Applied Analyses in Geotechnics, F Azizi.pdf

(1)

, , .:,( . , _{·i.··,,{,,"~}t · c" 11" _{.~_}.

_{... ,' ...}

...., ,¡rr.:. ',' .• _~.','~"-'u'"

_...

_~~J

, ':." ... ( '; .\ '.!

~:::4't.¡ií.,

" .• , ' . ( '-1. "" Jl;:,~I;-¡

,,"f:.;,.:

.' .. ' . .' .'

~ ?~

.·· ..

'·,.'1.~·l.J • • . ' ,. A.~ ,',/,., ' ' , ' .. • . ) ... ~ ... ¡.~

A

PRq~jjA

N

A~X~l~

.~

.:':·:<!'.r:~\',~:

.

JN

:

G

E.OJE~.~

NI

es

'''_' s..

/

.

-

:::'

"""C....

.,

--

,

._

-

--E & FN SPON

(2)

Applied Analyses

in Geotechnics

(3)

(4)

,\

Applied Analyses

in Geotechnics

u

Fethi Azizi

"

U

n

i

v

e

r

s

i

ty of

Pl

y

mouth

,

U

K

(5)

Slmultaneously published in the USA and C&IIIIda by E &: FN Spon

29 West 351h Slreet. New York, NY 10001

E & FN Spo" is mI impri", of,he: ro)'lor & FrlHlcis Group e 2000 E &: FN Spon

Pnnted.nd bound in Greal Briuin by TJ InterNIllonal LId. Padstow, Comwall

AH nghl$ Te$f'rved. No par! ofthls booIt lNIy be repnnted or rq¡roduced or unlJ.Sed In lIIy form or by In)' de<:trOmc. me<:hamcal, or other !mans. now known or hereafter lnvenled, LIIcludmg photocopyulg 100 recording, or In lIIy informalion Slongc or retoe"'. S)'SIen!. wllhoul pcmtlSSlOn In wTlllng from Ihc

pubhshcn.

Thc pubhshcr makcs no representallon, cxpre" or lmphed, wllh regard 10 ¡he accurac)' oribe mfonnahon conlalned in rhis book and cannOI accC'p1 In)' legal re!iponsiblhl)' or habl.n), for any CITOI'S or oml$SIOns Ihal lOa)' be made.

8ritlslr Librtlry Calaloguillg ill Publica/io" Dala

A catalogue record for!his book is a'nl.lable (rom !he Brillsh Llbrary Librar)' ofCoIIgress Curalogillg 111 Publlcalloll Data

Apphed Inalyses in geole<:hnics I Fethl A:t"!1 p. cm.

lnc.udcs blographocal refm:ncn and rndcx. 1. Engmeenllg gcolog), 1. T'II(>. TA105.A91 1999 624 .• ·5.-dc21 ISBN 0-4.9-2S340-8 (hb) ISBN 0-419-25350-5 (pb) 99-35.26 ClP

(6)

To

/he memory

o/

my

Fa/her

La raison nous commande bien plus impérieusemenl qu'un maÍ/re; car en désobéissanl ir ('un on est mafheureux. el en désobéissanl ir f'aulre on esl un sol. Blaise Pascal

(7)

(8)

PREFACE

LIST OF MAIN SYMBOLS CONVERSION FACTORS ACKNOWLEDGEMENTS

Preface

When writing this book, I was always guided by four things: (a) the book must be organised to suit different needs of undergraduate and postgraduate teaching programmes; (b) the content should stimulate the reader to think, thus avoiding it being a user's manual that describes procedures and methodologies; (e) the book must be innovative in that it should contain aspects of analyses that are only accessible through specialised books or research papers, and (d) the analysis related to each section should be thorough and well referenced so that the reader can concentrate on thinking and understanding.

I have al so endeavoured to make the book enjoyable to read, but with respect to this subjective aspect, only your judgement matters, and so, do not hesitate to compare this book with other textbooks in terms of style and content while remembering the words of Benjamin Disraeli ' ... this shows how mueh easier il is lo be eritieal than lo be eorreet'.

I am assuming that the reader has a working knowledge of engineering mechanics, especially of stresses and strains, Mohr's circle representation of stresses and strains, equilibrium equations in terms of moments and forces, elasticity and plasticity.

1 am aware that part of the analysis in sorne sections of the book may appear somewhat mathematically involved. In fact, you would be surprised to find how easy they are once you have re-read them: it is only by reading the analysis time and time again that an intellectual skill can be mastered. The book contains more than 65 worked examples and in excess of 450 illustrations which are presented in a clear and detailed way which can only enhance the understanding of the principIes involved.

By the time you have read through this book, you will have hopefully understood, learned and mastered its content. In addition, you will have been able to satisfy sorne of your curiosity through sorne detailed

(13)

derivations of forrnulae and equations which are presented as a fait accompli in other textbooks. Y ou will have al so noticed the rather lengthy process related to the detailed calculations of sorne worked examples. In practice, these calculations are almost exclusively undertaken numerically, because not only specialised software package s dealing with different themes in geotechnics are wide1y available, but a designer can always develop a software to suit his or her needs. So, do not despair if you find the hand calculations in sorne instances long and tedious because that was the feeling I had when I wrote them! However, as a learning experience, every example presents you with an opportunity to see how these details are handled and, more to the point, helps you develop your engineering judgement, in that the outcome of any computation always reflects the choice made vis

a

vis the soil parameters. Remember that geotechnics is not an exact science; rather, it can be described as an art in which the artist (Le. the engineer) has to rely sometimes, if only partly, on his or her intuition (Le. engineeringjudgement).

Moreover, you will be able to appreciate that, although in practice calculations may be undertaken in a different way, the fundamental thinking is similar to what you would have learned throughout the book. If anything, once you have assimilated the basic knowledge and principIes, you will find it much easier to make the appropriate assumptions with confidence. This reinforces the statement made earlier about mastering an intellectual skil!. So do not be put off if you do not fully understand what you have read; go through it again bearing in mind that the last reading is the one that matters.

This book is al so meant to be useful to a postgraduate student who seeks a deeper understanding and a specialised knowledge about different subjects in geotechnics. In addition, practising engineers may find the book valuable since most of the mathematical forrnalisms can be forgone, provided that the assumptions of the forrnulation are well understood, so that it is not used in a perfunctory way as a recipe. Remember, the book is meant to make you think, then acto

Bearing in mind what Sir Arthur Conan Doyle once said, through Sherlock Holmes, 'it is a capital mislake lo Iheorise withoul data', 1 went to great lengths to try to relate any theoretical aspects to practice, and this is reflected in the choice of the title for the book 'Applied Analyses in

(14)

PREFACE xiii

Geotechnics'. The emphasis is always on the practical side although, admittedly, sorne sections contain a fair amount of accessible mathematical formalismo There is no escape from it, an engineer must be able to understand how to translate a physical behaviour into non trivial mathematical equations; after aH, that is what design is about!

I have refrained from making any suggestions on how the book may be used by coHeagues to organise a course at whatever level. Had I done so, it would not only be presumptuous on my part, but it would al so contradict my earlier statement regarding the different needs of different programmes.

I would like to thank the many friends and coHeagues who contributed, in their own ways, to the improvement of the content of this book. In particular, I am grateful to Clive Dalton, John Davidson, Kenneth Fleming, Henri Josseaume, David Naylor, David Rowe Bottom, Robert Whittle and Dennis Wilkinson for taking the trouble to go through part or aH of the manuscript and make valuable constructive comments.

Finally, on a personal note, notwithstanding the sporadic late-night frustrations or the occasional social deprivation, I have had a tremendous pleasure writing this book. I hope you will enjoy reading it.

(15)

a A Ab As B B' h C Ch Cu Cv C_w Ce

C

g

C

s

C

u Ca CSL D,d de, dq , dy dw [D] e E Eb Ee Em

DE

Eplp

J

F

Jo

{F}

radius, area, amplitude

air content, activity, area, porewater pressure coefficient base area

shaft area

width, porewater pressure coefficient effective width

width

apparent cohesion

coefficient of horizontal consolidation undrained shear strength

coefficient ofvertical consolidation cohesion at soil/wall interface compression index

coefficient of gradation swelling index

coefficient of uniformity

slope of secondary consolidation graph critical state line

diameter, depth, depth factor, depth of embedment depth factors

drawdown elasticity matrix

void ratio, eccentricity, depth corresponding to zero net pressure

stiffness (elasticity modulus) soil secant modulus

stiffness modulus of pi le material pressuremeter modulus

work done by an extemalload flexural stiffness

motion frequency force, factor of safety correction factor vector of nodal forces

(16)

g G G*

G

s h, H ier ie 1 le, lq, ly

Ir

lp

la l/y J k

k'

_h k~ kt K [K] K* Ka Ke Ka Kp L M Ms m mu mv

LIST OF MAIN SYMBOLS xv

acceleration due to gravity, intercept of Hvorslev surface, depth corresponding to zero bending moment

shear modulus

modified shear modulus specific gravity

height, total head, length of drainage path, slope ofHvorslev surface

capillary rise elevation head pressure head

capillary saturation level hydraulic gradient critical hydraulic gradient exit hydraulic gradient

moment of inertia, rigidity index inclination factors

liquidity index plasticity index influence factor steel section modulus cross modulus permeability

equivalent horizontal permeability equivalent vertical permeability transformed permeability

ratio of horizontal to vertical effective stresses, bulk modulus

stiffness matrix modified bulk modulus

coefficient of active earth pressure length ratio

coefficient of active pressure at rest coefficient of passive pressure length

mass, moment

soil/shaft flexibility factor soil creep parameter slope constant

(17)

n N Ne,Nq,Ny Ni Nd [N] NCL OCR P

_,

Pe Pa Pe PI Pp PBP [P] q Q Qa Qb Qn Qs Qu Qug R, r ru Re Ro RC S Se S Se, Sq, Sy SBP SBS f..S t T Tv , Tr porosity

normal force, stability number bearing capacity factors number of flow channels number of equipotential drops shape functions

normal consolidation line overconsolidation ratio mean stress, perimeter

equivalent mean effective pressure active thrust creep pressure limit pressure passive thrust prebored pressuremeter permeability matrix

load, rate offlow, deviator stress load, rate of flow

safe working load ultimate base resistance negative skin friction ultimate shaft friction

ultimate loading capacity of a single pile ultimate loading capacity of a pilegroup radius

pore pressure ratio Reynolds number radius of influence relative compaction

degree of saturation, sensitivity, spacing, total settlement elastic shortening

reduced mean stress shape factors

self-boring pressuremeter state boundary surface stress error

time, reduced deviator stress

surface tension force, torque, shear force per linear metre time factors

(18)

u Ug U {U} v V Va Vs

V

w W WL Wp

Ws

W 8W z z¡ ~ y Y 8 I 8 f,K,A,M fe

fh

fv A,n V ~ P (J (J I LlST OF MAIN SYMBOLS

porewater pressure, lateral displacement pore gas pressure

degree of consolidation, thrust due to water pressure vector of nodal unknowns

velocity, specific volume, vertical displacement volume, model centrifuge in flight velocity volume of air

volume of solids volume ofwater

water content, weight, radial displacement liquid limit

plastic limit shrinkage limit weight

work dissipated per unit volume depth

neutral depth for negative skin friction soil slope behind a retaining wall unit weight

effective unit weight soil/structure friction angle strain

XVIl

dynamic viscosity, ratio oftotal heads, drainage

coefficient, ratio of deviator stress to mean effective stress, efficiency factor, optimum angle of failure behind a retaining wall

temperature

critical state parameters Coriolis acceleration horizontal shaking vertical acceleration wall inclinations Poisson's ratio

correction factor, strength ratio density, sheet pile stiffness total stress

effective stress active pressure horizontal stress

(19)

crp crv,cr¡ t 'tmob U co preconsolidation pressure vertical stress shear stress

mobilised shear stress angle of dilation

angle of shearing resistance, fram factor

peak, critical and residual angles of shearing resistance stress function, velocity potential

constant stream function angular velocity

(20)

- multiplication factors 109 giga 106 mega 103 _kilo 10-3 _mi/li 10-6 _micro 10-9 _nano -lengtb 1 cm = 0.3937 in 1 m = 3.28ft - area 1 m2 = 1O.76ft2 - volume G M k m J..L n 11 = 1000 cm3 = 61.02 in3 1 m 3 = 35.32ft3 - mass 19=0.0022p - density Conversion factors 1 in

=

2.54 cm 1ft

=

30.48 cm 1 in3 = 16.388 cm3 lft3 = 0.02832 m 3 lp=453.6g 1 g/cm 3 = 1 Mg/m 3 = 62.43 p/ft3 lp/ft3 = 0.01602 Mg/m 3 - force 1 N = 102g = 0.2248p - pressure 1 N/m2 = 1Pa 1 kN/m 2

=

20.89 p/ft2 1 kN/m 2 = 0.1450psi - angle 1 rad

=

57.296" - temperature

oc

= 0.555 °F -17.778 1p = 4.448 N 1 bar = 100 kPa 1 p/ft2 = 0.04787 kN/m 2 1 psi = 6.895 kN/m 2 10 = 0.017453 rad

(21)

We are very grateful to the following for granting us permission to use published material for which they are copyright holders: the American Society ofCivil Engineers (figures 8.32, 8.33, 4.24), the Institution ofCivil Engineers, London (figures 1.8, 4.35, 11.7), the National Research Council of Canada (figures 4.28, 10.4), Longman Scientific & Technical (figures 7.19, 9.14), the British Research Establishment (figure 8.27), Solétanche-Bachy (figure 9.8), la Revue Fram;aise de Géotechnique (figures 5.4, 5.26, 6.32), le Laboratoire Centrale des Ponts et Chaussées (figures 5.27, 5.31, 5.32, 9.12, 11.22, 11.31, 14.18), the Transportation Research Board, Washington DC (figure 2.20), Atlas Palen Franki Geotechnics, Belgium (figure 9.3), Kvaerner Cementation Foundations (figure 9.44), Cambridge Insitu (figures 12.18, 12.19), A. A. Balkema (tab/es 10.1, 10.2, figure 13.1), John Wiley VCH, Winheim (figure 3.45), Construction Industry Research and Information Association, London (figure 9.15), John Wiley, New York (figures 3.6, 7.20).

(22)

CHAPTER 1

Formation and physical properties of soils

1.1 Geological aspects of soil formation

Since its formation cirea 4600 Ma (million years) ago, the planet Earth was (and still is) subjected to continuous geological changes due mainly to volcanic, seismic and climatic activities. To understand the constitution of soils, which is a consequence of these activities, it is of paramount importance to relate it to the geological evolution at the Earth crust (i.e. the Earth upper layer).

The basic definition of a soil is that of a material having three components, namely solid particles, air and water. The mechanical behaviour of a soil mass, for instance its response to any externally applied load, is, in large part, governed by the size and proportion of the solid particles it contains as will be explained. However, the question to be addressed at this stage relates to the formation of these solid particles which have originated from roeks; thus one needs to explore, first, the formation of rocks.

Rocks comprise one or more minerals. The behaviour of a rock in terms of hardness, nature, and chemical reaction to external agents, depends on the minerals it contains. The most common minerals in rocks are silieates (such as feldspar, quartz and muscovite).

There are three main types of rocks:

- igneous roeks such as granite, dolerite and basalt are formed as a result of magma cooling either on or below the surface of the Earth. Rocks which have resulted from the cooling of magma on the ground surface are called extrusive, the basalt block s of the Giant's Causeway in Northern Ireland being a well known example. On the other hand, rocks made from the slow cooling of magma within the Earth crust, in other words below the ground surface, are known as intrusive, dolerite and granite being typical examples of this type of rocks.

(23)

- metamorphic rocks such as slate, schist and marble are products of existing rocks subjected to changes in temperature amI/or pressure leading to changes in mineral composition of the parent rock. In regional metamorphism, pressure rather than heat represents the main agent of change, whereas in thermal metamorphism, the changes in mineral composition occur in a relatively short time, that is in geological terms, and are mainly generated by an increase in temperature. For example, the process of thermal metamorphism can cause apure limestone to be re-crystallised as marble. A famous example is provided by sorne of the mountains in Tuscany, Italy, which are virtually solid marble. Slate on the other hand is a mildly metamorphosed shale.

- sedimentary rocks, such as sandstone, shale and limestone, are formed through compaction of rock fragments and organic debris. These fragments are usually cemented through the action of calcite and silica. The white cliffs oi Dover, England, provide a typical example of soft limestone (al so known as chalk).

The geological process of soil formation is based on rock weathering which can occur either chemically when the mineral s of a rock are altered through a chemical reaction with rainwater, or mechanically through climatic effects such as freeze-thaw cycles and erosion (see for instance Bell, 1993).

Soils thus formed consist of a mixture of solid particles ranging from boulders (rock fragments usually larger than 200 mm in size) to clay minerals (solid particles of less than 2 x 10-3 mm in size), and are usually classified according to the way they were deposited. Hence, if the present location of the soil is that in which the original weathering of the parent rock occurred, the soil is known as residual. If it is not the case, the soil is referred to as transported The process by which the soil is transported from one location to another can be due to gravity, wind, ice or water. - gravity deposits result generally from rock weathering at the face of a cliff that causes the rock fragments to fall to the foot of the cliff, thus forming apile of rock deposit known as talus. The final position of each fragment is entirely due to gravitational forces and no other transporting agent is involved in this process depicted in figure 1.1.

(24)

PHYSICAL PROPERTIES OF SOILS

3

-wind deposits are best illustrated by sand dunes which are shaped according to the size of sand particles and the velocity ofthe wind.

- most of the glacial deposits were

fonned throughout the glacial age which, in terms of geological timescale, corresponds to the Pleistocene epoch (between 1.5 and 2 million years ago). From a mechanical viewpoint, lhe flow of large ice blocks is such that it can erode soils and crush rocks. The debris thus transported, known as glacial drift, is formed of soil particles and rock fragments of various sizes and shapes and can be buried al the bottom of glaciers, forming in lhe process what is known as glacial tills which can contain large boulders.

lalus

~y-

:...--_/

/

Figure 1.1: Rock weathering and gravity deposils.

-a/luvial deposits are formed through a process of erosion during which the solid particles are transported by water and deposited as soon as the water velocity becomes minimal. During the process, a natural selection occurs in that the heavier grains will be deposited first and the lighter ones will travel further downstream and be deposited through sedimentation. The soil thus formed contains a layered structure.

1.2 Physical properties of soils

On the basis of the grain sizes of different soils given in table 1.1 (notice the logarithmic scale), one would expect, for instance, a c\ay with solid partic\es of a size smaller than 2 x 10-3 mm to be much more compressible than a sand; similarly, a gravel would be much more permeable than a silt. These logical conclusions are closely related to the soil composition, whose behaviour is dependent on the type and size of solid particles and their volumetric proportion with respect to water and air, as well as on the soil stress history (i.e. the way in which it was deposited and its subsequent loading-unloading cyc\es).

(25)

Table 1./: Graín size corresponding 10 differenl types

01

soj/s lOO mm 60 mm 'mm 0.06 mm 0.002 mm

boufders cobb/es gravel sond silt day

Consider the volume of soil depicted in figure 1.2a. Assume that a microscopic analysis reveaJed that water is nol filling all the voids between

solid partic1es and that sorne pockets of air exist within the soil matrix. Now imagine that the volume V of this unsaturated soil is rearranged in a (theoretical) way such that all solid particles are squeezed together 50 that

no gap is len for water or air to til! (figure 1.2b). The remainder of the volume is therefore constituted of water and air. hence :

WClI~r _ Ir bubble volume V ::;:;.-

---R

~~

---:::::: ...--:

-.;::~.-

---solid panic1es - --- _ ::...-::::::

:---.-

---:-=::::::

- - ----...-::

-

- - .-#

----

(a)

Figure J.}: (a) Elementary ~'oJume ollloil. (b) Iloil COmpollilion. (1.1)

1'. _ I

rb)

The volume VII in the above equation is that of salid particles and the quantity Vw+ Va corresponds to the vO/lIme 01 voids V". Let the total volume of soil V be selected so that V" corresponds to one unit. While this astute choice does not affect in any way the general aspect of the following fonnulation, it makes it easier to handle and to establish. Thus, with reference lO figure 1.2b, the following basic soil properties can now be defined in a straightforward way:

(26)

- the void ratio: Vv e=-Vs

PHYSICAL PROPERTIES OF SOILS

- the specific volume: V= Vs + Vv

=1+e

- the degree 01 saturation:

- the water content: Mw w = -Ms Vw 1 w = -Vs Gs G -~ s - Pw 5 (1.2) (1.3) (1.4) (1.5a) (1.5b) (1.6) The dimension1ess parameter Gs in equation 1.6 refers to the specific gravity and represents the relative density of solid particles with respect to water. Typical values of Gs for soils are in the range 2.65 (sands) to 2.75 (clays). Ms and Mw correspond to the mass of solids and that of water respectively, and pw refers to the density ofwater [ = 1 Mg/m 3 ].

Equation 1.4 indicates that a ful/y saturated soil has a volume of water identical to that of voids because no air is contained within the soil matrix, in which case the degree of saturation is S = 100%. On the other hand, a totally dry soil contains no water and therefore its degree of saturation according to equation 1.4, is S

=

0%.

In practice, the water content is calculated from equation 1.5a in which the quantities Mw and Ms are measured in the laboratory in a very simple way.

(27)

For a given volume of soil, the operator needs to determine the mass of soil in its natural state MI, then its mass Ms after being thoroughly dried in an oven at 105°C for long enough (usually 24 h) to ensure that all but the chemically bonded water has evaporated. The mass of water M w is therefore MI - Ms.

AIso, it is easy to show that, by substituting for Vv and Vs from equations 1.4 and 1.5b respectively, into equation 1.2, the following expression for water content can be established:

eS

w=-Gs

(1.7)

so that for a fully saturated soil (i. e. S

=

100%), the latter equation reduces to e = wGs . However, for a totally dry soil, equation 1.7 cannot be used to calculate the void ratio since, in this case, both w and S are zero. Typical values for the void ratio range from 0.5 to 0.8 for sands and between 0.7 and 1.3 for clays.

Another useful property related to the void ratio and known as porosity is defined as the proportion of the volume of voids (that is air and water) with respect to the total volume (refer to figure l.2b):

(1.8)

The air content in a soil mass corresponds to the ratio of the volume of air to the total volume and, referring to figure l.2b, it can be seen that:

A= Va

= Vv-Vw =

VV[l_ VwJ=_e-O-S)

V V V Vv 1 +e

Substituting for S from equation 1.7, it follows that: A = e-wGs

l+e

(1.9a)

(1.9b) A relationship between the degree of saturation S of a soil and its air content A can easily be found were the void ratio in the aboye equation to be replaced by its value from equation 1.7, in which case:

(28)

PHYSICAL PROPERTIES OF SOILS 7

S= l-A

1 +A/wGs

(1.10)

Equation 1.10 is markedly non-linear when the air content is larger than zero.

The soil density is defined as the ratio of the soil mass to its volume. Accordingly, the following relationships which will be most useful for soil compaction calculations (see section 1.6) are easily established using equations 1.2 to 1.6:

- the bulk density:

Ms+Mw Gs(1+w) P

=

V

=

1 +e pw - the dry density (zero water content: w

=

O):

Gs

Pd=--pw l+e

- the saturated density (w

=

e/Gs , equation 1.7 with S = 100%): Gs+e Psat = -1-- pw +e (1.11) (1.12) (1.13)

A combination of equations 1.11 and 1.12 yields the relationship between the bulk and dry densities of a soil:

Pd

=

p/(1 +w) (1.14)

Furthermore, it is most helpful during a compaction process (which will be explained shortly) to relate the dry density to the air content of the soil. Thus, rearranging equation 1.1:

(1.15)

(29)

v, __

1 _ _ Pd

...!...

V - l+e - pw G_s

On the other hand, both equations 1.2 and 1.4 can be exploited:

Hence, substituting for the degree of saturation S from equation 1.7 and rearrangmg:

Vw _ wGs _ WPd

V - l+e - pw

The last ratio in equation 1.15 corresponds to the air content (equation 1.9a). From whence, a straightforward substitution for the different ratios into equation 1.15 yields:

Pd _1 + W Pd + A = 1

pw G_s pw or

Gs(1-A)

Pd

=

1 +wG_s pw (1.16)

As will be se en later, the calculation of stresses are undertaken using the appropriate soil unit weight instead of the density, the two being related through the acceleration due to gravity g = 9.81 m/s2. Accordingly, the

bulk unit weight of a soil is established from equation 1.11: Gs(1 + w)

y=pg= Yw

l+e (1.17)

Yw being the unit weight ofwater [ = 9.81 kN/m3

J.

Both the dry and the saturated unit weights are obtained in a similar way from equations 1.12 and 1.13 respectively:

(30)

PHYSICAL PROPERTIES OF SOILS Gs+e Ysat = -1--Yw +e 9 (1.19)

When a soil is fully saturated (or submerged), its solid particles are subjected to a buoyancy due to the water in accordance with Archimedes' principIe. In that case, the effective unit weight of the soil corresponds to the difference between its saturated unit weight and the unit weight of water: I Gs- l y

=

Ysat -Yw

=

-1--Yw +e Example 1.1 (1.20)

A sample of sand occupying a total volume V = 1000 cm3 has a total mass M = 1960 g. Once dried in the oven, the mass of the sample was reduced to Ms = 1710 g. Assuming the specific gravity of the solid particles is Gs = 2.65, let us evaluate the different following quantities:

- the water content: using equation 1.5a, it is seen that:

=

M w = 1960 - 1710

=

14 6 o/

w Ms 1710 . 1'0

- the void ratio: equation 1.2 yields e =

~:'

where the volume of solids is caIculated from equation 1.6:

Ms 1710 645 3

Vs = GsPw = 2.65 x 1 = cm

Hence the volume ofvoids: V_v= V - V_s= 1000 - 645 = 355 cm3 ,

and therefore the void ratio: 355

e = 645 = 0.55

- the degree

01

saturation: equation 1.4 is now used:

S = Vw _V = 1960-1710 =7040/

(31)

- the bulk density: calculated from equation 1.11:

Al 1960 3

P

= V

= 1000 =

1.96A1g/m

- the air content: found from equation 1.9a: A = Va

= Vv - V w

=

355 - 250

=

105°1'

V V 1000 . /0

If the sample of sand were saturated, then obviously the air content would be reduced to zero, implying that the volume of voids is equal to that of water: Vv

=

Vw• Accordingly, the mas s ofwater becomes:

Alw = VvPw

=

355 xl = 355g

thus yielding a water content (for the saturated sand) of: - 355 _ 208°1'

W - 1710 - . /0

The void ratio is defined as the ratio of the volume of voids to that of solids, both of which are unchanged, and hence, the void ratio remains constant. Knowing that the degree of saturation is in this case S = 100%, equation 1.7 then yields:

e = wGs

=

0.208 x 2.65

=

0.55

which is the same value as the one calculated previously. The density of the sand, however, increases to reflect the saturation and is calculated from equation 1.11 in which the total mass is now:

AI=Als+Alw= 1710+355=2065g Therefore:

2065 3

(32)

PHYSICAL PROPERTIES OF SOILS 11

Example 1.2

A sample of compacted clay with a total volume V = 7.85 x lO-4m3 has a

moisture content w = 15%, an air content A = 8%, and a specific gravity Gs = 2.7. Required are: the degree of saturation, void ratio, porosity, and bulk, dry and saturated densities.

The volume of air is 8% of the total volume and thus : Va = 0.08 x 7.85 X 10-4 = 0.628 X lO-4m3

Moreover, the volume of water as a proportion of the volume of solids can easily be calculated from equation 1.5b:

Vw = wGsVs = 0.15 X 2.7Vs = 0.405Vs

Since the total volume is V

=

Va + V w + Vs , it follows that: V= 0.08V + 1.405Vs Hence: V = 0.92 V= 5 14 X 10-4 3 s 1.405· m and V w = 0.405 Vs = 2.08 X lO-4m3

The degree of saturation is evaluated from equation 1.4:

S = Vw = 2.08 -7701" Va+V_w 0.628+2.08- 1'0

the void ratio and the porosity being calculated from equations 1.7 and 1.8 respectively:

= wGs = 0.15x2.7 =053

e S 0.77 . , n= _e_ _l+e~0.35

Finally the bulk, dry and saturated densities are determined from equations 1.11, 1.12 and 1.13 respectively, in which the density of water is taken as

(33)

P=G(1+w)p s l+e w =2.7x1.15=203~f.1 1.53 . 1V1gm 3 Pd = Gs pw = 2.7 = 1.77 Mg/m3

1 + e 1.53

Gs+e 2.7+0.53 3

psat = --¡¡-;-Pw = 1.53 = 2.11 Mg/m

1.3 Particle size analysis

The mechanical behaviour of a given soil depends on the size of its solid particles, on the minerals it contains, as well as on its stress history. In particular, the resistance of a soil to any applied load or the ease with which water can flow through it is govemed by its granulometry (i. e. the range of solid particles it contains). Soils can be classified either as granular or as cohesive.

Granular soils such as sands and gravels have individual solid particles, most of which can be identified by sight. The strength of such materials results from the interlocking of solid particles which provides resistance through friction. Accordingly, the range of particle size present within the soil matrix and the way these particles interact are a key element to predicting the soil behaviour. The grain size distribution can be determined with the help of a technique known as sieving.

This old technique, which applies to granular soils (i.e. soils that do not contain silt or clay), has the advantage of being simple and cheap. During the standard procedure (a detailed description of which can be found in any standard laboratory testing manual), the soil is sifted through progressively finer woven-wire sieves, down to a mesh size of 31 ~m. The percentage by weight of material passing through each sieve is then plotted on a chart representing the particle size distribution such as the one depicted in figure 1.3.

The shape of the curve thus obtained gives an indication of the distribution by weight of different sizes of so lid particles within the soil. However, the graph in question does not correspond to the true weight distribution since the material retained in any sieve yields the weight of so lid particles with

(34)

PARTICLE SIZE ANALYSIS 13

an individual size exceeding that of the sieve mesh, including those (elongated) particles with only one dimension larger than the mesh size. Accordingly, the outcome of such a test depends on the time and method of operation: the longer it takes to undertake the test, the more likely that sorne (or in sorne instances aH) elongated solid particles will faH through the sieve because of a change in orientation.

Assuming that these shortcomings are acceptable, then the two typical curves in figure 1.3 are indicative of different soil grading: the soil corresponding to curve A is referred to as well graded because the graph reflects relatively similar proportions of different particle sizes. For curve

B, however, although there is no disproportion between the solid particles' size distribution, the graph is more compact than in case A, and the corresponding soil is known as uniformly graded.

0.002 mm 0.06 mm 2mm 60 mm 100 ~ 80 .:: .¡;;

'"

S, 60 ~ .':1 40 .::

'"

<..> .... El.. 20 O

e/ay silt sand gravel

Q

r

1 i

¿8-1 1 I 1 1 I 1 _I 1 1 I 1 _I 1 1 I 1 I

IJ

I 1 1 I 1 I I 1 1 I 1 I I 1 1 I 1 I I 1 1 i

V:/

1 1 _I 1 1 I 1 I 1 1 I 1 I 1 I 1 1 I 1

/i

}

I 1 1 I 1 _I 1 1 i 1 A I 1 1 1 _I _I 1 1 I ¿

:1.

1 : 1 1 1 1 1 I 1 I 1 1 1 1 lB 1 I 1 cobble sedimentation ? sieving

Figure 1.3: Partic/e size distributionfor granular soils.

Moreover, if dIO, d30, and d60 are the solid particle sizes corresponding to 10%, 30% and 60% respectively of percentage passing, then it is useful to calculate the two foHowing coefficients:

(35)

- the coefficient

01

gradation: (1.22)

The higher the value of C u, the larger the range of particle sizes contained

within the soil matrix. Also, the coefficient of gradation of a well graded sand is usually in the range 1 :$; C g :$; 3 .

Example 1.3

Calculate the coefficients of uniformity and gradation of the soils corresponding to graphs A and B in figure 1.3.

For soil A, it is seen that: dIO >::: 0.01 mm, d30 >::: 0.09 mm, d60 >::: 0.55 mm.

055 0.092

Hence: Cu _{= 0:01 = 55,} _Cg _{= 0.55}x 0.01 = 1.47.

ForsoilB: dlO>:::0.35mm, d3o>:::0.6mm, d6o>:::l.1mm,and

Cu

=

₀1

.3

15 = _3.1, 0.6 2

Cg₌_{1.1 x 0.35 =0.94.}

Soil A contains a wide range of particle sizes, from gravel to silt (hence the large value of C u), none of which is predominant. This is reflected in the value of the coefficient of gradation C g. Soil B on the other hand has a

uniform grading indicated by a grading coefficient value of slightly smaller than one.

Cohesive soils have solid particles with a size smaller than 0.063 mm, making the sieving technique impractical. Instead, the grain size distribution of such soils can be determined using other techniques such as sedimentation. This traditional method, which is only (realistically) applicable to grain sizes in the range 2 to 50 Ilm, is based on Stokes' law relating the terminal velocity v of a solid particle with an equivalent sphere diameter D s, falling in water of dynamic viscosity TI, to its weight (read diameter Ds ):

(36)

PARTICLE SIZE ANALYSIS 15

where g is the acceleration due to gravity, ps and pw are the densities of solid particles and of water respectively, and 11

=

1.005 x 10-6 _Mg/m.s_IS

the water dynamic viscosity at a standard temperature of 200

C.

Equation 1.3 can therefore be used in a straightforward way to calculate the settling time.

Example 1.4

Estimate the time t that will take a solid particle with an equivalent diameter Ds = Illm and a density ps = 2.5 Mg/m3 to settle a distance h = 1 cm in water at 20° C.

Rearranging equation 1.23, it follows that:

therefore:

_ b. _

(Ps-Pw) D2 v - t -1811 g. s t = _ _ 1_811....!.h_--::-(Ps - Pw)gD; t= 18 x 10-2 ,::::j 12 200s (2.5 -1) x 106 _{x 9.81}_{x 10-}12 _'

The time needed for this particle to settle a mere 1 cm (about 3.4 hours ) is an indication that sedimentation is an exceedingly slow process.

Knowing that Stokes' law uses an equivalent diameter Ds which assumes that the solid particle has a regular compact spherical shape, one therefore expects an irregularly shaped normal particle to have a larger surface area than its equivalent sphere, thus causing the settling to be even slower because of the increased drago Consequently, the time calculated using equation 1.23 is likely to deviate substantially from the actual settling time. Furthermore, theoretical and experimental evidence indicate that for particles smaller than 21lm in size, the gravitational settling calculated from Stokes' law is markedly affected by the Brownian movement (Le. the irregular oscillations of particles suspended in water) which can induce an error of more than 20%. On the other hand, Stokes' law is no longer applicable for particles larger than 50 Ilm in size, the settling being turbulent. Therefore the range 2 to 50 Ilm mentioned earlier within which equation 1.23 can be applied under a strict temperature control since the water viscosity 11 is a temperature dependent parameter (it is useful to

(37)

remember that a 1°C change in temperature will induce a 2% change in water viscosity).

To offset these shortcomings, other sophisticated techniques have been developed and are slowly being adopted in soil mechanics laboratories, a1though their use was restricted until recentIy to the fields of elay mineralogy and sedimentology. These techniques inelude photon corre/ation specstroscopy which can be applied to measure partiele sizes in the range 1 nm to 1 J-lm (that is 10-9 m to 10-6 m). Because of their small size, these partieles, known as colloíds, are subjected mainly to a Brownian motion that scatters light. The principIe of the method consists of relating the diffusion of partieles to the autocorrelation function of the scattered light, and more details can be found in MacCave and Syvitski (1991) and Weiner (1984).

Laser diffraction spectroscopy is another reliable technique that can be applied to measure partiele sizes in the range 0.1 J-lm to 2 mm. The method is based on the fact that the light diffraction angle is inversely proportional to partiele size. The technique itself consists of passing a laser beam through a suspension and focusing the diffracted light on to a ring detector which senses the angular distribution of scattered light intensity. This distribution is then related to the size distribution of the suspension through an appropriate mathematical expression, the details of which can be found in Weiner (1979) and Agrawal and Riley (1984).

1.4 Design of soil filters

Soil filters are used to drain water seeping out of a given soil surface in a controlled way so as to prevent the erosion of the natural soil. Examples inelude seepage through an earth dam, behind a retaining wall or around a pumping well. To fulfil these requirements, the grain size distribution of the filter material, relative to that of the soil it supports, must inelude enough large partieles to ensure free drainage of water, and an adequate proportion of small partieles to preelude the migration of the natural soil's finer material. AIso, to prevent any gaps in the grading of the filter material, it is suggested that the shape of its grain size distribution curve should be similar to that of the natural soil against which the filter is applied.

(38)

DESIGN OF SOIL FILTERS 17

Extensive experimental studies led to the establishment of empirical rules satisfying the previously mentioned requirements. In what follows, the design rules suggested by the Hong Kong Geotechnical Engineering Office (1993) are adopted, other rules suggested by different authors being marginally different. Note that subscripts

f

and s refer to the fi!ter and to the natural soil respectively.

1- to prevent the migration of the natural soil's fine particles through the filter:

dlSf"::; 5 x dsss

2- to ensure that the filter is more permeable than the soil:

3- to ensure a good performance ofthe filter:

dmaxf"::; 50 mm, (dmaxf being the maximum particle size of the filter material)

These design criteria can be extended to include the suggestions by Somerville (1986):

4- to ensure an adequate drainage of water: dSf'? 0.0750mm

5- to prevent any segregation ofthe filter material:

6- were the filter to be placed against a screen mesh (the case of a well for instance),

(39)

7- the grading curve of the natural soil should be limited to a maximum particle size:

d maxs ::; 19mm

Example 1.5

Consider the case of a well whose lining has maximum mesh size of 0.75 mm. Assuming that the soil in which the well is dug has a grading represented by curve A in figure lA, it is seen that:

dlSs = 0.07 mm, dsos = 0.15mm, d8Ss = 0.5 mm, dmaxs = 2mm.

Applying the aboye rules to design the soil filter to be placed around the lining, the following limiting values are easily computed:

l. dlSf=5xd8Ss=5xO.5=2.5mm, 2. d15f= 5dlSs = 5 x 0.07 = 0.35 mm, 3. dmaxf= 50mm, 4. _{d Sf=0.075mm,} 5. d sOf = 25 x dsos = 25 x 0.15 = 3.75 mm, 6. d8Sf=2xmeshsize =2xO.75= 1.5mm, 7. dmaxs

=

19mm.

These values are then plotted in the same figure yielding the shaded area within which should lie the grain size distribution curve of the filter material.

Although any graph within the shaded area meets the filter design requirements, the one shown in bold represents a good compromise because: (a) it has no grading gaps, (b) its shape is somewhat similar to curve A, and more importantly (e) it meets the design requirement represented by rule 3. In fact, taking the bold curve as that of the filter material, it follows that:

d60f = 2.5 = 5 dlOf 0.5

(40)

CLASSIFICATION OF COHESIVE SOILS

19 '"

o 0.01

'"

particl~ Jiz~ (mm) e

,

.

e

,

.

e

'silt

I

sand

I

g,av~J' C:COUfU

I

Figure J.4: Grain sile distribution graphsfor natural soil andfilter mmerial.

1.5 ClassificatioD of cohesive soils

Given the small size of its solid particles, a cohesive soil is characterised by a large specific surface (i.e. the surface area per volume of solid particles) and extremely small pores in comparison with sand. In fact,

taking the sand pores as a reference, the clay pares in figure 1.5 (depicting the arrangement of perfectly spherical solid particles for identical volumes

of sand and clay) are in reality at least five times smaller than the ones

shown in the figure.

e/ay

pores

(41)

Accordingly. clays have a high sur/aee lension which results in a high capillary rise as will be explained in details in scction 2.2; mast

importantly, Ihe velocity with which water can seep through lhe pores, known as permeahility is much lower for clays Ihan for sands.

Furthermore. Ihe behaviour of a clay can be markedly affected by lhe types of mineral that il contains and their reaction lO porewater. The three main groups of clay minerals are knolinite, illite and m011lmorillonite. Of these three, kaolinite is Ihe mast stable vis a vis water so that practically no volume change occurs

i

r

water content changes. 00 lhe other hand, ¡!lite

has a modcrate reaction lo water in that Ihe changes in volume due to a variation in water content can result in modest swelling or shrinkage.

However, montmorillonite is well known for its swelling and shrinkage

properties so that a clay containing this mineral is bound to be subject to large volume changes in the event ofwater content variation.

The water content of cohesive soils has therefore an effect on their

mechanical bchaviour, so much so that the empirical system of classification, based exclusively on the moisture conteot and known as the consistency limits (sometimes referred to as Auerberg limits) is widely

used to determine the type of cohesive soil. Hence, the ¡¡quid ¡¡mil WL

corresponds 10 the moisture conlenl oflhe soil as it changes from a plastic state to a slurry type material. The p/asfic ¡¡mi( Wp 00 the other hand is the water content when the soil changes from plastic to friable. The quantity Ws in figure 1.6 represents the shrillkage ¡¡mif which is the moisture content at which the soil volume starts 10 decrease.

C:~

pl

d

.

pfcutic S/alt

1

sfurry

w.o

w

,

w

,

W

,

WQttr contml

Figure 1.6: Consistency ¡¡milS lar cohesive soi/s.

Notice that these limits are measured using a sample of soil whose fabric has been totally destroyed.

(42)

CLASSIFICATION OF COHESIVE SOILS 21

Because the liquid limit is proportional to the specific surface, it follows that the greater the liquid limit, the higher the clay content, and the more compressible the soil.

These limits are most useful when applied in conjunction with the plasticity chart (due to Casagrande) represented in terms of variation of the liquid limit ofthe soil versus its plasticity index defined as:

Ip=wL-wp (1.24)

60 PLASTICITY

i

non plastic low medium high

.,

---,--I l> .... I I ~ 40 ___ L __ __J_ I I >< I I ~ ---.L--- ----+- t -.:; I I ,:-. 20 I I :~ 1 -§, 1 I - - - 1 - - - 1 -6 O 20 40 60 80 100

liquid limit (per cent)

Figure 1.7: Casagrande empirical plasticity charts lor soils.

The chart, shown in figure 1.7, provides a quick and accurate way of classification according to the moisture content of the soil and its plasticity indexo The skewed line in the figure separates clays from silts so that, for instance, a soil with a liquid limit WL

=

40%, a plastic limit Wp

=

20% corresponds to a day of intermediate plasticity (point B on the chart); similarly, a soil with WL

=

30%, Wp = 4% is a sUt of low plasticity (point Con the chart).

Highly plastic silts with liquid limits W L > 50% are organic soils whose natural moisture content at a saturated state can be as high as 1000% (bog peat for instance) corresponding therefore to void ratios in excess of lO.

(43)

Another useful relationship between the natural in situ moisture content w of a soil and its plasticity index Ip is given by the liquidity index

h:

W-Wp W-Wp

IL -_- _I -

---=--p -WL-Wp

It is clear from equation 1.25 that: h<O

05,h5,1

h>

1

=> soil is in a non-plastic state,

=> soil is in a plastic state,

=> soil is in a liquid state. Finally, the activity of a soil, defined as:

A= Ip

% e/ay partie/es

«

2 J..lm)

(1.25)

(1.26)

reflects the degree o/ plasticity of the clay content of the soil. Typical A values for the three main clay minerals are as follows:

- kaolinite - illite - montmorillonite A =0.5 0.5 5,A 5, 1 A> 1.25.

From a mechanical point of view, undisturbed clays with low plasticity are characterised by small deformations when subjected to external loading. Once disturbed, a clay can potentially exhibit a marked decrease in its resistance or shear strength, depending on its structure. Microscopic studies of such soils, though not conclusive, tend to link the structure of natural clays to the way in which they were formed; thus a glacial till does not have the same structure as a lacustrine clay. Accordingly, once disturbed or remoulded, through excavation for instance, part of the natural structure will be destroyed. The effect that this structural dislocation has on the resistance of the clay can be measured by the sensitivity, defined as:

S ensltlvlty = ... strength o/ undisturbed soil ' ' ' . . . : < . . -strength o/ remoulded soil

(44)

CLASSIFICATION OF COHESIVE SOILS

23

Hence, clays with a sensitivity larger than 16 are referred to as quick c1ays, and extra-sensitive clays are known to have a sensitivity in excess of 100.

The structure in this case is liable to total collapse, leading to a transfonnation ofthe clay from a plastic material to a viscous liquid almost

instantaneously. Quick clays can be activated by any type ofshock such as

vibration or earthquake activities, and are spread especially throughout

Scandinavia and Canada. Bjerrum's (1954) empirical scale can be used as a guide to clay sensitivity:

sensitivity <2 2-8 8-16 >16

c1ays insensilive sensilive highly sensilive quick

Bjerrum suggested that the sensitivity of the post-glacial Scandinavian clays (i.e. 10,000 to 15,000 year-old clays) is related to the salt content of their pares. This suggestion is supported by the graph in figure 1.8 showing a dramatic increase in sensitivity when the salt content beeomes

smaller than 10 gIl. One has to bear in mind, however, that Bjerrum's

suggestion is nol universal; in faet Sangrey (1972) reported that sorne of

the highly sensitive post-glacial Canadian elays were deposited in fresh

water. 16

""

~

-

20

"

~

"

o u

"

o

"

O 1 10 100 1000 sensitivity

Figure /.8: Relalionship be/Ween sensi/ivity and sal/ conlenl for Norwegianpos/-glacia/ c1ays (Bjerrum. /954. by permission

(45)

1.6 Soil compaction

The process of compaction applies to remoulded unsaturated soils, and consists of increasing mechanically the density by reducing the volume of air contained within the soil matrix, without any significant change in the moisture content. This process is generally used in conjunction with fill materials behind retaining structures or during the construction of embankments and earth dams. The increase in density, generated by a reduction in the void ratio, results in a substantial increase in shear strength of the soil and a marked decrease in its compressibility as well as permeability. The compaction is usually measured in terms of dry density

Pd, whose value depends on the level of compacting energy, the soil type and its natural moisture content. Accordingly, the maximum dry density Pdmax that can be achieved for a given soil increases with increasing level of compacting energy.

The maximum dry density is measured in the laboratory using compaction tests. The standard test is undertaken on a completely remoulded soil sample with a moisture content well below its natural value (ofien the sample is dried prior to testing). The test itself is conducted in stages, the first of which consists of mixing the dry soil sample with a small amount of water so that it becomes damp. An extended mould is then filled with moist soil in three equallayers, each one being compacted using 25 evenly distributed blows from a rammer weighing 2.5 kg and falling freely from a height of 305 mm. Once compaction is completed, the mould extension is then removed and the soil at the top of the mould is levelled off so that the remaining volume of compacted soil is precisely 1000cm3 ₍₁_litre)._The

bulk density P of the soil is then calculated by weighing the known volume of the sample. A small quantity of soil is then taken randomly from within the mould and placed in an oven for drying so that the moisture content w can be calculated. Knowing P and w, the corresponding dry density Pd is thence determined from equation 1.14:

Pd

=

p/(1 +w)

The sample is next removed from the mould, mixed with the remainder of the original soil to which an increment of water is added to in crease its moisture content. The compaction procedure is then repeated in precisely the same way as described previously. Usually four to five points are

(46)

SOIL COMPACTION 2S

enough to yield a compaction graph such as the ones depicted in figure 1.9, corresponding to a clay with a specific gravity Gs = 2.7.

The figure represents the variation of moisture content with the dry density al different energy levels:

-graph EFG corresponds to a standard compaction energy. - graph BCD represents a higher compaction energy.

In both cases, lhe maximum dry density Pdmax is achieved at a water

content known as the optimum mois/ure content w_{op •}and in the case of figure 1.9, it can be seen that:

-alongEFG: Pdmax:::::: 1.72Mglm3,

-along BCD: Pdmax :::::: 1.845 Mg/m3_• where pw = 1 Mglm3. 1.9 P<lmax Pd /.8

pw

PJmu 1.7 1.6 1.5 0.05 0.10 wop :::::: 15.7%, w_op::::::13%, 0.15 0.20 moiSlure contenr 0.25

(47)

Both graphs, which are similar in shape, indicate that initially the test yields a comparatively small dry density, because at a low moisture levels the relatively dry soil tends to form in lumps that have to be crushed before any significant reduction in void ratio takes place. As the moisture content increases, so do the soil workability and energy efficiency, leading thus to a gradual decrease in void ratio and a steady increase in dry density until Pdmax is reached at the optimum moisture content wop . Beyond the value wop , the build-up of porewater pressure within the soil matrix starts in eamest, increasing in the process the void ratio, thus decreasing the dry density of the soil.

On the other hand, the dry density equation can be expressed as follows:

Pdmax

=

p/(l + wop ) (1.27)

and so figure 1.9, together with equation 1.27 show clearly that the higher the compaction energy, the higher the maximum dry density, and the lower the optimum moisture content.

Example 1.6

The graphs in figure 1.9, corresponding to laboratory measurements, contain valuable information for a site engineer in charge of building an embankment, for instance, using the same material. Under such circumstances, the engineer has to make a decision as to what maximum dry density (and hence what compaction energy level) is required to minimise post-construction problems related to settlement and shear strength. If, for example, the material used corresponds to point P on the figure with a natural moisture content w

=

11 %, the choice consists of either increasing the moisture content of the soil by about 5% and using a standard compaction energy to achieve a maximum dry density similar to that ofthe graph EFG, or increasing the moisture content by 3% and using a high compaction energy leading to a maximum dry density comparable to that of the curve BCD.

If, on the other hand, the material used has a natural moisture content

Applied Analyses in Geotechnics, F Azizi.pdf

... ,' ...

...

, ':." ... ( '; .\ '.!

~:::4't.¡ií.,

,,"f:.;,.:

.' .. ' . .' .'

.·· ..

A

PRq~jjA

N

A~X~l~

.~

.:':·:<!'.r:~\',~:

.

JN

:

G

E.OJE~.~

NI

es

/

.

-

"""C....

.,

--

,

._

-

Applied Analyses

in Geotechnics

,\

Applied Analyses

in Geotechnics

u

Fethi Azizi

"

U

n

i

v

e

r

s

i

ty of

Pl

y

mouth

,

U

K

To

o/

my

Contents

Preface

a

C

C

C

DE

J

Jo

G

Ir

lp

k'

,

V

Ws

fh

=

=

=

=

oc

Formation and physical properties of soils

3

_{... ,' ...}

_...

_,