ratio, height and vertical effective stress e1; H1; σ'v1 final void ratio, height and vertical effective stress

Compressibility is the reciprocal of stiffness, and coefficient of volume compressibility is

related to the oedometer stiffness M by the relationvm

M 1= .

Compression (Cc) and swelling (Cs) indexes: Compression and swelling indexes are the slopes respectively of the normal compression line (NCL) and swelling lines (assumed to be linear in e-logσ plot).

Compression index = slope of the NCL, or )'/'log( 01

10

vvc

eeCσσ

−=

Swelling index = slope of the OCL, or )'/'log( 01

10

vvs

eeCσσ

−=

Pre-consolidation pressure: When testing undisturbed samples, the state of the soil will be overconsolidated either as a result of in-situ stress history or of the stress relief that goes hand-in-hand with removing the soil from its stress environment. In either case the pre-


consolidation pressure (s'c) or previous maximum vertical effective stress can be determined experimentally from the results of an oedometer test by the empirical Casagrande Method. Due to the effects of sampling and specimen preparation the specimen in an oedometer test will be slightly disturbed. Some adjustments have been proposed to account for this disturbance

Normally consolidated soil– Total stress increase : Use Cc: ⎟⎟⎠

⎞⎜⎜⎝

⎛+

=0

1

0 ''

log.1

.

v

vcc e

CHs

σσ

Normally consolidated soil – Total stress decrease : Use ⎟⎟⎠

⎞⎜⎜⎝

⎛+

=0

1

0 ''

log.1

.

v

vsc e

CHs

σσ

Normally consolidated / Overconsolidated soil – Total stress increase : Use Cs /Cc

If σ'v1< σ'c : ⎟⎟⎠

⎞⎜⎜⎝

⎛+

=0

1

0 ''

log.1

.

v

vsc e

CHs

σσ

; if σ'v1>σ'c

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=c

vc

v

csc CC

eHs

''

log''

log.1

1

00 σσ

σσ

Overconsolidated soil – Total stress decrease : Use Cs ⎟⎟⎠

⎞⎜⎜⎝

⎛+

=0

1

0 ''

log.1

.

v

vsc e

CHs

σσ

Table 8. Relations between indexes (CUR 195, 2000) Swelling index Compression index Creep index Modified Cam Clay κ* λ* μ* Cam Clay κ/(1+e) λ/(1+e) - Bjerrum ~(2.Cr)/(2,3(1+e0)) Cc/(2,3(1+e0)) ~Cα/(2,3(1+e0)) Dutch practice ~2/Cp 1/C’p The difference between the modified index and the original Cam Clay parameters is that the latter parameters are defined in terms of void ratio instead of the volumetric strain. There is no exact relation between the isotropic compression indices and the 1D swelling indices because the ratio of horizontal and vertical stress changes during 1D unloading. For the approximation it is assumed that the average stresses state during unloading is an isotropic stress state. b.Secondary compression parameters : Secondary compression (or creep) occurs in soil at constant vertical effective stress progressing with time. The creep index is :

ppp

pt tttt

ett

eeC >

Δ=

−−= ;

)/log()/log()(

α

The magnitude of secondary compression at a given time is generally higher for normally consolidated clays than for overconsolidated clays. Determination from the soil tests From the oedometer The compression versus vertical pressure characteristics of soil are usually obtained from a one dimensional consolidation test. In a e- logσ’a diagram (Figure 38a), the slope of the


virgin compression line is known as the compression index Cc which is calculated as Cc = Δe/Δ(logσ’a), and the slope of the swelling line is the swelling index. In a specific volume ν (=1+e) – mean effective stress (p’=(σ’a+2.K0.σ’a)/3) diagram (Figure 38b), the slope of the virgin compression line is denoted λ = Δν/Δ(lnp’) and the slope of the swelling line is denoted κ. These two parameters, λ and κ, are essential for critical state type models, such as Cam Clay or Modified Cam Clay.

(a) (b) Figure 38. Typical results for clay from the oedometer According to the Cam Clay theory, the reloading inclination is similar as the unloading inclination since both act below the preconsolidation pressure. However, oedometer tests may show a significant difference between unloading and reloading, in particular when comparing the first part of the test (reloading up to the initial preconsolidation pressure) with an unloading line. When using Cam Clay type models, it is preferred to use the unloading line to determine κ rather than the reloading line (CUR 195). For sand the behaviour in the oedometer test is more complex, as the initial density of the sample affects its behaviour. The magnitude of the vertical effective stress at which the virgin compression line is reached is much higher than for clay soils and is often considerably larger than the stress levels experienced in many practical situations. Behaviour is therefore characterised either as on a normal compression line, on the virgin compression line or on a swelling/reloading curve. These lines are often assumed to be straight in either e-log σ’a, ν-lnp’ or lnν-lnp’ space (Potts and Zdravkovic, 1999).

The creep index can be obtained by measuring the volumetric deformation on the long term and plotting it against the logarithm of time.

(a) (b) (c) Figure 39. (a), (b) Consolidation and creep behaviour in standard oedometer test, (c) Idealised stress-strain curve from oedometer test with division of strain increments into an elastic and a creep component

For a given soil the magnitude of the secondary compression at a given time, expressed as percentage of the total compression, increases proportionally to the decrease of the ratio between pressure increment and initial pressure: the magnitude of the secondary compression also increases when the depth of the sample placed in the oedometer is diminished and when the temperature increases. Thus, normally, the secondary compression characteristics deduced


from a sample tested in an oedometer can not be extrapolated to the case of a real scale foundation (Craig, 1997). From correlations and empirical relations For a rough estimate of the model parameters, one might use the following correlations:

- Correlation λ* ~ 0.3 Ip with Ip the plasticity index and wl the liquidity limit; - λ*/µ* is in the range between 15 to 25 ; - λ* /κ* ~ 5 to 10 (in general λ* /κ* ~ 5 for plastic clay).

Data collected by Engel (2001) contains modified compression indices for 21 clays and silts, with wl = 0.2 up to 1.1 and Ip between 0.03 and 0.7. These data show that the relation λ*~0.3 Ip has some shortcomings; it is nice for plastic clays, but not for less plastic silts. To include such silts one could better use the correlation: λ* ~0.2 (wl-0.1).

Table 9 – Compression indices – correlations Equation Reference Soil Cc=0.007(LL(%)-7) Cc=1.15(e0-0.27) Cc=0.3(e0-0.27) Cc=0.0115wn Cc=0.75(e0-0.5) Cc=0.156e0+0.0107 Cc=0.009(LL(%)-10)

38.2

02.1 1141.0 ⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=

ssc G

eGC

sc GLLC ⎥⎦⎤

⎢⎣⎡=

100(%)2343.0

ss GLLC ⎥⎦⎤

⎢⎣⎡=

100(%)0463.0

Skempton (1944) Nishida (1956) Hough (1957) Terzaghi et Peck (1967) Rendon-Herrero (1983) Nagaraj and Murty (1985) Nagaraj and Murty (1985)

Remoulded clay Clay Non organic cohesive soils : silt, silt clay, clay Organic soils, peats, organic silts and clay Low plasticity soils Clays Natural clays Natural clays Natural clays Natural clays

LL = limit of liquidity, wn = in situ water content, e0 = initial void ratio

Natural clays (generally) : Cs= 1/5 à 1/10 Cc Typical values : Cc = 0,1 à 0,8 ; Cr = Cc/5 à Cc/10 ; Cα/ Cc = 0,01 à 0,07

Table 10 – Compression indices – correlations (EPFL website) Soils USCS Ip (%) Grain shape Cc Cs Crushed rock Clean gravel Loam gravel Clay gravel Poorly graded clean sand Well graded clean sand Slightly loamy sand Slightly clayey sand Clay sand Loam Clay loam Loam clay Clays

GW GM GC SW SP SM SC SC-CL ML CL-ML CL CH CH CH CH

0 2-6 7-12 0 0 2-6 6-12 9-15 2-6 4-10 12-18 ~20 ~40 ~60 >100

angular all all all angular rounded all all all

0.1+/-0.07 0.15+/0.1 0.3+/-0.15 0.4+/-0.2 0.6-1 0.9-1.5

0.01+/0.007 0.025+/0.015 0.04+/-0.025 0.06+/-0.04 0.07-0.09 0.09-0.13

+/- = standard deviation


The compression and swelling parameters λ and κ are soil properties and the values depend on the nature of the soil. Typical values are given below (Table 11).

Table 11. Typical values of compression and swelling parameters. wL Ip λ Very high plasticity clay High plasticity clay Intermediate plasticity clay Low plasticity clay Quartz sand Carbonate sand

80 60 42 30

50 34 23 12

0.29 0.20 0.14 0.07 0.15 0.34

For clays λ ~Ip/170 and κ/λ is relatively large (eg.0.25 à 0.35) because clay particles can be bend and distort. For sands λ is relatively large due to particles crushing (but states only reach NCL at high pressure). The ratio λ/κ is relatively small (eg.0.1) because sand particles crush and rearranged during first compression. 3.5.3. SHEAR STRENGTH PARAMETERS Definition of the parameters Shear strength components in soils are generally given in Figure 40.

Figure 40. Shear strength components (Lambe, 1960) Concerning strength parameters most models implemented in commercial softwares involve the Mohr Coulomb’s failure criterion. This may lead to differences in ‘model’ strength and ‘real’ strength, depending on the type of stress path that is followed. For example the determination of a friction angle based on a drained triaxial loading test, a simple shear test or a triaxial extension test generally leads to different values. It is therefore advised for the determination of strength parameters to select the type of test that resembles most to the leading stress path in the application. Besides, a special attention should be paid on the differences between drained and undrained parameters. Some softwares allow to perform undrained calculations with drained parameters or drained calculations with undrained parameters and the way to deal with is software specific. Shear strength of clays :

The strength behaviour of a clay is highly dependent on its state of overconsolidation, i.e. normally consolidated NC or overconsolidated OC.


- For normally consolidated clays, there is a gradual increase in the shear resistance within the

sample with strain. If the loading is undrained, the pore pressure increases in response to the loading, conversely if loading is drained the sample volume will decrease. At some stage the maximum and ultimate or critical state strength of the sample is reached characterised by no more change in shear resistance or pore pressure or volume.

- Concerning heavily overconsolidated clays, the shear resistance of the sample increases rapidly at first initially by either positive pore pressure build-up or reduction in sample volume. Soon the pore pressure starts reducing even to the point of becoming negative, and in a drained test compression changes to dilation of the sample. At fairly low strain levels maximum shear strength is reached, at a maximum rate of negative pore pressure build-up or dilation. However, with continued strain the shear resistance reduces gradually accompanied by a reduction in the rate of pore pressure response or dilation. At fairly large strain levels the shear resistance reaches a constant level, indicative of the critical state with no more pressure or volume change responses. Two separate strengths have thus been defined for an overconsolidated clay soil, i.e. peak strength and critical state or ultimate strength.

- There is a third possibility in clay called the residual strength that is lower than the peak and critical state strengths. Residual strength is the result of particle re-alignment on a failure plane in a soil as a result of continued shear. The residual strength will only be mobilised with considerable movement on a slip surface.

Shear strength of sands :

Due to the fast draining (high permeability) properties of sands they are usually tested fully drained, i.e. either dry or saturated but with a slow enough loading rate so that excess pore pressures are fully dissipated. The concepts of normal and overconsolidation applies in theory to sands, but has very little practical significance. Most natural sands will behave as though they are heavily overconsolidated, irrespective of their stress history. It is also possible to build reconstituted sand samples at a range of densities or void ratios at exactly the same confinement pressure. However, relative density has a major influence on the shear behaviour of this material. Only under extremely loose densities, or very high confinement stresses do sands exhibit shear behaviour typical of normally consolidated clays, i.e. a steady increase in shear resistance coupled with volume reduction up to the failure shear strength. Under normal conditions sand will behave like overconsolidated clay, reaching a peak shear strength at fairly low strain, which then gradually reduces to the critical shear strength at a much higher strain level. Initially the sample will contract and reduce in volume, but soon starts to dilate and increase its volume at a maximum rate as the peak strength is mobilised, following peak strength the rate of dilation decreases to zero when the critical state strength is mobilised.

a .friction angle ϕ and dilatancy angle ψ The effective angle of internal friction relates to the frictional properties of the soil particles. The relative density or void ratio of a sand is critical to its shear behaviour. In order to shear, sand grains must physically ride over each other. This requires the sand to expand in the direction perpendicular to the shear. This expansion is known as dilation. When the soil is loose, the shearing process will actually cause contraction rather than dilation, as the sand particles readily bed in to a denser structure. Sands can display behaviour between these two extremes depending on the particular relative density. The angle of dilation, ψ (or α or ν), is expressed ν = α = ψ =Arctg (dy/dx) where dy and dx are incremental dilation and shear movements respectively (see Figure 41 & 42). The dilation and contraction behaviour of the soil during shear is directly associated with the influence of relative density on friction angle. For an assumed basic friction angle of the sand φb the following expression is approximately correct: τ ≈ σnArctg(φb + ψ). Test results (Bolton, 1986) show that for plane strain tests pcsp ψφφ ⋅+≈ 8.0'' .


Figure 41. Dilation and contraction during a direct shear test (Monash University, 2002) Dilation of dense sands implies that the sand becomes looser, as the volume increases. Conversely, the contraction of loose sands implies that the sand becomes denser as the volume decreases. It is also noted that dilation tends to be suppressed at high levels of normal stress, as more energy is required to overcome the high imposed stress level. In conclusion, the shear strength needed to shear a sand sample will be a function of the inherent sliding friction angle, φ, the dilation angle, ψ, and on the applied normal stress σn. For given sand at a constant normal stress, only the dilation angle varies. For dense sand, ψ starts negative, becomes zero, reaches a maximum ψmax, and then reduced again to zero as the maximum dilation is reached. For loose sand, ψ starts negative, and then gradually increases to a final maximum value ψmax of zero at the maximum sample contraction. During shear, the dilation of dense sand and the contraction of loose sand results in both approaching the same density. This is the specific density where shearing occurs without volume change, and this critical state is the basis of an advanced theory of soil behaviour.

Figure 42. Dilatancy concept illustration Apart from heavily overconsolidated layers, clay soils tend to show little dilation (~0). The dilation of sand depends on both the density and on the friction angle. For quartz sands the order of magnitude is ψ ~ ϕ – 30°. For ϕ-value of less than 30°, the angle of dilation is almost zero. A small negative value for the angle of dilation is only realistic for extremely loose sands (CUR 195). After extensive shearing, dilating materials arrive in a state of critical density where dilatancy has come to an end. This phenomenon can be included in the soil model by means of a dilatancy cut-off. In order to specify this behaviour, the initial void ratio einit and the maximum void ratio emax of the material must be entered as general parameters (Figure 43).


Figure 43 Resulting strain curve for a standard drained triaxial test when including dilatancy cut-off b. Undrained shear strength su: The shear strength of fine soils under undrained conditions is called undrained shear strength, su, it corresponds to the radius of the Mohr’s circle, expressed

in terms of total stresses( ) ( ) ( ) ( )

2''

23131 ffff

usσσσσ −

=−

= . The undrained shear strength

only depends on the initial void ratio or initial water content. Increases in initial normal stresses (confining pressure) lead to decreases of the initial void ratio and to more important change of water pressure when the soil is sheared under undrained conditions. As a result the Mohr’s circle expressed in terms of total stresses extends and the undrained shear strength increases (the undrained shear strength depends on the confining pressure).

Figure 44. Influence of the confining pressure on the undrained shear strength Volume changes occurring under drained conditions are suppressed under undrained conditions. As a result the soil which tends to compress under drained conditions will generate positive water pressures under undrained conditions, while a soil which tends to dilate under drained conditions will generate suction pressures under undrained conditions. c. effective cohesion c’ The effective cohesion parameter c’ loosely models the effects of : - Bonding – physical ‘glue’ between particles giving the soil some tensile strength - Aggregates interlock – may be the result of heavy compaction and overconsolidation (cfr particles interlocking with themselves due to their physical closeness) or locked sands for example where particles contacts have been transformed by for example the contact pressure to a type of jigsaw effect. - Pore water suction – from partial saturation or under undrained unloading of saturated dilative soils Null cohesion introduction can lead to numerical difficulties. Where one of the above phenomena is considered appropriate for the soil being modelled, a low value but credible c' value should be used to prevent numerical instability.


Note: The effective cohesion parameter c’ can be used to model the effects of bonding/cementation, aggregates interlock and pore water suction. Shear strength parameters may be derived from : - Cone penetration tests; - In situ vane test (undrained strength); - Standard penetration test; - Triaxial test (also dilation angle); - Translational shear box test; - Direct simple shear test; - Pressuremeter tests; - Dilatometer tests; - Correlations with index parameters (lower reliability). Determination from the soil tests From Cone penetration test (CPT) The applicability of CPT tests in estimating strength parameters, following Mitchell and Gardner (1975) is

- for clays: 1-2 (su) - for sands: 2

with 1. High reliability 2. High to moderate reliability

In literature, graphs can be found that show the cone resistance versus the friction ratio to determine the friction angle. Figure 49 shows a correlation based on 20 different data sets from tests on sands.

For clays: qc = Nc.su.σ0 where σ0: vertical, horizontal or mean stress (depending on the theory) and

Nc: theoretical cone factor.

Figure 45. Correlation between Friction angle and Cone resistance for sands (CUR 195)


In the following table from Lunne et al. , as well as in the following graph, some cone factors are presented. Table 12. Relationship between cone resistance and undrained shear strength (Lunne et al). Nc (ϕ=0) σi Remarks Reference 7.41 σv0 Terzaghi (1943)

7.0 σv0 Caquot and Kerisel (1956) 9.34 σv0 Smooth base Meyerhof (1951) 9.74 σv0 Rough base 9.94 σv0 De Beer (1977)

13

ln134

+⎥⎦

⎤⎢⎣

⎡+

u

t

sE

σv0

SCE, Et: initial tangent modulus

Meyerhof (1951)

1ln134

+⎥⎦

⎤⎢⎣

⎡+

u

s

sE

σv0

SCE, Es: secant modulus at 50% failure

Skempton (1951)

ϑcot3

ln134

+⎥⎦

⎤⎢⎣

⎡+

u

s

sE

σv0

SCE Gibson (1950)

ϑcotln134

+⎥⎦

⎤⎢⎣

⎡+

u

st

sE

σv0

SCE, finite strain theory

Gibson (1950)

[ ]rIln134

+ σv0

SCE Vesic (1972)

[ ]rIln134

+ +2.57 σv0

SCE Vesic (1975)

[ ]rIln1 + +11 σi CCE Baligh (1975)

Note: SCE: spherical cavity expansion, Ir: rigidity index=Gu/su=Eu/3su ; σmean : mean normal total stress = (σv0+2σh0)/3 ; θ : semiapex angle

Figure 46. Relationship between cone resistance and undrained shear strength (Lunne et al).


Empirical correlations are available between the CPT measured quantities and the angle of shearing resistance and relative density for sands and undrained shear strength for clays. These correlations should be used with care because they may only apply to a particular set of conditions (penetrometer dimensions, penetration rate, soil type). Sand: Robertson and Campanella (1983) presented relationships between qc and peak φ’ for non consolidated quartz sands at increasing vertical effective stress (fig 47). The angles of shearing resistance were obtained from triaxial tests performed at confining stresses approximately equal to the horizontal effective stress in the calibration chamber prior to cone penetration. Using the relationships given in figure 47 for overconsolidated sands will tend to slightly overestimate φ’.

Figure 47. Cone resistance vs peak ϕ’ for sand (Robertson and Campanella, 1983) Other relationships were proposed by Meyerhof (1956) and Olsen and Farr (1986), as follows: Table 13 - Meyerhoff (1956): sand, gravels qc (MPa) State Relative density Dr (%) Friction angle (°) < 2 Very loose < 20 < 30 2 – 4 Loose 20 – 40 30 – 35 4 – 12 Medium dense 40 – 60 35 – 40 12 – 20 Dense 60 – 80 40 – 45 > 20 Very dense 80 - 100 45

Olsen and Farr (1986): sand, gravels: ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞⎜

⎝⎛ ′

+=2

1

0

log116.17

a

v

a

c

tc

p

pq

σϕ

Clays: The measurement of clay properties using the CPT is highly dependent on the rate of cone penetration, because of the build up of excess pore water pressure. The CPT in clays is principally used to estimate the undrained shear strength su and the relationship with qc is expressed as qc = Nksu + σv0 where σv0 is the total overburden stress and Nk is the “cone


factor” analogous to the bearing capacity factor Nc. Nk is established using empirical correlations, with shear strength measured using other techniques. Lunne and Kleven (1981) found that for NC clays Nk was independent of Ip and suggested an average value of 15, with the majority of data being between 11 and 19. For OC clays Marsland and Quaterman (1982) presented cone factors based on the undrained shear strength from plate loading tests and showed that for stiff fissured marine clays the cone factor was as high as 30, with an average of about 27. Other common used correlations for clay: cu = (qc-p)/Nk where p is the overburden pressure and Nk = 10 – 15 for NC clays ; Nk = 15 – 20 for OC clays. From In situ vane test (CUR 195) This test only gives an estimate of the undrained shear strength of the soil from

su = T/K where K = π (D²H/2) and T = maximum torque K = constant depending on the dimensions and shape of the vane D = Diameter of vane H = Height of vane From standard penetration test The following table (table 13) gives an indication of the consistency of soil based on N-values and qc values. Table 13 – Consistency of soil based on N-values and and qc values N value blows/305mm (ft) Cone tip resistance [MPa] Consistency < 2 < 5 Very soft 2 to 8 5 to 15 Soft to medium 8 to 15 15 to 30 Stiff 15 to 30 30 to 60 Very stiff > 30 > 60 Hard Sands: Stroud (1989) replotted all the available data presented by Sempton (1989) and Bolton to obtain the variation of φ’ versus (N1)60 and (N1)60 vs Dr for different OCRs and different φ’cs as shown in figure 52. Table 14 - Based on Peck et al (1974) State N* (blows/300mm) Friction angle (°) Relative density (%) Very loose < 4 < 30 < 15 Loose 4 – 10 30 – 32 15 – 35 Medium dense 10 – 30 32 – 35 35 – 65 Dense 30 – 50 35 – 38 65 – 85 Very dense > 50 > 38 85 – 100

Clay : The undrained shear strength of overconsolidated clays can be related to SPT blow counts using the simple correlation proposed by Stroud (1974): su = f1.N60 where the variation of f1 with plasticity index is shown in figure 49.


Table 15 – Consistency of soil based on N-values and and qu values Consistency N qu (kPa) AS1726 Very soft < 2 < 25 Soft 2 – 4 25 – 50 Firm 4 – 8 50 – 100 Stiff 8 – 15 100 – 200 Very stiff 15 – 30 200 – 400 Hard > 30 > 400 Su = qu / 2

Figure 48a. Variation of ϕ’ and (N1)60 with ϕ’cs and OCR (after Stroud, 1989)

Figure 48b. Variation of ϕ’ and Dr with ϕ’cs and OCR (after Stroud, 1989)


Figure 49. Variation of f1 with plasticity index (after Stroud, 1989) From triaxial tests Triaxial tests are very suitable to determine the strength parameters c’ and φ’. Undrained tests give the undrained shear strength su. When more undrained tests are performed at different cell pressures, it is also possible to evaluate effective strength parameters. Table 16 - Determination of the strength properties from triaxial tests (CUR 195) CUS, CUC, CDS, CDU s’-t-plot p’-q-plot σ’1-σ’3-plot Inclination dy/dx=b (y=ax+b) Intersection y-axis=a φ’ c’

b=tan(θ) a=t0

sin(φ’)=b c’=a/cos(φ’)

b=tan(η)=(6sinφ’sc)/(3-sinφ’sc)=Msc a=q0 ;sc sinφ’sc=(3b)/(6+b) c’sc=a/[(6cosφ’sc)/(3-sinφ’sc)]

b=tan(ω)=(1+sinφ’sc)/(1-sinφ’sc) a=s1’0 ;sc sinφ’sc=(b-1)/(b+1) c’sc=a/[(2cosφ’sc)/(1-sinφ’sc)]

CUU, CUE, CDU s’-t-plot p’-q-plot σ’1-σ’3-plot Inclination dy/dx=b (y=ax+b) Intersection y-axis=a φ’ c’

b=tan(θ) a=t0

sin(φ’)=b c’=a/cos(φ’)

b=tan(η)=(6sinφ’ue)/(3-sinφ’ue)=Mue a=q0 ;ue sinφ’ue=(3b)/(6+b) c’ue=a/[(6cosφ’ue)/(3-sinφ’ue)]

b=tan(ω)=(1+sinφ’ue)/(1-sinφ’ue) a=s1’0 ;ue sinφ’ue=(b-1)/(b+1) c’ue=a/[(2cosφ’ue)/(1-sinφ’ue)]

* ‘sc’ for Standard and compression, ‘UE’ for unloading and extension Parameters of importance for modelling and design are values of the angle of shearing resistance at peak and at critical state, and they can be different in triaxial compression and triaxial extension. They are calculated from the known stress state at peak and critical state as φ’=sin-1((σ’1-σ’3)/(σ’1+σ’3)) (Potts and Zdravkovic, 1999). NB: The slope of the critical state line, M (cfr critical state models), obtained from a triaxial

compression test is related to the friction angle through: φ

φsin3

sin6−

=M .

The dilatancy angle can be obtained from a triaxial test pvertical

pv

pv

ddd

)(1

sinεε

εψ−

= .


From correlations with index parameters Mitchell (1976) has published some relations which show the relation of the (remoulded) undrained shear strength su(r) versus the liquidity index or the plasticity index (fig 50).

Figure 50. Different correlations of undrained shear strength from liquidity index and plasticity index (Potts and Zdravkovic, 1999). From pressuremeter test (Potts and Zdravkovic) During the plastic phase of a pressuremeter test, the measurements can be used to evaluate the shear strength. If the soil behaves as a Tresca material, yielding will occur when the shear stress in the cavity wall reaches the undrained shear strength of the soil: P = σho + su. If the cavity pressure increases above this yield value, an annulus of plastic soil will develop around the cavity, while the soil further away from the cavity will continue to behave elastically. The cavity pressure can then be expressed in the following form: P=σh0 + su [1+ln(G/su)]+su[ln(ΔV/V)] where V is the volume of the cavity occupied by the inflatable membrane. With further loading a limit condition will be reached where the cavity expands with no change in pressure, at which stage ΔV/V ~ 1 and the limiting cavity pressure, PL becomes PL=σh0+su[1+ln(G/su)]. The current cavity pressure P can therefore be re-expressed as P=PL+su ln(ΔV/V). If the cavity pressure P is plotted against ln(ΔV/V) for the duration of the pressuremeter test, it should give a straight line with a gradient equal to su during the later stages of the test. Undrained shear strength determined from pressuremeter tests are often significantly larger than values obtained by other means and therefore should be used with caution. While it is theoretically possible to obtain estimates of the angle of shearing resistance φ’ and dilation ψ from a


pressuremeter test in sand, it is difficult in practice due to the disturbance involved in the ground (see Mair and Wood 1987). Typical values of strength parameters For sands / gravels: values of ϕ depends on : - mineralogy - grain size distribution, shape, surface roughness - density of soil and typical range is : - very loose : ϕ < 28° - very dense : ϕ > 45° (50° max) Table 17 – Typical values Soil loose compacted loose or compacted Angle of friction cohesion

bulk density

Bulk 'weight density'

bulk density

Bulk 'weight density'

submerged density

submerged 'weight density' loose compacted

[kg/m³] [kN/m³] [kg/m³] [kN/m³] [kN/m³] [kN/m³] [°] [°] [kN/m²] fine sand 1750 17.2 1900 18.6 1050 10.3 30 35 0 coarse sand 1700 16.7 1850 18.2 1050 10.3 35 40 0 Gravel 1600 15.7 1750 17.2 1050 10.3 35 40 0 Peat - - 1300 12.8 300 3 - 5 5 river mud 1450 14.2 1750 17.2 1000 9.8 - 5 5 loamy soil 1600 15.7 2000 19.6 1000 9.8 - 10 10 Silt - - 1800 17.7 800 7.9 - 10 10 sandy clay - - 1900 18.6 900 8.8 - 0 15 to 40 very soft clay - - 1900 18.6 900 8.8 - 0 <20 soft clay - - 1900 18.6 900 8.8 - 0 20 to 40 firm clay - - 2000 199.6 1000 9.8 - 0 50 to 75 stiff clay - - 2100 20.6 1100 10.8 - 0 100 to 150very stiff clay - - 2200 21.6 1200 11.8 - 0 >150

Table 18 – Typical relations Cohesionless soils Relative density SPT 'N' CPT qs (MN/m²) phi' (°) very loose 0-4 2.5 25 Loose 4-10 2,5-7,5 28 med dense 10-30 7,5-15 30 Dense 30-50 15-25 36 very dense > 50 over 25 41 Cohesive soils Description Ip (%) undrained c (kN/m²) drained c' (kN/m²) phi' (°) very soft >80 <20 0 15 Soft 80 20-40 0 15 Firm 50 50-75 0 20 Stiff 30 100-150 0 25 very stiff 15 >150 0 30


Table 19 – Some empirical relations Soil type Equation Reference NC clay OC clay Clay Sands

( )( )

33.0'

).04.023.('

)(''

0037.011.0'

6.0

8.0

=

±=

=

+=⎟⎟⎠

⎞⎜⎜⎝

⎛

z

u

z

u

nczu

oczu

p

ncz

u

s

OCROs

OCRss

Is

σ

σ

σσ

σ

Φ’p=Φ’cs+3Dr(10-lnp’f)-3 p’f=middle effective stress at failure (kPa) equation only for 12>(Φ’p-Φ’cs)>0

Skempton (1957) Ladd and al (1977) Jemiolkowski and al (1985) Mesri (1975) Bolton (1986)

Table 20 – Typical values (Bolton, 1986) Soil type Φ’cs Φ’p Φ’r Gravel Graviers and sand mixing with fine soils Sand Silt or silty sand Clay

30-35 28-33 27-37 24-32 15-30

35-50 30-40 32-50 27-35 20-30

5-15

Fine soils apparent cohesion (first approximation) for undrained analysis w>wL wL>w>wp wp>w

cu < 10kPa 10 < cu < 50 kPa 50 kPa < cu

w = water content ; wL = liquidity limit (%) ; wp = plasticity limit (%) Table 21 – Typical values (EPFL website) Soil USCS Ip (%) Grain shape Φ’ (°) c’ (kPa) Crushed rock Clean gravel Loamy gravel Clayey gravel Clean sand poorly graded Well graded clean sand Slightly loamy sand Slightly clayey sand Clayey sand Loam Clayey loam Loamy clay Clay

GW GM GC SW SP SM SC SC-CL ML CL-ML CL CH CH CH CH

0 2-6 7-12 0 0 2-6 6-12 9-15 2-6 4-10 12-18 ~20 ~40 ~60 >100

angular all all all angular rounded all all all

47+/-7 40+/-5 36+/-4 34+/-4 40+/-4 36+/-6 34+/-4 32+/-3 27+/-3 33+/-4 30+/-4 27+/-4 20+/-4 15+/-4 11+/-4 <8

0 0 ~0 ~0 0 0 ~0 ~0 5+/-5 ~0 15+/-10 20+/-10 20+/-10 25+/-10 too variable too variable

+/- = standard deviation Undrained shear strength is often used as criterion for the classification of the clays. One example of classification is given below (BS 8004 :1986). Sensitivity of clays (ratio between undrained shear strength not remoulded and undrained shear strength remoulded for the same


water content of the original sample) is also classified as function of the undrained shear strength (clay sensitivity is a function of its structure). Table 22 – Typical values Consistence cu (kPa) Very stiff > 150 Stiff 100-150 Firm to stiff 75-100 Firm 50-75 Soft to firm 40-50 Soft 20-40 Very soft <20 Sensitivity Not sensitive <1 Slightly sensitive 1-2 Middle sensitive 2-4 Very sensitive 4-8 Slightly fast 8-16 Middle fast 16-32 Very fast 32-64 Very very fast >64… up to 100 3.5.4. STATE PARAMETERS (K0 AND OCR) The coefficient of earth pressure at rest is used when the soil is not subjected to any deformation: K0 = σ’h/σ’v in the levelled ground. To characterise a particular soft soil layer, it is also necessary to know the initial preconsolidation pressure σp0. This pressure can be calculated e.g. from a given value of the overconsolidation ratio OCR = σ’zc/σ’z0. From pressuremeter test

Under ideal circumstances, the radius of the cavity will only begin to deform when the pressure P equals the in-situ horizontal stress σh0. If this pressure is noted along with the in-situ pore water pressure, then it is possible to estimate the coefficient of earth pressure at rest K0 in the ground. In practice such measurements are only usually possible with the selfboring types of pressuremeter under perfect installation process, which is not easily achievable. From triaxial tests If σa and σr are increased together such that the radial strain εr=0, the sample will deform in one dimensional compression, similar to oedometer conditions, and K0

NC=σ’r/σ’a. If σa and σr are decreased together such that εr=0, the coefficient of earth pressure in overconsolidation K0

OC can be estimated for different OCRs. Although not common, the triaxial cell can be used to obtain an estimate of in-situ values of K0 in clay soils. If it is assumed that during the whole sampling and storage process, from extraction from the ground until placement in the triaxial cell, the sample remains undrained, then the sample will retain its mean effective stress p’k. The sample on placement in the triaxial cell will have a zero mean total stress applied to it and consequently the initial pore


water suction will be equal to p’k. If a cell pressure (σr) is then applied to the sample until a positive pore water pressure is recorded, the initial p’k can be calculated as the difference between the applied cell pressure and measured pore water pressure. This can then be used to determine the in-situ value of K0 using the following equation (Burland and Maswoswe 1982): K0 = [(p’k/σ’v)-As]/(1-As) where As account for the change in pore water pressure due to the reduction in deviatoric stress which occurs during sampling in the field. A value of As=1/3 corresponds to an isotropic elastic material, whereas a stiff clay typically has a value of approximately ½; σ’v is the original vertical effective stress in the sample in-situ and can be estimated knowing the depth of the sample, the bulk unit weights of the overlying materials and the pore water pressure. Accurate estimates of K0 from this approach require that the sample remains undrained at all times. The approach also neglects any changes in pore water pressure that occur when the sample is forced into and out of the sampling tube in which it is stored. From CPT K0 and OCR determination. The relations could only be used within a specific location or type of soil (see Lunne, Robertson and Powell, 1997). From dilatometer tests (DMT) The profile of KD (Horizontal Stress Index) is similar in shape to the profile of the overconsolidation ratio OCR. KD ~ 2 indicates in clays OCR = 1, KD > 2 indicates over-consolidation (G. Totani, S. Marchetti, P. Monaco & M. Calabrese).

Figure 51. Correlation KD-OCR for cohesive soils from various geographical areas (Kamei & Iwasaki 1995) General rules The following are several approaches used in determining Ko values for different types of soils.

K0NC

= 1-sinφ’ (Jacky formula) : Normally consolidated sands K0

0C = 1-sinφ’.(OCR)h with h = 0,4..0,5 up to 0,6 for very dense sands (Schmidt 1966 & 1975,

Alpan 1967, Al-Hussaini and Townsard, 1975) Natural soil deposits : K0~0,4…0,5 for sedimentary soils never pre-loaded, K0 up to 3 and more for pre-loaded deposits (cfr Holtz and Kovacs, 1981). Clays NC : K0

NC = 1-sinφ’+/- 0,05 (Ladd et al, 1977) ;

Clays OC : K0 depends on the stress state.


From the oedometer The original overconsolidation ratio of the sample can be calculated by dividing (σ’a)b by the effective stress existing in the field at the location the sample was taken from. For structured soils this approach may lead to an overestimate of OCR, as the process of ageing has the effect of increasing the stress associated with point ‘b’. These parameters can be determined from : - Selfboring pressuremeter test; - Triaxial tests; - Cone penetration tests (lower reliability); - Dilatometer tests; - Oedometer; - Correlations with ϕ‘ and OCR (lower reliability) 3.5.5. PERMEABILITY Soil permeability range is very large (The term permeability, used here, is replaced by hydraulic conductivity in some texts and contexts). The following characteristics can influence the permeability of the soils: size of the particles, void ratios, composition, structure, degree of saturation, homogeneity, layers successions, cracking etc..

These parameters can be determined from: - Triaxial tests; - Oedometer test (lower reliability); - Permeameter test (>10-7m/s) (lower reliability); - Pumping tests; - CPTU tests; - Dilatometer tests. Range of soil permeability ranges is very large. For homogeneous structural soils, it depends mainly on the fine particles. A low percentage of fine particles can fill the pores of a coarse material, what lead to a lower permeability. From triaxial test The triaxial cell can be used to obtain an estimate of permeability value. A sample can be subject to a cell pressure, σr, and then a difference in pressure applied between the drainage lines at the top and bottom plattens. This will cause the flow of water through the sample. By knowing this pressure difference and measuring the quantity of water flowing through the sample in a set time, once steady state conditions have been reached, enables the permeability to be estimated using Darcy’s law. By performing a series of permeability tests at different cell pressures enables a relationship between permeability and voids ratio (or mean effective stress) to be determined. As the time required for these tests depends on the drainage path length, which in turn is related to the sample height, short stubby samples are usually used. For soil with low permeability (k0 < 10-7m/sec) it is also possible to estimate the permeability k from a dissipation stage of a triaxial test. This involves applying a change in cell pressure quickly and then allowing consolidation during which time the decay in excess pore water pressure with time is monitored. One dimensional consolidation theory can then be used to determine permeability value k.


From oedometer test If, after each increment of load is applied, the change in sample height is monitored with time this data can be combined with one dimensional consolidation theory to estimate the coefficient of permeability in the vertical direction kv.

From Permeameter It is the most common means of measuring permeability in the laboratory. The types of equipment most generally used are:

- the constant head permeameter, for permeabilities k to about 10-4m/s - the falling head permeameter, for values of k between 10-4 and 10-7m/s

Below a value of 10-7m/s permeability can only be measured by indirect means in a consolidation test in the oedometer, or in dissipation test in the triaxial apparatus. Pumping test In the field permeability of a stratum is most commonly determined by measuring the discharge from a well. While pumping tests are applicable to relatively permeable soils, they are not necessarily appropriate for some clays. In these cases a borehole can be sunk into the clay and sealed over its length in contact with overlying soils. Over time water will flow from the clay into the borehole. By measuring the rate of increase in the height of the water level in the borehole and using Darcy’s law it is possible to estimate the permeability of the soil. Alternatively, the borehole can be filled to the ground surface with water and the rate of reduction in level is noted. Coupled with Darcy’s law this allows an estimate of k to be made.

From CPT tests The applicability of CPT tests in estimating the permeability following Mitchell & Gardner, is - for Clays: class 2-4 - for sands: - with

1. High reliability 2. High to moderate reliability 3. Moderate reliability 4. Moderate to low reliability 5. Low reliability

From dilatometer tests (DMT) The DMT allows the estimation of the horizontal coefficient of consolidation ch and permeability kh in clay by means of dissipation tests. Various procedures have been formulated (DMTC, Robertson et al. 1988; DMTA, Marchetti & Totani 1989). All methods are based on the decay with time of sh total against the membrane after stopping the blade at a given depth.

Determining ch and kh from DMT dissipations presents various advantages over the piezocone: (a) Lower distortion induced in the soil by the penetration of the blade; (b) Absence of problems of saturation, filter clogging, smearing; (c) "Averaged" - rather than "punctual" - measurement.


Typical ranges of permeability values Table 23 – Typical ranges of permeability values (Hsin-yu Shan, 2005)

Material Intrinsic Permeability (darcy) Hydraulic Conductivity (cm/s)

Clay 10-6 –10-3 10-9 –10-6

Silt, sandy silts, clayey sands, till 10-3 –10-1 10-6 –10-4

Silty sands, fine sands 10-2 –1 10-5 –10-3

Well-sorted sands, glacial outwash sands 1 – 102 10-3 –10-1

Well-sorted gravel 10 – 103 10-2 –1

For a homogeneous soil, the hydraulic conductivity k depends on the soil structure or on the grain arrangement. Several empirical relations have been proposed to relate k to the void ratio and to the particle size for coarse grained soils: 2

10sec)/( CDcmk = :Hazen (1930) with C =

constant expressed in ( cm⋅sec1 ), comprised between 1 and 42. For coarse and fine sands, C

≈ 1. D10 = effective size of the particles, in cm. Table 24 – Typical ranges of permeability values (BS 8004:1986)

Table 25 – Typical ranges of permeability values (Terzaghi et Peck, 1967) Formation Permeabilities (cm/sec) River deposits Rhône and Génissiat Low streams, Est of the Alpes Missouri Mississippi Glaciary deposits Outwash plains Esker, Westfield, Mass. Delta, Chicopee, Mass. Till Wind deposits Sand dunes Loess Loamy loess Lake and see deposits Fine uniforme sand U*=5-2 Clay

Up to 0.40 0.02-0.16 0.02-0 .20 0.02-0 .12 0.05-2.00 0.01-0.13 0.0001-0.015 < 0.0001 0.1-0.3 0.001 0.0001 0.0001-0.0064 < 0.0000001

U* = coefficient of uniformity Table 26- Soil classification as function of their coeff. of permeability (Terzaghi & Peck, 1967) Degree of permeability Coefficient of permeability k (cm/sec) High Moderated Low Very low Nearly impermeable

> 10-1 10-1 to 10-3 10-3 to 10-5 10-5 to 10-7 < 10-7


3.6 Models and parameters for structural elements The modelling of structural elements depends on the problem to be treated. Examples of recommendations are given below for retaining walls and shallow foundations. 3.6.1. RETAINING WALLS - A key feature of any numerical analysis of a retaining wall is the modelling of the interface between the soil and the structure. This interaction depends on the soil model and parameter selection and on the assumptions concerning the modelling of the structural elements (sheet piles, slurry walls, props, anchorages etc.). When modelling structural components, it is important to consider the mechanism by which the component contributes to the stability of the construction. For example a prop has a passive role for the stability of a wall, reducing displacements independently of the stresses mobilised in the ground, while an anchorage has an active role, its effect can be reduced because of modification of the soil conditions. Thus a prop can be modelled by simple fixed or spring conditions, while an anchor should be modelled in more details to simulate its interaction with the ground. - Choice of the elements for the modelling of the wall: embedded walls can be modelled using either solid or beam elements. The use of the latter type of element which has a zero thickness can influence the results, even if an equivalent thickness is calculated for the 1D elements, and the predicted horizontal wall displacements and associated bending moments are larger when modelling the wall with beam elements, particularly for low values of K0. The reason why there are such differences between the analyses modelling the wall with solid and beam elements is that the vertical shear stresses mobilised on the back of the wall provide a stabilising moment, because of the thickness of the wall. Another influence of the selection of the structural elements concerns the unrealistic uplift movement predicted under specific analysis conditions (see KirkebØ S., 1998), which can be more important when solid elements are used to model the wall. - Selection of the behaviour laws: steel and concrete elements are generally modelled by linear elastic laws. Tensile capacity of struts or unreinforced concrete elements should be limited. Concerning steel sheet piles design, Eurocode 3 allows the development of plastic hinges, which leads to a redistribution of the forces in the sheet pile with consequences on the movements, deformations and stresses in the ground. Steel behaviour should then be modelled by an elastic plastic law. Similarly, some degree of moment redistribution may be allowed in concrete structures. - Parameter: *It is common practice to assume that there is a reduction in stiffness in the long term for concrete walls, due to factors such as creep and cracking. In a finite element analysis, this can be modelled by changing the stiffness of the concrete after construction and prior to simulation of long term pore water pressure levelling (e.g. reduction of 50%). The deformation modulus of anchorage grouting can also be reduced by 40% from its initial value because of cracking. Stress relaxation should also be allowed when the stiffness is reduced in order to activate the proper structural response. *The horizontal displacements are systematically underestimated when the floor is represented by a fix end condition (theoretically infinite stiffness). This underestimation is still larger when a strut is replaced by a fix end condition. When the stiffness of the struts varies within a common range of values (between 1.10e4 and 1.10e5 kPa), its influence on the wall and ground movements are rather limited. *Concerning soil nailed structures, the determinant parameters are the stiffness of the nails and the relative stiffness between the wall and the nails. Details on the modelling of anchorages are given in CUR 178 (CUR, 1995). *Anchor or strut stiffness can be determined by the slope at the origin of the loading curve of the anchor, or by the ratio « A * E / L » with A, E respectively the section and elasticity modulus of the free part of the anchor, and L its free length. The example below (Potts et Zdravkovic, 2001) shows the bending moments and displacements of a retaining wall (Figure 52), the wall being modelled respectively using beam and solid elements. Figure 53a shows the calculation results with K0=2 and Figure53b with K0 = 0,5. The Figures show that the predicted horizontal wall displacements and associated bending moments are larger when modelling the wall with beam elements. This is particularly so for low values of K0. The reason why there are differences between the


analyses modelling the wall with solid and beam elements is that the vertical shear stresses mobilised on the back of the wall provide a stabilizing moment, because of the thickness of the wall. Figure 54 shows that the wall and adjacent ground horizontal displacements are systematically underestimated by about 50% when the floor is represented by a fix end condition (theoretically infinite stiffness). The difference is still larger when a strut (stiffness about 1/10 relative to the floor) is replaced by a fix end condition.

Figure 52 (Potts and Zdravkovic, 2001) Influence of the choice of the elements for the modelling of the wall

Figure 53 (a) (Potts and Zdravkovic, 2001) Influence of the choice of the elements for the modelling of the wall

Figure 53 (b) (Potts and Zdravkovic, 2001) Influence of the choice of the elements for the modelling of the wall


The effective stiffness of the support system can be significantly reduced by effects such as temperature, creep, bedding, concrete shrinkage, openings in floor slabs, packing between the prop and the wall etc (Potts & Zdravkovic, 2001). The following example (Hight and Higgins, 1994) shows the significant effect of variations in prop stiffness on wall and ground displacements during construction. The initial analyses was performed with a prop stiffness K = 50 MN/m/m (this value is typical of that provided by a concrete slab).

Figure 54 (Hight and Higgins, 1994) Influence of the strut elements stiffness. The following example corresponds to a deep excavation supported by a diaphragm wall and two levels of struts (Figure 27). When the stiffness of the struts varies within a common range of values (between 1.10e4 and 1.10e5 kN/m/m), this parameter influences the bending moment in the wall and the forces in the struts, but its influence on the wall and ground movements are less significant.

Figure 55. Influence of the stiffness of the struts – deep excavation (Bulletin Plaxis n°4)

The influence of the wall stiffness on its horizontal displacements and on the ground deformation at proximity is illustrated below (Figure 28, Hight and Higgins, 1994). The problem corresponds to the theoretical study of the behaviour at short term of a deep excavation (in London). The wall (1m width) has a stiffness of E = 28 GPa.


Figure 56 (Hight and Higgins, 1994) Influence of the stiffness of the walls Another study by Potts and Day (1990) shows the importance of the flexional rigidity of the wall on the bending moments distribution and on the lateral movements of the wall. The stiffness of the interface elements (between wall and soil) has no significant effect on the results (Day and Potts, 1998). The friction angle of the interface elements can have an important effect on the soil movements. - Attention should be paid when adopting 2D calculations for ground anchors because of the 3D nature of ground anchors. When ground anchors are simulated by plate elements, the soil-structure interaction is excessively simplified. 3.6.2. SHALLOW FOUNDATIONS

In theory, it is necessary to discretise both the soil and the foundation into finite elements, which enables the correct stiffness of the foundation to be included in the analyses. Interface elements can also be positioned between the underside of the foundation and the soil, so that the interface behaviour can be modelled more accurately. However, as the foundation is often very stiff compared to the soil, numerical problems can arise due to the large difference in stiffness between adjacent elements on each side of the interface. Therefore surface foundations are sometimes analysed using one of two extreme assumptions: perfectly flexible or rigid foundation. If the foundation is flexible, it is assumed that any loading is uniform and can be represented by a surface surcharge pressure applied to the surface of the soil immediately below the position of the footing. If the footing is rigid, analyses can be performed under either load or displacement control (for load control it is necessary to tie the vertical displacement of the nodes below the position of the footing). The results are less influenced by the stiffness of the interface between soil and foundation than by the roughness of the foundation. For shallow footings buried below the ground surface, it is important to correctly model the interface between the sides of the footing and the soil (Potts and Zdravkovic, 2001).


3.7 Analysis aspects 3.7.1. THE INITIAL CONDITIONS The initial conditions within the ground (K0, OCR/POP parameters, water conditions,…) are important to consider when using non linear constitutive models for the soil. Some guidelines are summarized below.

- For most soil models, the value of K0 is specified in the data and should be imposed by the program. It is recommended to run the program for this initial state in order to check vertical equilibrium. The strength limits of the material should be incorporate din the analysis to ensure that they are compatible with the stress states specified as data.

- In situations of non-horizontal soil layering or water tables, the program should compute a new equilibrium state. The user should check how this is done in the software in use, and be satisfied that it is appropriate.

- Instead of requiring the user to specify K0, some soil models compute the initial stress state from data specifying the geological history of the soil.

- Definition of the initial conditions should take into account the history of the soil (through assignment of proper OCR or POP), the soil water conditions (steady state) and if relevant the influence of existing constructions.

3.7.2. THE SEQUENCE OF CONSTRUCTION Almost all geotechnical finite element analyses are performed in steps that simulate a sequence of real events; the construction steps to consider are dependent on the type of constructions. The existence of adjacent structures will inevitably modify the state of stress within the ground and these effects must be accounted for in any analysis. Performing the analyses in steps has two important advantages for geotechnical problems o The geometry can be changed from one step to the next to simulate excavation or fill

placement, by removing elements or adding elements to the mesh ; o The properties of the soil can be changed from one step to the next to simulate the change

in behaviour that results from changes in the stresses within the soil mass. Excavations and retaining walls Construction process and time taken for construction is important: different sequences of construction could result in the soil experiencing different stress paths and additional forces might be imposed on the retaining structure. In many situations gravity walls are built on a foundation, with backfill placed behind them. Therefore it is necessary to consider not only the properties of the structure and the behaviour of the foundation, but also the nature of the backfill. In particular as most backfill are compacted in layers, stresses will be imposed on the retaining wall due to the compaction process. However, there are very few reliable measurements of these stresses from which suitable values may be deduced. Besides, although numerical procedures do exist for estimating compaction stresses, these are generally quite complicated. Hence it is much simpler to specify the horizontal stresses in layers of fill, as a proportion of the vertical stress, as they are constructed during an analysis. Sheet pile wall or slurry wall installation should be considered in the analysis. The sheet pile driving process is very difficult to model (impact driving, vibro-driving). Sheet pile driving lead to residual stresses in the sheet piles, soil remoulding and degradation of the mechanical


properties of the sheet piles, accompanied by an initial deformation of the sheet piles. These effects are rarely considered in the analyses and designer should be aware of such additional stress in the sheet pile. Slurry wall construction is three dimensional (2D modelling tends to exaggerate the effect of wall construction). Modelling of slurry wall construction allows for stress relaxation leading to lower coefficient of earth pressure values. One example of such modelling is given in the Plaxis Bulletin n°10, with the followings steps of modelling :

o Excavation of the trench and application of the bentonite pressure on the faces of the trench;

o Concrete filling : by increasing the lateral pressure (bi-linear law) ; o Concrete hardening : trench elements properties are replaced by the concrete

properties; o Execution of the other panels.

Details on the modelling of ground anchors and strut placement are given in CUR-178 (CUR, 1995). Details on the modelling of geotextile placement are given in the Potts and Zdravkovic, 1999. Deep foundations Ideally, pile installation process should be considered in the modelling. Displacement piles will tend to densify the soil, while bored piles will cause stress relief local to the pile shaft. These effects can have significant influence on pile behaviour. Embankments Staged construction allows embankments to be constructed to significant heights. However, any analyses must use a constitutive model that can account for changes in undrained strength during the consolidation periods. The limitations of computer resources require relatively thick layers to be used in the idealization of dam construction (this is not considered as a problem with the advances in computer technology). However, significant errors can occur if the layers are too thick, particularly when the effects of compaction are to be modelled 3.7.3. THE CONSOLIDATION (DRAINAGE) CONDITIONS Soil drainage conditions depend on the type of soil, on the geologic formation and on the loading rate. Under long term conditions, the excess porewater pressure developed during loading stage dissipates and drained conditions must be considered in the analysis. In clay soils, excess porewater pressure can take several years to dissipate because of the low permeability of the soil. During or immediately after construction (short term conditions, in soils with low permeability, or fine grained soils), the excess porewater pressure does not have the time to dissipate and undrained conditions must be considered. Permeability of coarse soils is high enough to rapidly dissipate the excess porewater pressure under static loads. As consequence, the undrained conditions do not apply to coarse grained soils under static load. Under dynamic loads (earthquake for example), because of the loading rate, even in coarse grained soils the excess porewater pressure does not have the time to dissipate and undrained conditions must be considered in the analysis: σ’ = σ-u =0 (liquefaction). For the design of a geotechnical system, drained and undrained conditions must both be considered. The choice of analysis drained/undrained may then be based on a calculation of the degree of consolidation (eg. from the dimensionless consolidation time, Vermeer, 1998). An analysis under drained conditions is based on the effective shear parameters φp’ and φcs’ (resolution in terms of effective stresses). The value of φcs’ is constant for a soil independently of the initial conditions and of the magnitude of the normal stress. But the value of φp’ depends on the normal effective stress (the normal stress can suppress the dilation). Under static loads, the analysis under undrained conditions is carried out in terms of total stresses


based on the undrained shear strength parameter su (defined for fine grained soils) (Note: shear strength is also governed by the effective stresses). Another approach is to take account of the whole consolidation process with a coupled consolidation FE analysis. In general, the coupled elastic plastic FE analysis (2 phase model) yields similar results for the long term behaviour of the foundation as the drained FE analysis with the 1 phase model, and the 1 phase model overpredicts the settlements during the loading/construction process (see example with abaqus: «Study of the influence of the consolidation process on the calculated bearing behaviour of a piled raft», Reul, NUMGE 2002).

Figure 57. Comparison between drained and undrained conditions

With respect to the choice of analysis, i.e. drained/undrained, for the modelling of deep excavations in clayey soils, numerous studies have been performed in particular by Puller (1996) and Vermeer (see Plaxis bulletins). According to these authors, the undrained shear strength cu is only correctly used when load is applied immediately and it is strictly illogical to use it as soon as pore pressures change. In clay soils, where the retaining wall structure deforms and attempts to move away from the retained soil, negative pore pressures are generated in the retained soil as excavation proceeds in front of the wall. In highly fissured or laminated overconsolidated clays the reduction in suction pressure may proceed relatively quickly and the original value of cu quickly becomes inapplicable. The use of cu in such analyses can therefore become over-optimistic. For soft soils with low permeability Puller states that it is prudent to undertake analysis for both drained and undrained soil conditions. Vermeer suggests deciding on the basis of the degree of consolidation. To judge the degree of consolidation, most textbooks provide the function U(T), where T is the dimensionless

consolidation time tD

EkT

w

oed

².

γ=

with k = soil permeability Eoed = oedometer modulus γw = specific water weight D = drainage length t = consolidation time Available function for U relate to one-dimensional consolidation, but we often have near-vertical drainage of deep layers to the bottom of the excavation and similar conditions for shallower clay layers behind an impermeable retaining wall. For example for T= 0,01, we have U~0,1 and thus little consolidation, as U=0,10 implies an average dissipation of excess pore pressures of only 10%. In such cases Vermeer suggests undertaking the undrained analysis. On the other hand, if construction takes a very long time, with T>0,4, giving U>0,7, we have nearly drained conditions and it is suggested undertaking only a drained analysis. For intermediate consolidation times, it is recommended to undertake a coupled analysis taking into account the soil consolidation.

Drained conditions Undrained conditions

Diminution of the effective stress and of the shear strength

Increase of the effective stress and of the shear strength


3.8 Supplementary guidelines specific for different types of construction EXCAVATIONS

- The initial stresses within the ground before construction of the retaining wall can have a significant influence on wall behaviour : presence of adjacent structure modifies the state of stress within the ground for example. The higher K0 is the higher the horizontal stresses becomes in the soil, hence the higher lateral movements and forces in structural elements are computed (on the other hand, soils with higher K0 are generally stiffer)..

- Construction process and time taken for construction is important: different sequences of construction may result in the soil experiencing different stress paths and additional forces might be imposed on the wall. In particular excavation should be modelled by removing elements in several layers.

- In many situations gravity walls are built on a foundation, with backfill placed behind them. Therefore it is necessary to consider not only the properties of the structure and the behaviour of the foundation, but also the nature of the backfill. In particular as most backfill are compacted in layers, stresses will be imposed on the retaining wall due to the compaction process. However, there are very few reliable measurements of these stresses from which suitable values may be deduced. Although numerical procedures do exist for estimating compaction stresses, these are generally quite complicated. Hence it is much simpler to specify the horizontal stresses in layers of fill, as a proportion of the vertical stress, as they are constructed during an analysis.

- Sheet pile wall or slurry wall installation should be considered in the analysis, however: - The sheet pile driving process is very difficult to model. Sheet pile hammering and driving lead

to residual stresses in the sheet piles, soil remoulding and degradation of the mechanical properties of the sheet piles, accompanied by an initial deformation of the sheet piles. These effects are rarely considered in the analyses. The soil is also displaced laterally by the thickness of the sheet piles.

- Slurry wall construction is three dimensional (2D modelling tends to exaggerate the effect of wall construction: in particular when simulating the execution of a slurry wall in 2D, the trench is not stable). Modelling of slurry wall construction allows accounting for changes to stress and K0 values. One example of such modelling includes the followings steps : o Excavation of the trench and application of the bentonite pressure on the faces of the

trench ; o Concrete filling : by increasing the lateral pressure (bi-linear law) ; o Concrete hardening : trench elements properties are replaced by the concrete properties ; o Execution of the other panels.

- For details on the modelling of ground anchors and strut placement it is referred to CUR-178 (1995).

- For details on the modelling of geotextile placement; refer to Potts and Zdravkovic (1999). CUT SLOPES

- In stiff «softening» soils, the earth pressure at rest K0 strongly influence the location of the shear surface and the time to collapse, but not the probability of collapse. There is a slight increase in the amount of progressive failure with increase of K0 up to a critical value, after which the base rupture surface extends beyond the final inclined rupture surface, which relieves the stress in the slip prior to its final formation and reduces the amount of progressive failure. At higher values of K0, the rupture surface is developed by progressive failure at a greater depth than the critical surface which would be determined by limit equilibrium analysis. As a consequence, the predicted operational strength on the rupture surface is less than that would be determined for the same slope with the same pore water pressures by limit equilibrium analysis, using a search technique to find the critical surface.

- The field experience shows considerable variability in time to collapse (depending on the soil drainage conditions…). There will be seasonal variation in superficial pore water pressures, not represented in many analyses. Thus collapse will tend to occur in winter, when suction is lower. The surface hydraulic boundary condition has a strong effect on stability (surface pore water pressures can be reduced by increasing evapo-transpiration through the controlled use of vegetation (influence of the tension cracks) or by surface drainage which reduces pore water pressure below the depth of the drains).


- The mechanism causing deep-seated delayed slips is progressive failure promoted by swelling. They enable the importance of the controlling variables to be established, and the effect of possible stabilising measures to be evaluated.

SHALLOW FOUNDATIONS

- Conventional calculations for the foundation bearing capacity are based on the assumption that the superposition principle is valid. In reality the failure mechanisms associated with Nq and Nγ differ as the magnitudes of the footing settlements required to mobilise full Nq and Nγ values. In practice, this could have significant implications, especially in situations where the soil strength degrades (i.e.ϕ’ reduces) with straining: for example at ultimate load the average strength associated with the first mechanism to form (Nγ) is likely to have decrease from its peak value, while that associated with the second mechanism (Nq) is unlikely to have reached its peak value. In such a case the superposition assumption may not be conservative and the residual strength value must be considered in the calculations (progressive failure). With the finite element analysis, it is only possible to model a progressive failure mechanism if the soil model accounts for soil softening (rarely the case for the soil models implemented in common commercial softwares).

- Within the framework of the Labenne experiments, described and commented by Mestat and Berthelon (Mestat and Berthelon, 1998), the essential role of the soil elasticity modulus on the settlement and bearing capacity predictions of shallow foundations has been shown. Laboratory values traditionally used correspond to the initial modulus, while a lower value such as obtained in situ is more realistic for shallow foundations.

DEEP FOUNDATIONS

- Elements placed adjacent to the pile shaft: - Type: eg. when analysing a single pile subject to axial loading, if solid elements are used and

they are not sufficiently thin, the analysis will over-estimate the pile shaft capacity - Properties:

* If interface elements are positioned adjacent to the pile shaft and if the normal and shear stiffness values are not sufficiently large, these values can dominate pile behaviour, while if the values are too large numerical ill-conditioning occurs. * When analysing a single pile subject to lateral loading the possibility of a crack forming down the back of the pile (i.e. grapping) should be considered. If this is likely to occur interface elements should be installed along the pile soil interface, which cannot sustain tensile normal stresses. * In order to model traction pile, an interface layer with very low axial stiffness can be used allowing forces and water pressures to act on the boundary, while limiting the resistance to the uplift pile movement. These elements do not allow to take into account the decrease in surface contact between pile and soil when pile uplift. Important shear deformations then appear at the base of the pile, and the modelling become not valuable.

- Soil parameters: Dilation can have a dominant effect on pile behaviour and consequently care must be exercised when selecting an appropriate constitutive model and its parameters. When analysing axially loaded piles using an effective stress constitutive model it is unwise to use a model which predicts finite plastic dilation indefinitely, without reaching a critical state condition (such analyses will not predict an ultimate pile capacity).

- Geometry: Piled raft design implies most often a three- dimensional modelling. The assumption of plain strain state can lead to large over-predictions of the settlements and loads carried by the piles (Van Impe, 2001).

- Analysis: Ideally, the pile installation method should be considered in the modelling. Displacement piles will tend to densify the soil, while bored piles will decompress the soil locally. These effects can have a great influence on the pile behaviour.

EMBANKMENTS

- Modelling can be performed by adding elements to the mesh or by applying load pressures. - Soil parameters

- Compacted fills (generally executed with sand) have a behaviour pattern similar to that of granular materials.

- Soft clays are usually lightly overconsolidated but often have a surface crust of higher


strength and stiffness, overconsolidated by climate. The surface crust can have a significant influence on the height to which an embankment can be constructed and on the shape and localisation of the final failure mechanism.

- The provision of reinforcement in the base of an embankment can significantly increase the height to which an embankment can be constructed. Both the stiffness and strength of the reinforcement affect the maximum embankment height.

- Short term behaviour of embankments is highly dependent on the initial pore water pressures. - Staged construction allows embankments to be constructed to significant heights. However,

any analyses must use a constitutive model for the soft clay that can account for changes in undrained strength during the consolidation periods. The limitations of computer modelling require relatively thick layers to be used in the idealization of dam construction. However, significant errors can occur if the layers are too thick, particularly when the effects of compaction are to be modelled.

- Anisotropic soil behaviour has a significant effect on embankment behaviour. Accounting for anisotropy observed in laboratory and field experiments enables a more accurate prediction of embankment behaviour. Complex constitutive models, such as MIT-E3, must be used to represent anisotropic behaviour.

- A constitutive model that requires the specification of stress history is unsuitable for modelling of added elements.

- A great variety of stress paths, accompanied by a rotation of principal stresses occurs in embankment dams during construction. ‘Variable’ elastic constitutive models cannot account for stress path dependency which is caused by non elastic components of soil behaviour.

- FE analyses of earth embankments are more complex because they are usually derived from clayey fill only and may sit on a foundation of varying strength and permeability. The short term behaviour of the embankment depends on the initial pore pressures.

- Progressive failure develops when the centre of an embankment is of weak fill and imposes a substantial increase in active force on the shoulder fill as the embankment is completed. The exact geometry of the core has little influence, provided the total trust is the same.

- Critical state constitutive models, such as modified Cam-Clay, are often used to model soft clay. However, care must be taken to ensure that the model gives a realistic undrained strength profile.

- Today’s computer resources allow for reasonably thin layers. TUNNELS

- Methods of simulating tunnel construction in plane strain require at least one assumption: the volume lost to be expected; the percentage of load removal prior to lining construction; or the actual displacement of the tunnel boundary.

- If severe distortions of the tunnel lining are expected then the model can be used which allows segmental linings to open or rotate at their joints, or allows sprayed concrete lining to crack.

- Permeability’s dependence on stress level leads to nonlinear models for permeability. Using such a model in place of a linear alternative will alter the long term pore water pressure regime, for the same hydraulic boundary conditions. This will alter the ground response during consolidation and swelling.

- The intermediate and long term behaviour is governed by many factors. In particular, whether the tunnel acts as a drain or is impermeable, and whether the initial pore water pressure profile is close to hydrostatic or not.

- It is important to select constitutive models capable of reproducing field behaviour. For example, in situation where pre-yield behaviour dominates the ground response, it is essential to model the nonlinear elasticity at small strains.

- Devices for improving settlement predictions can be developed. These questions must be asked: What is the influence of this adjustment on the soil behaviour? What are the knick effects? For example, if one is adopting a device to match a surface settlement profile, how does this alter any prediction of sub-surface movement, the pore pressure response, or the lining stresses and deformations?

- The FE method can be used to quickly assess the impact of different influences on tunnelling-induced ground movements. Parametric studies can prove extremely useful in the development of design charts and interaction diagrams. One of the great benefits of numerical analysis to tunnel engineer is that an analysis can incorporate adjacent influences. For example existing surface structures, or existing tunnels. It is also possible to reproduce the effects of compensation grouting to protect


surface structures during tunnelling projects. Remark: No commercially available model takes into account the effects of pure principal stress rotation. 4 Pitfalls Some pitfalls when using the Finite Element Methods are tabulated below (Table 27).


Pitfalls General recommendations - 2D in place of 3D - In any cases: it is essential to be conscious of the approximations made when interpreting the results

- With axis of symmetry: check that everything is axi-symmetric (geometry, loads…) - When ground anchors are simulated by plate elements (in 2D analyses), the soil-structure interaction is excessively simplified because of the 3D nature of ground anchors. - With plane strain conditions: check the arching effect, the lateral heterogeneity; if structural elements are included in the model, attention should be paid to the stiffness equivalence of the structure

- Stratigraphy - Attention should be paid when introducing thin layers or layers with sharp changes of properties (eg. stiffness) in the model due to risk of numerical oscillations and divergence in the calculations and/or inaccuracies in the results

Geometry

- Extent of the geometry

It is always advised to check for: - the independence of the results with the extent of the model boundaries; - the output given for the elements placed close to the model boundaries.

- Coarseness of the mesh The coarseness of the mesh should be checked by controlling the influence of refining (locally) the mesh on the output

- Element shape When elements with questionable interior angle or with very high aspect ratio are used, it should be verified that the results are realistic by trying different element shapes

- Element order Should be related to the type of analysis (coupled with flow, …), the constitutive model (degree of non linearity), the geometry (axi-symmetry…)

Discretisation

- Boundary conditions - Choice of the soil

constitutive model Should capture the relevant soil features for the problem considered (strength, deformations…) Soil model

- Definition of the parameters

- Should be adequate with regard to the soil model - Evaluation of the influence of the variation of the main parameters is always recommended (parametric study)

- Interaction with the soil - Need for specific (eg interface) elements? - Type of elements - Constitutive model

Structural elements

- Parameters Numerical aspects - Eg.convergence criteria Because most of discretisation aspects and numerical difficulties are software specific it is claimed for the use of

benchmarks (specific to the software) Stages of analysis - Initial stress generation;

stage of construction

Others - Time effects (drainage, creep) - Water effects (flow…) - …

Table 27 : pitfalls when using the Finite Element Methods


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5. Inventory of resources 5.1 Benchmarks In the Plaxis bulletins benchmarks are regularly proposed. Other benchmarks have been established by committees dealing with the finite element method, see for example:

- «Results from numerical benchmark exercises in geotechnics», Schweiger - «Recommendations for the verification of finite element geotechnical models», Mestat, Humbert, Dubouchet

5.2 Relevant committees: - Working group 1.6 of the German Society of Geotechnics - «Numerical group», group leader : Helmut F. Schweiger - COST Action C7, chairman Oddvar Kjekstad - Comité français de mécanique des sols et de géotechnique, Groupe de travail sur la «Modélisation numérique» - ETC 7 «Numerical methods in Geotechnics Engineering» de la Société Internationale de Mécanique des sols et de Géotechnique (SIMSG) - COSMOS - NAFEM - Interclay II (…) 5.3 Existing (validated) softwares 5.4 Internet resources

FINITE ELEMENTS BOUNDARY ELEMENT FINITE DIFFERENCE SPECIFIC METHODS

Abaqus TELSTA BEAN CHASM ESTAVEL*Adina VERSAT-S2D BEMFEM FLAC FESEEPAFENA WANFE PIESAnsys ZSOIL SCARPCESAR Z_SOIL.PC 2001 2D UPRESCRISP XSLOPEDACSAR GeoslopeDianaFE2DNLFEADAM84FEECONGeoFEAPGeoStressIntuitiveFEM/GEOLusasMARCMICROFINENFAPPLAXISRHEO-STAUBRosalie-LCPCSAFESage-CrispSEEP/WSIGMA-WSOILSTRUCTSWANDYNE


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On the finite elements http://www.comco.com/feaworld/fealist.html: Finite element analyses list (softwares) http://www.gearhob.com/eng/fe.htm Finite element software list http://www.afgc.asso.fr/groupes/PEEFGC/liens.htm Finite elements site directory http://www.engr.usask.ca/~macphed/finite/fe_resources/Internet Finite element resources http://ohio.ikp.liu.se/fe/index.html Information Retrieval on Finite Element Books http://skyscraper.fortunecity.com/copland/949/ The finite element method Software for Structural Analysis Availability, Features and User Experiences. http://www.nafems.org/ International resource for engineering analysis and simulation http://femur.wpi.edu Finite element method universal resource On numerical methods http://www.enssib.fr/Enssib/f_bibliofr.htm Catalogue des bibliothèques scientifiques francophones (ENSSIB) http://sunsite.berkeley.edu/Libweb/ Catalogue des bibliothèques mondiales http://conbio.rice.edu/vl/database/ Base de données des bibliothèques virtuelles http://www.eevl.ac.uk/ Bibliothèque virtuelle d'Edimbourg(2400 sites de grande qualité en sciences de l'ingénieur) http://liinwww.ira.uka.de/bibliography/index.html The Collection of Computer Science http://www.dtv.dk/ipg/5/ Bibliographies Internet Pointer Guide–Computer Science http://www.guideme.com/CivilEngineering.htm Guide des ressources en Génie Civil sur l'Internet http://www.usacm.org/U.S. Association for Computational Mechanics http://www.witcmi.ac.uk/isbe/isbe.html The International Society for Boundary Elements http://www.iabem.mines.edu/iabem.net/iabem.html The International Association for Boundary Element Methods Databases http://www.cosmos-eq.org/workshop_oct2001.html Workshop on Archiving and Web Dissemination of Geotechnical Data http://tecno.upc.es/cluster/cdubp/gelist.htm Index of geotechnical sites, include geotechnical softwares Software http://www.igt.ethz.ch/softer/ Software reviews for geotechnical engineering http://www.engsoftwarecenter.com/ Engineering software centre http://www.ejge.com/GVL/soft-gvl.htm Geotechnical Software Resources on the Net


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http://www.comtecresearch.com/ Tunnel analysis programs http://www.geocentrix.co.uk/ Web design and software for reinforced slopes and retaining wall design http://dutcgeo.ct.tudelft.nl/software/software_e.htm Soil mechanics and groundwater software http://gcagint.com/ Subsurface investigation software http://www.compulink.co.uk/~markz/ Geotechnical database management and slope stability software http://www.data-surge.com/ Geotechnical design, borehole log drafting, soil lab testing and rock mechanics design software http://www.flowpath.com Groundwater software http://www.gaea.ca/softIndex.html Borehole log and contaminant transport software http://www.geopak.com/ Civil design software http://www.geo-slope.com/ Slope stability, groundwater seepage, stress and deformation analysis, contaminant transport analysis and geothermal analysis software http://www.geosystemsoftware.com/ Software for geotechnical data processing http://www.mitresoftware.com/ Slope stability and inclinometer software http://www.pisa.ab.ca/ Program for incremental stress analysis of geotechnical structures http://www.rockeng.utoronto.ca/ Geomechanics software and research http://www.scisoftware.com/ Ground water, surface water, bioremediation, geotechnical, air pollution and environmental software http://www.tagasoft.com/ Slope stability, finite element analysis, pile and pile groups, settlement and consolidation, seismic analysis and foundation design software http://www.tapsoftware.com Bearing capacity, lateral support and standard penetration test software http://www.soilvision.com/ Knowledge based database systems for soil properties http://www.bossintl.com/ http://www.algor.com Centre for mechanical design technology, finite element analysis, simulation and optimization methods http://www.ansys.com Engineering simulation software http://www.oasys-software.com/products/geotechnical/finite_element_analysis/safe/ Oasys SAFE Finite Element program


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