Paper title · Web viewIn that case, most of the load is transmitted to the inclusion head through the compression of a shear cone volume (inverted cone or pyramid) centered on the

Sociedad Mexicana deIngeniería Geotécnica, A.C.

XXVI Reunión Nacional de Mecánica de Suelos e Ingeniería Geotécnica

Noviembre 14 a 16, 2012 – Cancún, Quintana Roo

Discusión sobre las diferencias de la metodología de diseño entre las inclusiones granulares y las inclusiones rígidas

Discussion of differences in design methodology between granular and grouted inclusions

Brandon BUSCHMEIER1, Frederic MASSE2

1Senior Design Engineer, Menard USA, Pittsburgh, Pennsylvania, United States of America2Vice-President of Engineering, Menard USA, Pittsburgh, Pennsylvania, United States of America

ABSTRACT: Inclusions are a type of ground improvement that provide an improvement of a soil mass by insertion of a material with better characteristics than the surrounding soils. There are primarily two types of inclusions: “Soft” inclusions which are constructed of granular material (Stone Columns, Rammed Aggregate Piers, VibroPiers...), and “rigid” inclusions which are constructed of grout inserted with or without pressure (VibroConcrete Columns, Controlled Modulus Columns, Soil Mixing Columns…). While the central principle of the inclusions is similar (i.e., overall improvement of characteristics of the soil mass below the structure), the load transfer mechanism is radically different for each technique. Therefore, the applicable design methodology is specific to the type of inclusions and there exists no universal method of design which spans across all inclusions. This paper will present an in-depth comparison between the standard design methods used for “soft” granular-type inclusions (Priebe modified, Balaam & Booker, Goughnour & Bayuk...) and the methods used for “rigid” inclusions which mainly rely on finite element analysis. The paper will explain why the design for soft inclusion methods cannot be applied to rigid inclusions and vice-versa. Each design method will be reviewed in detail to better emphasize its limitations and advantages.

RESUMEN: Un tratamiento por medio de inclusiones consiste en una mejora de la masa del terreno por medio de elementos cuya rigidez es mucho mayor que aquella del suelo que las contiene. Existen principalmente dos tipos de inclusiones: inclusiones flexibles constituidas por un material granular (columnas de grava, Geopiers, VibroPiers...), e inclusiones rígidas, constituidas por un material cementante con una puesta en obra con o sin presión (columnas de cemento vibrocompactadas, columnas de módulo controlado, columnas de soil mixing ...). Partiendo de la base de que el principio de ambos tipos de inclusiones es similar (es decir, la mejora en general de las características de la masa del terreno de cimentación), se tiene sin embargo que el mecanismo de transferencia de carga es radicalmente diferente en cada tipo de técnica. Por tanto, la metodología de diseño aplicable es específica para cada tipo de inclusión, no existiendo ningún método universal de diseño aplicable a todos los tipos de inclusiones. En el presente documento se despliega una comparación en profundidad entre los métodos de diseño estándar que se utilizan para las inclusiones granulares (Priebe modificado, Balaam y Booker, Goughnour y Bayuk...) y los métodos utilizados para las inclusiones rígidas, basados principalmente en el análisis por medio de elementos finitos. En el siguiente texto se explica por qué los métodos de diseño de las inclusión granulares no se pueden aplicar a las inclusiones rígidas y viceversa. Cada método de diseño será revisado en detalle con el fin de analizar y resaltar tanto sus limitaciones como sus ventajas.

1 INTRODUCTION1.1 General overview of ground improvement with inclusionsThe use of ground improvement methods to provide the appropriate amount of foundation support is a growing and constantly evolving industry. Traditional structural foundation methods including ‘shallow’ and ‘deep’ foundations do not adequately address the wide range of settlement tolerances, soil conditions, and loading considerations inherent to each unique potential project. In some instances, ‘shallow’ foundation solutions are simply not feasible due to soil and loading constraints. Furthermore, ‘deep’ foundation solutions are used to ‘bridge’ the soft compressible soils, ensuring that the entirety of the load in the structure is transmitted through individual

elements into competent layers below the ground surface (bedrock, dense residual soils). Highly concentrated structural loads are transferred into the individual elements by means of direct contact requiring the use of structurally rigid surficial components (structural slabs, grade beams, and pile caps).

Ground improvement bridges the gap between these two traditional methods, offering adequate foundation support while using more cost-effective techniques and design methodologies. Inclusions are a type of ground improvement used to provide a reinforcement of the soil mass through the vertical insertion of a material with better characteristics than the surrounding soils. Much like deep foundations, inclusions are installed through soil layers with high compressibility and/or low bearing capacity and

SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.

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terminated in a suitable bearing stratum. However, unlike deep foundations, ground improvement inclusions rely on a distribution of load between the soil and the inclusions. By doing so, some of the structural load is applied directly to the soil in a proportion equal to its bearing capacity and compressibility limits, while the rest of the load is shed into the vertical inclusions down to the bearing stratum. Load sharing is a critical aspect in the performance and design of the various ground improvement methods.

There are primarily two types of inclusions commonly used in various forms throughout the world: “Rigid” inclusions and “soft” inclusions. Both types are suitable for a wide range of applications and projects (industrial tanks, embankments, warehouses, buildings, etc.).

2 OVERVIEW OF INCLUSION TYPES2.1 “Soft” inclusionsInclusions are considered to be “soft” when the inclusion material has minimal cohesive properties. Soft inclusions are typically constructed with a granular material with stiffness 5 to 10 times larger than that of the surrounding soil. Soft inclusions require lateral confinement from the surrounding soil in order to provide effective vertical support. Some amount of lateral deformation must occur before the granular material is sufficiently engaged and confined by the surrounding soils.

Soft inclusions are often cylindrical in shape and installed on a regular grid pattern dependent on the soil properties and applied load. Both the diameter and spacing of the inclusions are considered, and the resultant replacement ratio (surface area of inclusion over unit area of treatment grid) is the most commonly understood design parameter. Typical replacement ratios for soft inclusions are on the order of 10 to 30% of the soil mass.

Soft inclusions are installed using a variety of methods which penetrate, vibrate, or auger through the unimproved soil mass to create a cavity for the introduction of granular materials. As the cavity is filled with granular material, vibratory probes and/or ramming components are used to densify the granular materials. Depending on the technique, the installation methods may greatly modify the properties of the in situ soil mass, thereby modifying the performance of the soft inclusion.

2.2 “Rigid” inclusionsInclusions are considered to be “rigid”’ when the inclusion material displays a significant permanent cohesion. The stiffness of the inclusion is much larger than that of the surrounding soil, thereby attracting a larger portion of the applied loads. Instead of truly improving the soil, the rigid inclusion acts as a reinforcement of the soil mass. Due to the

strong cohesion of the material, rigid inclusions do not rely on the confinement of the soil for stability and performance.

Rigid inclusions are often smaller in diameter than soft inclusions ranging from 200 mm to around 800 mm. Nevertheless, much larger elements have been used in large scale projects such as the Rion-Antirion bridge foundations in Greece (Pecker, 2004). Typical replacement ratios are much smaller than soft inclusions; on the order of 2% to 12% of the soil mass.

The technique has many different appellations depending on the countries and the inclusions themselves can be built with various materials and installation techniques: Deep Soil Mixed Columns, Jet Grouting Columns, Vibro-Concrete Columns can all part of the rigid inclusion category. One installation technique that has gained popularity, first in Europe and now in North America, is the use of displacement techniques of installation to form inclusions of mortar called Controlled Modulus Columns (CMCs).

CMCs are drilled in the ground using a drilling rig with a displacement auger attached to a kelly bar therefore displacing the ground laterally instead of bringing spoils to the surface as the auger is pushed and screwed into the soils with a high torque – high pull down rig. The drilled hole is filled with a lean cementitious mix (usually a sand-based mix) under moderate pressure through the hollow stem auger and kelly as the drill withdraws the tools (Figure 1).

Figure 1. Installation of Controlled Modulus Columns

2.3 Differences between inclusion typesAside from the variation in the stiffness and cohesion of the two inclusion types, there exist multiple differences in the load transfer mechanisms, behavior, and assumptions specific to their respective installation methods and design methodologies.


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As a result of the soft inclusions reliance of confinement of the surrounding soil, the plasticity of the soft inclusion must be carefully considered in the design process. Because the material used to build rigid inclusions is usually a cementitious mix or steel, each inclusion usually exhibits small and nearly elastic deformations, whereas soft inclusions will often exhibit significant plastic deformations under loading dependent on the inclusion stiffness and lateral confinement available. While the large deformations may be prohibitive for strict settlement criteria, such behavior has its advantages when considering the concentration of load at the interface of the structural components. The ‘soft’ behavior of the inclusion allows for direct contact load transfer while preventing the highly concentrated stress points at the inclusion head and the provision of heavily reinforced structural components to accommodate them.

Rigid inclusions are generally designed without a direct, mechanical connection to the surface structure. The absence of this rigid linkage is an important conceptual difference between a traditional deep foundation and the rigid inclusion foundation solution, which greatly simplifies the structural components and improves the overall economy of the project. The high stiffness and minimal deformation components of the rigid inclusion at the surface are mitigated through the use of a ‘cushion’ load transfer layer between the top of the inclusion and bottom of the structural components. This load transfer layer is commonly referred to as a ‘Load Transfer Platform’ or (LTP) and is made up of a well-compacted dense-graded aggregate. As settlement of the compressible soil occurs between the inclusions, shear mechanisms engage in the LTP, creating an effective ‘arching’ of the load into the inclusions. Furthermore, a properly designed LTP serves to minimize the concentrated stress points that would otherwise exist directly above the inclusions, while maintaining an equal settlement plane at an elevation within the LTP to prevent induced moments and shear forces within the rigid structure.

The higher component stiffness of the rigid inclusion, combined with the complex load transfer platform mechanism located above the inclusion head, provides an application capable of adequately supporting higher vertical loading conditions with minimized deformations as compared to a soft inclusion system. Horizontal settlement profiles across the unit grid of the rigid inclusion system are only equal at some point above the inclusion head within the LTP and at some point below the tip of the inclusion. This is in sharp contrast to the assumptions inherent for a soft inclusion system, where settlements across the unit mesh are considered to be equal in all horizontal planes.

3 COMMON DESIGN METHODOLOGIES FOR “SOFT” INCLUSIONS

3.1 IntroductionSeveral design methods will be presented to gain a principle understanding of the current standards of practice. Within the framework of each method, various assumptions and simplifications are proposed. In most cases, a unit cell of an inclusion and tributary soil area (axisymmetric model) is selected for consideration; assuming there exists an infinite and uniformly loaded area of study. This assumption greatly simplifies the calculation methods and reduces the complexity of the analytical and/or numerical model.

Each method is further delineated by the assumption proposed regarding the behavior of the inclusion material within the unit cell. Early design methodologies assume the inclusion material exhibits perfectly elastic behavior, while later methods attempt to identify the exact layers of the inclusion that exhibit plastic behavior.

Understanding the behavior of the inclusion along its entire length is necessary in accurately estimating the interaction between the inclusion and the soil. The primary modes of failure for soft inclusions are ‘bulging,’ where the inclusion material is not laterally confined by the soil, and ‘punching’, where the inclusion toe penetrates the bearing stratum. The susceptibility of soft inclusions to laterally deform under vertical loading is critical to the chosen design approach, as this lateral bulging is the predominate mode of load transfer in the soft inclusion method. Careful consideration must be given to determine the most applicable design method. The performance and behavior (elastic and/or plastic) of the inclusion under vertical load is highly dependent on the passive resistance of the surrounding soil and the magnitude of the imposed loads.

3.2 Homogenization methodOne of the simplest methods to estimate soil improvement with soft inclusions, the homogenization method describes a perfectly elastic system where the stress taken by the soil and inclusion are in direct proportion to their respective stiffness. This elastic approach is conservative when considering the behavior of the soil; however, if the soils are significantly less stiff than the inclusion, stresses within the soft inclusion may be overestimated causing the potential for unrealistic settlement reduction ratios (improvement ratios). This method may be more reliable for limited load cases where plasticity is not expected within the inclusion.



3.3 Balaam and Booker methodBalaam and Booker (1981) developed an exact analytic solution using the theory of elasticity to calculate the magnitude of settlement of rigid foundations supported by soft inclusions. Similar in the approach of distribution of stress and load sharing like the aforementioned homogenization method, this solution appears to be more reliable when plasticity is not anticipated in the soft inclusion.

Figure 2. Simplified soil profile used for elastic methods

3.4 Priebe methodThe Priebe (1995) method considers the inclusion material to be incompressible with a plastic behavior over the full length of the inclusion, with the soil exhibiting an elastic behavior. Priebe considers first that the soil is displaced during inclusion installation to such an extent that its initial resistance corresponds to the liquid state (i.e. the coefficient of earth pressure is given as: k=1). Therefore, any settlement of the unit cell results in a corresponding bulging of the inclusion which remains constant over its length. Deformations and stresses can be systematically calculated for the elastic soil region using the conservation of load and equality of settlement assumptions.

Since the inclusion is known to be made of a compressible material, corrections are then made to the replacement ratio using the stiffness ratio between the inclusion and soil. This method also neglects the bulk density of both the inclusion and soil, but later corrects this simplification using a compatibility control referred to as the ‘depth factor.’ Although this factor is not connected mathematically to the rest of the design method, it ensures that no load is assigned to the inclusions so that the settlement of the inclusions resulting from their inherent compressibility does not exceed the settlement of the composite system. In essence, this depth factor takes into account the additional lateral constraint of the earth pressure at depth.

The improvement factor (no) which gives the magnitude of settlement reduction can be directly correlated to the following parameters which are illustrated in the well-known diagram in ‘Figure 3’:

- A/AC - the reciprocal of the area replacement ratio; where A = unit area of grid and AC = area of inclusion

- φC - friction angle of inclusion material- μS – Poisson ratio of soil

Figure 3. Priebe – Design chart for vibro replacement

3.5 Goughnour and Bayuk methodGoughnour and Bayuk (1979) proposed an elasto-plastic method to describe the behavior of soft inclusions in a soil matrix subjected to vertical load. This method proposes that if the inclusion is designed with sufficient strength, equilibrium will be reached without plastic deformation. However, if the load on the inclusion is sufficiently large, bulging will occur, and the inclusion will reach a state of plastic equilibrium. The method allows for the selection of the initial earth pressure coefficient, which is advantageous when considering the improving effects of the particular installation process.

Two simplifying assumptions are proposed: During the consolidation of the soil, shear stresses will be generated between the inclusion and the soil, which this method will ignore (conservative); and zero relative movement will occur between the inclusion and the soil (equal vertical strain). Furthermore, stresses and deformations for each incremental element will be solved for conditions of consolidated equilibrium. For this final condition, the total vertical load across all elements must be the same; however the stress distribution between the inclusion and the soil will vary over each increment.

Each vertical increment is examined, first assuming that the inclusion has followed plastic deformation, and then assuming the inclusion has remained elastic up to the completion of consolidation (based on Terzaghi’s theory). The final solution for a given vertical increment is given by the greater of the two methods. The components of strain which contribute to the composite behavior are detailed below:



Figure 4. Combined strain behavior approach

3.6 Comparison of “soft” inclusion design methodsThe following practical example provides a comparison of the soft inclusion design methods previously described. At the time of this publication, field-measured settlement results were not yet made available to the author for comparison purposes.

3.6.1Project overviewPreliminary geotechnical investigations were conducted at a proposed tank site within a tank terminal located in Pittsburgh, PA, USA directly adjacent to the Allegheny River. Investigations concluded that the site subsurface materials were inadequate to provide proper settlement control and bearing capacity for the installation of the proposed 19 meter diameter tank. The tank was detailed with a slab-on-grade structural base to provide a level and rigid bearing surface for the steel tank floor.

Vibro-replacement stone columns were recommended to provide adequate settlement control while avoiding the excessive costs associated with a traditional deep foundation alternative. Expected tank loads were on the order of 144 kPa and high settlement tolerances were acceptable by the tank contractor. As such, stone inclusions with a 0.76 meter diameter were proposed to be installed on a 1.71 meter grid beneath the tank.

3.6.2Soil informationThe project site is located in close proximity to the Allegheny River, providing subsurface soils which are alluvial in nature; consisting of silt, silty to clayey gravel, and silty to clayey sand. Test borings indicated that the site contained up to 1 meter of heterogeneous man-made fill, underlain by 3 meters of alluvial/fill deposits. The fill/alluvial deposits displayed soft consistencies with high silty contents. Dense to medium dense clayey sand and gravel alluvial soils were located directly below the upper fill/alluvial deposits down to a depth of approximately 7 meters. Beneath the dense alluvial soils the project borings encountered very dense residual soils which were not examined due to their high

stiffness and low compressibility. Preliminary calculations indicated that up to 122 millimeters of settlement could be expected at the center of the tank under the design load without soil improvement.

3.6.3Results of the methodsEach of the methods previously described were utilized in an effort to provide comparisons on the magnitude of the settlement reduction ratio per calculation technique. The results of the analysis are highlighted below in ‘Table 1’.

Table 1. Calculation results for each method

Elastic Methods Elasto-plastic Method

Homoge-nization

Balaam & Booker Priebe Goughnour&

BayukSettlement of soil sur-rounding the columns 2.6 cm 3.3 cm 3.8 cm 5.0 cm

Settlement reduction ratio 3.0 2.4 2.1 1.6

Settlement of soil be-low columns 4.5 cm

TOTAL SETTLE-MENT OF SOIL + STONE COLUMNS

7.1 cm 7.8 cm 8.3 cm 9.5 cm

Calculated settlement reduction ratios demonstrate the potential wide deviation between the specific methods. As previously discussed, the elastic methods generally overestimate the load in the soft inclusion, resulting in substantially less stress in the soil matrix. In contrast, the Priebe and Goughnour methods assume a certain degree of inclusion plasticity, which increases the stress in the soil matrix, resulting in less improvement. ‘Figure 5’ below details the stress (kPa) in the inclusion (given as Δqc) versus the stress in the soil (given as Δqs) over the depth of the inclusion:

Figure 5. Stress in soft inclusion and stress in soil matrix vs. depth of inclusion.



3.6.4ConclusionsThe estimations of the improved settlement of the tank using the various soft inclusion methods provide a first-order estimation of stress distribution and soil settlement. With high tank loads of 144 kPa, careful consideration should be given when using purely elastic methods, as plastic deformations of the inclusions are anticipated to some extent.

Overall, each of the methods discussed use certain assumptions and simplifications to provide reasonable design estimations. Selection of the most appropriate method should be carefully weighed by giving ample consideration to the project constraints including the loading and soil conditions present at the proposed location. Furthermore, the use of Finite Element Modeling could greatly benefit the designer in providing a more in-depth analysis when a greater accuracy and understanding are desired and/or required.

4 DESIGN METHODOLOGY FOR “RIGID” INCLUSIONS (GROUTED INCLUSIONS)

Although similar to soft inclusions in the fact that they are vertical inclusions usually installed on a regular grid pattern across compressible soft soil layers to a more competent bearing layer, rigid inclusions differ in the way they attract loads and release these loads in the substratum.

4.1 IntroductionRigid inclusions are still an emerging technique although the use of rigid elements such as timber piles covered by a layer of stones can be traced back several hundred years ago.

The understanding of the mechanism and behaviors governing the design of rigid inclusion solutions has only started to be developed in the last 20 to 30 years and the recent successes of these technologies (particularly in Europe) has accelerated the pace of research and development around rigid inclusions not only of new improved equipment and techniques but also in the optimization of the design methods.

One of the better ways to define rigid inclusions is to use the definition adopted by the French National Program (ASIRI, 2012): the rigid inclusion concept assumes that inclusion stability is achieved without any lateral confinement of the surrounding soils (Simon, 2012). In other words, rigid inclusions encompass all columns showing a strong permanent internal cohesion.

4.2 Basis of the design / Load Transfer PlatformThe general concept behind the design of Rigid Inclusions is to combine them with a Load Transfer Platform (LTP) made of compacted granular material

in most cases placed atop the head of the inclusions. The structure and the rigid inclusions are therefore never in direct contact or rigidly connected to each other.

While it is easy to confuse rigid inclusions and deep foundations, the presence of this LTP is the key difference that leads to completely different load paths, philosophy, and design approach. Although they may appear equivalent because they use similar installation techniques and equipment, their design and “inner workings” are substantially different and should not be confused with one another.

Contrarily to soft inclusions, the assumption of an equal plane deformation at any horizontal level of the column / soil mass is not true for rigid inclusion systems. The load of the structure is transferred to the inclusions through shear. This is the direct consequence of the differential deformations that develop within the LTP due to the large difference (several orders of magnitude) in compressibility (Modulus of Deformation) between the inclusion (low compressibility – high modulus - rigid) and the surrounding soils (high compressibility – low modulus – soft).

Figure 6. Typical section of a rigid inclusions system

4.2.1Under an embankment

Under an embankment of sufficient thickness, as the soil and the LTP settles around the rigid elements (inclusions are “punching” into the LTP), there is a gradual rotation of the principal stresses. From a major principal stress fully vertical at the top of the embankment to a 90 degrees rotation with a horizontal principal stress. This is similar to what happens with a perfect arch that would be supported on top of the inclusions.

As a result, under an embankment, a full arch can develop if the thickness of the LTP or the span between the inclusions is adequately designed. Above the arch, the settlement becomes uniform (equal plane settlement) at some distance above the head of the inclusions. Below the arch, as the soils



settle more than the shaft of the inclusions, friction develops along the inclusions (negative skin friction). This causes additional load to be transferred to the elements. The negative skin friction is in the case of rigid inclusions a beneficial, necessary effect that improves the efficiency of the system unlike deep foundations for which negative skin friction is detrimental to the design.

As load is transferred from the soil to the rigid elements, the relative deformation between soil-inclusions decreases with depth until it reaches equilibrium where the soil and the inclusion show equal strain (no differential settlement, no relative movement, and therefore no friction). At this plane called the neutral plane, the inclusion and the soil move together similarly to what occurs in soft inclusions.

Below the neutral plane, the inclusion settles more than the soil surrounding it and therefore load is now transferred from the inclusion to the soil through positive skin friction. The remainder of the load in the inclusion is then released in the bearing layer at the tip of the inclusions (end-bearing).

As a consequence, the load in the rigid inclusion varies with depth starting at the top of LTP below the structure transferred though arching, increasing with depth to the maximum load at the neutral plane and decreasing until the tip load in the bearing layer. As far as deformations are concerned, the movement of the inclusion is the sum of the movement at the tip and the elastic compression of the column material itself. The soil has a gradual deformation from a maximum at the top of the LTP to zero displacement some distance below the tip of the inclusion.

There are therefore only equal settlement planes at certain locations along the profile (Figure 7):

- At the neutral plane- At some distance above the arch and for

every plane above that.- At some distance below the tip of the inclusion

Figure 7. Neutral planes, negative and positive skin friction

4.2.2Under a slab / raft / footing

Under a slab, the tendency of the designers is to reduce the thickness of the LTP which is often too thin to allow full development of an arch and of equal settlement planes within the LTP. In that case, the structure (i.e. slab / raft / footing) plays an important part in the load transfer mechanism.

Indeed, the presence of the relatively rigid slab lowers the plane of equal strain to coincide with the under face of the slab. Since equal strain may not be achieved within the LTP itself, the slab may have to sustain a non-uniform reaction from the ground improvement system. If the LTP is not thick enough and this phenomenon occurs, the rotation of the major principal stresses is not fully possible and therefore a complete arch cannot form.

In that case, most of the load is transmitted to the inclusion head through the compression of a shear cone volume (inverted cone or pyramid) centered on the inclusion heads and through bending of the slab. If that is the case, it is therefore necessary to take these additional bending moments into account in the design of the slab. The designer shall evaluate the economic and technical considerations between thickening the LTP or strengthening the slab (thickness, reinforcement) to accommodate these increases in the moment amplitude.

Figure 8. Load transfer mechanism under slab with thin LTP

4.3 Failure mechanisms developing in the LTPThe behavior of the LTP depends on many factors:

- Clear span between columns- Presence or not of a slab- Compaction of LTP (i.e. modulus of

deformation)- Friction angle of the material (i.e. shear

strength)Both centrifuge and field tests (Dias, 2012 & Chevalier, 2011) have shown that the efficiency of



the LTP is limited by two ultimate failure mechanisms:

- Prandtl’s failure mechanism- Inverted shear cone punching failure

mechanismThe actual equilibrium diagram depends on the geometry and nature of the loading. The Prandtl’s mechanism will be predominant in cases where the LTP is covered by a rigid structure (slabs on grade, raft, footing) or in the case of a thick embankment. The shear cone punching failure mechanism corresponds to the formation of a shear cone centered on the rigid element. This would likely occur within thin embankments on rigid inclusions.

4.3.1Prandtl’s failure mechanism

Without going into the details of the Prandtl’s mechanism which is amply described in the literature, the basic mechanism associates three main domains:

- A Rankine active limit state domain (Domain I) right above the inclusion head

- A domain limited by a logarithmic spiral arc (Domain II)

- A Rankine passive limit state domain (Domain III) within the LTP outside the influence of the inclusion

Figure 9. Prandtl’s failure mechanism diagram

It is possible to determine the ultimate stress on the soil and on the inclusion through the application of the formula of the Prandtl mechanism as well as the conservation of the load.

4.3.2Shear cone punching failure

The second failure mechanism in the LTP can be modeled by a vertical cone from the edge of the inclusion to the top of the LTP. Establishing the relation of limit pressure is fairly straightforward and well documented (ASIRI 2012).

Figure 10. Shear Cone failure diagram for thin embankments

These limit state relations define different allowable domains that coupled with the load conservation equation allow to determine the allowable stresses for the system.

4.4 Modification of slab design / Method of additional moments

4.4.1Context

To illustrate the reasons why the design of slabs-on-grade might need to be adapted when supported by a rigid inclusions system, let us consider a typical warehouse slab. There are 3 very different loading cases:

- LC1 : uniform distributed load over a large area of the slab

- LC2 : alternate loading : succession of loaded and non-loaded bands or strips of slab corresponding at alleys between uniformly loaded strips of slabs

- LC3 : rack storage with non-continuous punctual load under each rack

Concrete slabs-on-grade are typically between 150 mm and 250 mm thick. Hinged construction joints usually on a 6 m or 7 m square grid are also included in the typical design process. The rigid inclusions and the LTP are under the slab. The location of the rigid inclusions with regards to the joints, the racks, and the loaded/unloaded areas can vary across the slab. We are therefore confronted with an increasingly complex three dimensional problem and need to add further assumptions to transform this multi-variable problem into a more manageable one. Two methods are proposed:

- Coefficient of Subgrade Reaction method- Additional moments methodBoth methods are based on results of

axisymmetrical finite elements analyses that need to be performed prior to the design of the slab and have been validated by 3D Finite Element calculations (ASIRI 2012).



4.4.2Coefficient of subgrade reaction

This method is an iterative, simplified method to estimate the bending moments in the slab due to the presence of rigid inclusions.

Initially, a finite element axisymmetrical model is performed where the load is assumed to be uniformly spread over the whole area of the slab. This calculation gives a reference maximum bending moment directly above the inclusion (upper fiber in tension) and a maximum negative bending moment at mid-span between two inclusions (lower fiber in tension). The model is then discretized using two different subgrade reaction coefficients: ki over a length rk (influence of the inclusion) from the center of the inclusion; and ks over the remainder of the area of each unit cell. We of course have ki > ks.

Figure 11. Coefficient of Subgrade Reaction Method

ks, the homogenized modulus of subgrade reaction between inclusions, is therefore given by the ratio of the average uniform load on slab divided by the average settlement of the slab.

ki, the homogenized modulus of subgrade reaction is calculated for a given rk (influence of the inclusion) by writing the conservation of load and assuming that ki is the ratio of average uniform load on slab over the length rk divided by the average settlement over the length rk.

The moments in the slab are therefore evaluated using the two reactions ki and ks. The influence of the inclusion rk is changed until there is a close match between the value calculated with the subgrade reaction coefficients and the finite element reference calculation.

Once the correct values of ki and ks are fixed, all the different load and configuration cases need to be studied one by one for each of the lower and the upper fibers of the slab.

4.4.3Method of additional bending moments

A few definitions and notations are required:- Soed(NJ) : bending moment distribution in a

continuous slab (without any hinged joints) over an equivalent homogeneous soil profile

- Soed(WJ) : bending moment distribution in a slab with hinged joints over an equivalent homogeneous soil profile

- IR(NJ) : bending moment distribution in a continuous slab (without any hinged joints) over a soil reinforced with rigid inclusions

- IR(WJ) : bending moment distribution in a slab with hinged joints over a soil reinforced with rigid inclusions

Regardless of the type of loading, one can write the following trivial equality:

IR(WJ) = Soed(WJ) + [ IR(NJ) - Soed(NJ)] +[(IR(WJ) - Soed(WJ)) – (IR(NJ) - Soed(NJ))]

Let’s look at each term of this equation:

- Soed(WJ) is the classical moment diagram of a slab with joints on homogeneous ground (uniform reaction) and should be therefore provided by the slab designer based on an equivalent homogenized modulus for the soil + rigid inclusion system (given by Finite element calculation)

Figure 12. Method of additional moments – calculation of Mupper and Mlower

- [IR(NJ) - Soed(NJ)] represents the contribution of the rigid inclusions on a slab without taking any joints into account. This is the same model as the reference model for the coefficient of subgrade reaction method. This bending moment distribution does not depend on the loading type as it is based on uniform



equivalent loads. Let’s call [mb] = [Mupper;Mlower] the envelope of moments in this configuration between the maximum moment in the upper fiber and the maximum moment in the lower fiber

- [(IR(WJ) - Soed(WJ)) – (IR(NJ) - Soed(WJ))] represents the influence of the interaction between the rigid inclusions and the hinged joints. The main effect of the hinged joints is to redistribute the moments in the slab since at a joint, the moment is zero. In the worst case, a joint and a rigid inclusion are aligned. In that configuration the moment directly above an inclusion goes from Mupper to 0, a net shift of - Mupper. Similarly, if the joint is located exactly at the mid span between two inclusions, then the shift in moments is from -Mlower to 0 (i.e. a shift of -Mlower). We can therefore define an envelope of moments [mc] = [-Mlower; - Mupper]. This allows assigning a safe and wide range of values that can be refined by calculation taking into account the actual worst configuration for the location of the rigid inclusions with regards with the joints.

It is now possible to write:

IR(WJ) = [mclassical] + [mb] + [mc]≤ [mclassical] + [Mupper ; Mlower ] + [ - Mlower;-Mupper ]≤ [mclassical] + [Mupper ; Mlower ] + [ -Mlower;-Mupper ]

And therefore:

IR(WJ)≤[mclassical]+[(Mupper -Mlower );-(Mupper - Mlower)]

As shown above, it is therefore possible to determine an envelope of moments that bounds the actual complex moment distribution for all configurations (load cases, rigid inclusions and hinged joints) using values calculated with very simple methods: first, the moment distribution in a slab on homogeneous soil without joints followed by the moment distribution in a slab on rigid inclusions without joints that is given by a simple axisymmetrical finite element calculation.

Since this is an envelope method, it will yield conservative values and can therefore lead to over-conservative designs.

4.5 Calculation of deformations

While analytical solutions are available for simple geometries (Curia, 2009), the best way to determine

the deformations of a ground improvement rigid inclusion system is through the use of Finite Element or Finite Difference Analysis.

While axisymmetrical calculations are used mainly in symmetrical configurations (for example, a pattern of inclusions in the central part a very large slab), many situations require more advanced modeling. It is not unusual to model rigid inclusion supported embankments using 2D plane strain models. This allows modeling the non-uniform conditions at the edge of the model while keeping the amount of elements and calculation time manageable with regular personal computers.

With the recent advances of computer hardware and the improvement in computational power, the use of 3D modeling is becoming more widespread. For highly complex geometries (i.e. full-scale model of a rectangular footing), 3D modeling may be the only way to evaluate settlement accurately.

It should be noted that in addition to the selection of the modeling technique, the selection of parameters and the behavior laws used to model the soft compressible soils and the LTP are paramount to obtaining accurate estimates. Since it has been shown earlier that in the upper part of the inclusions, the predominant phenomena is load transfer through shear, the selection of parameters should be focused on obtaining accurate shear parameters (shear modulus, friction angle...) from the soil investigation. In the mid-section of the model, there is no more rotation of the principal stresses and the main behavior is vertical deformation and consolidation. The parameters should therefore be selected with that behavior in mind (modulus of deformation, coefficient of consolidation...). Finally, in the lower part, the behavior is again governed by shear and shear parameters should govern selected properties.

5 CONCLUSIONS

In this paper, we established the similarities and the differences between rigid and soft inclusions. We described some of the methods used to design these techniques while showing that these design methods were not interchangeable depending on the types of inclusions. For the soft inclusions, it has been shown that while some of the methods are in good agreement with elasto-plastic calculations by finite element software, the choice of the coefficient ko has a great influence on the results of the calculations. We would therefore suggest that further research be done to measure the actual value of this parameter after installation of the soft inclusions. Displacement installation methods are definitely favorable to an increase of ko in certain types of soils as compared to methods of installation using removal and replacement of the soils.

The rigid inclusions have gained popularity in the recent years following several nationally funded


Moment on homogenized ground

/ no joint

Additional Moment envelope given by FEM calculation


research programs (FHWA, ASIRI, AMGISS, NO RECESS). The Load Transfer Platform is a key part of the system but there is not yet any universally recognized method to efficient design the Load Transfer Platform. Depending on local habits and building codes, several methods and guidelines are used (ASIRI, BS8006, EBGEO 2010, FHWA...) and these methods can lead to very different designs in terms of spacing of the inclusions, thickness and quality of the Load Transfer Platform, and presence of several layers of geo-grid.

REFERENCES

ASIRI, Multiple authors (2012), “ASIRI 2012, Amélioration des sols par inclusions rigides”, Presses des Ponts et Chaussées, ISBN 978-2-85978-462-1

Balaam, N.P. & Booker, J.R. (1981), “Analysis of rigid rafts supported by granular piles” International Journal for Numerical and Analytical in Geomechanics, Vol 5: 379-403

Buschmeier, B., Masse, F., Swift, S. & Walker, M., (2012), “Full scale instrumented load test for support of oil tanks on deep soft clay deposits in Louisiana using Controlled Modulus Columns” ISSMGE - TC 211 International Symposium on Ground Improvement IS-GI, Brussels

Chu J., Varaksin S., Klotz U. & Menge P. (2009), “State of the Art Report, Construction Processes, TC17”, 17th ICSMGE, Alexandria

Combarieu O. (1988). “Amelioration des soils par inclusions rigides verticales. Application a l’edification des remblais sur sols mediocres”, Revue Francaise de Geotechnique #44, pp. 57-79

Cuira F. & Simon B. (2009) “Two simple tools for evaluation the complex interactions in a soil reinforced by rigid inclusions” 17th ICSMGE, Alexandria, pp. 1163-1166

Dias D. & Simon B. (2012) “Spread Foundations on rigid inclusions subjected to complex loading: Comparison of 3D numerical and simplified analytical modeling”, ISSMGE – TC 211 International Symposium on Ground Improvement, Brussels

EN 1997-1, 2004, Eurocode 7: “Geotechnical design Part 1: General rules”, European Committee for Standardization

Goughnour, R. R. & Bayuk, A. A. (1979), “Analysis of stone column soil-matrix interaction under vertical load”, C.R. Colloque int. sur le renforcement des sols: Terre armée et autres méthodes, Paris. Vol. 1: 271-277

Simon B. (2012). “General Report – Session 5 – Rigid Inclusions and Stone Columns”, ISSMGE – TC 211 International Symposium on Ground Improvement, Brussels

Okyay U.S., Dias D., Thorel L. & Rault G. (2012) “Centrifuge modeling of a pile-supported granular earth-platform over soft soils” Journal of Geotechnical and Geoenvironmental Engineering, ASCE

Pecker A. (2004) “Design and construction of the Rion Antirion Bridge”, Proc. ASCE Geo-trans 2004 on Geotechnical Aspects of Transportation Engineering, ASCE

Plomteux, C. & al (2003) - "Controlled Modulus Columns (CMC): Foundation system for Embankment support: a case history" Geosupport 2004, Orlando, USA, pp. 980 - 992

Priebe, H.J., (1995), “The design of vibro replacement”, Ground Engineering, pp. 31-46


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Paper title · Web viewIn that case, most of the load is transmitted to the inclusion head through the compression of a shear cone volume (inverted cone or pyramid) centered on the