XXVI Reunión Nacional de Mecánica de Suelos
e Ingeniería Geotécnica
Sociedad Mexicana de
Ingeniería Geotécnica, A.C.
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 America
of Engineering, Menard USA, Pittsburgh, Pennsylvania, United States of America
2Vice-President
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 INTRODUCTION
1.1 General overview of ground improvement with
inclusions
The 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
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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 TYPES
2.1 “Soft” inclusions
Inclusions 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” inclusions
Inclusions 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 types
Aside 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 wellcompacted 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
3 COMMON DESIGN METHODOLOGIES FOR
“SOFT” INCLUSIONS
3.1 Introduction
Several 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 method
One 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.
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3.3 Balaam and Booker method
Balaam 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.
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 method
Figure 2. Simplified soil profile used for elastic
methods
3.4 Priebe method
The 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.
Goughnour and Bayuk (1979) proposed an elastoplastic 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:
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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.
3.6.3 Results of the methods
Each 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
Figure 4. Combined strain behavior approach
Elastic Methods
3.6 Comparison of “soft” inclusion design methods
Settlement of soil surrounding the columns
The 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.
Settlement reduction
ratio
Settlement of soil below columns
TOTAL SETTLEMENT OF SOIL +
STONE COLUMNS
3.6.1 Project overview
Preliminary
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.
3.6.2 Soil information
The 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
Elasto-plastic
Method
Goughnour&Bayuk
Homogenization
Balaam &
Booker
Priebe
2.6 cm
3.3 cm
3.8 cm
5.0 cm
3.0
2.4
2.1
1.6
8.3 cm
9.5 cm
4.5 cm
7.1 cm
7.8 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.
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3.6.4 Conclusions
The 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.
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).
4.1 Introduction
Rigid 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 Platform
The general concept behind the design of Rigid
Inclusions is to combine them with a Load Transfer
Platform (LTP) made of compacted granular material
Figure 6. Typical section of a rigid inclusions system
4.2.1 Under 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
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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 soilinclusions 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
7
4.2.2 Under 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 LTP
Figure 7. Neutral planes, negative and positive skin friction
The 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
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the LTP is limited by two ultimate failure
mechanisms:
- Prandtl’s failure mechanism
- Inverted shear cone punching failure
mechanism
The 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.
4.3.1 Prandtl’s failure mechanism
Figure 10. Shear
embankments
Without going into the details of the Prandtl’s
mechanism which is amply described in the
literature, the basic mechanism associates three
main domains:
These limit state relations define different
allowable domains that coupled with the load
conservation equation allow to determine the
allowable stresses for the system.
-
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.
4.3.2 Shear 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).
Cone
failure
diagram
for
thin
4.4 Modification of slab design / Method of
additional moments
4.4.1 Context
To illustrate the reasons why the design of slabson-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 method
Both 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).
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BUSCHMEIER B. et al.
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4.4.2 Coefficient of subgrade reaction
4.4.3 Method of additional bending moments
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.
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 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.
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
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
10
Discusión sobre las diferencias de la metodología de diseño entre las inclusiones granulares y las
inclusiones rígidas
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)]
Moment on
homogenized ground
/ no joint
Additional Moment
envelope given by
FEM calculation
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 overconservative 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
SOCIEDAD MEXICANA DE INGENIERÍA GEOTÉCNICA A.C.
BUSCHMEIER B. et al.
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.
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