Effect of uncertainties in soil data on settlement of soft columnar inclusions

Engineering Geology 121 (2011) 123–134

Contents lists available at ScienceDirect

Engineering Geology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e n g g e o

Effect of uncertainties in soil data on settlement of soft columnar inclusions Julien Dubost a, Alain Denis b, Antoine Marache b,⁎, Denys Breysse b a b

SNCF, Pôle Régional Ingénierie, Bordeaux, France Université de Bordeaux, UMR 5295 I2M - Environmental and Civil Engineering Department, Av. des facultés, 33405, Talence cedex, France

a r t i c l e

i n f o

Article history: Received 2 April 2010 Received in revised form 19 October 2010 Accepted 6 May 2011 Available online 24 May 2011 Keywords: Inclusions Uncertainties Settlement effects Statistical analysis Ground improvement Numerical modelling

a b s t r a c t This paper deals with the assessment of settlements of soft columnar inclusions when uncertainties in soil properties and geometric features are considered. Two common design methods, the analytical homogenisation method and the finite element method, are then combined with Monte Carlo simulations in order to find, using probability distributions of soil and geometrical characteristics, the probability of settlements exceeding a given value. The results show that length and modulus of granular piles are the two major sources of uncertainty and that the model uncertainty obtained from the interpretation of relevant sensitivities is smaller with the finite element model. From the outcome, this paper demonstrates that when geotechnical uncertainties and their effects on soil responses are appropriately accounted for, they can lead to an improved understanding and modelling of a practical geotechnical problem such as the ground improvement by columnar inclusions. © 2011 Elsevier B.V. All rights reserved.

1. Introduction In urban and peri-urban areas, potential sites for new infrastructures are becoming increasingly scarce. Sites with unfavourable geological and geotechnical conditions are thus frequently used. In such cases, the owner, the contractor and/or the engineers may have to modify their specifications because of unpredictably poor ground conditions, which can induce disorders due to settlements effects. In these complex geotechnical conditions ground improvement techniques have to be used. Amongst the various techniques for improving in situ ground conditions, columnar inclusions are considered as one of the most versatile and cost effective ground improvement method. Different empirical and theoretical approaches concerning the analysis and behaviour of columnar inclusions reinforced ground have been proposed over 30 years. Almost all of the approaches are based on the standpoint of foundation analysis that the ground reinforced by granular pile is a composite ground consisting of a relatively stiffer and stronger column and softer surrounding soil. An empirical chart for evaluating settlement reduction ratio in terms of spacing of the granular piles has been first proposed by Greenwood (1970). Balaam and Booker (1985) presented an analytical solution to estimate the settlement of rigid foundation on soft clay stabilised by large number of fully penetrating stone columns. Schweiger and Pande (1986) or Canetta and Nova (1989) presented different detailed numerical ⁎ Corresponding author at: Université de Bordeaux, UMR 5295 I2M - Environmental and Civil Engineering Department, Avenue des facultés, 33405, Talence cedex, France. Tel.: +33 5 40 00 88 27; fax: +33 5 40 00 31 13. E-mail addresses: [email protected] (J. Dubost), [email protected] (A. Denis), [email protected] (A. Marache), [email protected] (D. Breysse). 0013-7952/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2011.05.004

method for the analysis of ground improved by columnar inclusion. Alamgir et al. (1996) proposed a simple theoretical approach to predict the deformation behaviour of uniformly loaded soft ground reinforced by soft inclusions based on the unit cell concept and “free strain” theory. Shashu et al. (2000) extended this one dimensional model in order to consider the effect of granular mat present on the top of the soft soil reinforced with granular piles. Homogenisation techniques have also been proposed by Liausu (1984) and Cognon et al. (1991). However, most of the above models do not incorporate the effect of uncertainties in soil properties and geometrical characteristics of columnar inclusions on settlement analysis. It is now well recognised that uncertainties in the characterisation of the soil's properties must be considered in geotechnical engineering. The uncertainties associated with geotechnical data are divided into two main types: random (i.e. physical) uncertainties and epistemic (i.e. related to imperfect knowledge) uncertainties (Phoon and Kulhawy, 1999; Baecher and Christian, 2003; DNV, 2007). The techniques usually used to analyse geotechnical data, in order to obtain an estimation of its uncertainty, include traditional statistical methods and spatial statistical methods (such as random field theory (Vanmarcke, 1983) and geostatistics (Chilès and Delfiner, 1999)). Numerous research programmes have been carried out: for example, Fenton and Griffiths (2002) and Nour et al. (2002) analysed foundation settlements on spatially random soil; Elachachi et al. (2004) studied soil–pipe interactions, Low (2005), Fenton et al. (2005) and Zevgolis and Bourdeau (2010) analysed the reliability of a retaining wall. El-Ramly et al. (2005) and Srivastava and Sivakumar Babu, 2009 studied slope stability, and Niandou and Breysse (2007) carried out reliability analysis of a piled raft. In these types of study, either Monte-Carlo simulations or the

124

J. Dubost et al. / Engineering Geology 121 (2011) 123–134

stochastic finite element method are currently used to take soil uncertainties into account. In spite of such recent developments, these approaches are not commonly used by engineers, and very few cases exist in which they have been developed starting from the initial soil conditions, and taken through to the structural design. Reliability analyses are seldom performed to obtain the probability of exceeding certain design criteria. This paper presents a detailed uncertainty analysis which is carried out using two common design methods (the analytical homogenisation method and the finite element method) combined with Monte Carlo simulations to evaluate the influence of each parameter on the overall response of the reinforcement ground and the probability to exceeding design criteria. Data, from a real case of a freight railway platform, showing significant settlement effects despite the use of soil reinforcement, are used to obtain probability distributions of soils properties and geometrical parameters of soil layers and granular columns. Most of them are obtained from soil data investigations, others from engineering judgement. Results allow to investigate whether the use of a statistical–probabilistic method to compute the probability of exceeding a given settlement or differential settlement value is relevant to the analysis of a practical geotechnical problem such as the ground improvement by columnar inclusions. The paper framework is the following: Section 2 presents the soil reinforcement method and the two design methods used, Section 3 presents the uncertainty modelling and the soil investigation from a case example and Section 4 is devoted to the analyses and discussion of the different analytical and numerical simulations. 2. Ground reinforcement by soft columnar inclusions The dynamic substitution or replacement method is a technique derived from the dynamic compaction and stone column techniques (Kruger et al., 1980; Gambin, 1984), which involve the creation of large granular columns (DR pillars) approximately 2.5 m in diameter, driven to a maximum depth of 5–6 m in order to transfer structural loads to a suitable bearing stratum (Figure 1). Such granular piles are designed for embankment stabilisation and the reduction of settlement effects. The settlement calculation of a soil reinforced by granular pillars can be made analytically, using the ‘homogenisation method’ (Liausu, 1984; Cognon et al., 1991; COPREC and SOFFONS, 2005), which first requires the calculation of the equivalent modulus (E) of the reinforced soil (Eq. (1)), followed by the use of elasticity theory (Eq. (2)). E=

EC :AC + ES :AS AC + AS

ð1Þ

and s=

q:HS E

ð2Þ

where EC and ES are respectively the pillar and compressible soil moduli, AC is the area of a pillar and AS its area of influence, q is the distributed load applied to the pillar, and HS is the thickness of the compressible soil (HS = HC with HC is the length of a pillar in the compressible layer) (Figure 2). Note that the bearing layer is considered to be totally incompressible. Eqs. (1) and (2) can also be written as follows: s=

q:HS a:EC + ð1−aÞ:ES

ð3Þ

with: a=

AC A

ð4Þ

where a is the substitution ratio. Cognon et al. (1991) showed that this method overestimates settlement, by about 35% in the case of soil supporting a large petrol tank, if an elastic modulus (for ES) is used in Eq. (1). Having compared the predicted outcome with observed settlements after construction, these authors proposed to directly use the oedometer modulus in expression 1 (in the following, we assume ES = Eoed). This choice was later confirmed by Dhouib and Blondeau (2005). Moreover, improvements in compressible soil properties between pillars are possible, depending on the nature of the soil (Liausu, 1984; Emrem et al., 2001). These were not taken into consideration in the design. An alternative method, based on the Finite Element Method (FEM), is more and more commonly used, because the problem in hand can then be considered in a more versatile manner than in the case of analytical calculations. Nevertheless, in current geotechnical engineering practise, FEM computations, in the case of the dynamic substitution method, are fairly straightforward. For simplicity, in this proposed approach only the elastic solution is considered. Moreover, Balaam and Booker (1985) suggested that since the settlement values obtained from elastic and elasto-plastic analysis differ very little, the relative effects of various change in a column-soil unit could be determined by means of 2D axi-symmetric elastic analysis. Settlement calculations were made using the Finite Element Method (software: CASTEM®), according to the standard procedure used by the geotechnical designer, for the settlement estimation of soil reinforced with granular pillars (Dubost et al., 2007b). A mesh

Fig. 1. Principle of the dynamic replacement pillar method.

J. Dubost et al. / Engineering Geology 121 (2011) 123–134

125

Fig. 2. Geometry of the problem — a) Layout of the pillars — b) Cross section.

density of 0.1 m with 4-node square elements was used, and the interface between soil and columns was considered to be perfect. The bearing layer of the columnar inclusions is taken into consideration with a modulus EIS. The load is applied as a uniform pressure at the top where compacted fillings can be considered. The boundary conditions for the mesh are shown on Fig. 3. 3. Uncertainty modelling Soil conditions often exhibit considerable uncertainty, in terms of their geotechnical parameters, whereas geotechnical designs are generally based on deterministic constitutive relationships. The use of appropriate uncertainties in the analysis of complex geotechnical systems is an important topic. For Parry (1996), it is essential to maintain the distinction between different categories of uncertainty. The most widely recognised distinction is that made between random

and epistemic uncertainties. The first of these arises from natural, unpredictable variations. Additional knowledge may be used to provide an improved characterisation of such a variability, but cannot reduce it. The terms ‘irreducible’, ‘inherent’ or ‘objective’ are also used to describe this type of uncertainty. An epistemic uncertainty is associated with incomplete, inaccurate, or total lack of knowledge. An increase in knowledge or information can reduce this type of uncertainty, also referred to as ‘ignorance’ or ‘subjective’ uncertainty. Because the same mathematical frame (probability theory) is used to characterise and quantify both types of uncertainty, they are often confused. In soil investigations used for geotechnical analysis and foundation design, it is well known that standard field data is often too limited to allow a real distinction to be made between random and epistemic uncertainties. New measurement technologies and new data sources with improved accuracy would be necessary. Moreover, the same

Fig. 3. Finite element modelling of a unit cell of reinforced ground — a) Used geometry — b) Example of computed results.

126

J. Dubost et al. / Engineering Geology 121 (2011) 123–134

source of uncertainty may be labelled as ‘random’ in one study, and ‘epistemic’ in another. The distinction is relevant, but not of primary importance for a practical geotechnical design. Model uncertainty is commonly regarded as being of the epistemic type, and of great importance. In general, a model is comprised of (more or less) simplified representations of selected aspects of the real system, such that the performance predicted by the model and the actual performance of the system often differ at certain levels. Inaccuracies can arise from limitations in the designer's knowledge of the phenomena and data, or from deliberate simplifications introduced into the mathematical model or with respect to the choice of probability distributions. As approximations in the model represent one of the sources of uncertainty, the designer should provide a means of evaluating this uncertainty. To this intent, the differences between the model and the real system should be quantified, in terms of the mean and variance of the stochastic errors. For the present investigation, the limited number of practical cases under study makes this procedure very difficult. An alternative method would be to perform a pertinent sensitivity analysis, followed by an uncertainty analysis. Usually, a sensitivity analysis is used to identify those sources of uncertainty which have the greatest influence on the outputs, and to quantify the model's behaviour in response to changes at its input. This approach is of fundamental importance, to ensure correct use of the two models (analytical and numerical) evaluated in our study. Firstly, these uncertainties need to be identified and modelled (Emeriault et al., 2004), before being incorporated into an engineering design (Houy et al., 2005; Dubost et al., 2007a; Lacasse and Nadim, 2007). Here, the spatial dependence of this variability is not taken into account, despite its likely influence on geotechnical designs and performance. This point will be discussed in the last section of this paper. The variables under study are: ES and EC, the soil and column moduli; HS and HC, the soil thickness and the column length; RC and RS, the radius of the column and of the improved soil; and q, the load. Each of the random variables has to be represented by its first two order moments (mean value and standard deviation or coefficient of variation) and modelled using different probability distribution functions. Statistical analysis of a real case, which is a railway platform suffered damage from large settlement effects, despite the use of the replacement technique to reinforce weak soils, is performed to obtain from soil data investigation, expert judgement and previous studies, the different probability distributions modelling the random variables in hand. 3.1. Case study The French railway company needed to build an important, new 110 000 m 2 freight platform near to Bordeaux (France). Available land was located, approximately 1 km from the Garonne River, on its West bank. The first zone selected for this project has a surface area of more than 400 000 m². 3.1.1. Geological setting The city of Bordeaux is located on the North West border of the Aquitaine sedimentary basin (France). The geological formations which can be encountered in the greater Bordeaux area, within the first fifteen metres below the surface, are composed of Quaternary alluviums and Cenozoïc deposits, which form a limestone substratum (Bourgine et al., 2006). Several phases of erosion and deposition have led to the formation of seven staircase terraces, the basis of which ranges in altitude between 22 m below sea level and 65 m above sea level. These are composed of sands and gravels with more or less clay. Finally, during the Flandrian (early Holocene) period, recent alluviums filled the alluvial river plain (Figure 4).

It should be pointed out that the Flandrian alluviums (Fyb) are very heterogeneous and can have a thickness of more than 15 m. Dubreuilh (1976) described them as being composed of clays, peaty clays and peat, including more or less sand or sandy clay layers, thus explaining that they can be mistakenly attributed to the terrace formations, because they contain similar materials. For this reason, the interpretation of boring samples is often difficult. 3.1.2. Geotechnical investigations Three different geotechnical site investigation programmes were carried out in 1992, 1994 and 1999; more than 240 boreholes were drilled. They highlighted the presence of peat and very compressible mud–clays with a thickness of several metres. The engineering report distinguished three soil layers over the substratum: – recent embankments covering approximately 80% of the site, made mainly of waste material, since the area had been used as a waste disposal site. Their thickness varies between 2 and 3 m. These layers had to be partially removed before construction. – from the South to the North, 1 to 5 m of compressible, highly heterogeneous soils (Flandrian alluviums). These soils are made of silty-clays and more or less organic mud–clays, giving way to peat in the southern third of the site. – a sandy layer (terrace formation), capping a marly-limestone substratum, which is assumed to be incompressible. 3.1.3. Properties of the compressible layer (Flandrian alluvium) Of the 240 boreholes drilled at the site, 108 reach the compressible layer (Flandrium alluvium) and only 79 traverse this layer completely. From these boreholes, a large number of laboratory tests were carried out: water content, unit weight, Atterberg limits and oedometric tests. The results of these tests are presented in Table 1. The values obtained for the liquid limit LL, the plasticity index PI and the liquidity index LI confirm the plastic, to very plastic, nature of the Flandrian alluvia. The maximum values for these indices indicate the presence of clayey to peaty soils on the study site. With a mean value of 1.11, and minimum and maximum values of respectively 0.2 and 3.92, the compression index confirms the compressible characteristics of this layer, as well as its heterogeneity. The latter characteristic is also confirmed by the unit weight values obtained for soil samples in this layer. The water content is a parameter, which can provide information about the nature of soils, i.e. their physical and mechanical characteristics. On the study site, it is the most commonly measured parameter, with 603 measurements distributed over the 108 boreholes which reached the compressible layer. The water content is a parameter which is well correlated with the liquid limit, the plasticity index and the compression index, and which is inversely correlated with the unit weight. For the studied site, we determined correlation coefficients of −0.92 with the unit weight γ, 0.95 with the liquidity limit Wl, 0.94 with the plasticity index Ip, and 0.98 with the compression index Cc. The latter relation is illustrated in Fig. 5. It should be noted that these correlations confirm the association of the strongest water content values with the most unfavourable mechanical characteristics, as indicated by the highest values of compression index. In order to estimate the compression, using the two methods described above (see Section 2), two characteristics of the compressible layer must be determined from geotechnical surveys and laboratory tests: the thickness of the compressible layer (Hs), and the oedometric modulus (Eoed) of this layer. 3.1.3.1. Estimation of the thickness Hs of the compressible layer. For the entire site, we have only 79 boreholes which fully traverse the compressible layer, reaching the sandy–gravely alluvial terrace and the calcareous substratum. We can thus determine the altitude of the

J. Dubost et al. / Engineering Geology 121 (2011) 123–134

127

Fig. 4. Synthetic W–E geological cross-section of Greater Bordeaux.

base of the Flandrian alluvium and its thickness, at 79 points. Fig. 6a and b present the spatial distribution of these two quantities, obtained using interpolation methods (Dubost et al., 2007b). The altitude of the base of the compressible layer is highly variable over the study site; it should however be noted that this layer tends to become deeper towards the north-east (Figure 6a). It can also be observed that its altitude is greater at the north-west extremity and to the south, which is in agreement with the information indicated on the geological map of this area, in which the Terrace is shown to be sub-outcropping in these zones. Beneath the platform, the altitude of the Flandrian/terrace interface varies irregularly, by several metres. Over the entire site, the thickness of the compressible layer is highly variable (Figure 6b). It can also be seen that the thicknesses increase markedly towards the north-east of the site, and that an additional thickness is present to the north of the platform. Below the remainder of the platform, the thickness of the compressible layer varies between 1 and 5 m. In the longitudinal direction of the platform (north/south), along the rails, it can be seen that 2–3 m thick zones of compressible soil alternate with zones with a thickness of 3–5 m. The thickness of the compressible soil beneath the platform is 3.1 m on average, with a standard deviation of 1.4 m. 3.1.3.2. Estimation of the oedometric modulus (Eoed = Es). The oedometric modulus is determined by combining the Terzaghi expression, given in Eq. (5), with the linear expression which relates the oedometric modulus, the soil deformation and the constraint q induced by the loading, as given in Eq. (6). Δh =

Hs :Cc 1 + e0


lg

 σ00 + q 0 σ0

Eoed =

Δh

q 

ð6Þ

Hs

where Δh is the settlement of the compressible layer, according to Terzaghi's consolidation theory (Cassan, 1978), e0 is the void ratio, σ′0 is the pre-consolidation pressure (also equal to the in situ stress, for a normally consolidated soil), Cc is the compression index and Hs is the thickness of the compressible soil. Over the full extent of the site, we have 21 core drillings from which 41 samples of intact soil were taken, for oedometric testing. For each of these soundings, we know: the thickness of the compressible layer Hs, the void ratio e0, the compression index Cc and the in situ stress σ′0. For the whole site, the compressible layer is considered to have been normally consolidated. The oedometric tests carried out on samples taken from the 21 core drillings allowed us to determine 21 values of oedometric modulus, using Eqs. (5) and (6), with a mean value of 660 kPa and a standard deviation of 393 kPa. This strong variability of the oedometric modulus can be attributed to the greater significance of high or low values, when only small quantities of data are present. In an attempt to improve the estimation of the oedometric modulus over the site, we proposed to use the linear relationship observed between the water content and the compression index (Figure 5). The linear relationship between these two parameters thus allowed us to determine more than 600 values of

ð5Þ

Table 1 Summary of laboratory test results relating to the compressible layer.

Number of values Mean Maximum Minimum Standard deviation

W%

γ (kN/m3)

LL(%)

PI(%)

LI(%)

CC

603 71.96 376.0 5.00 57.82

150 16.08 21.4 10.2 2.32

39 87.03 280 16 40.97

38 49.87 182 9 28.79

36 0.61 1.4 0.2 0.29

41 1.11 3.93 0.2 0.95

Fig. 5. Correlation between the water content W and the compression index Cc.

128

J. Dubost et al. / Engineering Geology 121 (2011) 123–134

Fig. 6. a) Altitude of the Flandrian alluvium bottom — b) Thickness of the compressible layer.

compression index Cc over the whole site. By using the same methodology as in the case of 21 core drillings to calculate the oedometric modulus, we obtained 108 values for this parameter. The mean of these 108 values is 573 kPa, with a standard deviation of 143 kPa. Table 2 shows the results obtained, following interpretation of the geotechnical survey, for the estimation of the thickness and the oedometric modulus of the compressible layer. It can be noticed that the statistical parameters describing the oedometric modulus, over similar areas, are very close for the two methods. In addition, the standard deviation found with the second method is considerably smaller than that obtained with the first method, as a consequence of the considerably greater number of samples in the latter case.

3.1.4. Projected platform and observed settlements after construction The projected platform was a 2 m thick, wide embankment, with a surface area of 110 000 m² and a unit weight of 22 kg/m 3, inducing a load of about 45 kPa. The final, one-dimensional settlement calculations were then performed based on the thickness of the compressible soil and the oedometric properties of the soil layers. The predicted settlement values vary from 5 cm to 50 cm (Figure 7), and it can be seen that there are only three zones, corresponding to three boreholes, where settlement values are of real significance. Because rolling cranes, used to move heavy containers, require the total and differential settlement values of their railway tracks to be very small, the dynamic replacement technique was chosen to reinforce the compressible soils over the platform area. Following the conclusions of the engineering report, approximately 2500 2.5 m diameter pillars were built on the site, with a 6 × 6 m² grid spacing. The FEM model predicted maximum settlement values of about 2 cm, and differential settlements of less than 2 cm per 10 m (i.e. an angular distortion of 1/500). Table 2 Thickness and oedometric modulus of the compressible layer.

Number of values Mean Maximum Minimum Standard deviation a b

Hs (m)

Eoeda(kPa)

Eoedb(kPa)

79 3.1 7.1 0.45 1.4

21 662 1450 170 393

108 573 1200 210 143

Determined using oedometric tests. Determined from water content values.

Fig. 7. Initial settlement calculation: interpolation map.

The railway platform was levelled one year after the ground reinforcement work. Fig. 8a presents an interpolated map of the measurements, showing two zones in which significant settlement had occurred (Figure 8a and b): – the northwest of the platform, where a surface area of approximately 8000 m² suffered the greatest settlements (maximum settlement of 23 cm). – the central zone, which is also the widest area (approximately 10 500 m²), where significant, but lower settlement values occurred (8 cm maximum). This shows that the observed settlement differs strongly from the initial estimations. Extensive checking of the stone pillars was carried out, showing that no error had been made during construction. If it can be assumed that the observed settlement is a random variable independent of its spatial location, the probability to exceeding design criteria is about 65%, (Figure 8c). 3.2. Data set from case example With Monte-Carlo simulations, deterministic calculations (homogeneous and finite element methods) are implemented, for different sets of input parameter values. Each calculation, for which the input data of the different variables is simulated using the assumed distributions, leads to one outcome: the settlement value in the present case. A large number of deterministic calculations will result in a large number of settlement values, from which the mean, standard deviation and coefficient of variation can be calculated. Firstly, each parameter (columnar inclusion geometry, soil and inclusion modulus, and compressible layer thickness) must be modelled with a probability distribution. 3.2.1. Data from the geotechnical report (Dataset 1) The statistical distributions of HS and ES are assessed from geotechnical data (see Section 3.1.3), to which theoretical distributions can be fitted, in order to model their uncertainty. Fig. 9a shows the cumulative modulus distribution for normally consolidated compressible soil, and Fig. 9b shows its thickness. A lognormal distribution provides the best fit for the oedometer modulus Eoed (which is considered to be the soil modulus ES). Because the

J. Dubost et al. / Engineering Geology 121 (2011) 123–134

129

Fig. 8. Observed settlement 7 years after construction.

distribution for compressible soil thickness cannot be very well fitted to common statistical distributions, a normal score transformation was performed (Deutsch and Journel, 1992) (Figure 9b). Before the soil reinforcement work began, unsuitable filling material was partly removed from the site, leaving a 1 m thick layer, which was then compacted until its modulus reached 50 MPa. Pillars were driven through this compacted layer. The pillar lengths (HC) are considered to be equal to the soil thickness (HS), although their length is bounded to a theoretical maximum of 5 m (1 m in the embankment layer + 4 m in the compressible layer): – if the compressible soil thickness is less than 4 m, the pillar length and soil thickness have the same values, i.e. HC = HS if HS ≤ 4 m; – if the compressible soil thickness is greater than 4 m, the pillar length is bounded by its maximum theoretical length, even though it has not reached the bearing layer, i.e. HC = 4 m if HS N 4 m.

Fig. 9. Characteristics of the compressible soil — a) Oedometer modulus probability distribution — b) Compressible soil thickness probability distribution.

The above assumptions are close to the reality: compressible soils were found underneath the basis of the pillars after their construction, in some places where there was a local increase in compressible soil thickness. The parameters which could not be identified by soil investigations are the geometrical parameters (RC and RS), the load (q) and the column modulus (EC). The mean values retained in the engineering report were used (Table 3 — Dataset 1). The pillar radius RC is usually taken to be equal to the drop-weight radius, of about 1.25 m; this is the value retained for the present study. We also assumed that the distance between two neighbouring pillars (i.e. the soil radius RS used for design calculations), the column modulus and the platform load could vary. They have thus been modelled with Gaussian distributions, and appropriate coefficients of variation are chosen in order to model their uncertainty: CV of 10% for RS, RC an EC; and CV of 5% for q. (Table 3 — Dataset 1).

130

J. Dubost et al. / Engineering Geology 121 (2011) 123–134

Table 3 Statistical characteristics of parameters: datasets 1 and 2. Parameters DATASET 1–Initial values

DATASET 2–Modified values

a

Soil investigations Engineering report Distribution function Mean CV

q

RS

RC

HC

HS

Origin

Soil investigations Expert judgement Distribution function Mean CV

EC

x x Gaussian 45 kPa 5%

x Gaussian 3.4 m 10%

x Gaussian 1,25 m 10%

x Gaussian 45 kPa 5%

x Gaussian 3.4 m 10%

x Gaussian 1,00 m 10%

–HC = HS a –

Origin

Normal score transformation 3.1 m 45% x

– HC = HS a –

Normal score transformation 3.1 m 45%

ES x

x Gaussian 55 MPa 10% x x Beta 40 MPa 60%

Log normal 0.6 MPa 25% x Log normal 0.6 MPa 25%

HC = HS if Hs ≤ 4 m or HC = 4 m if HS N 4 m.

3.2.2. Data modified on the basis of expert judgement (Dataset 2) Many test results, taken from various studies, reveal the high variability of the stiffness in the pillars (EC), with a coefficient of variation as high as 80% (Kruger et al., 1980; Liausu, 1984; Emrem et al., 2001; Marcu, 2004). A specific analysis was thus made, using pressuremeter tests made inside the columns of the site. Roughly 87 tests were made at 22 pillar locations, providing values for the Menard pressuremeter modulus (EMC) in the columns. The outcome of this analysis shows that substantial differences in modulus exist between different columns, and inside any given column (depending on the depth at which the test is realised). We assume that EMC ranges between the extremes of the measured values. The probability distribution appropriate for this type of bounded variable is the Beta distribution. Fig. 10 shows the experimental data distribution for EMC, with a mean value of approximately 13.5 MPa, and the fitted theoretical Beta distribution, based on a standard deviation equal to 8 MPa and measured extreme values of 2 MPa and 50 MPa. The elastic modulus of the columns (EC) can be assessed from pressuremeter moduli, using common correlations (Poisson ratio of 0.33 and α Menard structure coefficient of 0.25). A mean value for EC of about 40 MPa, and a standard deviation of 24 MPa were determined, which are considerably lower than the previously assumed initial value of 55 MPa. The mean of EC was reduced, its coefficient of variation increased, and its distribution was bounded between 6 MPa and 120 MPa in order to avoid unrealistic extreme values. Similarly, as the periphery of the columns can be polluted by the surrounding soils, Kruger et al. (1980) recommend considering only 80% of the pillar radius. We followed this recommendation, and retained a column radius of one metre in dataset 2 (Table 3). All of the parameters used in the following sections are summarised in Table 3, with their characteristics, their origin, and the statistical inference used for modelling their uncertainties. Two sets of parameters can be distinguished: dataset 1 is based on the

statistical analysis of initially available parameters (data from engineering reports), whereas dataset 2 takes all the available knowledge into account (modified data from expert judgement and a review of the literature). 4. Results and discussion Soil properties uncertainties and geometrical parameters can now be used in a geotechnical design. Their influence on the estimation of settlement values, for the case of the site studied here, can be analysed in order to assess the probability of the design not achieving the required performance. The two models used in this paper (analytical and FEM methods) are now associated with Monte-Carlo simulations (Fishman, 1995). Each parameter (material modulus, model geometry or position of the selected layers) is modelled with a probability distribution as specified above. Following the recommendations of Fishman (1995), 2400 values are needed to obtain accurate results. It should be noted that all parameters are assumed to be uncorrelated. From the Monte-Carlo simulations, two types of calculation can be performed: a global calculation, in which all of the parameters are simulated from the assumed distributions, or a local calculation, in which only one parameter is simulated from its assumed distribution, and the other parameters have a deterministic value, i.e. their mean. The latter method is also referred to as a sensitivity analysis. These two types of calculation are used in the following, and are presented in Tables 4 and 5, which provide the statistical parameters of the settlement effect, when either all parameters, or only one of the parameters, is simulated. 4.1. Analytical settlement calculations The analytical homogenisation method is often used, and provides good settlement estimations for the case of a uniform load, such as a platform. However, significant differences can arise if inappropriate parameters are used. As the pillar shapes are highly uncertain, their length can differ from the thickness of the compressible soil: the latter can be thicker, or the pillars shorter, according to soil conditions (i.e. HS ≥ HC). In such a case, the substitution ratio a (Eq. (4)) decreases. Conversely, lateral expansion of a pillar during pile-driving can lead to a more voluminous pillar (an increase in its radius), thereby producing an increase in this ratio. In order to correctly account for the uncertainty of the geometrical parameters (RC, RS, HC, HS), the substitution ratio (a), which is a surface ratio as defined in Eq. (4), is modified and expressed as a volume ratio (av), written as: 2

av = Fig. 10. Probability distribution of the pressuremeter modulus EMC.

VC R :H = C2 C V RS :HS

ð7Þ

J. Dubost et al. / Engineering Geology 121 (2011) 123–134

131

Table 4 Settlement results from the homogenisation method. Observed settlements

Initial parameters (Dataset 1) Modified parameters (Dataset 2) a b

Mean = 3.34 cm CV = 105%

Assessed settlements values (mean and CV) All parameters

RC

RS

HSa

HCa

HS = HC b

EC

ES

q

1.83 cm 75.49% 4.99 cm 96.83%

1.57 cm 22.40% 3.28 cm 19.50%

1.54 cm 22.11% 3.25 cm 18.73%

1.77 cm 84.33% 3.66 cm 80.36%

1.86 cm 43.37% 3.80 cm 37.15%

1.69 cm 61.69% 3.53 cm 59.68%

1.59 cm 21.10% 4.43 cm 59.61%

1.52 cm 1.28% 3.21 cm 2.93%

1.52 cm 4.92% 3.21 cm 4.92%

HS and HC are independent. HC = HS if HS ≤ 4 m or HC = 4 m if HS N 4 m.

to a measured value of 3.34 cm. The parameters of dataset 2 lead to a 75% probability for the settlements exceeding the maximum allowable value of 2 cm, and a 10% probability for them to exceed 10 cm (Figure 11). These probabilities are larger than those obtained from observed settlements after construction.

Firstly, it can be of interest to identify, by testing each parameter independently, those parameters which make the strongest contribution to the scatter in settlement values. This approach also reveals the degree of model uncertainty. The geometrical parameters (radii RC and RS, and thickness HS) have the greatest influence (CV of 22% and 62% respectively) (Table 4 — Dataset 1) on settlement scatter, whereas variations in compressible soil thickness lead to the highest settlement values (m = 1.77 cm and CV = 84%, when only HS varies). By varying only one of the two parameters HS or HC, which allow for the fact that the shape of the columns can deviate from the expected outcome, significant settlement values and large coefficients of variation are found. This justifies the use of expression (7), which more accurately takes the influence of the substitution ratio (av) into account, and shows that the pillars may in some cases not reach the bearing layer. The pillar modulus (EC) also influences the settlement effect, with a CV of about 21%. The uncertainties in the soil modulus (ES) and the applied load (q) have less influence on the outcome of the calculations (CV of 1% and 5% respectively). When all parameters are taken into account, the settlement results derived using dataset 1 (Table 4) show that although the computed mean settlement value (equal to 1.83 cm) is underestimated, when compared with the mean of the measured values (3.34 cm) (Figure 8c), it is close to the value of 2 cm calculated by the geotechnical designer. A coefficient of variation of 75% is obtained, which is lower than the value of 105% found with the in situ measurements. Nevertheless, our simulations (Figure 11) predicted there was a 25% probability of the settlements exceeding 2 cm (the maximum allowable value). This is an initial indication of the possibility of undesirable settlement effects being reached. The use of the second dataset confirmed these results: the influence of soil thickness (HS), as well as that of the column modulus (EC), is also highlighted. A reduction in the value of EC and RC leads to significantly larger settlement values: between 1.6 cm and 4.4 cm, and between 1.6 cm and 3.3 cm respectively, with the coefficients of variation being approximately 60% and 20% respectively (Table 4 — Dataset 2). The other parameters (ES, RS, HC, q) have approximately the same influence as that found for dataset 1. As a consequence of the high variability of EC (CV = 60% in dataset 2), when all parameters are allowed to vary the settlement variability increases significantly, to CV = 96.8%. This value is close to the observed in situ variability. Nevertheless, under these conditions, the settlements are overestimated on average: a value of 4.99 cm is computed, as opposed

4.2. FEM settlement calculations The finite element modelling is also combined with Monte-Carlo simulations, in order to include the effects of uncertainties in the soil properties, soil layer geometry and geometrical characteristics of columnar inclusions. The same datasets used for the previous analytical simulations (RC, RS, EC, ES, HC, HS) were taken for these computations (Table 3 — Datasets 1 and 2). In the soil investigation report, the terrace formation, with a modulus EIS = 50 MPa, was initially taken to be the bearing layer of the pillars, and the column length was kept equal to the soil thickness, down to a maximum depth of 5 m (1 m through the filling layer, and 4 m through the compressible layer). A sensitivity study was carried out first, in order to highlight the most influential parameters and to quantify the model's uncertainty (Figure 12). It can be seen that a reduction in the pillar moduli or radii, below their initial mean values (respectively EC = 50 MPa and RC = 1.25 m), leads to an increase in the settlement values, whereas an increase in these parameters does not reduce the settlement in the same proportion (Figure 12a and b). Fig. 12c and d illustrate the influence, on settlement values, of the compressible soil thickness and the soil modulus (ES) below the pillar base (assuming there is 1 m of compressible soil at this level). It is important to note that the presence of a one metre layer of compressible material (of 1 MPa modulus) underneath the base of a pillar multiplies the settlement at the top of the column by a factor of four (Figure 12c). The results of the numerical Monte Carlo simulations made using datasets 1 and 2 are presented in Table 5. When all parameters are taken into account, the cumulative frequency distribution functions of the settlement values can be computed (Figure 13). Both datasets highlight the sensitivity of the ground reinforcement technique to uncertainties in soil properties and geometrical parameters. Uncertainties in the column properties (RC, EC) have a significant influence, leading to an increase in settlement and its variability. As an example, a 25% reduction in column modulus multiplies the average settlement value by a factor of 2.5 (from 1.9 cm in dataset 1 to 4.7 cm

Table 5 Results of FEM settlement computations. Parameters

RC EC ES HC , HS a All parameters a

Settlement values from Dataset 1

Settlement values from Dataset 2

Mean

Max.

Min.

CV

Mean

Max.

Min.

CV

1.88 cm 1.87 cm 1.84 cm 3.54 cm 3.69 cm

3.93 cm 2.96 cm 2.00 cm 22.81 cm 41.70 cm

1.05 cm 1.21 cm 1.62 cm 0.00 cm 0.0 cm

25.0% 19.6% 4.91% 165.3% 170.2%

3.72 cm 4.75 cm 3.67 cm 4.93 cm 5.73 cm

5.97 cm 11.36 cm 4.24 cm 24.85 cm 61.23 cm

2.35 cm 1.84 cm 2.88 cm 0.00 cm 0.00 cm

19.87% 46.01% 8.11% 125.5% 127.09%

HC = HS if HS ≤ 4 m or HC = 4 m if HS N 4 m.

132

J. Dubost et al. / Engineering Geology 121 (2011) 123–134

Fig. 11. Probability distributions of assessed settlements (homogenisation method-unit cell).

in dataset 2). Its variability is also significantly increased, from 20% to 46% (Table 5). The variability of the applied load (q) and the compressible soil modulus (ES) leads to the same conclusions as those found with analytical calculations: q and ES have a small influence on settlement values and variability. Nevertheless, ES becomes significant when HS N HC. In this case, it becomes responsible for a large proportion of the settlement effects and their resulting variability (165% and 125% for the two datasets). The influence of the compressible layer under the pillar base can be seen to be very significant: when such a layer is present, and all parameters can vary, it can lead to an increase of up to 40 cm in settlement values, and to coefficients of variation of about 170% and 127%. The bearing soil modulus (ESI) has only a small influence on settlement. However, the initial value assumed for this parameter was high, and the real properties of this layer would merit closer analysis.

Fig. 13. FEM settlements resulting from variations in all parameters.

Both datasets show that there is a high probability for unwanted settlement values to occur (Table 5 — all parameters; Figure 13). Using the initial parameter values, there is a probability of about 25% that the settlements can exceed 2 cm, and of about 12% that they can exceed 10 cm. When the parameters in dataset 2 are used, the probability of these settlements exceeding 2 cm becomes greater than 60%, and the probability of them exceeding 10 cm is approximately 18%. 4.3. Differential settlement analysis In many cases, the effect of critical importance is not total settlement, but the differential settlement effects, which in the case studied here are responsible for relative movements between two adjacent tracks. If it is assumed that the differential settlement Δu is normally distributed, and that the two displacements are

Fig. 12. Influence of geotechnical parameters on FEM settlement results — a) Influence of pillar modulus (EC) — b) Influence of pillar radius (RC) — c) Influence of compressible soil thickness (HS) — d) Influence of soil modulus below the pillar basis (ES).

J. Dubost et al. / Engineering Geology 121 (2011) 123–134

uncorrelated, the two first moments of the absolute differential settlement are (Frantziskonis and Denis, 2003):
E½jΔuj =

V ½jΔuj =

2 :V ½Δu π

1 = 2


 2 :V ½Δu 1− π

ð8aÞ

ð8bÞ

However, from these two first moments, we cannot estimate the probability of exceeding a given design criteria, since the distribution is unknown. Moreover, a differential settlement should refer to a constant distance between two points. As an alternative, one can compute the angular distortion δ, which is the differential settlement (Δu) between two points divided by the horizontal distance between them (L). For the angular distortion, we obtain for the first two moments: E½δ = 0 V ½δ =

V ½Δu 2:V ½u = L2 L2

ð9aÞ ð9bÞ

The probability of unsatisfactory performance (Pr) is often estimated from the second moment reliability analysis index β, using the following equation, which for angular distortion becomes: Prðjδj N δMax Þ = 2:ð1–Φ ðβÞÞ

ð10Þ

with β=

δMax :L pffiffiffi V ½u1 = 2 : 2

where Φ is the distribution of the standard normal variant and δMax is the angular distortion design criteria. In our case study, where δMax is assumed to be 1/500, for L = 10 m and V[u] = 50.10 −4 m 2 (the average variance derived from the FEM calculations, using datasets 1 and 2 (Table 5)), we compute a probability of unsatisfactory performance equal to 0.84. 4.4. Specific remarks Firstly, the sensitivity analysis shows that ES is a secondary parameter, and that RC and RS have the same influence on both the homogenisation and the FEM methods. Following these parameters, it appears that the next two most significant sources of uncertainty in the model output are HS (HC = HS if HS ≤ 4 m), and EC. The possibility of the stone columns not reaching a bearing stratum can be accounted for in both the analytical and numerical calculations. If such a case is suspected, the FEM model is far more strongly influenced by this parameter than the homogenisation model. Moreover, model uncertainty should be less significant with the FEM method, in which the terrain geometry is represented more realistically. Sensitivity analysis from analytical and numerical calculations shows that the efficiency of the dynamic substitution technique depends on the knowledge of soil layer geometry and on the assessment of the pillars' moduli. The modulus of the compressible soil under consideration is low, and a slight increase in its value would significantly reduce the settlement variations (particularly when it is present below the pillar basis). Uncertainties in other parameters have less influence on the outcome. Such a probabilistic approach leads to a range of possible settlement values, and illustrates the role of uncertainties in settlement calculations. The classical homogenisation method needs to be modified, to take into consideration the volume of reinforced soil rather than its surface area (Eq. (7)). With the FEM model, a more complex geometry can be

133

considered and different characteristics can be attributed to the model components. These calculations give results which are distributed at the scale of a unit cell. They show that most of the settlement effects occur at the base of the pillars, and that lateral deformations of the pillars are not particularly relevant. In both calculation methods, predicted settlement results are close to the observed effects (Figure 14), but the largest values are overestimated. Nevertheless, settlement is probably still taking place over the platform, meaning that the computed results may be close to the final settlement values of the site. The uncertainties of the geotechnical conditions (geometrical and mechanical properties), as well as the workmanship applied in the construction of the stone columns (spacing between pillars in our case), have to be incorporated into both models. The pillar length cannot be directly controlled during construction; it is assessed indirectly from the volume of incorporated material in the compressible layer. It is important to correctly characterise the thickness of the compressible layer, since local deepening, due to a geological anomaly (such as local erosion of the Terrace formation), could induce a significant increase in settlement in such a zone. Moreover, the following general remarks can be made about the calculations: – The dynamic substitution process used to create the pillars can modify the soil properties under the base of the pillar. Special soil investigations should be carried out in order to characterise those properties. – Concerning the FEM model, the axi-symmetric mode does not take into consideration the arching effect, which may occur between two pillars (Chen and Martin, 2002); the FEM models could possibly be improved in this way. Afterwards, if sufficient data is available, spatial analysis of the soil variability can be carried out. A geostatistical approach, characterising a site's geological and geotechnical conditions, can be implemented in order to estimate the spatial variability of the geometry and properties of soil layers (Chilès and Blanchin, 1995; Marache et al., 2009a, 2009b). These results could also be associated with the homogenisation method and/or the FEM method, in order to map settlement values and the probability of exceeding design criteria. In summary, for the site under study, and if it can be assumed that the soil properties are homogeneous within the project area, the probability of unacceptable settlement has been evaluated: on average 60% of the computed settlement values are greater than the maximum permissible total of 2 cm. In terms of differential settlement, we find a probability of unsatisfactory performance equal to 0.84, which could be reduced if the spatial connectivity of the settlement were weighted. If these results had been made available to the platform design engineers at the beginning of the project, they would have understood that the chosen ground reinforcement technique was inappropriate, and suitable adaptations or new techniques could then have been implemented.

Fig. 14. Settlement probability distribution from site measurements and Dataset 2.

134

J. Dubost et al. / Engineering Geology 121 (2011) 123–134

5. Conclusions In this paper, the settlement of soft columnar inclusions was analysis following an uncertainty approach. Soils and granular piles properties, soil layers thicknesses and geometrical characteristics of granular piles were considered random variables following different probability distributions. Two common design methods, the analytical homogenisation method and the finite element method were used and the possibility of the granular pile not reaching a bearing stratum was taken in consideration. Then, they were combined with Monte Carlo simulations in order to find, from these different probability distributions, the probability of settlements exceeding a given value. Spatial variability was not included in this analysis and random variables were considered independent to each other. Statistical analysis of soil data of a real site, showing significant settlement effects despite the use of soil reinforcement, gives a more objective characterisation of the uncertainties involved. It appears that the modulus and the length of the granular piles are two most significant sources of uncertainty, whilst the soil modulus is a second parameter. The latter can become significant when the columnar inclusions are not reaching a bearing stratum. In this case the finite element analysis is far more strongly influenced than the homogenisation method. Moreover, model uncertainty should be less important for the finite element analysis where the geometrical characteristics of soil layers and columnar inclusions are represented more realistically. Finally, it can be showed that, when used in the initial stages of a geotechnical engineering project, a simple uncertainty approach can be highly useful in providing convincing risk evaluation. If necessary, a second level of analysis could be undertaken, whose goal would be to characterise the spatial correlation of geotechnical parameters, cross-correlations and joint probability distributions between the involved random variables. The analysis carried out clearly contributes towards a more rigorous form of geotechnical engineering analysis, where risks are evaluated instead of being ignored. References Alamgir, M., Miura, N., Poorooshasb, H.B., Madhav, M.R., 1996. Deformation analysis of soft ground reinforced by columnar inclusions. Computers and Geotechnics 18 (4), 267–290. Baecher, G.B., Christian, J.T., 2003. Reliability and statistics in geotechnical engineering. Wiley. Balaam, N.P., Booker, J.R., 1985. Effect of stone column yield on settlement of rigid foundation in stabilized clay. International Journal of Numerical and Analytics Methods in Geomechanics 9, 331–351. Bourgine, B., Dominique, S., Marache, A., Thierry, P., 2006. Tools and methods for constructing 3D geological models in the urban environment: the case of Bordeaux. 10th International Congress of the IAEG, paper 72, Nottingham, UK. Canetta, G., Nova, R.A., 1989. A numerical method for the analysis of ground improved by columnar inclusions. Computers and Geotechnics 7, 99–114. Cassan, M., 1978. Les essais in situ en mécanique des sols. Tomes 1 et 2. Eyrolles. Chen, C.Y., Martin, G.R., 2002. Soil-structure interaction for landside stabilizing piles. Computers and Geotechnics 29, 363–386. Chilès, J.P., Blanchin, R., 1995. Contribution of geostatistics to the control of the geological risk in civil-engineering projects: the example of the Channel Tunnel. In: Lemaire, M., Favre, J.L., Mébarki, A. (Eds.), Applications of statistics and probability — Civil Engineering Reliability and Risk Analysis, vol 2. A.A. Balkema, Rotterdam, Netherlands, pp. 1213–1219. Chilès, J.P., Delfiner, P., 1999. Geostatistics: modeling spatial uncertainty. Wiley series in Probability and statistic. Wiley. Cognon, J.M., Liausu, P., Juillie, Y., 1991. Comportement de réservoirs fondés sur sols améliorés. 10th European Conference on Soils Mechanics and Foundation, pp. 377–380. COPREC, SOFFONS, 2005. Recommandations sur la conception, le calcul, l'exécution et le contrôle des colonnes ballastées sous bâtiments et ouvrages sensibles au tassement. Revue Française de Géotechnique 11 (2), 3–16.

Det Norske Veritas, 2007. Recommended Practice for Statistical Representation of Soil Data. DNV-RP-C207, Hovik, Norway. April. Deutsch, C.V., Journel, A.G., 1992. GSLIB. Oxford University Press0-19-507392-4. Dhouib, A., Blondeau, F., 2005. Colonnes Ballastées. Presses de l'ENPC. Dubost, J., Denis, A., Chanson, M., La Borderie, C., 2007a. Analyse de désordres affectant un remblai sur sols renforcés. Apport de méthodes statistiques et de simulations numériques. Revue Européenne de Génie Civil 11 (4), 477–492. Dubost, J., Denis, A., Marache, A., Chanson, M., 2007b. Probabilistic FEM analysis of railway-platform settlements. In: Pande, G.N., Pietruszczak, S. (Eds.), International Symposium on Numerical Model in Geomechanics, NUMOG X. Taylor & Francis, Rhodes, Greece, pp. 525–530. Dubreuilh, J., 1976. Contribution à l'étude sédimentologique de système fluviatile Dordogne-Garonne dans la région bordelaise. Thèse de doctorat, Université Bordeaux 1, France. Elachachi, S.D., Breysse, D., Houy, L., 2004. Longitudinal variability of soils and structural response of sewer networks. Computers and Geotechnics 31, 625–641. El-Ramly, H., Morgenstern, N.R., Cruden, D.M., 2005. Probabilistic assessment of a cut slope in residual soil. Geotechnique 55 (1), 77–84. Emeriault, F., Breysse, D., Kastner, R., Denis, A., 2004. Geotechnical survey and mechanical parameters in urban soils: modelling soil variability and inferring representative values using the extension of Lyon subway lined as case study. Canadian Geotechnical Journal 41, 773–786. Emrem, C., Spaulding, C., Durguno, H.T., Varaksin, S., 2001. A case history for soil improvement against liquefaction, Carrefoursa Shopping Center, Izmir, Turkey. Proceeding of the 15th International Conference on Soil Mechanics and Geotechnical Engineering, Istanbul, Turkey, pp. 1737–1742. Fenton, G.A., Griffiths, D.V., 2002. Probabilistic foundation settlement on spatially random soil. Journal of Geotechnical and Geoenvironmental Engineering 128 (5), 381–390. Fenton, G.A., Griffiths, D.V., Williams, M.B., 2005. Reliability of traditional retaining wall design. Geotechnique 55 (1), 55–62. Fishman, G.S., 1995. Monte Carlo: Concepts, Algorithms, and Applications. Springer. Frantziskonis, G., Denis, A., 2003. Complementary entropy and wavelet analysis of drilling-ability data. Mathematical Geology 35 (1), 89–103. Gambin, M.P., 1984. Puits ballastés à la Seine sur Mer. Actes du Colloque International : Renforcement en place des sols et des roches. Paris, pp. 139–144. Greenwood, D.A., 1970. Mechanical improvement of soils below ground surface. International Conference on Ground Engineering, London, pp. 9–20. Houy, L., Breysse, D., Denis, A., 2005. Influence of soil heterogeneity on load redistribution and settlement of a hyperstatic three-support frame. Geotechnique 55 (2), 163–170. Kruger, J.J., Guyot, C., Morizot, J.C., 1980. The dynamic substitution method. Colloque International sur le Compactage, Paris, pp. 339–343. Lacasse, S., Nadim, F., 2007. Probabilistic geotechnical analyses for offshore facilities. Georisk 1 (1), 21–42. Liausu, P., 1984. Renforcement de couches de sol compressible par substitution dynamique. Actes du Colloque International : Renforcement en place des sols et des roches. Paris, vol. 1, pp. 151–155. Low, B.K., 2005. Reliability-based applied to retaining walls. Geotechnique 55 (1), 63–75. Marache, A., Dubost, J., Breysse, D., Denis, A., Dominique, S., 2009a. Understanding subsurface geological and geotechnical complexity at various scales in urban soils using a 3D model. Georisk 3 (4), 192–205. Marache, A., Breysse, D., Piette, C., Thierry, P., 2009b. Geological and geotechnical modelling at the city scale using statistical and geostatistical tools: the Pessac case. Engineering Geology 107, 67–76. Marcu, A., 2004. Réalisation d'un hypermarché dans une zone sismique, sur une décharge améliorée par plots et colonnes ballastées. In: Dhouib, A., Magnan, J.P., Mestat, P. (Eds.), International Symposium on Ground Improvement ASEP GI 2004, pp. 215–223. Paris. Niandou, H., Breysse, D., 2007. Reliability analysis of piled raft accounting for soil horizontal variability. Computers and Geotechnics 34, 71–80. Nour, A., Slimani, A., Laouami, N., 2002. Foundation settlement statistics via finite element analysis. Computers and Geotechnics 29, 641–672. Parry, G.W., 1996. The characterisation of uncertainty in probabilistic risk assessments of complex systems, Reliability Engineering and System Safety 54 (2), 119–126. Phoon, K.K., Kulhawy, F.H., 1999. Characterization of geotechnical variability. Canadian Geotechnical Journal 36, 612–624. Schweiger, H.F., Pande, G.N., 1986. Numerical analysis of stone column supported foundations. Computers and Geotechnics 2, 347–372. Shashu, J.T., Madhav, M.R., Hayashi, S., 2000. Analysis of soft ground-granular pilegranular mat system. Computers and Geotechnics 27, 45–62. Srivastava, A., Sivakumar Babu, G.L., 2009. Effect of soil variability on the bearing capacity of clay and in slope stability problems. Engineering Geology 108, 142–152. Vanmarcke, E.H., 1983. Random fields: analysis and synthesis. M.I.T. Press, Cambridge, Mass. Zevgolis, I.E., Bourdeau, P.L., 2010. Probabilistic analysis of retaining walls. Computers and Geotechnics 37, 359–373.