ABSTRACT:

Since decades vibro stone columns are used to improve the bearing capacity and the settlement behaviour of soft soils. The design is based on empirical or semi-analytical formulae, most of which using the unit cell approach. Such design procedures, e.g. the Priebe method, proved their reliability in many cases. Stone columns are also used to improve the ground below embankments constructed for infrastructural measures. In such cases the columns act predominantly in order to enhance the slope stability. To give a realistic picture of the actual situation the design of the stone column pattern needs to take into account the stress distribution between columns and soil. All approaches not considering the stress concentration in the columns gain results which are over-conservative and thus lead to uneconomical solutions.

In the paper the results of numerical analyses considering the spatial nature of the problem are presented. The results are compared with those of analytical approaches. In addition to the numerical analysis the results of long term monitoring of the stresses at a case history are presented. The knowledge of the stress distribution offers the possibility of a realistic approach for the calculation of the slope stability of embankments on improved ground.

1 VIBRO REPLACEMENT METHOD

Stone columns can be used to improve soft layers under dams and embankments in order to reduce the settlements, accelerate the consolidation process and increase the stability.

They are installed using either the vibro replacement or the vibro displacement process. Figure 1 depicts the different construction stages. More detailed descriptions of the equipment and the procedure itself can be found in Moseley & Priebe (1993) or Kirsch & Sondermann (2003).

Stone supply

Filling of material lock

Pene-tration

Displa-cement

Comple-tion

Stone supply

Filling of material lock

Pene-tration

Displa-cement

Comple-tion

Figure 1. Dry bottom feed vibro displacement method

Field Measurements and Numerical Analysis of the Stress Distribution below Stone Column Supported Embankmentsand their Stability

Fabian Kirsch Dipl.-Ing., Institute for Soil Mechanics and Foundation Engineering, TU Braunschweig Wolfgang Sondermann Dr.-Ing., Keller Grundbau GmbH, Offenbach

Int. Workshop on Geotechnics of Soft Soils-Theory and Practice. Vermeer, Schweiger, Karstunen & Cudny (eds.) 2003 VGE

Infrastructural measures are the most common field of application for stone column supported embankments (e.g. Sondermann (1996), Sondermann & Jebe (1996), Raju & Hoffmann (1996)). Usually the columns are placed in a regular pattern improving the weak layers below the embankment with vertical zones of improved soil.

2 CURRENT DESIGN PROCEDURES

2.1 Settlement Reduction

In order to assess the settlement reduction factor numerous analytical and semi-analytical approaches exist. One of the most common design procedures is the method developed by Priebe (1995). An example of his design is given below, where an embankment of 15 m height is being constructed for the dam close to a bridge abutment at a highway crossing in Kuala Lumpur. Figure 2 shows the geometry.

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Stone columns Ø1.1m @1.7m Stone columns Ø1.1m @2.1m

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± 0,0

Figure 2. Cross section of a stone column supported embankment

Adopting the Priebe method to this type of structure calls for several idealisations in order to calculate the stresses at the base of the embankment. In this case a total settlement of 192 cm was calculated for the centre of the dam.

2.2 Slope Stability

The calculation of the slope stability is usually performed using analytical procedures like the Bishop method. In order to use these analytical procedures some sort of homogenisation of shear parameters has to be adopted. Generally it is proposed to use a weighted average for the shear parameters of the unimproved soil and the stone column material. It seems however to be over-conservative to calculate an average on the basis of the area ratio of columns and soil AS/A since the columns concentrate the vertical stresses and therefore create a higher resistance against slope failure. Priebe suggests a weighting of the angle of inner friction and the cohesion by the stress distribution, which is a result of his analysis. Weighting the cohesion by the stress distribution has no physical justification but accounts for conservatism.

In figure 3 the results of stability analyses using different mean friction values are shown. Column 3 shows the result of an approach in which the shear parameters are weighted as an average of area ratio and Priebe's method. The subsoil was divided into 14 different soil compounds to compute the indicated factors of safety. One should emphasize, that the stress concentration factor calculated by the Priebe method cannot simply be taken at the dam base, but needs to be calculated for each and every layer. This stands because the stress distribution between column and soil is by no means constant along the column depth.

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Slope stability using analytical procedures

1 2 3

Average type Area-Ratio Priebe Average of 1 and 2

η 1,12 1,77 1,42

Figure 3. Factors of safety by analytical procedures

There is quite a scatter in the results and since none of the methods has thorough physical justification further investigation was deemed necessary.

3 STRESS DISTRIBUTION 3.1 Measurement

In order to gain more information about the stress distribution between column material and surrounding soil pressure cells can be installed. When applied at the dam base usually the measured stress distribution values n=σCol/σSoil vary between 2 and 3. The stress concentration is depending on various parameters such as the loading type (soft or rigid), the surcharge, the material parameters of column and soil and the geometrical dimensions.

Figure 4 shows the result of a measurement at another embankment site in Kuala Lumpur, which shows approximately the same conditions as the example above. The columns were installed using a square pattern at 2,2 m. The result of approx. n=2,6 compares well with the results of an in situ trial field reported by Gruber (1994). There values of n=2,8 were measured for the same column pattern and a surcharge of 120 kN/m².

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ss d

istr

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Figure 4. Measurement of stress concentration below an embankment

3.2 Numerical Simulation In order to calculate the stress distribution and to give an assessment on the slope stability

within the same model a numerical analysis of the spatial problem is advantageous. The following section shows the approach, in which the finite element analysis can be used to design embankments such as the example above.

According to the column and embankment geometry calculations can make use of several planes of symmetry. Modelling of the ground was done by employing the Drucker-Prager yield criterion using a non associated flow rule. Division of the continuum was done using isoparametric brick elements with square interpolation functions and three degrees of freedom at each node. Obviously it is important to perform a 3-D analysis in order to model the stress distribution correctly.

The chosen constitutive law is of a linear-elastic ideal-plastic type, but cannot take care for hardening. To overcome these restrictions a cap model was proposed by Vittinghoff, Plaßmann & Schmitt (1997). In this model a multi-surface Drucker-Prager criterion with isotropic strain hardening is combined with a hydrostatic cap, which accounts for volumetric hardening (figure 5). Unfortunately however the cap parameters could not be determined for the example presented here. Therefore some care must be taken when interpreting the results of the numerical simulations.

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'1σ

'3σ

residual strength

elastichardeningsoftening

peak strength'2σ

'1σ

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residual strength

elastichardeningsoftening

peak strength

Figure 5. Constitutive model

Figure 6 shows the results of a simplified FEM analysis in which the typical stress distribution between columns and soil can be seen. The calculated stress concentration results in n=3,1.

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column no.

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Figure 6. Numerical analysis of the stress distribution below a simplified embankment

4 SLOPE STABILITY

4.1 Numerical Simulation The calculation of safety factors in finite element analyses is not a straight forward procedure.

From the practical viewpoint a method, in which the strength parameters of the soil are reduced stepwise until failure, can give a reasonable assessment of the stability problem (e.g. Cai & Ugai (1999), Naylor (1999)). The calculation of a factor of safety is done by the use of the Fellenius rule:

actual shear strenghtnecessary shear strengh

η =

The strength parameters tan ϕ' and c' are reduced in parallel till system failure occurs. Failure can be observed by monitoring the displacement of several key points. When the shear reduction-displacement graph shows a horizontal asymptote failure is about to begin (figure 7).

One of the shortcomings of this ϕ-c-reduction method is, that the influence of ϕ and c remains constant during the analysis. Thus possible changes in the failure mechanisms during the analysis are not included. The results however reveal valuable information about the stability and the failure mechanisms of embankments on stone column improved ground.

Figure 7 shows the results of the analysis for the embankment introduced in chapter 2. The finite element analysis yields a settlement of 240 cm at the centre of the dam. The following stepwise reduction of the shear parameters revealed a factor of safety of 1,36.

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Countours of horizontal displacementand deformed model

Figure 7. Numerical model and results

Focus should be given to the fact, that there is obviously a second failure mechanism below the berm at the embankment toe. This was not included in the analytical analysis and is probably one reason why the safety factor in the numerical simulation is smaller than in the analytical procedures.

4.2 Comparison of results The comparison of both, the calculated settlements and the established factors of safety are

summarised in table 1. The match of the settlement results between the Priebe method and the finite element analysis appears satisfactory.

Table 1. Comparison of results

FEMSettelment of

embankment base - centre [cm]

240

Homogenisation method 1. Area-Ratio 2. Priebe (stress

ration)Average of 1.

and 2. none

safety factor η 1,12 1,77 1,42 1,36

192

Analytical procedures

The results of the stability analyses indicate however, that the analytical method using the area ratio homogenisation approach for the shear parameters may be somewhat over-conservative. Also the Priebe method leads to factors of safety, which are greater than those obtained by the numerical model adopting the ϕ-c-reduction method.

However no general trend can be conducted from this observation since in this specific case the numerical analysis revealed a potential second failure mechanism below the berm at the embankment toe, which rests partially on improved and unimproved ground. Comparative studies with reduced berm lengths revealed that the factors of safety according to the Priebe method remain higher than those obtained from the spatial numerical analyses, which compare favourably with the safety factors obtained by an analysis with the mean of the shear strength calculated by the area ratio and the stress ratio homogenisation method.

It is instructive to compare the results with a 2-D numerical analysis conducted by Indraratna, Balasubramaniam & Sivaneswaran (1997). They calculated a normalised deformation β2 being the ratio of the maximum settlement to the corresponding filling height of β2=0.097 for a test embankment on Malaysian soft clay. The study presented here provides a normalised deformation of β2=0.149.

5 SUMMARY AND CONCLUSION The deformations and the stability of embankments on stone column improved ground can be

analysed using either analytical or numerical methods. Whilst the analytical models contain simplifications which are not completely justified, the finite element analysis of stability problems also need approximations like the ϕ-c-reduction method. The complex load bearing mechanisms call for a three dimensional analysis, which until now is not suitable for day to day design purposes.

Numerical analyses can account for the correct reproduction of the stress distribution between stone columns and surrounding soil. Useful conclusions on approximate factors of safety and failure mechanisms can be drawn from the results of these numerical simulations.

The comparison of the results from both numerical and analytical computations leads to a practical approach when analysing the slope stability with homogenised shear parameters using the average of area ratio and stress ratio according to Priebe's method. Then analytical methods like Bishop's can be adopted to calculate factors of safety.

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Pietruszczak, Schweiger (eds.) NUMOG VII. Rotterdam: Balkema. 541-546. Gruber, F.J. 1994. Verhalten einer Rüttelstopfverdichtung unter einem Straßendamm. Diss. TU Graz. Indraratna, B., Balasubramaniam, A.S. & Sivaneswaran, N. 1997. Analysis of settlement and lateral defor-

mation of soft clay foundation beneath two embankments. I. J. Num. Anal. M. Geom. 21. 599-618 Kirsch, K. & Sondermann, W. 2003. Ground improvement. In U. Smoltczyk (ed.), Geotechnical Engineering

Handbook. Vol. 2: 1-56. Berlin: Ernst & Sohn. Moseley, M.P. & Priebe, H.J. 1993. Vibro techniques. In M.P. Moseley (ed.), Ground Improvement: 1-19.

Glasgow: Blackie. Naylor, D.J.. 1999. On the use of the F.E.M. for assessing the stability of cuts and fills. In Pande,

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Schnellbahnbau. 3. Darmstädter Geotechnik Kolloq. TU Darmstadt. Heft 35. 147-164. Sondermann, W. & Jebe, W. 1996. Methoden zur Baugrundverbesserung für den Neu- und Ausbau von

Bahnstrecken auf Hochgeschwindigkeitslinien. Vorträge der Baugrundtagung 1996 in Berlin. 259-280. Vittinghoff, T., Plaßmann, B. & Schmitt, J. 1997. Programmentwicklungen im ANSYS-Open-System für

Anwendungen in der Geotechnik. 15th CADFEM Users' Meeting. Fulda. Part 1-30. 1-17.