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SOIL MECHANICS
A. Verruijt
Delft University of Technology, 2001, 2012
This is the screen version of the book SOIL MECHANICS, an elementary textbook for students of civil engineering.
It can be read using the Adobe Acrobat Reader. Bookmarks are included to search for a chapter.
The book is also available in Dutch, in the file GrondMechBoek.pdf.
Exercises and a summary of the material, including graphical illustrations, are contained in the file SoilMex.ZIP.
All software can be downloaded from the website http://geo.verruijt.net/.
CONTENTS
- Introduction
- Classification
- Particles, water, air
- Stresses in soils
- Stresses in a layer
- Darcy’s law
- Permeability
- Groundwater flow
- Floatation
- Flow net
- Flow towards wells
- Stress strain relations
- Tangent-moduli
- One-dimensional compression
- Consolidation
- Analytical solution
- Numerical solution
- Consolidation coefficient
- Creep
- Shear strength
- Triaxial test
- Shear test
- Dutch cell test
- Pore pressures
- Undrained behaviour of soils
- Stress paths
- Elastic stresses and deformations
- Boussinesq
- Newmark
- Flamant
- Deformation of layered soil
- Lateral stresses in soils
- Rankine
- Coulomb
- Tables for lateral earth pressure
- Sheet pile walls
- Blum
- Sheet pile wall in layered soil
- Limit analysis
- Strip footing
- Prandtl
- Limit theorems for frictional materials
- Brinch Hansen
- Vertical slope in cohesive material
- Stability of infinite slope
- Slope stability
- Soil exploration
- Model tests
- Pile foundations
- Appendix A. Stress analysis
- Appendix B. Theory of elasticity
- Appendix C. Theory of plasticity
- Answers to problems
- Literature
- Index
PREFACE
This book is the text for the introductory course of Soil Mechanics in the Department of Civil Engineering of the Delft University of Technology,
as I have given from 1980 until my retirement in 2002. It contains an introduction into the major principles and methods of soil mechanics, such
as the analysis of stresses, deformations, and stability. The most important methods of determining soil parameters, in the laboratory and in
situ, are also described. Some basic principles of applied mechanics that are frequently used are presented in Appendices. The subdivision into
chapters is such that one chapter can be treated in a single lecture, approximately.
Comments of students and other users on the material in earlier versions of this book have been implemented in the present version, and
errors have been corrected. Remaining errors are the author’s responsibility, of course, and all comments will be appreciated.
An important contribution to the production of the printed edition, and to this screen edition, has been the typesetting program TEX, by
Donald Knuth, in the L
A
TEXimplementation by Leslie Lamport. Most of the figures have been constructed in L
A
TEX, using the PICTEXmacros.
The logo was produced by Professor G. de Josselin de Jong, who played an important role in developing soil mechanics as a branch of science,
and who taught me soil mechanics.
Since 2001 the English version of this book has been made available on the internet, on the website . Several users,
from all over the world, have been kind enough to send me their comments or their suggestions for corrections or improvements. In recent
versions of the screenbook it has also been attempted to incorporate the figures better into the text, using the macro wrapfigure, and colors. In
this way the appearance of many pages seems to have been improved.
Upon the suggestion of Prof. Emmanuel Detournay of the University of Minnesota, the problems at the end of chapters have been supple-
mented in the versions after 2010 by worked examples, as a further aid to students. Additional sets of exercises and problems are available in
the file SoilMex.ZIP.
Delft, March 2012 A. Verruijt
A. Verruijt, Soil Mechanics : 1.2. History 7
1.2 History
Figure 1.1: Landslide near Weesp, 1918.
Soil mechanics has been developed in the beginning of the 20th century. The
need for the analysis of the behavior of soils arose in many countries, often
as a result of spectacular accidents, such as landslides and failures of founda-
tions. In the Netherlands the slide of a railway embankment near Weesp, in
1918 (see Figure 1.1) gave rise to the first systematic investigation in the field
of soil mechanics, by a special commission set up by the government. Many
of the basic principles of soil mechanics were well known at that time, but
their combination to an engineering discipline had not yet been completed.
The first important contributions to soil mechanics are due to Coulomb, who
published an important treatise on the failure of soils in 1776, and to Rank-
ine, who published an article on the possible states of stress in soils in 1857.
In 1856 Darcy published his famous work on the permeability of soils, for
the water supply of the city of Dijon. The principles of the mechanics of
continua, including statics and strength of materials, were also well known
in the 19th century, due to the work of Newton, Cauchy, Navier and Boussi-
nesq. The union of all these fundamentals to a coherent discipline had to
wait until the 20th century. It may be mentioned that the committee to
investigate the disaster near Weesp came to the conclusion that the water
levels in the railway embankment had risen by sustained rainfall, and that
the embankment’s strength was insufficient to withstand these high water
pressures.
Important pioneering contributions to the development of soil mechanics
were made by Karl Terzaghi, who, among many other things, has described
how to deal with the influence of the pressures of the pore water on the be-
havior of soils. This is an essential element of soil mechanics theory. Mistakes
on this aspect often lead to large disasters, such as the slides near Weesp,
Aberfan (Wales) and the Teton Valley Dam disaster. In the Netherlands
much pioneering work was done by Keverling Buisman, especially on the
deformation rates of clay. A stimulating factor has been the establishment of the Delft Soil Mechanics Laboratory in 1934, now known as
Deltares. In many countries of the world there are similar institutes and consulting companies that specialize on soil mechanics. Usually they
also deal with Foundation engineering, which is concerned with the application of soil mechanics principle to the design and the construction
of foundations in engineering practice. Soil mechanics and Foundation engineering together are often denoted as Geotechnics. A well known
8 A. Verruijt, Soil Mechanics : 1. INTRODUCTION
consulting company in this field is Fugro, with its head office in Leidschendam, and branch offices all over the world.
The international organization in the field of geotechnics is the International Society for Soil Mechanics and Geotechnical Engineering, the
ISSMGE, which organizes conferences and stimulates the further development of geotechnics by setting up international study groups and by
standardization. In most countries the International Society has a national society. In the Netherlands this is the Department of Geotechnics
of the Royal Netherlands Institution of Engineers (KIVI), with about 800 members.
1.3 Why Soil Mechanics?
Soil mechanics has become a distinct and separate branch of engineering mechanics because soils have a number of special properties, which
distinguish the material from other materials. Its development has also been stimulated, of course, by the wide range of applications of soil
engineering in civil engineering, as all structures require a sound foundation and should transfer its loads to the soil. The most important
special properties of soils will be described briefly in this chapter. In further chapters they will be treated in greater detail, concentrating on
quantitative methods of analysis.
1.3.1 Stiffness dependent upon stress level
Many engineering materials, such as metals, but also concrete and wood, exhibit linear stress-strain-behavior, at least up to a certain
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Figure 1.2: Pile foundation.
stress level. This means that the deformations will be twice as large if the stresses are twice
as large. This property is described by Hooke’s law, and the materials are called linear elastic.
Soils do not satisfy this law. For instance, in compression soil becomes gradually stiffer. At the
surface sand will slip easily through the fingers, but under a certain compressive stress it gains
an ever increasing stiffness and strength. This is mainly caused by the increase of the forces
between the individual particles, which gives the structure of particles an increasing strength.
This property is used in daily life by the packaging of coffee and other granular materials by a
plastic envelope, and the application of vacuum inside the package. The package becomes very
hard when the air is evacuated from it. In civil engineering the non-linear property is used to
great advantage in the pile foundation for a building on very soft soil, underlain by a layer of
sand. In the sand below a thick deposit of soft clay the stress level is high, due to the weight of
the clay. This makes the sand very hard and strong, and it is possible to apply large compressive
forces to the piles, provided that they are long enough to reach well into the sand.
10 A. Verruijt, Soil Mechanics : 1. INTRODUCTION
1.3.4 Creep
The deformations of a soil often depend upon time, even under a constant load. This is called creep. Clay and peat exhibit this phenomenon.
It causes structures founded on soft soils to show ever increasing settlements. A new road, built on a soft soil, will continue to settle for many
years. For buildings such settlements are particular damaging when they are not uniform, as this may lead to cracks in the building.
The building of dikes in the Netherlands, on compressible layers of clay and peat, results in settlements of these layers that continue for
many decades. In order to maintain the level of the crest of the dikes, they must be raised after a number of years. This results in increasing
stresses in the subsoil, and therefore causes additional settlements. This process will continue forever. Before the construction of the dikes the
land was flooded now and then, with sediment being deposited on the land. This process has been stopped by man building dikes. Safety has
an ever increasing price.
Sand and rock show practically no creep, except at very high stress levels. This may be relevant when predicting the deformation of porous
layers from which gas or oil are extracted.
1.3.5 Groundwater
A special characteristic of soil is that water may be present in the pores of the soil. This water contributes to the stress transfer in the soil. It
may also be flowing with respect to the granular particles, which creates friction stresses between the fluid and the solid material. In many cases
soil must be considered as a two phase material. As it takes some time before water can be expelled from a soil mass, the presence of water
usually prevents rapid volume changes.
In many cases the influence of the groundwater has been very large. In 1953 in the Netherlands many dikes in the south-west of the
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Figure 1.5: Overflowing dike.
country failed because water flowed over them, penetrated the soil, and then flowed through
the dike, with a friction force acting upon the dike material. see Figure 1.5. The force of the
water on and inside the dike made the slope slide down, so that the dike lost its water retaining
capacity, and the low lying land was flooded in a short time.
In other countries of the world large dams have sometimes failed also because of rising water
tables in the interior of the dam (for example, the Teton Valley Dam in the USA, in which water
could enter the coarse dam material because of a leaky clay core). Even excessive rainfall may
fill up a dam, as happened near Aberfan in Wales in 1966, when a dam of mine tailings collapsed
onto the village.
It is also very important that lowering the water pressures in a soil, for instance by the production of groundwater for drinking purposes,
leads to an increase of the stresses between the particles, which results in settlements of the soil. This happens in many big cities, such as
Venice and Bangkok, that may be threatened to be swallowed by the sea. It also occurs when a groundwater table is temporarily lowered for the
construction of a dry excavation. Buildings in the vicinity of the excavation may be damaged by lowering the groundwater table. On a different
scale the same phenomenon occurs in gas or oil fields, where the production of gas or oil leads to a volume decrease of the reservoir, and thus
A. Verruijt, Soil Mechanics : 1.3. Why Soil Mechanics? 11
to subsidence of the soil. The production of natural gas from the large reservoir in Groningen is estimated to result in a subsidence of about
50 cm in the production time of the reservoir.
1.3.6 Unknown initial stresses
Soil is a natural material, created in historical times by various geological processes. Therefore the initial state of stress is often not uniform,
and often even partly unknown. Because of the non-linear behavior of the material, mentioned above, the initial stresses in the soil are of great
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Figure 1.6: Stresses.
importance for the determination of soil behavior under additional loads. These initial stresses depend upon
geological history, which is never exactly known, and this causes considerable uncertainty. In particular, the initial
horizontal stresses in a soil mass are usually unknown. The initial vertical stresses may be determined by the weight
of the overlying layers. This means that the stresses increase with depth, and therefore stiffness and strength also
increase with depth. The horizontal stresses, however, usually remain largely unknown. When the soil has been
compressed horizontally in earlier times, it can be expected that the horizontal stress is high, but when the soil is
known to have spread out, the horizontal stresses may be very low. Together with the stress dependency of the
soil behavior all this means that there may be considerable uncertainty about the initial behavior of a soil mass.
It may also be noted that further theoretical study can not provide much help in this matter. Studying field history, or visiting the site, and
talking to local people, may be more helpful.
1.3.7 Variability
Figure 1.7: Pisa.
The creation of soil by ancient geological processes also means that soil properties may be rather different on different
locations. Even in two very close locations the soil properties may be completely different, for instance when an
ancient river channel has been filled with sand deposits. Sometimes the course of an ancient river can be traced on
the surface of a soil, but often it can not be seen at the surface. When an embankment is built on such a soil, it
can be expected that the settlements will vary, depending upon the local material in the subsoil. The variability of
soil properties may also be the result of a heavy local load in the past.
A global impression of the soil composition can be obtained from geological maps. These indicate the geological
history and character of the soils. Together with geological knowledge and experience this may give a first indication
of the soil properties. Other geological information may also be helpful. Large areas of Western Europe have, for
instance, been covered by thick layers of ice in earlier ice ages, and this means that the soils in these areas have been
subject to a preload of considerable magnitude, and therefore may be rather dense. An accurate determination of
soil properties can not be made from desk studies. It requires testing of the actual soils in the laboratory, using
samples taken from the field, or testing of the soil in the field (in situ). This will be elaborated in later chapters.
Chapter 2
CLASSIFICATION
2.1 Grain size
Soils are usually classified into various types. In many cases these various types also have different mechanical properties. A simple subdivision
of soils is on the basis of the grain size of the particles that constitute the soil. Coarse granular material is often denoted as gravel and finer
material as sand. In order to have a uniformly applicable terminology it has been agreed internationally to consider particles larger than 2 mm,
but smaller than 63 mm as gravel. Larger particles are denoted as stones. Sand is the material consisting of particles smaller than 2 mm, but
larger than 0.063 mm. Particles smaller than 0.063 mm and larger than 0.002 mm are denoted as silt. Soil consisting of even smaller particles,
smaller than 0.002 mm, is denoted as clay or luthum, see Table 2.1. In some countries, such as the Netherlands, the soil may also contain
Soil type min. max.
clay 0.002 mm
silt 0.002 mm 0.063 mm
sand 0.063 mm 2 mm
gravel 2 mm 63 mm
Table 2.1: Grain sizes.
layers of peat, consisting of organic material such as decayed plants. Particles
of peat usually are rather small, but it may also contain pieces of wood. It is
then not so much the grain size that is characteristic, but rather the chemical
composition, with large amounts of carbon. The amount of carbon in a soil
can easily be determined by measuring how much is lost when burning the
material.
The mechanical behavior of the main types of soil, sand, clay and peat,
is rather different. Clay usually is much less permeable for water than sand,
but it usually is also much softer. Peat is usually is very light (some times
hardly heavier than water), and strongly anisotropic because of the presence
of fibers of organic material. Peat usually is also very compressible. Sand is
rather permeable, and rather stiff, especially after a certain preloading. It
is also very characteristic of granular soils such as sand and gravel, that they can not transfer tensile stresses. The particles can only transfer
compressive forces, no tensile forces. Only when the particles are very small and the soil contains some water, can a tensile stress be transmitted,
by capillary forces in the contact points.
The grain size may be useful as a first distinguishing property of soils, but it is not very useful for the mechanical properties. The quantitative
data that an engineer needs depend upon the mechanical properties such as stiffness and strength, and these must be determined from mechanical
tests. Soils of the same grain size may have different mechanical properties. Sand consisting of round particles, for instance, can have a strength
that is much smaller than sand consisting of particles with sharp points. Also, a soil sample consisting of a mixture of various grain sizes can
have a very small permeability if the small particles just fit in the pores between the larger particles.
14 A. Verruijt, Soil Mechanics : 2. CLASSIFICATION
The global character of a classification according to grain size is well illustrated by the characterization sometimes used in Germany, saying
that gravel particles are smaller than a chicken’s egg and larger than the head of a match, and that sand particles are smaller than a match
head, but should be visible to the naked eye.
2.2 Grain size diagram
The size of the particles in a certain soil can be represented graphically in a grain size diagram, see Figure 2.1. Such a diagram indicates the
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.................................................................................................................................................................
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.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
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.........
0.01 mm 0.1 mm 1 mm 10 mm
0 %
100 %
Figure 2.1: Grain size diagram.
percentage of the particles smaller than a certain diameter, mea-
sured as a percentage of the mass (or weight). A steep slope
of the curve in the diagram indicates a uniform soil, a shallow
slope of the diagram indicates that the soil contains particles
of strongly different grain sizes. For rather coarse particles, say
larger than 0.05 mm, the grain size distribution can be deter-
mined by sieving. The usual procedure is to use a system of
sieves having different mesh sizes, stacked on top of each other,
with the coarsest mesh on top and the finest mesh at the bot-
tom, see Figure 2.2. After shaking the assembly of sieves, by
hand or by a shaking machine, each sieve will contain the par-
ticles larger than its mesh size, and smaller than the mesh size
of all the sieves above it. In this way the grain size diagram can
be determined. Special standardized sets of sieves are available,
as well as convenient shaking machines. The example shown in
Figure 2.1 illustrates normal sand. In this case there appear to be no grains larger than 5 mm.
The grain size distribution can be characterized by the quantities D
and D
. These indicate that 60 %, respectively 10 % of the particles
(expressed as weights) is smaller than that diameter. In the case illustrated in Figure 2.1 it appears that D
≈ 0 .6 mm, and D
≈ 0 .07 mm.
The ratio of these two numbers is denoted as the uniformity coefficient C
u
C
u
D
D
In the case of Figure 2.1 this is about 8.5. This indicates that the soil is not uniform. This is sometimes denoted as a well graded soil. In a
poorly graded soil the particles all have about the same size. The uniformity coefficient is than only slightly larger than 1, say C
u
16 A. Verruijt, Soil Mechanics : 2. CLASSIFICATION
Although the interaction of clay particles is of a different nature than the interaction between the much larger grains of sand or gravel, there
are many similarities in the global behavior of these soils. There are some essential differences, however. The deformations of clay are time
dependent, for instance. When a sandy soil is loaded it will deform immediately, and then remain at rest if the load remains constant. Under
such conditions a clay soil will continue to deform, however. This is called creep. It is very much dependent upon the actual chemical and
mineralogical constitution of the clay. Also, some clays, especially clays containing large amounts of montmorillonite, may show a considerable
swelling when they are getting wetter.
As mentioned before, peat contains the remains of decayed trees and plants. Chemically it therefore consists partly of carbon compounds.
It may even be combustible, or it may be produce gas. As a foundation material it is not very suitable, also because it is often very light and
compressible. It may be mentioned that some clays may also contain considerable amounts of organic material.
For a civil engineer the chemical and mineralogical composition of a soil may be useful as a warning of its characteristics, and as an
indication of its difference from other materials, especially in combination with data from earlier projects. A chemical analysis does not give
much quantitative information on the mechanical properties of a soil, however. For the determination of these properties mechanical tests, in
which the deformations and stresses are measured, are necessary. These will be described in later chapters.
2.4 Consistency limits
For very fine soils, such as silt and clay, the consistency is an important property. It determines whether the soil can easily be handled, by soil
moving equipment, or by hand. The consistency is often very much dependent on the amount of water in the soil. This is expressed by the
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Figure 2.3: Liquid limit.
water content w (see also chapter 3). It is defined as the weight of the water per unit weight of solid
material,
w = W
w
/W
k
When the water content is very low (as in a very dry clay) the soil can be very stiff, almost like a
stone. It is then said to be in the solid state. Adding water, for instance if the clay is flooded by rain,
may make the clay plastic, and for higher water contents the clay may even become almost liquid. In
order to distinguish between these states (solid, plastic and liquid) two standard tests have been agreed
upon, that indicate the consistency limits. They are sometimes denoted as the Atterberg limits, after
the Swedish engineer who introduced them.
The transition from the liquid state to the plastic state is denoted as the liquid limit, w
L
. It represents the lowest water content at which the
soil behavior is still mainly liquid. As this limit is not absolute, it has been defined as the value determined in a certain test, due to Casagrande,
see Figure 2.3. In the test a hollow container with a soil sample may be raised and dropped by rotating an axis. The liquid limit is the value
A. Verruijt, Soil Mechanics : 2.4. Consistency limits 17
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Figure 2.4: The fall cone.
of the water content for which a standard V-shaped groove cut in the soil, will just close
after 25 drops. When the groove closes after less than 25 drops, the soil is too wet,
and some water must be allowed to evaporate. By waiting for some time, and perhaps
mixing the clay some more, the water content will have decreased, and the test may be
repeated, until the groove is closed after precisely 25 drops. Then the water content must
immediately be determined, before any more water evaporates, of course.
An alternative for Casagrande’s test is the fall cone, see Figure 2.4. In this test a steel
cone, of 60 grams weight, and having a point angle of 60
, is placed upon a clay sample,
with the point just at the surface of the clay. The cone is then dropped and its penetration
depth is measured. The liquid limit has been defined as the water content corresponding
to a penetration of exactly 10 mm. Again the liquid limit can be determined by doing the
test at various water contents. It has also been observed, however, that the penetration
depth, when plotted on a logarithmic scale, is an approximately linear function of the
water content. This means that the liquid limit may be determined from a single test, which is much faster, although less accurate.
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0
10
20
30
0 100 %
w
P
w
L
Figure 2.5: Water content.
The transition from the plastic state to the solid state is called the plastic limit, and denoted as
w
P
. It is defined as the water content at which the clay can just be rolled to threads of 3 mm diameter.
Very wet clay can be rolled into very thin threads, but dry clay will break when rolling thick threads.
The (arbitrary) limit of 3 mm is supposed to indicate the plastic limit. In the laboratory the test is
performed by starting with a rather wet clay sample, from which it is simple to roll threads of 3 mm. By
continuous rolling the clay will gradually become drier, by evaporation of the water, until the threads
start to break.
For many applications (potteries, dike construction) it is especially important that the range of the
plastic state is large. This is described by the plasticity index PI. It is defined as the difference of the
liquid limit and the plastic limit,
PI = w
L
− w
P
The plasticity index is a useful measure for the possibility to process the clay. It is important for
potteries, for the construction of the clay core in a high dam, and for the construction of a layer of low
permeability covering a deposit of polluted material. In all these cases a high plasticity index indicates
that the clay can easily be used without too much fear of it turning into a liquid or a solid.
In countries with very thick clay deposits (England, Japan, Scandinavia) it is often useful to deter-
mine a profile of the plastic limit and the liquid limit as a function of depth, see Figure 2.5. In this
diagram the natural water content, as determined by taking samples and immediately determining the
water content, can also be indicated.
Chapter 3
PARTICLES, WATER, AIR
3.1 Porosity
Soils usually consist of particles, water and air. In order to describe a soil various parameters are used to describe the distribution of these three
components, and their relative contribution to the volume of a soil. These are also useful to determine other parameters, such as the weight of
the soil. They are defined in this chapter.
An important basic parameter is the porosity n, defined as the ratio of the volume of the pore space and the total volume of the soil,
n = V
p
/V
t
For most soils the porosity is a number between 0.30 and 0.45 (or, as it is usually expressed as a percentage, between 30 % and 45 %). When
the porosity is small the soil is called densely packed, when the porosity is large it is loosely packed.
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Figure 3.1: Cubic array.
It may be interesting to calculate the porosities for two particular cases. The first case is a very
loose packing of spherical particles, in which the contacts between the spheres occur in three mutually
orthogonal directions only. This is called a cubic array of particles, see Figure 3.1. If the diameter of
the spheres is D, each sphere occupies a volume πD
/6 in space. The ratio of the volume of the solids
to the total volume then is V
p
/V
t
= π/6 = 0.5236, and the porosity of this assembly thus is n = 0.4764.
This is the loosest packing of spherical particles that seems possible. Of course, it is not stable: any
small disturbance will make the assembly collapse.
A very dense packing of spheres can be constructed by starting from layers in which the spheres form a pattern of equilateral triangles, see Fig-
ure 3.2. The packing is constructed by packing the layers such that the spheres of the next layer just fit in the hollow space between three spheres
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Figure 3.2: Densest array.
of the previous layer. The axial lines from a sphere with the three spheres that support it from below
form an regular tetrahedron, having sides of magnitude D. The height of each tetrahedron is D
Each sphere of the assembly, with its neighboring part of the voids, occupies a volume in space of
magnitude D × (D
3 /4) × (D
2 /3) = D
1 /2. Because the volume of the sphere itself is πD
the porosity of this assembly is n = 1 − π/
18 = 0.2595. This seems to be the most dense packing of
a set of spherical particles.
Although soils never consist of spherical particles, and the values calculated above have no real
meaning for actual soils, they may give a certain indication of what the porosity of real soils may be. It can thus be expected that the porosity
n of a granular material may have a value somewhere in the range from 0.25 to 0.45. Practical experience confirms this statement.
20 A. Verruijt, Soil Mechanics : 3. PARTICLES, WATER, AIR
The amount of pores can also be expressed by the void ratio e, defined as the ratio of the volume of the pores to the volume of the solids,
e = V
p
/V
s
In many countries this quantity is preferred to the porosity, because it expresses the pore volume with respect to a fixed volume (the volume of
the solids). Because the total volume of the soil is the sum of the volume of the pores and the volume of the solids, V
t
= V
p
+ V
s
, the porosity
and the void ratio can easily be related,
e = n/(1 − n), n = e/(1 + e). (3.3)
The porosity can not be smaller than 0, and can not be greater than 1. The void ratio can be greater than 1.
The void ratio is also used in combination with the relative density. This quantity is defined as
RD =
e
max
− e
e
max
− e
min
Here e
max
is the maximum possible void ratio, and e
min
the minimum possible value. These values may be determined in the laboratory. The
densest packing of the soil can be obtained by strong vibration of a sample, which then gives e
min
. The loosest packing can be achieved by
carefully pouring the soil into a container, or by letting the material subside under water, avoiding all disturbances, which gives e
max
. The
accuracy of the determination of these two values is not very good. After some more vibration the sample may become even denser, and the
slightest disturbance may influence a loose packing.
It follows from eq. (3.4) that the relative density varies between 0 and 1. A small value, say RD < 0 .5, means that the soil can easily be
densified. Such a densification can occur in the field rather unexpectedly, for instance in case of a sudden shock (an earthquake), with dire
consequences.
Of course, the relative density can also be expressed in terms of the porosity, using eqs. (3.3), but this leads to an inconvenient formula, and
therefore this is unusual.
3.2 Degree of saturation
The pores of a soil may contain water and air. To describe the ratio of these two the degree of saturation S is introduced as
S = V
w
/V
p
Here V
w
is the volume of the water, and V
p
is the total volume of the pore space. The volume of air (or any other gas) per unit pore space then
is 1 − S. If S = 1 the soil is completely saturated, if S = 0 the soil is perfectly dry.
