Ground Improvement with Stone Columns - Methods of Calculating

Settlement Improvement Factor

Isabela Dellalibera Piccinini

Research advisor: Prof. Simon Wheeler

University of Glasgow, School of Engineering, MSc. Civil Engineering, Glasgow, 2014

Note: the present research has been realized supported by CNPq-Brazil (Conselho Nacional de Desenvolvimento

Científico e Tecnológico).

1. Introduction

Stone Column is a technique used in civil engineering to improve and stabilize soils

considered weak as soft clays or silts and loose sands, enabling the construction of highway

facilities, storage tanks, embankments, bridge abutments and so on. This technique uses columns

filled with a well compacted coarse grained material, which are allocated all over in the in situ

soil. Because material of the columns is stiffer, more permeable and has a higher shear strength

then the natural soil, we end up with an improvement of the soil properties: increase of the bearing

capacity due to shear strength increase; reduce of total and differential settlements due to stiffness

improvement; decrease time for the settlements to occur and reduce of liquefaction potential of

cohesive soils due to increase of the soil mass permeability acting as a vertical drain.

In this research we are interested mainly in the potential of Stone Columns to reduce

settlement. When it comes to constructions which the main problem is reducing the settlement,

this technique is a very common solution applied by Ground Engineering Companies for being

considered as a low cost alternative, effective and ease of installation. Once this technique is pretty

much common among industry, there are many studies to improve the design method in order to

make the calculation process simpler and to get more accurate results helping engineers to predict

the behaviour of the soil due to the insertion of stone columns realistically and driving to reliable

conclusions about the degree of improvement achieved.

The design process for Stone Columns allows to estimate the reduction in the settlement in

terms of a Settlement Improvement Factor (η) which can be obtained by dividing the settlement

of the soil without columns by the settlement of the soil with columns. The inverse of this value is

the Settlement Reduction Factor (β). There are four published methods to obtain these factors:

1) Simple Elastic Method, described by Castro & Sagaseta (2008), although originally

developed much earlier;

2) Elastic Method, published by Ballam & Booker (1981);

3) The Priebe’s Method, published originally in german by Priebe (1976) and described

by Priebe (1995);

4) The Elasto-Plastic Method, published by Pulko & Majes (2005) with some slight

changes published by Pulko, Majes & Logar (2011).

In all four methods the calculations are based on the area ratio (plan area of stone columns

divided by the total plan area), the in situ soil properties and the properties of the fill in material

used in the columns. So we can expect that the results are implicitly influenced by the origin of

the parameters values used for the soil and by the use of the correct equipment and procedures on

the installation of columns as specified in the designing to guarantee that the columns properties

correspond to the values used, meaning that the reliability of data used for the soil and column as

input parameters will imply the reliability we can get in the results obtained for the Settlement

Improvement Factor.

Another aspect that has great influence over the accuracy of results is the constraints and

assumptions made for the calculations, and once we have four different methods, each one with

different assumptions and constraints, we have then four different results for the Settlement

Improvement Factor, even when considering the same values for the input parameters. One of the

aims of this research is to compare those results and try to identify conditions that may converge

the results of two different methods in order to try to point out one of them as a general method to

be used with a good level of reliability and that is also in favour of safety. Meanwhile, we expect

to show in which cases some of the methods have similar results and which ones have totally

different results in order to make clear for the users the implications of each method’s results in

the designing in terms of safety and costs.

We also intend to investigate and understand, for the four existing methods individually,

how each one of the input parameters considered in the equations for the Settlement Improvement

Factor calculation affect the output results.

As mentioned before, Stone Column is a common technique used and one of the goals of

designers is to simplify the calculations, that’s why one of the aims of this particularly paper is to

search for a way to simplify the representation of the equations used for the Settlement

Improvement Factor calculation, in order to facilitate their understanding and application in

practice, using dimensionless input parameters and pointing out which of these parameters must

be calibrated with a high precision for having a great influence on the final results. Doing so we

expect to help improving the designing process by pointing out which of these parameters are

worthwhile spending time and money for investigations and laboratory tests and for which ones

can be set a standard value without losing reliability and significant precision on final results.

By the end of this paper we’ll analyse a case of an embankment construction over a soil

improved with Stone Columns using fictional data for the input parameters in order to be able to

compare the methods and how the results would impact the design process. We’ll also analyse a

case with real field data to try to illustrate and understand how each method influences the final

results and which one wold get closer to the value obtained in practice for η.

2. Stone Columns Technique

This chapter is to give a brief introduction of how the Stone Columns technique is used to

improve the soil: how the columns are built, what its advantages and disadvantages and the

concepts used in the various methods for calculating Settlement Improvement Factor.

2.1 Construction and feasibility

As mentioned before, Stone Columns construction involves the partial replacement of

unsuitable weak soils from the more superficial layers with vertical columns filled with coarse and

well compacted aggregates of various sizes, penetrating the weak strata to create structural

elements to be based on a stiffer layer. The columns are considered not to affect significantly the

properties of surrounding soil, acting mainly as inclusions with higher stiffness, shear strength and

permeability than the natural soil (Castro & Sagaseta, 2008). As the column material is stiffer than

the natural soil, the load applied on the surface is no longer equally distributed, but a major part is

taken by the columns, relieving the load on the compressible soil and thus reducing the settlement

of the soil.

Soil improvement can be achieved using Stone Columns with a gain of bearing capacity,

reduction and acceleration of settlements and mitigating the potential of liquefaction, making

possible to replace the deep foundations with shallow ones, thus facilitating design and reducing

costs. The effectiveness of this technique may be compromised when the layer of soft soil grows

too thick and amounts of organic materials get too high, leading to a situation where the excessive

compressibility and low strength of the natural soil results in too little lateral support for the

columns and extremely large vertical deflections of columns resulting a local bulging failure of

the structure (Figure 1).

Figure 1 – Local bulging failure due to very thick soft and/or organic layer. (Barksdale & Bachus, 1983).

Stone Columns in general are feasibly and most economically attractive for sites requiring

column lengths between 4-10 m length with an area ratio of 15-35% (which means that up to 35%

of the area will be replaced with stone columns). Stone Columns depths greater than about 10m

are usually not economically competitive with conventional deep foundations. Furthermore,

construction of very deep stone columns is considered by many to pose serious construction

problems including stabilization of the hole and ensuring that uncontaminated stone gets to the

bottom and is properly densified (Barksdale & Bachus, 1983). The columns must be based on a

high strength layer to avoid a bearing failure of the structure by punching.

Basically there are two processes to build the stone columns, both of them using vibratory

method to compaction of the coarse-grained materials. When jetting water is used to aid the

penetration of the ground by the vibrator the process is named vibro-replacement (Figure 2).

Otherwise, when the process is dry (no jetting water is needed once the soil is partially saturated)

the process is named vibro-displacement (Figure 3). The controlled volumes of aggregates can be

put into the hole by the ground surface allowing it to fall under gravity to the bottom side (top feed

process) or the aggregates may be fed directly to the bottom of the hole by an inner tube along the

same equipment that does the digging (bottom feed process).

Figure 2 – Top feed vibro-replacement process of

installation (Babu, Lecture notes).

Figure 3 – Bottom feed vibro-displacement process of

installation (Babu, Lecture notes).

Basically, the construction of Stone Columns has 3 steps: penetration, replacement

(delivery and compaction) and completion. During construction it is important to monitor the

progressing of each step, because no matter how much field and laboratory exploration data is

available, the unknowns and uncertainties for soil will always be greater than for steel, concrete

and other construction materials. That’s why during the installation microprocessors (Figure 4)

must be used for real-time monitoring during installation. The outputs (Figure 5) for this

monitoring process must include: depths, power input during the process of penetration and

expansion of the hole in the soil and compacted around it, total time spend on the stone column

point and the total compaction time, diameter increases and quantity of stone backfill used. Also,

load tests of the stone columns must be made to assess the immediate settlement for verification

with the values predicted during designing process. Post-treatment penetration testing can be

performed to measure the improvement achieved in granular soils

Figure 4 – Example of real-time monitoring using

microprocessors (Babu, Lecture notes).

Figure 5 – Example of real-time monitoring output

(Babu, Lecture notes).

2.2 Design concept considerations

To analyse the performance of Stone Column reinforced foundations, the “unit cell”

approximation is used.

The Stone Columns are arranged as a grid with a regular spacing all over the area, most

typically forming a triangular or a square pattern as shown in Figure 6.

Doing so we can divide the total area in many “unit cells” which are formed by the stone

column and its surrounding area of influence (Figure 7). The delimitation of the areas of “unit

cells” form a boundary (hexagonal for the triangular pattern and square for the square pattern),

that can be approximated by an equivalent circle with an equivalent diameter (𝐷𝑒) in both cases.

The stone column is concentric to the exterior boundary of the “unit cell”.

Figure 6 – Example of triangular pattern of

arrangement (A) or square pattern of arrangement (B)

(Ballam & Booker, 1981).

Figure 7 – Unit cell idealization (Barksdale & Bachus,

1983).

By considering this cylindrical “unit cell” and attributing an equivalent diameter, we can

get the ratio between the area of the column (𝐴𝑐) and the area of the zone of influence (𝐴𝑒) which

represents the replacement area ratio (𝐴𝑟) used in the designing process:

𝐴𝑟 = 𝐴𝑐

𝐴𝑒= (

𝐷𝑐

𝐷𝑒)

2

(Eq. 1)

The bigger the area ratio value is, the more soil is replaced by the Stone Column and the

greater will be the effect on performance. Typical values range from 0.10 to 0.40.

“Because it has been assumed that the site has been stabilized by a large number of columns

it follows that each column and the area surrounding it will respond in virtually the same fashion

as those adjacent”, Ballam & Booker (1981). So we can state that the unit cell model will represent

the soil with stone column improvement for the following analytical calculation methods.

The application of the “unit cell” concept is relevant only to relatively wide foundations,

where we will have a large number of Stone Columns disposed side by side forming a grid (the

idealization states that this grid should be infinite). It’s necessary that the Stone Columns form a

plan with representative extensions for both directions, wider than the depth of columns installed.

This is necessary to guarantee that the settlement calculated for the “unit cell” will represent the

major portion of the site, with the exception of the foundation’s outer boundary, where settlements

may be lower.

3. Calculation methods of the Settlement Improvement Factor

This chapter will explain briefly the process of calculation of the Settlement Improvement

Factor used for each of the four methods, pointing out the assumptions made in each case.

The list of symbols used for the equations presented below are shown in Appendix A.

3.1 Basic assumptions

There are some basic assumptions applied to all four methods:

1) Bottom of the cylinder based on a rigid material (bedrock);

2) Zero radial displacement on the outer boundary of the “unit cell”;

3) Zero shear stress on the cylindrical outer boundary due to symmetry of loading and

geometry of the unit cell considered;

4) Top (surface) and bottom (bedrock) of the cylinder boundaries are considered as

smooth (zero shear stress);

5) Top (surface) of the cylinder with uniform settlements.

Figure 8 illustrates the idealization of the unit cell with the foundation pressure on top (𝑞𝑎),

the smooth rigid cylinder boundaries (top, bottom and outer boundary) and the direction of uniform

vertical displacement.

Figure 8 – Unit cell considered (modified from Barksdale & Bachus, 1983).

As we move on presenting each one of the four methods, we’ll discuss the particular

assumptions made for each case.

3.2 METHOD 1: Simple Elastic Method (Castro & Sagaseta, 2008)

It’s important to highlight that the Simple Elastic Method was developed long before 2008

but it’s not assigned to any specific author(s). According to Castro & Sagaseta (2008) this method

aims to develop a simplified and closed-form analytical solution for estimating the Settlement

Improvement Factor using the Elastic Theory.

Main assumptions:

1) Both soil and column act as elastic materials;

2) Horizontal strain is zero in both materials, resulting in zero radial displacement (∆𝜀𝑟,𝑐

= ∆𝜀𝑟,𝑠 = 0).

The second assumption would not actually be true in a unit cell consisting of two elastic

material (see next method).

According to the assumptions and boundaries given, the equation of vertical equilibrium is

used to develop the solution.

Vertical equilibrium:

𝑞𝑎 = 𝐴𝑟∆𝜎𝑧,𝑠 (Eq. 2)

Compatibility of vertical strain for both soil and column:

∆𝜀𝑧,𝑐 = ∆𝜀𝑧,𝑠 (Eq. 3)

From assumption 2) and considering we have the deformation occurring just in one

direction, we have:

∆εz,s = ∆𝜎𝑧,𝑠

𝐸𝑚,𝑠⁄ (Eq. 4) and ∆εz,c =

∆𝜎𝑧,𝑐𝐸𝑚,𝑐

⁄ (Eq. 5)

Replacing the equations 4 and 5 in equation 3 and then inserting the resulting equation

from these substitution in equation 2:

∆𝜎𝑧,𝑠 =𝑞𝑎

1 + 𝐴𝑟(𝐸𝑚,𝑐

𝐸𝑚,𝑠 ⁄ − 1) (Eq. 6)

By considering the elastic relation between strain and tension (Eq. 4) and replacing it in

equation 6, we have the equation for the vertical strain of the soil with the stone columns

improvement:

∆εz,s =𝑞𝑎

𝐸𝑚,𝑠 ( 1 + 𝐴𝑟(𝐸𝑚,𝑐

𝐸𝑚,𝑠 ⁄ − 1))

(Eq. 7)

The vertical strain for the soil without columns is given by:

∆εz,s =𝑞𝑎

𝐸𝑚,𝑠 (Eq. 8)

Dividing equations 8 for 7 we end up with the Simple Elastic Method solution for the

Settlement Improvement Factor:

𝜼 = 𝟏 + 𝑨𝒓 (𝑬𝒎,𝒄

𝑬𝒎,𝒔 ⁄ − 𝟏) (Eq. 9)

As we can see from equation 9, the improvement factor for the Simple Elastic Method

depends on only two non-dimensional parameters: 𝑨𝒓 (area ratio) and 𝑬𝒎,𝒄

𝑬𝒎,𝒔 ⁄ (elastic constrained

modulus ratio). We`ll be able to see the influence of each one of these parameters later in section

4.1 in this same report.

As we may see later on, the Elastic Method gives quite good approximations when dealing

with low loads and a soil that is not too soft and that can provide support for the columns.

According to Castro & Sagaseta (2008): “Linear elastic behaviour is a reasonable assumption for

the soil, because the partial lateral confinement provided by the columns produces small shear

strains. However, the columns are surrounded by a softer material, and even for moderate loads

they can reach the active limit condition. So, elastic behaviour is regarded only as first

approximation for the columns.”

For main cases, simple elastic analysis can highly overestimate the effect of Stone Columns

on settlement reduction. This is due to two effects. Firstly, the radial column deformation (not

considered by this method) has a significant influence on the distribution of stresses between the

soil and the column. Secondly, this method does not consider the plastic deformation of the

columns, which makes the columns softer, reducing its capacity to carry. We`ll be able to see both

the effects of the radial column and the plastic deformations later on the comparison of the methods

in section 5 of this report.

3.3 METHOD 2: Elastic Method (Ballam & Booker, 1981)

Main assumption:

1) Both soil and column act as elastic materials.

Unlike the Simple Elastic Method, the horizontal strains in the soil and the column are not

assumed to be zero. Instead, the full pattern of stress and strains in both materials is correctly

calculated by imposing the necessary conditions that the radial stresses in column and soil must

be equal at the column/soil interface and the radial displacements in column and soil must be equal

at the column/soil interface.

According to the assumptions and boundaries given, the equations of equilibrium are used

to develop the solution for the elastic method.

Vertical equilibrium:

𝑞𝑎 = 𝐴𝑟∆𝜎𝑧,𝑠 (Eq. 10)

Compatibility of vertical strain for both soil and column:

∆𝜀𝑧,𝑐 = ∆𝜀𝑧,𝑠 (Eq. 11)

Equilibrium of radial stress:

(∆𝜎𝑟,𝑐)𝑟 =𝐷𝑐

2= (∆𝜎𝑟,𝑠)𝑟 =

𝐷𝑐

2 (Eq. 12)

Compatibility of radial displacement:

(∆𝜀𝑟,𝑐)𝑟 =𝐷𝑐

2= (∆𝜀𝑟,𝑠)𝑟 =

𝐷𝑐

2 (Eq. 13)

In this case, we`ll have to use the Lamé’s parameters for homogenous isotropic linear

elastic materials:

λ = 𝜈𝐸

(1−2𝜈)(1+𝜈) (Eq. 14) and G =

𝐸

2(1+𝜈) (Eq. 15)

And it will be required the transformation of Young’s modulus (E) to the constrained

Young’s modulus (Em):

Em = 𝐸(1−𝜈)

(1−2𝜈)(1+𝜈) (Eq. 16)

Ballam & Booker (1981) end up with a parameter F:

F = (𝜆𝑐−𝜆𝑠)(𝐷𝑒

2−𝐷𝑐2)

2[𝐷𝑐2(𝜆𝑠+𝐺𝑠−𝜆𝑐−𝐺𝑐)+𝐷𝑒

2(𝜆𝑐+𝐺𝑐+𝐺𝑠)] (Eq. 17), where 𝐴𝑟 =

𝐷𝑐2

𝐷𝑒2 as Eq. 1 states.

According to Ballam & Booker (1981), the relation between the strain (ε) and the average

stress (q𝑎) can obtained by:

q𝑎𝐷𝑒2 = [(𝜆𝑐 + 2𝐺𝑐)𝐷𝑐

2 + (𝜆𝑠 + 2𝐺𝑠)(𝐷𝑒2 − 𝐷𝑐

2) − 2𝐷𝑐2(𝜆𝑐 − 𝜆𝑠)𝐹]𝜀 (Eq. 18)

The next follow steps were made to represent a final equation for the Settlement

Improvement Factor in terms of dimensionless parameters, as one of the requirements of this

research.

Replacing equation 16 in equations 14 and 15 we end up with:

λ = 𝜈𝐸𝑚

(1−𝜈) (Eq. 19) and G =

(1−2𝜈)𝐸𝑚

2(1−𝜈) (Eq. 20)

Replacing equations 19 and 20 in equation 17 and inserting 𝐴𝑟 in the place of 𝐷𝑐2

𝐷𝑒2⁄ , we

get:

F = [

𝜈𝑐(1−𝜈𝑐)

𝐸𝑚,𝑐𝐸𝑚,𝑠

− 𝜈𝑠

(1−𝜈𝑠)] (1−𝐴𝑟)

1

(1−𝜈𝑐)

𝐸𝑚,𝑐𝐸𝑚,𝑠

(1−𝐴𝑟) + 1

(1−𝜈𝑠) (𝐴𝑟+1−2𝜈𝑠)

(Eq. 21)

Replacing equations 19, 20 and 21 in equation 18, and once again inserting 𝐴𝑟 in the place

of 𝐷𝑐2

𝐷𝑒2⁄ , we get the equation for the vertical strain of the soil with the stone columns improvement:

∆𝜀𝑧,𝑠 =𝑞𝑎/𝐸𝑚,𝑠

1+𝐴𝑟 ( 𝐸𝑚,𝑐𝐸𝑚,𝑠

−1)− 2𝐴𝑟(1−𝐴𝑟)[

𝜈𝑐(1−𝜈𝑐)

𝐸𝑚,𝑐𝐸𝑚,𝑠

− 𝜈𝑠

(1−𝜈𝑠)]2

1(1−𝜈𝑐)

𝐸𝑚,𝑐𝐸𝑚,𝑠

(1−𝐴𝑟) + 1

(1−𝜈𝑠) (𝐴𝑟+1−2𝜈𝑠)

(Eq. 22)

The vertical strain for the soil without columns is given by:

∆εz,s =𝑞𝑎

𝐸𝑚,𝑠 (Eq. 23)

Dividing equations 23 for 22 we end up with the Elastic Method solution for the Settlement

Improvement Factor:

𝜼 = 𝟏 + 𝑨𝒓 ( 𝑬𝒎,𝒄

𝑬𝒎,𝒔− 𝟏) −

𝟐𝑨𝒓(𝟏−𝑨𝒓)[𝝂𝒄

(𝟏−𝝂𝒄) 𝑬𝒎,𝒄𝑬𝒎,𝒔

− 𝝂𝒔

(𝟏−𝝂𝒔)]

𝟐

𝟏

(𝟏−𝝂𝒄)

𝑬𝒎,𝒄𝑬𝒎,𝒔

(𝟏−𝑨𝒓) + 𝟏

(𝟏−𝝂𝒔) (𝑨𝒓+𝟏−𝟐𝝂𝒔)

(Eq. 24)

As we can see from equation 24, the improvement factor for the Elastic Method depends

on four dimensionless parameters: 𝑨𝒓 , 𝑬𝒎,𝒄

𝑬𝒎,𝒔 ⁄ , 𝝂𝒔 and 𝝂𝒄. We`ll be able to see the influence of

each one of these parameters later in section 4.2 in this same report.

As the previous method, this solution tends to overestimate the effect of Stone Columns,

resulting in unrealistically low values of Settlement Improvement Factor, due to similar reasons

as the Simple Elastic Method. Because the influence of any yielding and plastic straining of the

columns is ignored, the overestimation is, however, not as severe as with the Simple Elastic

Method, because the Elastic Method of Ballam & Booker (1981) does not include the second

problem of ignoring any effects of horizontal strain in column and soil.

In order to try to overcome the under design of Stone Columns due to the overestimated

results obtained with both the Simple Elastic Method and the Elastic Method, in practice designers

tend to assume unrealistically low values of 𝐸𝑚 ratio when applying these methods. The values

used for the 𝐸𝑚 ratio are based on the company records and experience.

3.4 METHOD 3: Priebe’s Method (Priebe, 1995 with original publication in german by

Priebe, 1976 )

Main assumptions:

1) Soil acts as elastic material (no yielding);

2) Column acts as a plastic material (yields as a purely frictional material when 𝜎3′

𝜎1′⁄ =

𝐾𝑎𝑐);

3) The bulk density of column and soil is neglected (initial stresses, 𝜎0′ , are considered

null) and hence the column yields over full depth (H) as soon as any foundation load is

applied and the radial deformation is uniform with depth (meaning that stresses and

strains vary with radius but not with depth);

4) Column material is assumed to be incompressible (no change of volume), including the

dilation during plastic shearing once there`s no change in volume in the column

yielding (ψ𝑐 = 0°).

In this method, by assuming the Poisson`s ratio of the column material (ν𝑐) equals 0.5

(value that represents perfectly incompressible materials deformed elastically), from equation

15, Em,c → ∞, which means that we`re dealing with a rigid material if the material is loaded with

strain prevented in the lateral direction.

In this method, we`ll have to use the Lateral Active Earth Pressure coefficient (Kac), that

measures the ratio of the horizontal effective stress (𝜎3′) to the vertical effective stress (𝜎1

′) at

failure, given by the Rankine theory as:

Kac = (1−𝑠𝑖𝑛𝜙`)

(1+𝑠𝑖𝑛𝜙`) (Eq. 25)

As the column is much stiffer than the soil, the vertical stress increment in the column is

much greater than the vertical stress increment in the soil. At the same time, the horizontal radial

stress increment must be the same in both materials (at least at the soil/column interface). As a

consequence, the vertical stress is the major principal stress in the column whereas the horizontal

radial stress is the major principal stress in the soil. It also transpires that the ratio of major principal

stress to minor principal stress is much greater in the column than in the soil. This means that it is

always the column rather than the soil that yields first (hence the assumption that the soil is an

elastic material whereas the column acts as a plastic material).

Priebe (1995) end up with an equation for the Settlement Improvement Factor given by:

η = 1 + Ar [0.5+ f(νs,Ar)

𝐾𝑎𝑐 f(νs,Ar)− 1] (Eq. 26)

Where the function f(νs, Ar) is given by:

f(νs, Ar) = (1−νs)(1−Ar)

1−2νs+Ar (Eq. 27)

Replacing equation 27 in equation 26 we end up with the Priebe`s Method solution for the

Settlement Improvement Factor:

𝜼 = 𝟏 − 𝑨𝒓 + 𝑨𝒓 [(𝟑−𝑨𝒓)+𝟐𝝂𝒔(𝑨𝒓−𝟐)

𝟐𝑲𝒂𝒄(𝟏−𝝂𝒔)(𝟏−𝑨𝒓)] (Eq. 28)

As we can see from equation 28, the improvement factor for Priebe`s Method depends on

three dimensionless parameters: 𝑨𝒓 , 𝝂𝒔 and 𝑲𝒂𝒄 (function of the friction angle of the column 𝜙𝑐` ).

We`ll be able to see the influence of each one of these parameters latter in section 4.3 in this same

report.

One of the downsides of this method is that is not considering the range of values for the

load applied, so we can`t be sure about the consideration that the yielding is occurring over the

whole length of the column. In practice, this yielding gradually extends down the column as the

foundation loading is increased.

3.5 METHOD 4: Elasto-Plastic Method (PULKO, MAJES & LOGAR, 2011)

Main assumptions:

1) Soil acts as elastic material (no yielding);

2) Column acts as a elasto-plastic material;

3) The initial stresses (𝜎0′) are taken into account;

4) Constant dilatancy during plastic shearing of the column (𝜓𝑐 value is constant).

So, for very small vertical distributed load (𝑞𝑎) the initial response of both soil and column

will be elastic but as the vertical load grows, the column begins yielding through its length acting

as a plastic material. In this case, the method assumes that the soil remains elastic throughout the

range of applied load and the settlement evolution will be governed by the progressive yield of the

column as loading increases (Pulko, Majes & Logar, 2011). So, the final settlement will be a

combination of the settlement due to elastic behaviour and the settlement due to plastic behaviour,

which means that the final Settlement Reduction Factor (β) depends on the Elastic Settlement

Reduction Factor (β𝑒𝑙), the Plastic Settlement Reduction Factor (β𝑝) and the final yield depth (𝑧𝑦)

reached depending on the range of load applied.

First of all, from the Elastic solution, the same as developed by Ballam & Booker (1981),

we have the parameters 𝐶1, 𝐶2 and 𝐶3 given by:

𝐶1 = 2𝜈𝑠 𝐴𝑟

(1−𝜈𝑠)(1−𝐴𝑟) (Eq. 29), 𝐶2 =

1− 2𝜈𝑠+𝐴𝑟

(1−𝐴𝑟)(1−𝜈𝑠) (Eq. 30),

, 𝐶3 = 1− 2𝜈𝑠+𝐴𝑟

(1−𝐴𝑟)(1−𝜈𝑠)−

2𝜈𝑠2 𝐴𝑟

(1−𝜈𝑠)2(1−𝐴𝑟) (Eq. 31)

The parameter F is exactly the same as shown in equation 21 and the Elastic Settlement

Reduction Factor is given by:

β𝑒𝑙 =𝐸𝑚,𝑠

(𝜆𝑐+2𝐺𝑐)𝐴𝑟 + (𝜆𝑠+2𝐺𝑠)(1−𝐴𝑟)−2𝐴𝑟(𝜆𝑐−𝜆𝑠)𝐹 (Eq. 32)

Replacing the equations 19 and 20 in equation 32 we end up with:

𝛽𝑒𝑙 =1

1+𝐴𝑟 (𝐸𝑚,𝑐𝐸𝑚,𝑠

−1)−2𝐴𝑟[𝜈𝑐

(1−𝜈𝑐) 𝐸𝑚,𝑐𝐸𝑚,𝑠

− 𝜈𝑠

(1−𝜈𝑠)]𝐹

(Eq.33)

Now, for the Elasto-Plastic solution we have 𝐾𝑖𝑛𝑖, 𝐾𝜓 and 𝐾𝑝𝑐 given by:

𝐾𝑝𝑐 = (1+𝑠𝑖𝑛𝜙𝑐

′ )

(1−𝑠𝑖𝑛𝜙𝑐′)

(Eq. 34), 𝐾𝜓𝑐=

(1+𝑠𝑖𝑛𝜓𝑐)

(1−𝑠𝑖𝑛𝜓𝑐) (Eq. 35)

and 𝐾𝑖𝑛𝑖 = (1 − 𝑠𝑖𝑛𝜙𝑠′)𝑂𝐶𝑅𝑠𝑖𝑛𝜙𝑠

′ (Eq. 36)

The parameters 𝐶4, 𝐷 and 𝐶5 are given by:

𝐶4 = (𝐾𝑝𝑐𝐾𝑖𝑛𝑖 − 𝛾𝑐

′

𝛾𝑠′⁄ )

(𝜆𝑐+2𝐺𝑐)𝐴𝑟 + (𝜆𝑠+2𝐺𝑠)(1−𝐴𝑟)−2𝐴𝑟(𝜆𝑐−𝜆𝑠)𝐹

2𝐺𝑐(1+𝐹𝐾𝑝𝑐)+𝜆𝑐(1−2𝐹)(1−𝐾𝑝𝑐) (Eq. 37)

𝐷 = 𝐸𝑐

2+𝐾𝜓𝐾𝑝𝑐−2𝜈𝑐(1+𝐾𝑝𝑐+𝐾𝜓) (Eq. 38)

𝐶5 = 𝐸𝑚,𝑠(1 − 𝐴𝑟)𝐶3 + 𝐷 {(1 − 𝐴𝑟)(𝐶1𝐾𝜓 + 2) + 𝐴𝑟𝐾𝑝𝑐 (𝐾𝜓𝐶2 +2𝜈𝑠

(1−𝜈𝑠))} (Eq. 39)

Replacing the equations 19 and 20 in equation 37 we get:

𝐶4 = (𝐾𝑝𝑐𝐾𝑖𝑛𝑖 − 𝛾𝑐

′

𝛾𝑠′⁄ )

𝐴𝑟 (𝐸𝑚,𝑐𝐸𝑚,𝑠

−1)+1−2𝐴𝑟[𝜈𝑐

(1−𝜈𝑐) 𝐸𝑚,𝑐𝐸𝑚,𝑠

− 𝜈𝑠

(1−𝜈𝑠)]𝐹

1

(1−𝜈𝑐)

𝐸𝑚,𝑐𝐸𝑚,𝑠

[(1−2𝜈𝑐)(1+𝐹𝐾𝑝𝑐)+𝜈𝑐(1−2𝐹)(1−𝐾𝑝𝑐)] (Eq. 40)

Replacing the equation 17 in equation 38 we get:

𝐷

𝐸𝑚,𝑠=

(1−2𝜈𝑐)(1+𝜈𝑐)𝐸𝑚,𝑐𝐸𝑚,𝑠

(1−𝜈𝑐)[2+𝐾𝜓𝐾𝑝𝑐−2𝜈𝑐(1+𝐾𝑝𝑐+𝐾𝜓)] (Eq. 41)

Dividing the equation 39 for 𝐸𝑚,𝑠 we get:

𝐶5

𝐸𝑚,𝑠= (1 − 𝐴𝑟)𝐶3 +

𝐷

𝐸𝑚,𝑠{(1 − 𝐴𝑟)(𝐶1𝐾𝜓 + 2) + 𝐴𝑟𝐾𝑝𝑐 (𝐾𝜓𝐶2 +

2𝜈𝑠

(1−𝜈𝑠))} (Eq. 42)

The Elasto-Plastic response the Settlement Reduction Factor is given by:

β𝑝 =2𝐷+𝐸𝑚,𝑠 𝐶2

𝐶5 (Eq. 43)

Dividing the equation 43 for 𝐸𝑚,𝑠 we get:

βp =2

D

Em,s+ C2

C5Em,s

(Eq. 44)

For a given uniform load (𝑞𝑎) the yield of the column will reach a depth (𝑧𝑦), given by:

𝑧𝑦 = 𝑞𝑎

𝐻 𝛾𝑠′

1

𝐶4 (Eq. 45)

Finally, we can get the final Settlement Reduction Factor for the Elasto-Plastic Method:

As we can see from equation 46, the improvement factor for Elasto-Plastic Method depends

on nine dimensionless parameters: 𝑨𝒓, 𝐄𝐦,𝐜

𝐄𝐦,𝐬 ⁄ , 𝝂𝒔, 𝝂𝒄, 𝑲𝒑𝒄 (depending on the friction angle of

the column, 𝜙𝑐` ), 𝑲𝒊𝒏𝒊 (depending on the friction angle of the soil, 𝜙𝑠

` , and its OCR), 𝑲𝝍

𝛃𝒆𝒍 𝒇𝒐𝒓 𝒛𝒚

𝑯≤ 𝟎;

𝛃 = 𝛃𝒆𝒍 (𝟏 − 𝒒𝒂

𝑯 𝜸𝒔′ ×

𝟏

𝟐𝑪𝟒) + 𝛃𝒑 (

𝒒𝒂

𝑯 𝜸𝒔′ ×

𝟏

𝟐𝑪𝟒) 𝒇𝒐𝒓 𝟎 <

𝒛𝒚

𝑯≤ 𝟏; (Eq. 46)

𝛃𝒆𝒍 [ (𝒒𝒂

𝑯 𝜸𝒔′)

−𝟏

× 𝑪𝟒

𝟐] + 𝛃𝒑 [𝟏 − (

𝒒𝒂

𝑯 𝜸𝒔′)

−𝟏

× 𝑪𝟒

𝟐] 𝒇𝒐𝒓

𝒛𝒚

𝑯≥ 𝟏.

(depending on the dilatancy angle of the column, 𝜓𝑐), 𝜸𝒄′

𝜸𝒔′⁄ and 𝒒𝒂

𝑯 𝜸𝒔′⁄ . We`ll be able to see the

influence of each one of these parameters latter in section 4.4 in this same report.

This method tends to refine the solutions imperfections of the previous methods, once now

it’s taking into account not only the plastic deformation of the columns due to yielding (not

considered by methods 1 and 2), but it’s also considering the volumetric deformation caused by

this same yielding according to the dilatancy theory (not considered by method 3). Method 4 is

also considering the initial stress state in the soil by using the parameter 𝐾𝑖𝑛𝑖.

A new factor introduced by this method is the dimensioless load factor (𝑞𝑎𝐻 𝛾𝑠

′⁄ ), which

influences the extension of yielding throughout the column length (see Eq. 45), starting at the soil

surface and reaching a certain depth 𝑧𝑦. As the yield depth has been determined, the vertical

displacement can be calculated as a combination of elasto-plastic zones (where the distributed load

is higher than the yield load, in other words, for depths lower than the value 𝑧𝑦 found) and elastic

zones (for depths where the column remains elastic, in other words, for depths higher than the

value 𝑧𝑦 found for the load applied). So, now we are considering the fact that the range of load

applied on the top of surface will influence the behaviour of the improved site by changing the

state of yielding of the columns, consideration that will impact on the amount of load the column

structure will be capable to absorb leading to a higher or lower settlement. Thus the depth of

yielding has a direct impact on the effectiveness of Stone Columns, being reasonable then to expect

results for Settlement Improvement/Reduction Factor more close to reality.

4. Parameters investigation

We expect to investigate and understand, for the four existing methods individually, how

each one of the input parameters considered in the equations for the Settlement Improvement

Factor calculation affect the output results in order to be able to give some orientations about the

relevance on designing process by pointing out which of these parameters are worthwhile spending

time and money for investigations and laboratory tests and for which ones can be set a standard

value without losing reliability and significant precision on final results. Also, we expect to show

in which cases some of the methods have similar results and which ones have totally different

results in order to make clear for the users the implications of each method’s results in the

designing in terms of safety and costs.

By considering the area ratio and Settlement Improvement/Reduction Factor as variables

for designing and fixing the other parameters according to the characteristics of the in situ soil and

the material used for the columns, we`ll now illustrate with graphs of 𝜂 against 𝐴𝑟 and/or 𝛽 against

𝐴𝑟, the influence of these parameters in the parametric equations found for each method mainly

aiming to analyse what happens with the efficiency of the Stone Columns (by observing the factors

of improvement and/or reduction of settlement) in a range of 10-30% for area replacement, which

are typical values used for designing.

4.1 Method 1, equation 9: 𝜂 = 1 + 𝐴𝑟 (𝐸𝑚,𝑐

𝐸𝑚,𝑠 ⁄ − 1)

As we can see, on the Settlement Improvement Factor equation formulated by method 1

the only parameter of materials that affects the η value is the constrained modulus ratio, which can

easily be obtained through the oedometer test.

As we can see in Figure 9, the relation between η and 𝐴𝑟 is linear, with a slope defined by

𝐸𝑚,𝑐𝐸𝑚,𝑠 ⁄ – 1. If

𝐸𝑚,𝑐𝐸𝑚,𝑠 ⁄ values used are high (in a range of 10-50), we`ll end up with substantially

high values of improvement factor (η) that won`t be consistent with the actual resulting reduction

values for settlement. Consistent values can be obtained only using extremely low modular ratio

(𝐸𝑚,𝑐

𝐸𝑚,𝑠 ⁄ ≈ 5), which is why in practice designers apply low values of 𝐸𝑚 ratio in their

calculations, based on previous experience using the Simple Elastic Method. This discrepancy of

theoretical and practical values are commonly attributed to the lack of consideration of lateral

deformation and column yielding.

As β is the inverse of η, a non-linear relation is predicted between β and 𝐴𝑟, as we can see

in Figure 10. The graph shows that the major reduction on settlements occurs until an area ratio of

30%.

Analysing Figure 10 makes clear that for a fixed value of 𝐴𝑟, the 𝐸𝑚 ratio increase is not

proportional to the increase in the efficiency of settlement reduction once curves are getting closer

as 𝐸𝑚 ratio increases by increments of 10 units. That means that in the designing process

sometimes is more effective and worthwhile to increase the area replacement instead using a higher

𝐸𝑚 ratio for further reductions in settlements.

Figure 9 – Effect of 𝑬𝒎 ratio on Settlement Improvement Factor using method 1.

Figure 10 – Effect of 𝑬𝒎 ratio on Settlement Reduction Factor using method 1.

4.2 Method 2, equation 23: 𝜂 = 1 + 𝐴𝑟 ( 𝐸𝑚,𝑐

𝐸𝑚,𝑠− 1) −

2𝐴𝑟(1−𝐴𝑟)[𝜈𝑐

(1−𝜈𝑐) 𝐸𝑚,𝑐𝐸𝑚,𝑠

− 𝜈𝑠

(1−𝜈𝑠)]

2

1

(1−𝜈𝑐)

𝐸𝑚,𝑐𝐸𝑚,𝑠

(1−𝐴𝑟) + 1

(1−𝜈𝑠) (𝐴𝑟+1−2𝜈𝑠)

Now, on the Settlement Improvement Factor equation formulated by method 2 we have

two more parameters of materials that affects the η value besides the constrained modulus ratio,

being the Poisson’s ratio of the soil and the column.

As we can see in equation 23 shown above, the solution obtained with method 2 includes

the exact equation 9 obtained by method 1, thus another term subtracting, resulting now a non-

linear solution for both η and β, as shown in Figure 11 and Figure 12.

The behave of 𝐸𝑚 ratio on settlement reduction is the same as the previous method with

the curves now “eased”, meaning that the values for the settlement reduction are lower for method

2 in comparison with method 1 (as shown in Figure 12), and therefore giving a more conservative

response in favour of safety. The similarity between the two methods are reasonable because both

of them take into account the elastic theory for the deformation of both the column and soil, with

the difference that the solution proposed by Ballam & Booker (1981) is more rigorous by

considering the influence of horizontal components of the deformation. Even so, the results

obtained with method 2 are not realistic once the calculated settlements are not close enough to

the values obtained by field tests for the improved site. This is because method 2 still takes no

account of any yielding of the columns. As discussed previously in method 1, to get acceptable

accuracy of solutions using method 2, in practice designers use to apply lower values of 𝐸𝑚 ratio

in their calculations, based on previous experience.

Figure 11 – Effect of 𝑬𝒎 ratio on Settlement Improvement Factor fixing 𝝂𝒔 = 𝝂𝒄 = 𝟎. 𝟑 for method 2.

Figure 12 – Effect of 𝑬𝒎 ratio on Settlement Reduction Factor fixing 𝛎𝐬 = 𝛎𝐜 = 𝟎. 𝟑 for method 2 in comparison

with the effect on method 1.

The influence of the Poisson’s ratio of the soil on the settlement reduction is extremely

small using Stone Columns as we can see in Figure 14 by the curves for usual values of 𝜈𝑠, which

are almost completely overlapped. In the other hand, the Poisson’s ratio of the column material

itself has a significant effect on solutions with more spaced curves (Figure 13). As we can see in

the η function (Eq. 23), 𝜈𝑐 is always multiplied by the 𝐸𝑚 ratio (which has a high value), making

the results of the terms involving 𝜈𝑐 more significant than the terms involving 𝜈𝑠, which explains

the behaviour of curves in Figure 13 and Figure 14.

The Poisson’s ratio is a difficult parameter to measure with confidence with laboratory

tests and both Poisson’s ratios (for soil and column) affect β relatively little compared to the effect

caused by 𝐸𝑚 ratio, which probably makes the time and costs to estimate the values of 𝜈𝑐 and 𝜈𝑠

not worthwhile. Therefore it’s better to fix an intermediate value of ν for each material based on

typical practical range of values obtained for soft soils and fill in materials used for stone columns

and design based on 𝐴𝑟 and 𝐸𝑚 ratio, simplifying the designing process. In order to maintain the

results in favour of safety, a typical value adopted for both materials is 𝜈𝑠 = 𝜈𝑐 = 0.3.

Figure 13 – Effect of 𝝂𝒄 on Settlement Reduction Factor fixing 𝝂𝒔 = 𝟎. 𝟑 and 𝑬𝒎,𝒄 𝑬𝒎,𝒔 = 𝟐𝟎⁄ for method 2.

Figure 14 – Effect of 𝝂𝒔 on Settlement Reduction Factor fixing 𝝂𝒄 = 𝟎. 𝟑 and 𝑬𝒎,𝒄 𝑬𝒎,𝒔 = 𝟐𝟎⁄ for method 2.

4.3 Method 3, equation 28: 𝜂 = 1 − 𝐴𝑟 + 𝐴𝑟 [(3−𝐴𝑟)+2𝜈𝑠(𝐴𝑟−2)

2𝐾𝑎𝑐 (1−𝜈𝑠)(1−𝐴𝑟)]

For method 3, we can see from the equation above that the Settlement Improvement Factor

depends basically of two parameters of materials, besides the area ratio: the Poisson’s ratio of the

soil and the friction angle of the column (implicit in the coefficient of active lateral earth pressure).

From Figure 15 we see that the friction angle affects quite substantially the settlement reduction

while Figure 16 shows very close curves for different values of 𝜈𝑠, meaning that this last parameter

has a low influence over β. So it’s reasonable to set a default value for 𝜈𝑠 = 0.3, as adopted for

method 2. As a matter of fact, a similar simplification is made by Priebe (1995). He assumed 𝜈𝑠 =

1 3⁄ , leading to a simplified expression for the Settlement Improvement Factor, given by:

η0 = 1 − 𝐴𝑟 + 𝐴𝑟 [(5−𝐴𝑟)

4𝐾𝑎𝑐(1−𝐴𝑟)] (Eq. 47)

Priebe’s Method is the most used by designers nowadays but even still its results for the

calculated settlements do not always match the values obtained by field tests for the improved site.

Figure 15 – Effect of friction angle on Settlement Reduction Factor fixing 𝝂𝒔 = 𝟎. 𝟑 for method 3.

Figure 16 – Effect of 𝝂𝒔 on Settlement Reduction Factor fixing 𝝓′ = 𝟒𝟎° for method 3.

4.4 Method 4, equation 46:

As we may see above, the equation for the Settlement Reduction Factor for method 4 is the

most complex one involving explicitly one parameter, the load factor (𝑞𝑎

𝐻 𝛾′

𝑠⁄ ) but implicitly it

involves other seven parameters, besides the area ratio: the Em ratio, the Poisson’s ratio for both

the soil and the column, the friction angle of the column, the dilatancy angle of the column, the

natural weight ratio and the friction angle of the soil (implicit in K𝑖𝑛𝑖 value). As the depth of yield

varies, the equation for β varies as well, as we can see in equation 46.

This method is responsible for the introduction of a very important parameter for modelling

the impact of Stone Columns in settlement improvement, the loading factor (𝑞𝑎

𝐻 𝛾′

𝑠⁄ ). It is

reasonable to assume that as the range of loading on the top of the ground grows, the column will

be responsible to receive a higher portion of stresses to relieve the soil, making both the vertical

and horizontal deformations higher and hence, increasing the effect of yielding on the column and

getting it deeper into the soil. So now we can see the effect of this new parameter on the Settlement

Reduction Factor with the increase of area ratio in Figure 17 . For small values of dimensionless

loads (𝑞𝑎

𝐻 𝛾′

𝑠⁄ ), the effect in β value is more effective as the reduction on settlement will be very

high using small values of area ratio once the column is not yielding, making it works at “full

capacity”. Because the load increases, there’s a tendency to stabilize β into a lower value

comparing to the situation where the load is zero, what is reasonable if we think that for a load

greater than a limit value, the yield will reach the whole column depth. With the increase of yield

depth, the column will be more “weakened” as it will deform more easily and so, it will be able to

support considerably less stresses until it reaches the rupture. This way, since the yield is occurring

in all length of column for a certain load, the structure will be on its limit. So for higher loads, the

column behaviour will be similar and hence, β will be similar as well tending to a minimum value

that can be achieved for the settlement reduction.

The load factor (𝑞𝑎

𝐻 𝛾′

𝑠⁄ ) is important once it influences the behaviour of the column

leading to a plastic, elastic or elasto-plastic deformation depending both on the range of load

applied and the column length. If the column is long and the load applied is small, we’ll have low

values for the load factor so the column will stay in the elastic state thus having a pronounced

β𝑒𝑙 𝑓𝑜𝑟 𝑧𝑦

𝐻≤ 0;

β = β𝑒𝑙 (1 − 𝑞𝑎

𝐻 𝛾′𝑠

× 1

2𝐶4) + β𝑝 (

𝑞𝑎

𝐻 𝛾′𝑠

× 1

2𝐶4) 𝑓𝑜𝑟 0 <

𝑧𝑦

𝐻≤ 1;

β𝑒𝑙 [ (𝑞𝑎

𝐻 𝛾′𝑠)

−1 ×

𝐶4

2] + β𝑝 [1 − (

𝑞𝑎

𝐻 𝛾′𝑠)

−1 ×

𝐶4

2] 𝑓𝑜𝑟

𝑧𝑦

𝐻≥ 1.

reduction of settlement as β ≈ β𝑒𝑙 calculated for the Elastic Method. As the load increases, the

column yields and plastic deformations dominate over the elastic ones.

Figure 17 – Effect of load factor on Settlement Reduction Factor fixing 𝝂𝒔 = 𝝂𝒄 = 𝟎. 𝟑, 𝑬𝒎,𝒄 𝑬𝒎,𝒔 = 𝟐𝟎⁄ , 𝝓𝑪′ =

𝟒𝟎°, 𝝍𝑪 = 𝟎°, 𝜸𝒄′ 𝜸𝒔

′ = 𝟎. 𝟕⁄ , 𝑲𝒊𝒏𝒊 = 𝟎. 𝟔 for method 4.

The Em ratio is not as important for method 4 as it is for methods 1 and 2 once the curves

for different ranges of Em ratio are very close to each other when considering usual values adopted

in practice for Ar (10-30%), as can be seen in Figure 18. The main reason for that is because upon

yielding the deformation process is no longer controlled by the stiffness of the column but by the

loading applied, as we’ve discussed above. So as the load increases, the initial influence of Em

ratio on settlement reduction becomes negligible (Pulko, Majes & Logar, 2011).

Figure 18 – Effect of 𝑬𝒎 ratio on Settlement Reduction Factor fixing 𝝂𝒔 = 𝝂𝒄 = 𝟎. 𝟑, 𝒒𝒂 𝑯𝜸𝒔′ = 𝟐⁄ , 𝝓𝑪

′ = 𝟒𝟎°,

𝝍𝑪 = 𝟎°, 𝜸𝒄′ 𝜸𝒔

′ = 𝟎. 𝟕⁄ , 𝑲𝒊𝒏𝒊 = 𝟎. 𝟔, for method 4.

As for the Poisson’s ratio of soil, we can see from Figure 19 that its influence on results

are not very significant as it happens for the other methods as well.

As for the column, the characteristic that influences β the most is the friction angle (Figure

21) once the Poisson’s ratio will have very little impact (Figure 20), even smaller if we consider

low values of Ar.

Figure 19 – Effect of 𝝂𝒔 on Settlement Reduction Factor fixing 𝝂𝒄 = 𝟎. 𝟑, 𝑬𝒎,𝒄 𝑬𝒎,𝒔 = 𝟐𝟎⁄ , 𝒒𝒂 𝑯𝜸𝒔′ = 𝟐⁄ , 𝝓𝑪

′ =

𝟒𝟎°, 𝝍𝑪 = 𝟎°, 𝜸𝒄′ 𝜸𝒔

′ = 𝟎. 𝟕⁄ , 𝑲𝒊𝒏𝒊 = 𝟎. 𝟔 for method 4.

Figure 20 – Effect of 𝝂𝒄 on Settlement Reduction Factor fixing 𝝂𝒔 = 𝟎. 𝟑, 𝑬𝒎,𝒄 𝑬𝒎,𝒔 = 𝟐𝟎⁄ , 𝒒𝒂 𝑯𝜸𝒔′ = 𝟐⁄ , 𝝓𝑪

′ =

𝟒𝟎°, 𝝍𝑪 = 𝟎°, 𝜸𝒄′ 𝜸𝒔

′ = 𝟎. 𝟕⁄ , 𝑲𝒊𝒏𝒊 = 𝟎. 𝟔 for method 4.

Figure 21 – Effect of friction angle of the column on Settlement Reduction Factor fixing 𝝂𝒔 = 𝝂𝒄 = 𝟎. 𝟑,

𝑬𝒎,𝒄 𝑬𝒎,𝒔 = 𝟐𝟎⁄ , 𝒒𝒂 𝑯𝜸𝒔′ = 𝟐⁄ , 𝝍𝑪 = 𝟎°, 𝜸𝒄

′ 𝜸𝒔′ = 𝟎. 𝟕⁄ , 𝑲𝒊𝒏𝒊 = 𝟎. 𝟔 for method 4.

Figure 22 – Effect of dilation angle of the column on Settlement Reduction Factor fixing 𝝂𝒔 = 𝝂𝒄 = 𝟎. 𝟑,

𝑬𝒎,𝒄 𝑬𝒎,𝒔 = 𝟐𝟎⁄ , 𝒒𝒂 𝑯𝜸𝒔′ = 𝟐⁄ , 𝝓𝑪

′ = 𝟒𝟎°, 𝜸𝒄′ 𝜸𝒔

′ = 𝟎. 𝟕⁄ , 𝑲𝒊𝒏𝒊 = 𝟎. 𝟔 for method 4.

The dilation angle affects considerably the results and as we can see in Figure 22, its effect

is more pronounced at the usual range of values used for Ar (10-30%). As the dilation angle grows

β reduces, meaning that settlements reduction will be more effective for higher angles. That’s

because dilatancy increases the column stiffness once the volume of columns will be increased

during yielding, so it’s expected that the dilatancy effect is more pronounced at higher loads when

the column yields over the entire length (Pulko, Majes & Logar, 2011). An important aspect that

must be pointed it out is that the dilation angle won’t be constant during loading, and after the

column achieves the peak shear strength it is unrealistic to expect that the material will retain its

ability to increase volume. So we should choose a dilation angle in accordance with the actual

behaviour of the column material during the range of expected deformations. That measurement

is very difficult to obtain and wouldn’t be precise nor constant, so it is more precautious to adopt

ψ = 0º which leads to more conservative predictions for the settlement reduction and meanwhile

will simplify the calculation process.

As we can see in Figure 23, the weight ratio doesn’t influence the results too much, the

curves are almost completely overlaid. This is reasonable once this parameter appears by itself just

in equation 40 to calculate C4. The same thing happens to the parameter K𝑖𝑛𝑖 as we can see in

Figure 24. So, in practice for designing we can adopted a medium value for 𝛾𝑐′

𝛾𝑠′⁄ = 0.7 and a

conservative value of K𝑖𝑛𝑖 = 0.6 getting values for β in favour of safety.

Figure 23 – Effect of 𝜸𝒄′ 𝜸𝒔

′⁄ ratio on Settlement Reduction Factor fixing 𝝂𝒔 = 𝝂𝒄 = 𝟎. 𝟑, 𝑬𝒎,𝒄 𝑬𝒎,𝒔 = 𝟐𝟎⁄ ,

𝒒𝒂 𝑯𝜸𝒔′ = 𝟐⁄ , 𝝓𝑪

′ = 𝟒𝟎°, 𝝍𝒄 = 𝟎°, 𝑲𝒊𝒏𝒊 = 𝟎. 𝟔 for method 4.

Figure 24 – Effect of 𝐾𝑖𝑛𝑖 on Settlement Reduction Factor fixing 𝝂𝒔 = 𝝂𝒄 = 𝟎. 𝟑, 𝑬𝒎,𝒄 𝑬𝒎,𝒔 = 𝟐𝟎⁄ , 𝒒𝒂 𝑯𝜸𝒔′ = 𝟐⁄ ,

𝝓𝑪′ = 𝟒𝟎°, 𝝍𝒄 = 𝟎°, 𝜸𝒄

′ 𝜸𝒔′ = 𝟎. 𝟕⁄ for method 4.

So we end up with the conclusion that for method 4 the parameters that influences the most

on designing and that are more important to be measured precisely are the load factor (𝑞𝑎

𝐻 𝛾′

𝑠⁄ )

and the friction angle of the column (𝜙𝐶′ ).

5. Methods comparison

By comparing the results obtained for the four different methods analysed, we intend to

identify conditions that may converge results of different methods in order to try to point out one

of them as a general method that can be used more comprehensively ensuring a good level of

reliability and that is also in favour of safety. The main goal of this research is to come up with the

conclusion that one of the four methods can be used in general providing results more close to real

behaviour of settlement reduction in the soil than the others methods, allowing flexibility for the

designing once you can use the mathematical model to make reliable predictions without the need

for manipulation of data available.

As discussed previously, for methods 1 and 2 the main design parameters are the area ratio

and the Em ratio, so we can see how this both methods respond to changes of these key parameters

(Figure 25). Method 2 gives more conservative values for the improvement achieved with the

installation of stone columns, once the curves are more marked while method 1 gives a linear

response with coincident coordinates for Ar = 0 and Ar = 1. The impact for usual values of Ar

(10-30%) can be better visualized in Figure 12.

Figure 25 – Effect of 𝑬𝒎 ratio on Settlement Improvement Factor fixing 𝝂𝒔 = 𝝂𝒄 = 𝟎. 𝟑 for method 2 in

comparison with the effect on method 1.

As we discussed previously in items 4.1 and 4.2, in practice designers use to apply low

values of 𝐸𝑚 ratio in their calculations based on previous experience, when using methods 1 or 2.

Table 1 gives an idea of the magnitude of the reduction necessary on 𝐸𝑚 ratio using method 1 to

get similar results as method 2 when we fix as target 𝛽 = 0.5. As can be seen in the table, when

method 1 is chosen we get a reduction around 75-80% of the value of 𝐸𝑚 ratio to be able to get

the same area ratio obtained from method 2.

Table 1 – Correspondent 𝑬𝒎 ratio obtained for method 1 when fixing the area ratio obtained for a determined

𝑬𝒎 ratio using method 2 (fixing 𝝂𝒔 = 𝝂𝒄 = 𝟎. 𝟑) when targeting 𝜷 = 𝟎. 𝟓.

𝑬𝒎,𝒄 𝑬𝒎,𝒔⁄

(Method 2) 𝑨𝒓

𝑬𝒎,𝒄 𝑬𝒎,𝒔⁄

(Method 1)

10 0,142 8

20 0,070 15

30 0,045 23

40 0,033 31

As the results obtained with method 2 are more conservative and both methods assume

simply the elastic behaviour for both materials, the imperfections of modelling will be very alike,

being preferable to use values from method 2 to favour safety.

Comparing methods 2 and 3, we end up getting different key parameters for designing,

because while the second method is mainly based on Em ratio, the third is mainly based on the

friction angle of the column, as discussed on item 4 of this same report. As we can see from Figure

26, the two methods give completely different range of values for β and it’s evident that Priebe’s

Method is much more conservative. Fixing Ar = 0.1, for example, the results for the most

optimistic case using method 3 (higher 𝜙𝑐′ value) matches the result for the worst case using

method 2 (lower Em ratio). This happens because Priebe’s Method is considering the reduction in

capacity of columns to handle stresses due to yielding.

For security reasons, once again it’s reasonable to adopt for designing the method giving

more conservative results, which is method 3. Actually, Priebe’s Method is the mathematical

modelling most used nowadays by ground engineering companies when applying Stone Columns

as a soil improvement solution.

Figure 26 – Effect of 𝑬𝒎 ratio on Settlement Reduction Factor fixing 𝝂𝒔 = 𝝂𝒄 = 𝟎. 𝟑 for method 2 in comparison

with the effect of the friction angle fixing 𝝂𝒔 = 𝟎. 𝟑 for method 3.

Finally, we are able to compare all four methods in Figure 27. The first interesting highlight

is that the curves obtained for method 2 and method 4 (when 𝑞𝑎 𝐻𝛾𝑠′ = 0⁄ ) are exactly overlaid and

as the load factor grows, the curve for method 4 deviates from the curve for method 2 giving more

conservative values. That means that the Elastic Method will present good accuracy in situations

where the load applied on stone columns are very little, tending to zero. This condition is not usual

in practice because Stone Columns are most used in cases that you need to improve the soil so you

can guarantee that it will support a considerable amount of load applied on top. We can see in the

graph (Figure 27) that the curve for method 1 is offset down the overlaid curves. So, once in reality

we usually are dealing with non-zero values for the load ratio, is not reasonable to choose neither

methods 1 or 2 for these situations.

Another situation where a load ratio tending to zero is possible is when the load applied is

quite low compared to a very high column length (𝐻) and/or a high effective weight of soil (𝛾𝑠′),

giving a very high value for the denominator and a very low value for the numerator,

approximating the load ratio to zero. But economically speaking, this situation wouldn`t be very

efficient once we`ll have very high efforts and costs to construct very deep columns that will be

used to support very low loads, so maybe would be more reasonable to apply another improvement

method in situations like this.

Figure 27 – Comparison of the effect on Settlement Reduction Factor using method 1 (fixing 𝑬𝒎,𝒄 𝑬𝒎,𝒔 = 𝟐𝟎⁄ ),

method 2 (fixing 𝑬𝒎,𝒄 𝑬𝒎,𝒔 = 𝟐𝟎⁄ and 𝝂𝒔 = 𝝂𝒄 = 𝟎. 𝟑 ), method 3 (fixing 𝝂𝒔 = 𝟎. 𝟑 ) and method 4 (fixing 𝝂𝒔 =

𝝂𝒄 = 𝟎. 𝟑, 𝑬𝒎,𝒄 𝑬𝒎,𝒔 = 𝟐𝟎⁄ , 𝝓′ = 𝟒𝟎°, 𝝍 = 𝟎°, 𝜸𝒄′ 𝜸𝒔

′ = 𝟎. 𝟕⁄ , 𝑲𝒊𝒏𝒊 = 𝟎. 𝟔).

Looking to low values of area ratios (0-15%), Priebe`s Method is quite close to the results

obtained using load factors of 0.5 to 1 for method 4, while for area ratio usual values (15-30%) the

curves for 𝜙𝑐` equals 35º, 40º and 45º using method 3 approximates the curves with load ratios of

0.5, 1 and 2 respectively, using method 4. So we can conclude that for small load pressures applied,

Priebe’s Method leads to good results and that the higher the friction angle of the column material

used, the higher can be the range of loads applied and still get accuracy using method 3 to analyse

the Settlement Reduction Factor.

Meanwhile we can see that compared to method 4, Priebe’s results sometimes are

underestimating β and sometimes are overestimating β, so when designing a structure, it would be

difficult to state for sure in which situation you are in. So if you choose to adopt method 3, in

general, your solution would be fine as the values obtained for β would be close to or higher than

the ones for method 4, assuring that the columns won’t fail. In this case you could be

underestimating your structure as well, meaning that you would be spending more money than

necessary to build your structure. But also, there would be even a tiny chance that your solution

will lead to a β value lower than you would get from method 4, and in this case your structure

would be overestimated and it wouldn’t work well or it could even collapse. So, it’s more

reasonable to adopt a more accurate and conservative method as method 4.

So, as we can see from the comparison of the four methods, the last one introduces a very

important key parameter for designing, the load factor (𝑞𝑎

𝐻 𝛾′

𝑠⁄ ), neglected by methods 1, 2 and 3.

The range of load applied is responsible for a change in the response of the column in terms of

deformations: for small loads, the column will tend to present an elastic behaviour with no yielding

but as the load grows, the column will tend to present a plastic behaviour once the column yields.

That explains why the results for small values of load ratios get closer to the results obtained with

the Elastic Method while for medium load ratios the results get closer to Priebe’s Method, which

considers the plastic behaviour of column as uniform for the whole column depth.

As we’ve seen before on item 3, the only method that considers this elasto-plastic state on

column deformation is method 4, so the calculations get more accurate solutions as the

mathematical modelling is refined. Therefore, we can imply that method 4 is the most generic

since it includes both elastic and plastic behaviours without the need for prior knowledge,

producing more reliable solutions in favour of safety for all ranges of loads.

6. Cases of study

Now that we have a good understanding of how the methods are affected by each one of

the parameters and how each method influences the solutions obtained for the Settlement

Improvement/Reduction Factor, the aim is to explore some fictional cases of an embankment

construction and one real case that uses Stone Columns as a solution. Doing so, we can now

compare how the methods would respond to a practical engineering problem.

It would be ideal to analyse as many real cases as possible, so we could get a good range

of applications and their respective results, so then we could get more reliable and general

conclusions. To be able to analyse a real case of construction using Stone Columns, we would

need 3 types of information about the specific site being analysed:

1) The values for the soil and column material properties obtained with laboratory tests;

2) Data of real settlement in the middle of foundations measured on field for a

construction with the same specificities WITHOUT Stone Columns;

3) Data of real settlement in the middle of foundations measured on field for a

construction with the same specificities after the installation of Stone Columns.

Unfortunately, it was very hard to find published papers with real cases of Stone Columns

implementation that provided all those characteristics. As noticed, normally the ground

engineering companies adopt the same material for Stone Columns, so they don`t necessarily do

the laboratory tests each time, instead they may have a record to help on setting the values for

designing. The same thing may happen to the soil properties. Based on the data obtained from the

boreholes, they can choose values from the company records and previous experience.

In the end, we could pick just one real case (see section 6.2), based on a paper published

by McCabe, Nimmons and Egan (2009) and indicated by Balfour Beatty, a local Ground

Engineering company that operates in the UK.

6.1 Fictional cases of study

Figure 28 illustrates a construction of an embankment of height h over a layer of soft soil

improved with Stone Columns with depth H. On the next few pages we’ll be analysing what

happens with the Settlement Reduction Factor when we change two main parameters for design:

the load factor (𝑞𝑎

𝐻 𝛾′

𝑠⁄ ) which in this case will change as we change the height of the embankment

constructed; and the 𝐸𝑚 ratio, which will change as we change the values for the constrained

modulus of the soil and column. As for the remaining parameters, it will be assigned fixed values

(see Table 2), based on standard values used in practice and the design criteria discussed before in

this same paper.

Figure 28 – Fictional case of embankment supported over Stone Columns.

Table 2 – List of parameters used for the fictional case.

Embankment fill material

𝜸𝒆𝒎𝒃𝒂𝒏𝒌𝒎𝒆𝒏𝒕 (𝑲𝑵/𝒎𝟑) 20

Columns

𝜸𝒄 (𝑲𝑵/𝒎𝟑) 20

𝜸′𝒄 (𝑲𝑵/𝒎𝟑) 10

𝑬𝒎,𝒄 (𝑴𝑷𝒂) 40

𝝓𝒄` (𝒅𝒆𝒈𝒓𝒆𝒆𝒔) 40

𝝍𝒄 (𝒅𝒆𝒈𝒓𝒆𝒆𝒔) 0

𝝂𝒄 0.3

𝑯 (𝒎) 10

Soft soil

𝜸𝒔 (𝑲𝑵/𝒎𝟑) 17

𝜸′𝒔 (𝑲𝑵/𝒎𝟑) 7

𝝂𝒔 0.3

𝑲𝒊𝒏𝒊 0.8

Meanwhile, we’ll vary 𝐸𝑚,𝑠 from 1 to 4 MPa to represent a very soft clay (lower value)

and a soft clay (higher value). Doing so, we’ll get 𝐸𝑚 ratio varying from 𝐸𝑚,𝑐 𝐸𝑚,𝑠 = 40 1⁄⁄ = 40

until 𝐸𝑚,𝑐 𝐸𝑚,𝑠 = 40 4⁄⁄ = 10, which represent the typical maximum and minimum values used

in practice for designing.

As for the embankment height, we’ll vary it from 2m until 10m, which represents

respectively a considerably low and high embankment to make sure we’ll cover a good range of

values for the load factor (𝑞𝑎

𝐻 𝛾′

𝑠⁄ = (2 × 20) (10 × 7) = 0.571⁄ for a 2m embankment until

𝑞𝑎

𝐻 𝛾′

𝑠⁄ = (10 × 20) (10 × 7) = 2.875⁄ for a 10m embankment).

It’s important to remember that all four methods quoted in this study are just valid for

situations where we have a very wide base for the embankment if compared to the depth that stone

columns reach, which approximates the situation to an infinite array of unit cells as we’ve

explained before in section 2.2.

Once we’ve stablished values for all parameters we are able to construct 4 different

possibilities:

1) A 2 meters high embankment with 𝐸𝑚 ratio equals 40;

2) A 2 meters high embankment with 𝐸𝑚 ratio equals 10;

3) A 10 meters high embankment with 𝐸𝑚 ratio equals 40;

4) A 10 meters high embankment with 𝐸𝑚 ratio equals 10.

Finally, fixing a target of 𝛽 = 0.5 we can see what would be the area ratio in each situation

using all four methods for analysis (see Table 3).

Table 3 – Values for area ratios obtained when targeting 𝛽 = 0.5 for all 4 possibilities of embankment.

SITUATION 1

(ℎ = 2𝑚 and

𝐸𝑚,𝑐 𝐸𝑚,𝑠 = 40)⁄

SITUATION 2

(ℎ = 2𝑚 and

𝐸𝑚,𝑐 𝐸𝑚,𝑠 = 10)⁄

SITUATION 3

(ℎ = 10𝑚 and

𝐸𝑚,𝑐 𝐸𝑚,𝑠 = 40)⁄

SITUATION 4

(ℎ = 10𝑚 and

𝐸𝑚,𝑐 𝐸𝑚,𝑠 = 10)⁄

METHOD 1 0.027 0.111 0.027 0.111

METHOD 2 0.034 0.142 0.034 0.142

METHOD 3 0.171 0.171 0.171 0.171

METHOD 4 0.121 0.170 0.212 0.243

As we can see in Table 3, for all situations the area ratios found for method 1 are lower

than the area ratios for method 2 which are much lower than the area ratios for method 4. Also,

the results for methods 1 and 2 are exactly the same for situations 1 and 3, and for situations 2 and

4. This is expected once method 2 just take into account the Poison’s ratio of both column and soil

and the 𝐸𝑚 ratio while method 1 just considers the 𝐸𝑚 ratio, and all these parameters are

maintained exactly the same for these pars of situations (1/3 and 2/4). So, we can presume that for

all situations, methods 1 and 2 will be excessively overestimating the reduction of settlements,

even for small loads (represent by the short embankment). Hence, it will result in much less

substitution of soil than necessary to assure the capacity of the improved subsoil to receive the

posterior loads on top.

One thing that highlights it`s that Priebe`s results are exactly the same for all situations,

which was expected once the it`s solution is influenced only by the friction angle of the column

and the Poison`s ratio of the soil, both unchanged for this analysis.

The most unpredictable result shown with this study is that there`s not a uniform pattern

for solutions obtained from methods 3 and 4. Sometimes, method 3 gives values for the area ratio

lower than the values obtained from method 4, sometimes they end up giving pretty close values,

and sometimes method 3 gives values higher than method 4. Based on Table 3 we can say that

seems like for low loads and high 𝐸𝑚 ratio, method 3 is more conservative than method 4. As 𝐸𝑚

ratio decreases, maintaining the load, seems like method 3 solutions gets closer to method 4.

Meanwhile, when it comes to large loads, method 4 seems to be always more conservative than

method 3. Nevertheless we can state that as rule once we would need a more extensive study of all

possibilities (changing all parameters, one at a time to see if we get the same pattern obtained with

this simple study) and comparing the solutions obtained with real cases of stone columns

implemented.

Even not being able to stablish a definitive rule to define in which cases is better to use

method 3 or method 4, this study showed that in general cases method 4 is the most safe to adopt.

We could also get some guidelines to understand the implications of choosing each one of the

methods presented.

6.2 Real case of study

In this section we`ll analyse a real case of an embankment founded on Stone Columns

which was published by Munfakh, Sarkar and Castelli (1983) and was posteriorly investigated by

Mccabe, Nimmons and Egan (2009). The project called Jourdan Road Terminal, consists in a port

facility constructed on the north bank of the Mississippi river in New Orleans, U.S.A. The selected

scheme consists of a relatively narrow pile supported deck, with its overall stability improved by

the soft cohesive soil behind the deck which was improved with the installation of Stone Columns.

The site contains approximately 18m of soft soil (soft clay and silty clays with some percentage

of organic material due to overlaying deltaic alluvial deposits) with a primary bearing layer of

medium to very dense sand. On top of the Stone Columns, a reinforced earth-embankment of 4.3m

high was constructed. Figure 29 illustrates the selected scheme.

The Stone Columns were installed using the vibro-replacement technique with top feed

process, using crushed limestone as fill in material. The grid of columns was constructed in a

triangular pattern with a space 𝑠 = 2,1𝑚 center to center, with an average diameter 𝑑 = 1.11𝑚

resulting an area ratio 𝐴𝑟 = 0.25. The columns go through all depth of soft soil with 𝐻 = 18𝑚.

Figure 29 – Selected scheme used on the real case study (Munfakh, Sarkar and Castelli, 1983).

Before apply the selected solution, Munfakh, Sarkar and Castelli (1983) studies results

obtained from an implementation of a full scale prototype field test, which includes:

Soil profile of the area;

Undrained shear strength test;

Water content in the soil;

Direct shear test;

Monitoring magnitude and rate of settlements at surface and various depths during

constructions and continuously during 3 months later;

Monitoring magnitude and pattern of subsurface lateral deformations;

Measure of vertical load distribution at the top of both soil and columns.

Therefore it was possible to obtain the internal friction angle of the soil from the direct

shear test, so the final value recommended by Munfakh, Sarkar and Castelli (1983) was 𝜙𝑐` = 42°.

As for the vertical load distribution, Munfakh, Sarkar and Castelli (1983) presented a final

value of 0.32kg/cm² on top of the soil and a range of 0.85-1.1 kg/cm² on top of the column. Once

the area ratio is known, we can obtain the weighted average of uniform load pressure on the system

soil-column, given by:

𝑞𝑎̅̅ ̅ = 0.32 × (1 − 0.25) + (0.85+1.1

2) × 0.25 = 0.485

𝑘𝑔𝑐𝑚2⁄ = 47,58 𝑘𝑁

𝑚2⁄ (Eq. 47)

This value found may be lower than the one obtained by considering the load applied on

top as the weight of the reinforced earth used for embankment multiplied by its height (𝑞𝑎̅̅ ̅ = 20 ×

3.4 = 68 𝑘𝑁𝑚2⁄ ) because the measurements were made on field, so they may be considering the

relieve caused on overburden due to the excavation made previously to construct the earth

reinforced embankment.

At last, from data obtained from the settlements monitoring in the untreated soil and them

in the improved soil, Munfakh, Sarkar and Castelli (1983) obtained that the Stone Columns

reduced total settlements by about 40%, leading to 𝛽 = 0.6 and 𝜂 = 1.7.

The only downside of this paper it`s that we can`t get enough data about the soil and column

material properties. Nevertheless, as we`ve seen in sections 5 and 6.1, method 4 is the most general

to be used and for this method, as we`ve seen in section 4.4, the most two influent parameters are

the load factor (which can be obtain once we know the uniform load pressure) and the internal

friction angle of the column (which has been determined as well). So, once we know that the soil

is a soft clay and the column fill material is limestone, we can set typical values for the others

parameters (see Table 4). As for the 𝐸𝑚 ratio, we`ll analyse what happens for typical values: 10,

20 and 40.

Table 4 – List of parameters used for the real case.

Columns

𝜸𝒄 (𝑲𝑵/𝒎𝟑) 21

𝜸′𝒄 (𝑲𝑵/𝒎𝟑) 11

𝝓𝒄` (𝒅𝒆𝒈𝒓𝒆𝒆𝒔) 42

𝝍𝒄 (𝒅𝒆𝒈𝒓𝒆𝒆𝒔) 0

𝝂𝒄 0.3

𝑯 (𝒎) 18

Soft soil

𝜸𝒔 (𝑲𝑵/𝒎𝟑) 17

𝜸′𝒔 (𝑲𝑵/𝒎𝟑) 7

𝝂𝒔 0.3

𝑲𝒊𝒏𝒊 0.8

Finally, fixing a target of 𝐴𝑅 = 0.25 we can see what would be 𝛽 in each situation using

all four methods for analysis (see Table 5).

Table 5 – Values for 𝛽 obtained when targeting 𝐴𝑅 = 0.25.

SITUATION 1

(𝐸𝑚,𝑐 𝐸𝑚,𝑠 = 10)⁄

SITUATION 2

(𝐸𝑚,𝑐 𝐸𝑚,𝑠 = 20)⁄

SITUATION 3

(𝐸𝑚,𝑐 𝐸𝑚,𝑠 = 40)⁄

METHOD 1 0.308 0.173 0.093

METHOD 2 0.361 0.216 0.120

METHOD 3 0.358 0.358 0.358

METHOD 4 0.375 0.254 0.182

As we can notice from Table 5, even for method 4 which we consider the most refined one,

gives β values much lower than the one obtained by measured settlements (𝛽 = 0.6). And as we

can see either, method 3 gives results close to the ones obtained for method 2, meaning that we

have a situation where the column behaviour is tending to elastic rather than plastic. This may be

due to the very deep column being used (𝐻 = 18𝑚) and low load 𝑞𝑎̅̅ ̅ = 47,58 𝑘𝑁𝑚2⁄ ) being

applied, what makes the load factor low (𝑞𝑎

𝐻 𝛾′

𝑠⁄ = 0.378) and also may be due to a very high

friction angle used for the column (𝜙𝑐` = 42°), all leading to a situation where yield doesn’t reach

a high depth along the column.

Even if me change this parameters to more conservative values by considering the load

applied on top as the weight of the reinforced earth used for embankment multiplied by its height

(𝑞𝑎̅̅ ̅ = 20 × 3.4 = 68 𝑘𝑁𝑚2⁄ , which leads to a load factor 𝑞

𝑎𝐻 𝛾′

𝑠⁄ = 0.54) and using a very low

value for the friction angle of the column such as 𝜙𝑐` = 35°, by maintaining the others parameters

and using 𝐸𝑚,𝑐 𝐸𝑚,𝑠 = 10⁄ (which, by the way is a very low ratio), we still gets a maximum 𝛽 =

0.415 for an area ratio of 0.25, which is still much lower than the value measured (𝛽 = 0.6).

Now, if we also use a very low value of 𝐾𝑖𝑛𝑖 = 0.3 to make it closer to 𝐾𝑝𝑐, which is the

maximum value that can be reached before rupture, we can get a a maximum 𝛽 = 0.551 which

gets closer to the measured value. Do, to get a 𝛽 = 0.6 we would have to either get a very strange

combination of unusual low values for the parameters such as the friction angle and the 𝐸𝑚 ratio

and 𝐾𝑖𝑛𝑖, combined with a much higher load factor (𝑞𝑎

𝐻 𝛾′

𝑠⁄ ). This scenario would be very unlikely

but not impossible as we had to assume some values for parameters that weren`t specified by the

case analysed by Munfakh, Sarkar and Castelli (1983).

Also, we don`t have much information about the soil profile much deeper than the base

layer (𝐻 = 18𝑚) so it may happen that we would find again a soft layer deeper that would be

highly contributing for the total settlement, and once it is not being improved by the installation

of Stone Columns, this would explain why the 𝛽 measured on field is higher than the ones obtained

with method 4.

So we can conclude that even using a more refined modelling, we need to make sure the

assumptions made by this model suits the specific case being analysed, otherwise we will get

predictions that won`t correspond to reality and that may overestimate the capacity of columns, as

happed in this particular case, making the structure doesn’t work well or even reaches failure.

So we can see the importance of monitoring during construction and first of all, of

obtaining reliable field tests to determine the real characteristics of the soil and column materials

being used to determine either or not the mathematical modelling being used is the adequate or not

and to set realistic values for parameters.

7. Conclusions

We note that the inaccuracy to determine the Settlement Improvement/Reduction Factor is

due to three main reasons:

1) Lack of knowledge of precise values for the materials properties (soil and/or column)

used specifically for the construction being analysed;

2) Lack of proper monitoring during the construction phase to ensure that columns

geometry (diameter, depth and spacing grid pattern) properly match the design

specifications and that the construction itself is not disturbing the adjacent natural soil

(for example, the walls of the columns should remain stable during the digging so the

natural soil don’t end up mixed to the column material, what can affect the behaviour

of the column by lowing its resistance an hence the settlements can be higher than the

ones predicted by calculations);

3) Lack of precision of the mathematical modelling used (the model makes unrealistic

assumptions and/or fails to take into account important factors), leading to results that

have to be empirically calibrated to assume values that can be used with minimal

reliability (in this case, we’ve seen that in practice designers are used to adjust the

values used for 𝐸𝑚 ratio when adopting methods 1 or 2 in order to address the flaws in

these models. When it comes to method 3, Priebe come up with some steps to correct

the value for η that weren’t mentioned in this report but can be found in Priebe (1995)).

There is not a preponderant imperfection among the three of them mentioned before when

it comes to its impact over the final results. Therefore, in a first approach we can assume that each

one contributes equally to the accuracy of final results obtained for the Settlement

Improvement/Reduction Factor. Thus, by correcting one of this imperfections, we would be

improving the accuracy of results by 33,3% in theory, which is a reasonable amount. So, it is

worthwhile investing in a mathematical modelling more refine as method 4, for example.

As dully noted before, method 4 end up being the most complex one once it involves many

parameters ignored previously by the others methods mentioned, besides it corrects some of the

wrong assumptions made before. Method 4 is more refine once it considers the elasto-plastic

behaviour of the column material and the effect of dilation over it. Thereafter, this method is able

to consider the loss of efficiency of columns being used due to deformations, what leads to

predictions of settlement improvement more close to reality behaviour. Even though, in some

situations (as we’ve seen in the real case analysed during this research in section 6.2) even method

4 may not represent a good approximation of the problem, leading to β values much lower than

reality, that’s why we should be careful when using a mathematical modelling to be very aware

the restrictions made by its assumptions.

From the analyses presented in this report comparing the solutions obtained using all four

known methods (see section 5) and by comparing the solutions obtained applying the methods to

a very common construction with typical values for material parameters and geometric dimensions

(see section 6), we could note that method 4 can be used as a general solution with more reliability.

Its greater correspondence with reality is pretty much due to the fact that we know the magnitude

of load being applied on the top of the system soil/column. This defines the way each material will

behave to the forces being applied, how the materials are going to deform and therefore how much

the whole soil/column system will help reducing settlements.

We could also demonstrate that is possible to simplify the utilization of method 4 by

building a custom spreadsheet and setting some default values for parameters without significant

loss of accuracy (see section 4.4).

Once method 4 has proven a great potential to predict Settlement Improvement/Reduction

Factor in an easy and reliable way, we would recommend further studies based on real field data

so we could get a good range of applications and their respective results and then we could get

more precise and general conclusions to guide the designing process.

APPENDIX A - NOTATION

Subscripts/superscripts:

S Soil.

C Column.

el Elastic.

P Plastic.

` (Apostrophe) Effective value.

Symbols used for parameters:

Ar Area replacement ratio. Ar = Ac/Ae

Ac Area of stone column in the “unit cell”. Ac = πDc²/4

Ae Total equivalent area of the “unit cell”. Ae = πDe²/4

De Equivalent diameter of the cylinder representing the “unit cell”.

E Young’s modulus.

Em Oedometric (constrained) modulus. Em = [E(1-ν)]/[(1-2ν)(1+ν)]

G Shear modulus. G = E/[2(1+ν)]

H Stone Column vertical length (depth).

K Bulk (volumetric) modulus. K = E/[3(1-2ν)]

Kac Lateral active earth pressure coefficient (minimum value). Kac = (1-sinϕ)/(1+sinϕ)

Kini Initial lateral earth pressure coefficient. Kini = (1-sinϕ`)(OCR^sinϕ’)

Kpc Lateral passive earth pressure coefficient (maximum value). Kpc = (1+sinϕ)/(1-sinϕ)

Kψ Dilation deformation coefficient. Kψ = (1+sinψ)/(1-sinψ)

qa Uniform vertical load applied.

ϕ Friction angle.

ψ Dilatancy angle.

λ Lamé’s constant. λ=(νE)/[(1-2ν)(1+ν)]

ν Poisson`s ratio.

γ Natural weigth.

σ Tensions.

ε Displacements.

η Settlement Improvement Factor.

β Settlement Reduction Factor.

References:

CASTRO, J.; SAGASETA, C. (2008) Consolidation around stone columns. Influence of column

deformation. International Journal for numerical and analytical methods in geomechanics, vol.

33, issue 7, p. 851–877.

BABU, G.L.S. Ground Improvement. Lecture 9, NPTEL course. Department of Civil Engineering,

Indian Institute of Science.

BALLAM, N.P.; BOOKER, J.R. (1981) Analysis of rigid rafts supported by granular piles.

International Journal for numerical and analytical methods in geomechanics, vol. 5, p. 379-403.

BARKSDALE, R.D.; BACHUS, R.C. (1983). Design and construction of Stone Columns.

Research Development and Technology, US Department of Transportation, vol I and II. Report

FHWA/RD-83/026-027.

MANI, K.; NIGEE, K. (2013). A study on Ground Improvement using Stone Column Technique.

International Journal of Innovative Research in Science, Engineering and Technology, vol. 2, issue

11, ISSN: 2319-8753.

McCABE, B.A.; NIMMONS, G.J.; EGAN, D. (2009). A review of field performance of stone

columns in soft soils. Proceedings of the Institution of Civil Engineerings, Geotechnical

Engineering, 162, paper 900019, issue GE6, p. 323-334.

MUNFAKH, G.A.; SARKAR, S.K.; CASTELLI, R.J. (1983). Performance of a test embankment

founded on stone columns. Proceedings of the International Conference on Advances in Piling and

Ground Treatment for Foundations, London, paper 20, p. 259-265.

PRIEBE, H. J. (1976). Abschätzung des Setzungsverhaltens eines durch Stopfverdichtung

verbesserten Baugrundes. Die Bautechnik 53, H.5, p. 160-162.

PRIEBE, H. J. (1995) The Design of Vibro Replacement. Ground Engineering, Technical paper

GT 12-61E.

PULKO, B.; MAJES, B. (2005). Simple and accurate prediction of settlements of stone column

reinforced soil. In: Proceedings of the 16th International Conference on Soil Mechanics and

Geotechnical Engineering, vol. 3, p. 1401-1404.

PULKO, B.; MAJES, B.; LOGAR, J. (2011) Geosynthetic-encased stone columns: Analytical

calculation model. Geotextiles and Geomembranes, vol. 29, p. 29-39.