(Developments in Geotechnical Engineering 75) Sven Hansbo (Eds.) - Foundation Engineering-Academic Press, Elsevier (1994) PDF
(Developments in Geotechnical Engineering 75) Sven Hansbo (Eds.) - Foundation Engineering-Academic Press, Elsevier (1994) PDF
(Developments in Geotechnical Engineering 75) Sven Hansbo (Eds.) - Foundation Engineering-Academic Press, Elsevier (1994) PDF
Janbu, ., 1965. Consolidation of clay layers based on non-linear stress-strain. Proc. 6th
Int. Conf. Soil Mech. Found. Eng., Montreal, Vol. 2, 83-87.
Karol, R. H., 1960. Field tests for evaluating the effectiveness of a grouting operation. Am.
Cyan. Co Expl. and Min. Chem. Dep.
Larsson, R., 1975. Konsolidering av lera med elektroosmos. (Consolidation of clay by
means of electro-osmosis). Byggforskningen R45: 1975.
Larsson, R., 1977. Basic behaviour of Scandinavian soft clays. Swedish Geotech. Institute,
Report No. 4.
Liedberg, S., 1991. Earth pressure distribution against rigid pipes under various bedding
conditions. Ph. D. Thesis, Chalmers University of Technology, Gothenburg.
Littlejohn, G. S., 1992. Chemical grouting. In: M. P. Moseley (Editor), Ground improvement, Blackie Academic & Professional, CRC Press, Inc., 100-129.
Maag, E., 1938.ber die Verfestigung und Dichtung des Baugrundes (Injektionen).
Erdbautechnik, .
Mesri, G. & Godlewski, P. M.,1977. Time- and stress-compressibility interrelationship.
ASCE, J. Geotech. Eng. Div., GT 5, 417-430.
Olander, H. C , 1950. Stress analysis of concrete pipe. US Bureau Reel. Eng. Monographs,
No. 6.
Pramborg, B. & Albertsson, B., 1992. Underskning av kalk/cementpelare. (Investigation
of lime/cement columns). SBUF-Anslag Projekt 1075.
Pusch, R., Hansbo, S., Berg, G. & Henricson, E., 1974. Brighet och sttningar vid
grundlggning p berg. (Bearing capacity and settlement when founding on rock).
Svenska Byggnadsentreprenrfreeningen, Report No. 11.
Schneider, P. J., 1963. Temperature response charts. New York/ London, Wiley.
Schmertmann, J. H., 1955. The undisturbed consolidation behaviour of clay. Transactions
ASCE, Vol. 120.
Schmertmann, J. H., Hartmann, J. P. & Brown, P. R., 1978. Improved strain influence
factor diagrams. ASCE, J. Soil Mech. Found. Div., No. GT 8.
Schulter, A. & Wagener, H., 1989. Improvement of clay and silt by dewatering with a new
anchoring technology. Proc. 12th Int. Conf. Soil Mech. Found. Eng., Vol. 2,1409-1414.
Simonini, P. & Sorenzo, M, 1987. Design and performance of piles driven into a soft
cohesive deposit. Proc. Int. Symp. on Geot. Eng. of Soft Soils, Mexico, Vol. 1,371-378.
Smith, W. W., 1978. Stresses in rigid pipe. ASCE, Transp. Eng. J., Vol. 104,No. TE 3.
Spangler, M. G., 1948. Underground conduitsAn appraisal of modem research. ASCE,
Trans., Vol. 113.
Svensson, P. L., 1991. Soil-structure interaction of foundations on soft clayExperience
during the last ten years. Proc. 10th European Conf. Soil Mech. Found. Eng., Florence,
Vol II, 583-586.
Szchy, ., 1965. Der Grundbau, Vol. 2, Part 1Die Baugrube. Springer-Verlag, Wien/
New York.
Sllfors, G., 1975. Preconsolidation pressure of soft high-plastic clays. Ph. D. Thesis,
Chalmers University of Technology, Gothenburg.
Terzaghi, K., 1923. Die Berechnung der Durchlssigkeitsziffer des Tones aus dem Verlauf
der hydrodynamischen Spannungserscheinungen. Akad. Wissensch., Wien, Sitzungsber.
Bd. 132, H. 1, Mat.-Naturwissensch. Klasse.
Terzaghi, K., 1925. Erdbaumechanik. Leipzig-Wien.
Terzaghi, K., 1943. Theoretical soil mechanics. New York.
stedt, ., Weiner, L. & Holm, G., 1990. Friktionsplar. Brfrmagans tillvxt med tiden.
(Friction piles. Increase in bearing capacity with time). Preprint, Swedish Geotech. Inst.,
Linkping.
CORRIGENDA
Cross - refe rences :
Delete:
p. 247, bottom
'Furthermore, the maximum
established.'
Add:
p. 113, after (Fig. 82).
Count the number of tetragons covered by
the loaded area. The stress is obtained by the
expression = 0.00\nq.
can be
p. 248, top
(298)
(Poulus, 1990)
Other corrections:
page
77,3+
79, 1125, Eq. (107)
(Schmertmann, 1955)
Mesri et al (1990)
^-ds + Ipia^'^-ds
+ /sin(v + ')
os
as
125, Fig. 92.
dv
,dv
Replace 2pds by 2/?tan</> ds
ds
os
, ^ dv . , _
,,v .
and ipdrby 2/?tan0 dr
or
or
i38;Ex. 13
1 (vert.): 1.5 (hor.)
195; Fig. 139
qc (in MPa)
214, Eq. (250)
da
=o
217,9+
(Davisson et ai, 1965)
220; Fig. 157
equal to 4bp
221; Table 26
Initial ID%
223; 12+
60 mm by 60 mm
223; Fig. 159
equal to 4bp
224; Fig. 160
equal to Sbp
228; 3+
Hansbo et ai, 1973
231, 4 Randolph and Clancy, 1993
266; 2+
The displacement amplitude at D is obtained
when the force vector is pointing in the
direction.
342, Eq. (447)
M s = Rlrl = RB(c'+ a'tantfO/^,
344,6F C ^ 1.05-1.06= 1.11
353,5FCQ= 1.05-1.06= 1.11
479, 5 The standardised form of normal distribution
(p(x)= = exp(- )
ERRATUM
Janbu, ., 1965. Consolidation of clay layers based on non-linear stress-strain. Proc. 6th
Int. Conf. Soil Mech. Found. Eng., Montreal, Vol. 2, 83-87.
Karol, R. H., 1960. Field tests for evaluating the effectiveness of a grouting operation. Am.
Cyan. Co Expl. and Min. Chem. Dep.
Larsson, R., 1975. Konsolidering av lera med elektroosmos. (Consolidation of clay by
means of electro-osmosis). Byggforskningen R45: 1975.
Larsson, R., 1977. Basic behaviour of Scandinavian soft clays. Swedish Geotech. Institute,
Report No. 4.
Liedberg, S., 1991. Earth pressure distribution against rigid pipes under various bedding
conditions. Ph. D. Thesis, Chalmers University of Technology, Gothenburg.
Littlejohn, G. S., 1992. Chemical grouting. In: M. P. Moseley (Editor), Ground improvement, Blackie Academic & Professional, CRC Press, Inc., 100-129.
Maag, E., 1938.ber die Verfestigung und Dichtung des Baugrundes (Injektionen).
Erdbautechnik, .
Mesri, G. & Godlewski, P. M.,1977. Time- and stress-compressibility interrelationship.
ASCE, J. Geotech. Eng. Div., GT 5, 417-430.
Olander, H. C , 1950. Stress analysis of concrete pipe. US Bureau Reel. Eng. Monographs,
No. 6.
Pramborg, B. & Albertsson, B., 1992. Underskning av kalk/cementpelare. (Investigation
of lime/cement columns). SBUF-Anslag Projekt 1075.
Pusch, R., Hansbo, S., Berg, G. & Henricson, E., 1974. Brighet och sttningar vid
grundlggning p berg. (Bearing capacity and settlement when founding on rock).
Svenska Byggnadsentreprenrfreeningen, Report No. 11.
Schneider, P. J., 1963. Temperature response charts. New York/ London, Wiley.
Schmertmann, J. H., 1955. The undisturbed consolidation behaviour of clay. Transactions
ASCE, Vol. 120.
Schmertmann, J. H., Hartmann, J. P. & Brown, P. R., 1978. Improved strain influence
factor diagrams. ASCE, J. Soil Mech. Found. Div., No. GT 8.
Schulter, A. & Wagener, H., 1989. Improvement of clay and silt by dewatering with a new
anchoring technology. Proc. 12th Int. Conf. Soil Mech. Found. Eng., Vol. 2,1409-1414.
Simonini, P. & Sorenzo, M, 1987. Design and performance of piles driven into a soft
cohesive deposit. Proc. Int. Symp. on Geot. Eng. of Soft Soils, Mexico, Vol. 1,371-378.
Smith, W. W., 1978. Stresses in rigid pipe. ASCE, Transp. Eng. J., Vol. 104,No. TE 3.
Spangler, M. G., 1948. Underground conduitsAn appraisal of modem research. ASCE,
Trans., Vol. 113.
Svensson, P. L., 1991. Soil-structure interaction of foundations on soft clayExperience
during the last ten years. Proc. 10th European Conf. Soil Mech. Found. Eng., Florence,
Vol II, 583-586.
Szchy, ., 1965. Der Grundbau, Vol. 2, Part 1Die Baugrube. Springer-Verlag, Wien/
New York.
Sllfors, G., 1975. Preconsolidation pressure of soft high-plastic clays. Ph. D. Thesis,
Chalmers University of Technology, Gothenburg.
Terzaghi, K., 1923. Die Berechnung der Durchlssigkeitsziffer des Tones aus dem Verlauf
der hydrodynamischen Spannungserscheinungen. Akad. Wissensch., Wien, Sitzungsber.
Bd. 132, H. 1, Mat.-Naturwissensch. Klasse.
Terzaghi, K., 1925. Erdbaumechanik. Leipzig-Wien.
Terzaghi, K., 1943. Theoretical soil mechanics. New York.
stedt, ., Weiner, L. & Holm, G., 1990. Friktionsplar. Brfrmagans tillvxt med tiden.
(Friction piles. Increase in bearing capacity with time). Preprint, Swedish Geotech. Inst.,
Linkping.
CORRIGENDA
Cross - refe rences :
page; line
65; 8 91; 1099; 3 127 11 +
134 6+
154 7 162 7 192 7+
201 16229 7 234 2+
241 5+
248 6+
251 11259 2 264 15+
280 11+
294 18305 12+
314 15+
320 8+
321 11323 8 323 4 329 11+
334 1+
376 3 389 8+
400 5+
401 9 434 9 444 6 445 12449 4 488 17+
489 1490 11+
Add:
p. 113, after (Fig. 82).
Count the number of tetragons covered by
the loaded area. The stress is obtained by the
expression = 0.00 \nq.
Delete:
p. 247, bottom
'Furthermore, the maximum
established.'
p. 248, top
mzQuP- +cz0co +kzo = QQ
p. 227, 2 -
can be
(298)
(Poulus, 1990)
Other corrections:
page
77, 3+
79, 1125, Eq. (107)
(Schmertmann, 1955)
Mesri et al (1990)
72
ERRATUM
Authors corrections after printing:
Foundation Engineering by S. Hansbo
ISBN: 0.444.88549.8
R E F E R E N C E LIST; C O M P L E M E N T A R Y A D D I T I O N
Alphan, I., 1967. The empirical evaluation of the coefficient K0 and K0R. Jap. Soc. Soil Mech.
Found. Eng., Soil and Found., Vol. 7, No. 1, 31-40.
American Concrete Pipe Association, ACPA, 1988. Concrete pipe handbook. ISBN 0-9038681-6, Vienna, USA.
Bergado, D. T., Chai, C. T., Alfaro, M. C. & Balasubramaniam, A. S., 1992. Improvement
techniques of soft ground in subsiding and lowland environment. Asian Inst, of Technology,
Bangkok.
Berggren, B., 1981. Gravplar p friktionsjordsttningar och brfrmaga (Bored piles on
non-cohesive soilssettlement and bearing capacity). Ph. D. Thesis, Chalmers University
of Technology, Gothenburg.
Bustamante, M. G. & Gianeselli, L., 1981. Readjustment des paramtres des calculs des
pieux. Proc. 10th Int. Conf. Soil Mech. Found. Eng., Vol. 2, 643-646.
Cambefort, H., 1967. Injection des sols. Eyrolles, Vol. 1 and 2.
Chambosse, G. & Dobson, T., 1982. Stone columns IEstimation of bearing capacity and
expected settlement in cohesive soils. GKN Keller Inc., Tampa, Florida.
Caquot, ., Kerisel, J. & Absi, F., 1973. Tables de bute et de pousse. Gauthier-Villars,
Paris-Bruxelles-Montral
Esrig, M. J., 1968. Pore pressures, consolidation, and electrokinetics. Proc. ASCE, J. Soil
Mech. Found. Eng., Vol. 94, SM 4.
Hansbo, S. & Jendeby, L., 1983. A case study of two alternative foundation principles:
conventional friction piling and creep piling. Vg- och Vattenbyggaren, N o . 7 - 8 , 2 9 - 3 1 .
Hansbo, S. & Kllstrm, R., 1983. Creep pilesa cost-effective alternative to conventional
friction piles. Vg- och vattenbyggaren No. 7-8, 23-27.
Hansbo, S., Pramborg, B. & Nordin, P. O., 1977. The Vnern terminal. Illustrative example
of dynamic consolidation of hydraulically placed fill of organic silt and sand. Sols Soils,
No. 25, 5 - 1 1 .
Hardin, B. O. & Black, W. L., 1969. Vibration modulus of normally consolidated clay.
(Closure). Proc. ASCE, J. Soil Mech. Found. Div., Vol. 95, No. SM 6, 1531-1537.
Hardin, B. O. & Richart, F. E. Jr., 1963. Elastic wave velocities in granular soils. Proc.
ASCE, J. Soil Mech. Found. Div., Vol. 89, No. SM 1, 35-65.
Karol, R. H., 1960. Field tests for evaluating the effectiveness of a grouting operation. Am.
Cyan. Co Expl. and Min. Chem. Dep.
Larsson, R., 1977. Basic behaviour of Scandinavian soft clays. Swedish Geotech. Institute,
Report No. 4.
Liedberg, S., 1991. Earth pressure distribution against rigid pipes under various bedding
conditions. Ph. D. Thesis, Chalmers University of Technology, Gothenburg.
Littlejohn, G. S., 1992. Chemical grouting. In: M. P. Moseley (Editor), Ground improvement, Blackie Academic & Professional, CRC Press, Inc., 100-129.
CORRIGENDA
Cross-references:
Delete:
page; line
65; 8 91; 1 0 99; 3 201; 16229; 7 234; 2+
241; 5+
248; 6+
251; 1 1 259; 2 264; 15+
280; 11+
294; 1 8 305; 12+
314; 15+
320; 8+
321; 1 1 323; 8 323; 4 329; 11+
334; 1+
376; 3 389; 8+
400; 5+
401; 9 434; 9 444; 6 445; 1 2 449; 4 488; 17+
489; 1 490; 11+
p. 247, bottom
'Furthermore, the maximum
established.'
Add:
p. 113, after (Fig. 82).
Count the number of tetragons covered by
the loaded area. The stress is obtained by the
expression = O.OOlnq.
can be
p. 248, top
Other
(Poulus, 1990)
corrections:
page
77, 3+
79, 1 138; Ex. 13
195; Fig. 139
217, 9+
220; Fig. 157
221; Table 26
223;12+
223; Fig. 159
224; Fig. 160
231,4214, Eq. (250)
dX ~
(298)
(Schmertmann, 1955)
M e s n e / / . (1990)
1 (vert.): 1.5 (hor.)
qc (in MPa)
(Davisson et al, 1965)
equal to 4bp
Initial ID %
60 mm by 60 mm
equal to 4b
equal to Sbp
RandolphandClancy, 1993
%/
266; 2+
The displacement amplitude at D is obtained
when the force vector is pointing in the
direction.
D e v e l o p m e n t s in G e o t e c h n i c a l E n g i n e e r i n g , 7 5
Foundation Engineering
Sven Hansbo
Lyckov2,
Stocksund,
S-18274,
Sweden
ELSEVIER
Amsterdam - London - New York - Tokyo
1994
66.
67.
68.
69.
70.
71.
72.
73.
74.
Library
Hansbo,
Sven,
ISBN
1.
engineering
cn.
Includes
Congress
Catalog1ng-1n-PublIcatIon
Data
1924-
Foundatlon
p.
of
Sven
(Developments
bibliographical
Hansbo.
In
g e o t e c h n 1 ca1
references
and
engineering
75)
Index.
0-444-88549-8
Foundations.
TA775.H36
I.
T i t l e .
I I .
Series.
1994
93-43290
CIP
ISBN 0-444-88549-8
1994 Elsevier Science B.V. All rights reserved.
No part of this publication may be reproduced, stored in a retrieval system of transmitted in any
form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the
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Special regulations for readers in the U.S.A. - This publication has been registered with the
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refered to the publisher.
No responsibility is assumed by the publisher for any injury and/or damage to persons or property
as a matter of products liability, negligence or otherwise, or from any use or operation of any
methods, products, instructions or ideas contained in the material herein.
This book is printed on acid-free paper.
Printed in The Netherlands
xvii
Preface
PREFACE
agreement with practice. Of course, other methods of tackling the foundation design than
those included here m a y be just as reliable and my choice cannot be regarded as an
intention to belittle possible alternative solutions. Thus, local experience of a certain
design m e t h o d can certainly justify its application.
D u e to increasing urbanisation there is an increasing d e m a n d for building sites, and
ground with very p o o r soil properties m a y have to be utilised for building purposes with
heavy initial capital costs. T h e costs, however, can generally be greatly reduced by the
use of ground modification techniques. In consequence, these have b e c o m e an integral
part of foundation engineering and have to be considered as possible m e a n s of reducing
capital costs in the building industry. In this textbook, the most important design and
practical aspects of soil i m p r o v e m e n t h a v e been included. M u c h of the material included
is based on m y personal experience.
T h e b o o k can b e used as a textbook for senior undergraduate and graduate students.
It can also serve as a c o m b i n e d text- and handbook for professional engineers working
in the field of geotechnical engineering.
All the line d r a w i n g s in the b o o k are d r a w n by hand. Photos presented in the b o o k h a v e
been a s s e m b l e d from time to time since the beginning of the 1960s and it is n o w impossible for m e to give credit to all the photographers in question. M a n y photos h a v e been
received from S w e d i s h and European contractors. A m o n g Swedish contractors, not
particularly m e n t i o n e d in the text, w h o have contributed photos I w o u l d like to
xviii
Preface
Hansbo
Introduction
INTRODUCTION
(i) Soil-structure
interaction.
Introduction
(ii) Creating a reliable basis for design. One of the most important parts in the design is
to establish a reliable picture of the soil conditions at the building site, both from a
geotechnical and a geological point of view. Therefore, the planning and the realisation
of the site investigation are vital for successful design. This fact is often neglected. From
the client's viewpoint, the money spent on soil investigations is unprofitable and should
therefore be minimised. This has entailed a common procedure of inviting tenders for soil
investigations, which doubtless hazards the information needed. It is but natural that the
economic pressure exerted on the field crew in an investigation received by tender can
have a negative influence on the results obtained. This may be the case even where the
skill of the field personnel is beyond question. The extent of the investigation, and even
the method of investigation, may also have to be modified with regard to the results
obtained. These facts speak against tendering; they also speak for a very close cooperation between the geotechnical expert on the one hand and the field investigation
crew on the other. In reality, for liability reasons, the field crew and the geotechnical
expert, who is responsible for the interpretation of the results obtained, should preferably
belong to the same organisation.
A correct determination of the strength and deformation properties of the soil is in fact
one of the most difficult tasks in geotechnical engineering. Their determination has to be
coupled with just the type of problem that is encountered in the design. In most cases in
practice, in situ investigations are preferable to laboratory investigations. However, for
determination of long-term deformation properties of cohesive soils, laboratory investigations are preferable to in situ ones. Sampling and laboratory investigations are also
required for the reason of soil classification which is necessary for the final assessment
of the geomechanical properties of the soil. Unfortunately, the classification systems vary
in different parts of the world, and this situation most probably will persist due to local
tradition and local soil conditions.
(iii) Execution.
The execution of the foundation often entails the need for excavations
being carried out to a great depth. The problems connected with excavation for foundation purposes often represent the most difficult and dangerous part of the j o b . Deep
excavations, with regard to slope and bottom stability, support of vertical cuts, etc. are
therefore just as important a part of the design as that of the foundation itself.
Quite often, provisional structures, necessary for the support of vertical cuts, can also
be utilised as structural members of the building itself. This is, for example, the case with
diaphragm walls. The borderline between the method of foundation in itself and the
execution of the excavation for the foundation has more or less vanished.
Evidently, to be a competent foundation engineer, broad knowledge of soil and structural mechanics is imperative. Furthermore, knowledge of geology is extremely important
for a correct assessment of the possible variation in geotechnical properties to be
expected at a building site.
Introduction
(i ) International
problems depend on local cost of labour, tradition, available building material, level of
geotechnical education and codes of practice. It is therefore impossible to give, in a text
book, a complete list of foundation and foundation design m e t h o d s that w o u l d satisfy all
readers with their various backgrounds.
In this text book, m o d e r n geotechnical investigation m e t h o d s and their interpretation
are exemplified. T h e foundation m e t h o d s are representative of the developed part of the
world. T h e theoretical approach is influenced by the results of research carried through
at Swedish universities and research institutes and by experience gained as a geotechnical
consultant for m o r e than 35 years.
T h e design of foundations has to b e carried out in accordance with the prescriptions
presented in the building code of the country concerned. In Europe, a new code for geotechnical design, the so-called E u r o c o d e 7, is now ready for publication. T h e philosophy
behind this code will b e presented in the last chapter of this book. In order not to create
confusion, safety aspects will only be treated in exceptional cases and then in a m o r e
traditional way. However, the safety philosophy, in the Eurocode, for e x a m p l e , can be
easily introduced without affecting the essence of the text.
(v)Aim
of the book. In this textbook, those parts of soil mechanics are included which are
Fundamentals
FUNDAMENTALS
1.
Micro-structure
Fundamentals
Fig. 1. Scanning microscope pictures of clay particles. Glacial quick clay (top) and lacustrine clay.
Fundamentals
illite but the attraction between the crystal mono-layers of montmorillonite is weaker and
therefore allows expansion of the interspace between the layers in connection with water
uptake. Montmorillonite clay therefore behaves differently from other clays in that it is
prone to swelling w h e n unloaded or exposed to water.
As a result of their crystalline structure, particles of clay and m i c a minerals have a m o r e
or less flat, flake-like form with very irregular contours. T h e edges of the particles are thin
Fig. 2. Ultrathin cuts of marine (top) and lacustrine clays photographed using a transmission electron
microscope and schematic pictures of their structure (after Pusch, 1970).
Fundamentals
Fig. 3. Microphotograph of sand showing cementation in the contact surface between grains.
Fundamentals
rock-forming
materials.
fruits, leaves, roots, seeds, etc., w e find microscopic remainders of spores, pollen, micro-
Fig. 4. Clay-size quartz and feldspar particles in sand from a sand pit in Sweden.
Fundamentals
scopic animals, algae, germs and viruses as well as organic molecules and c o m p o u n d s .
Living as well as dead organisms exist.
T h e organic matter is b o u n d to the surface of the clay particles via hydrophilic groups
which, depending on the electrolytic composition, m a y give rise either to a protecting
coat or to the formation of a cementing gel complex. Generally speaking, the organic
substance and its bonds are a m o r p h o u s in nature, which implies a strong tendency
towards creep deformations on loading. This effect is amplified by the fact that the degree
of order of the water in the region of contact b e t w e e n the grains is low. Organic matter
contributes to the creation of a very large v o l u m e of voids in the soil. This is also the case
as to coarse-grained soils. Organic soils are therefore generally very compressible.
1.2
Macro-structure
In a geological formation every type of stratum forming an integral part of the soil mass
has its specific mechanical characteristics depending upon grain size, grain shape, grain
origin, v o l u m e of voids, etc. T h e revelation of those structural features that m a y have a
decisive influence on foundation design is one of the foremost aims of soil investigations.
T h e extent and choice of m e t h o d of soil investigation is thus very m u c h dependent on the
geological situation at the site.
(i) Structural
anisotropy.
versa.
Undetected layers with different characteristics from those of the soil mass as a whole are
often the cause of unsuccessful foundation design and even of disastrous events. For
10
Fundamentals
e x a m p l e , the landslide at Furre in the centre of N o r w a y took place along a thin quick clay
layer, e m b e d d e d in a deep deposit of mainly sand and silt and sloping at an angle of about
6 degrees towards the river N a m s e n (Hutchinson, 1961). This thin clay layer was not
discovered until the slide had taken place.
Impervious layers in coarse-grained soil m a y serve as a watertight lid and cause
problems in connection with deep excavations (for e x a m p l e hydraulic uplift).
Coarse-grained layers e m b e d d e d in clay deposits are of great i m p o r t a n c e for the
overall permeability in the horizontal direction. In hilly surroundings, continuous coarsegrained layers e m b e d d e d in clay deposits are often fed with water under artesian pressure,
which leads to a condition of h y d r o d y n a m i c equilibrium.
So-called erratic strata, representing very irregular and unpredictable stratification,
are of particular concern in foundation engineering. Disturbed or distorted stratification
is quite c o m m o n . An e x a m p l e is given in Fig. 5.
(ii) Non-homogeneities.
fully
Voids
(i) Content of matter. T h e voids in the soil are filled with either water or gas or both. T h e
groundwater always contains dissolved elements, both ionised or non-ionised, and salts
as well as suspensions of mineral particles, h u m u s gel or other organic matter, gases, etc.
T h e content of organic matter can be roughly determined by vaporisation of the groundwater after which the remainders are oven-dried at 180C for one hour and then weighed.
T h e result is c o m p a r e d with the electrical conductivity of the groundwater.
11
Fundamentals
T h e substances usually dissolved in the water are different kinds of gases, such as
oxygen, nitrogen, h y d r o g e n sulphide, m e t h a n e and carbon dioxide; different elements,
such as silicate, iron, calcium, m a g n e s i u m , sodium and potassium; different kinds of
salts, such as carbonates and bicarbonates, sulphates, chlorides, nitrates, h u m t e s and
tannins. T h e concentration of dissolved matter can b e subjected to strong variations, from
about 0.01%o in rain water and snow to over 3 0 % in certain salt lakes. T h e total
concentration of suspended matter is very rarely above 0.5%c (0.005%o in respect of
mineral particles and, except for organic soils, 0.15%o in respect of organic matter). T h e
influence of dissolved and suspended matter on the unit weight of g r o u n d w a t e r m a y have
to be taken into consideration in geotechnical analysis.
T h e presence of gas in the soil voids is of great importance in geotechnical design. T h e
most c o m m o n gases are carbon dioxide above the groundwater level and m e t h a n e below.
T h e solubility of gases in the water is directly proportional to the water pressure and
inversely proportional to the temperature. T h e concentration of dissolved gases in the
groundwater varies generally from 0.001 to 0.1%o.
(ii) Volume of voids. In soil m e c h a n i c s , the v o l u m e of voids in the soil is either expressed
in terms of porosity
of voids to total v o l u m e and is designated by the symbol , while the void ratio is defined
as the ratio of v o l u m e of voids to v o l u m e of solids and is designated by the symbol e.
Expressed in t h e terms given in Fig. 6, w e thus have:
n = VpIV
(1)
e = Vp/Vs
(2)
volume
mass
Va
ma
gas
density pa
water
density pw
density'^','"
'." ' '''"",'
ms
Fig. 6. A schematic picture of a soil element divided into its constituents: solids, water and gas.
Fundamentals
12
(3)
l-n
Water
content
(4)
mwlms
water content.
weighing the specimen, first in its natural state and then again after having kept it in a
drying oven at 110 5C for 24 hours, or, alternatively, in a m i c r o w a v e oven (for details,
see Gilbert, 1991). T h e determination of the water content is part of the routine
investigation of fine-grained soils.
(ii) Degree of water saturation.
(5)
eSrpwlps
(6)
13
Fundamentals
1.5
Density
(i) Specific
density. The specific density (i.e. the density of solid material) is defined by
the relation:
V
(7)
Ps = s' s
to 2.70 t / m for coarse-grained soils, and from 2.70 to 2.80 t / m for fine-grained mineral
soils (clay).
(ii) Grain (particle)
density. The grains themselves are seldom completely solid but have
a certain porosity and, therefore, from a practical viewpoint, it is better to make use of
grain density instead of specific density. The grain density is defined by the relation:
P g
= mg/Vg
(8)
Amphibole
Biotite
Calcite
Quartz
Feldspar
Mica
Muscovite
Pyrite
Pyroxene
Illite
Kaolinite
Montmorillonite
Chlorite
2.8-3.4
2.7-3.1
2.7
2.65
2.5-2.9
2.8-2.9
2.8-3.0
5.0-5.1
3.1-3.6
2.6-2.8
2.6-2.7
2.4-2.8
2.6-3.0
Comments
Rock-forming minerals, mainly constituting coarser
grains. However, quartz and feldspar sometimes constitute more than 50% of clay fraction.
Remarks: According to Jelinek (1966), the average value of pg can be assumed equal to 2.65 t/m for
3
sand and gravel and 2.75 t / m for clay and clayey silt
14
Fundamentals
TABLE 2.
Typical bulk density values
3
Soil type
Peat
Dy and gyttja
Clay and silt
Sand and gravel
Till
Rock fill
Density (t/m )
Water-saturated
1.0-1.1
1.2-1.4
1.4-2.0
2.0-2.3
2.1-2.4
1.9-2.2
Often water-saturated
1.6-2.0
1.8-2.3
1.4-1.9
coarse grains of sandstone and limestone, for example, there may be a noteworthy
difference between the respective numerical values.
Typical grain densities are presented in Table 1.
(iii) Bulk density. The bulk density is defined by the relation:
= m/V
(9)
(10)
+ Srnpw
()
mass that the soil would have per unit volume if the water in its voids were removed
without volume change taking place. Its numerical value can be obtained by either of the
relations:
Pd=~f
= P*(l-n)
1+ w
(12)
1+ e
15
Fundamentals
As can b e seen, the dry density is directly correlated to the porosity (void ratio) of the
soil. It is therefore c o m m o n l y used as a m e a s u r e of the result achieved by compaction of
soil (p. 400).
Example 1: Determine the dry density, the degree of water saturation and the water content of a soil
3
with bulk density = 1.7 t/m and void ratio e = 0.8.
3
2.
= eSrpw/pg
It is often quite difficult to distinguish b e t w e e n soil and rock. T h u s the rock surface m a y
be subjected to severe weathering, converting rock to soil and m a k i n g the border line
between r o c k and soil indistinct. Since excavation costs for soil is generally m u c h lower
than for rock, this uncertainty m a y entail controversies between contractors and clients.
It is obviously important to h a v e a clear definition of what is m e a n t with rock in civil
engineering. T h e definition accepted in S w e d e n can serve as an e x a m p l e . Accordingly,
rock is defined as that part of the earth 's crust which is characterised by high hardness and
low porosity and which cannot normally be dislodged by excavation.
2.1
Micro-structure
T h e micro-structural features of rock concern the matrix of individual crystals and their
atomic arrangement. Investigations h a v e indicated that the strength and deformation
properties of the r o c k material are governed by micro-structural anisotropy. T h u s , microstructural features m a y be important for the shear strength of small rock samples tested
on a laboratory scale, but they are not significant of strength and deformation properties
of a rock m a s s . T h e s e are governed by macro-structural features, such as planes of
weakness, w e a k zones, joints, etc.
2.2
Macro-structure
16
Fundamentals
Fig. 7. Clay-altered zone has become fluid after being exposed to air.
17
Fundamentals
the rock cover (Fig. 8) and to long-term subsidence of the ground due to creep in the rock
material.
3.
INFLUENCE OF GROUNDWATER
3 . 1 Natural
groundwater
condition
18
Fundamentals
groundwater level. T h e thickness of the zone is dependent on the capillary rise of the soil
in question (grain size and density).
T h e capillary rise b e c o m e s smaller in the case w h e r e the g r o u n d w a t e r level rises than
in the case w h e r e it falls. T h e reason is that the upper part of the capillary zone contains
air-filled pockets which set b o u n d s for the m o v e m e n t of water. In the lower part of the
capillary zone, however, the voids are completely filled with water. Here the water m o v e ment is governed by the same physical laws as b e l o w groundwater level. T h e height of
the capillary z o n e in the case of a sinking groundwater surface is given in Table 3.
T h e water in the capillary zone, as well as in the unsaturated zones above, is bound
mainly by surface tension in the interfaces between water, soil particles and gas. Other
types of binding forces, such as sorption forces, and chemical bindings also contribute.
TABLE 3.
Approximate capillary rise, in m (sinking groundwater level)
Soil type
Coarse sand
Medium sand
Fine sand
Silt
Clay
2-0.6
0.6-0.2
0.2-0.06
0.06-0.002
< 0.002
Loose state
Dense state
0.03-0.12
0.12-0.50
0.30-2.0
1.5-10
0.04-0.15
0.35-1.10
0.40-3.5
2.5-12
> 10
the level w h e r e the hydrostatic pressure equals the atmospheric pressure (the lower
b o u n d of the capillary zone). T h e physical m e a n i n g of the given definition is illustrated
in Fig. 9.
Naturally, the groundwater level is influenced by the natural supply of water to the
ground. Since there is a great seasonal variation in the supply of rain-water to the ground
and loss of groundwater due to transpiration and evaporation, the g r o u n d w a t e r level can
be expected to vary with climatic changes during the years. T h e r e are also minor
influences on the g r o u n d w a t e r level resulting from changes in atmospheric pressure,
variations in gravity (tides, earthquakes), etc. which are not often taken into account.
The climatological variations, of course, have a dominating influence on the groundwater
situation. A n illustrative e x a m p l e of the seasonal variations that m a y take place in a clay
deposit is s h o w n in Fig. 10. As can b e seen from Fig. 10, the hydraulic heads observed
at different levels in the aquifer below the clay, and in the clay layer itself, are almost
completely synchronised. W e also notice that there is a time lag b e t w e e n rainfall and its
effect on the g r o u n d w a t e r level.
C o m p l e t e saturation is probably rare in natural soils. A certain a m o u n t of gas s e e m s
always to b e present, although its quantity from a practical point of view is negligible.
T h e ion content of p o r e water is of great i m p o r t a n c e for the m e c h a n i c a l b e h a v i o u r of
19
Fundamentals
Fig. 9. Definition of groundwater level. In the capillary zone, the pore pressure differs from the
atmospheric pressure by the capillary suction hc - a. Below the groundwater level, the pore pressure
differs from the atmospheric pressure by the pressure exerted by a water column of height h.
-2
Uta L J fil
1977
IT
rJhrrr _
1978
JlilLf
Precipitation, m m
Fig. 10. Seasonal variation of pore water pressure at different depths in a clay deposit in Gothenburg,
Sweden. (After Berntson, 1983).
Fundamentals
20
stresses
T h e concept of total and effective stresses constitutes the basis of soil mechanics,
although it does not always seem to b e fully appreciated.
By definition, the normal and shear stresses acting on a section surface of a material
are a s s u m e d to b e evenly distributed over a solid area. However, a section surface of a
soil consists partly of intersected grains and partly of intersected voids filled with water
or gas, or a combination of both. T h e s u m of all internal stresses on a unit area of the
section surface (intersected grains as well as intersected voids) is k n o w n as total
stress.
T h e latter can be considered as carried partly by direct contact be tw e en the grains and
partly by pore water or pore gas, or both combined. T h e part of the total normal stress a
that is carried in the points of contact between grains is called effective
part carried by pore water (gas) is called pore water pressure
ua ) .
In the case of water-saturated soils, the word pore water pressure is usually replaced by
pore pressure and the symbol uw is replaced by the symbol u.
T h e correlation between total stress and effective stress can be expressed by the
relation:
='+
u x+u (\-x)
w
(13)
(14)
Since water and gas cannot take up shear stress, total shear stress is equal to effective
shear stress.
Example 2: Determine the effective vertical stress 3 m below the ground surface.The capillary rise in
the soil is 2 m and the groundwater level is at 4 m depth. The soil in the capillary zone is assumed to
3
be water saturated. The density of non-saturated soil above the capillary zone is 1.8 t/m and of saturated
3
soil 2.1 t/m .
Solution: The total vertical stress at 3 m depth is equal to = 9.81(1.8-1 + 2.1-2) = 58.9 kPa and the pore
water pressure uw = - 10 kPa. Thus, the effective stress becomes equal to '\, = 58.9 + 10 = 68.9 kPa.
21
Fundamentals
3.3
Hydrostatic
and hydrodynamic
condition
In a hydrostatic condition, the pore water pressure at any depth is equal to the hydrostatic
pressure m e a s u r e d from the groundwater level. Moreover, the p o r e water pressure at one
and the s a m e level is the s a m e all over the place. T h u s , in both the vertical and the
horizontal directions w e h a v e a state of equal hydraulic head, i.e. the water is in a state
of static equilibrium and consequently n o water flow is taking place. This m a y be thought
of as the natural state in a virgin area but frequently is not. In reality, the pore water
pressure distribution quite often deviates from the hydrostatic o n e and has d o n e so for
ages (Fig. 11). In this case w e h a v e to deal with a h y d r o d y n a m i c condition; a condition
characterised by a steady state flow. T h e direction of flow d e p e n d s upon whether the
hydraulic head increases or decreases in the direction considered. T h e flow is governed
by the hydraulic gradient:
= //
(15)
GW
GW
clay
clay
m
G
clay
J3
10
\
/
15
20
10
\
1
- %-:
<
sand * ^ -~ -
15
Aug.-76 to May-82
pore pressure, Aug. 21 - 7 9
Fig. 11. Equilibrated pore pressure distribution with depth in two clay layers after groundwater lowering
in underlying sand, taking place around 4000 years ago (left) and 1000 years ago (Torstensson, 1975).
22
Fundamentals
= ki
(16)
[Laminar flow can be expected when the hydraulic gradient / < 0.1 / J H , , where dw = grain
diameter in a soil having one single grain size and the s a m e total grain surface area as in
real condition (normally dl0<dw<d40,
have been found to occur in clay soils at small hydraulic gradients. T h u s , in some
investigations, a threshhold gradient has been noticed below which no flow is taking
place. Considering the character of the internal forces in pore water (electric double layer
forces, sorption forces, etc.), it is logical to assume that the porosity of the clay would
seemingly increase, up to a certain point, with increasing h y d r o d y n a m i c forces. This fact
has in reality been observed in several experiments.
T h e rate of flow according to this concept could then be assumed to follow the relationship (Hansbo, 1960; Dubin and Molin, 1986), see Fig. 12:
n
= k] i
when / < ii
(17)
nkx
11
if ' ),
n> 1,
i0= ii (
(ii) Seepage
-\)ln.
pressure.
direction of seepage equal to igpw per unit v o l u m e of the soil, w h e r e i is the hydraulic
gradient vector and pw is the density of water. T h u s , in the case of vertical seepage
(hydraulic gradient in the vertical direction equal to i ), the effective density of the soil
p' is either increased by ipw ( d o w n w a r d seepage) or decreased by ipw (upward seepage).
Obviously, an u p w a r d flow of water will lead to liquefaction w h e n ipw>
Example 3: The groundwater level in a sand deposit is at 0.5 m depth. Determine the effective vertical
3
pressure at 3 m depth in a soil with porosity = 30% and grain density pg = 2.65 t/m if the pore pressure
is increasing linearly with depth below groundwater level to 70 kPa at 5 m depth. The degree of water
saturation above groundwater is equal to 10%.
Solution: There are two alternative solutions, one based on seepage pressure and the other on the basic
relation between total and effective stress.
Using seepage pressure as a basis of analysis w e find:
i = (70/9.81 - 4 . 5 ) / 4 . 5 = 0.59
The bulk density above groundwater is:
23
Fundamentals
k2(i -
i0)
Hydraulic gradient i
Fig. 12. Observed deviation from Darcy's law at low hydraulic gradients in clay (Hansbo, 1960).
3
induced
by
loading
T h e stress increments induced in the soil due to loading can b e considered as total stress
increments. In soils with a low hydraulic conductivity the i m m e d i a t e effect of loading is
the creation of an excess pore pressure that will carry part of the load until the soil skeleton
has had time to rearrange itself and b e c o m e strong enough to carry the load on its own
(the effective stress increase is b e c o m i n g equal to the total stress increase). Several pore
pressure equations h a v e been suggested for the prediction of excess p o r e pressure induced in the soil due to loading. A m o n g these, the ones k n o w n as S k e m p t o n ' s and
H e n k e l ' s are probably best k n o w n . T h e results obtained by the different equations are
very nearly the s a m e irrespective of the loading condition. A c c o r d i n g to S k e m p t o n ' s pore
pressure equation, w h i c h is less complicated than the others, the excess pore pressure Au
is obtained b y the relation:
Au = B [ 3 + (
- 3) ]
(18)
where A and are the so-called S k e m p t o n pore pressure coefficients (Skempton, 1954),
and 3 are the major and m i n o r principle stress i n c r e m e n t s due to loading.
For water-saturated soil, = 1 while for non-saturated soils < 1. T h e A value depends
Fundamentals
24
30
4
depth 4 m
depth 5.5r
"
depth 5 m
03
Fig. 13. Results of excess pore pressure observations during water-filling of a steel tank on watersaturated glacial medium-sensitive clay (left) and organic postglacial low-sensitive clay (Sllfors,
1975). Dimensions of tank: diameter 9.6 m; height 5 m. Observations yield A values between the limits
0.14 below and 1.3 above the preconsolidation pressure for the glacial clay and 0.58 below and 1.07
above the preconsolidation pressure for the postglacial clay.
a m o n g other things on the preloading history of the soil. Generally speaking, A = 1/3 for
an elastic material, A > 1/3 for a contractive
material.
25
Fundamentals
4.
DEFORMATION CONCEPTS
4.1
Deformation
moduli
E=
1+G/3K
1-2G/3K
=
(20)
;
2 + 2G/3K
Equations (19) and (20) yield:
= 2G(l+v)
(21)
For a m o r e or less incompressible material (in which case the value b e c o m e s very
large), for instance water-saturated clay in undrained condition (mineral particles and
water can be considered incompressible in comparison with the soil skeleton), w e find
= 3 G a n d = 0.5.
In the case of highly compressible soils, the deformation m o d u l u s usually applied in
settlement analysis is determined by m e a n s of compression tests in which lateral strains
are prevented, so-called o e d o m e t e r tests (Fig. 14b).
In classical soil m e c h a n i c s (Terzaghi, K., 1923, 1925), t w o moduli derived from the
results of o e d o m e t e r tests w e r e introduced, namely the coefficient of theoretical
compressibility av = Ae/Aav\
= / ' ,
where e0is the initial void ratio and is the relative compression. As the c h a n g e in void
ratio of clay in the virgin state, i.e. for a load a b o v e the past m a x i m u m load on the clay,
was found to b e directly proportional to the logarithm of effective vertical stress increase,
the primary c o m p r e s s i o n index Cc was later introduced, defined as:
Cc = AelA{\o%av')
(22)
These " m o d u l i " are all foreign to the traditional moduli used in neighbouring branches
Fundamentals
26
Fig. 14a. Stress/strain conditions for determination of bulk modulus (left) and shear modulus G.
v = 0).
T h e correlation between M and the basic moduli is given by the expression:
4G
M=K(\+
(23)
2G(l-v)
or, alternatively:
M=
l-2v
4.2. Strain
(24)
dependence
T h e shear m o d u l u s of soil depends to a high degree on the shear strain level the lower
the strain, the higher the m o d u l u s of deformation. T h e strains occurring in static loading
27
Fundamentals
TABLE 4
Soil parameters and ( = number of loading cycles)
Type of soil
-0.5
-0.21og(AO
l+0.251og(AO
0.16
0.16
1.3
are generally of another order of m a g n i t u d e than in d y n a m i c loading (Fig. 15). This strain
dependence has to b e taken into account in the analysis of foundation p r o b l e m s .
T h e relation b e t w e e n secant m o d u l u s G and shear strain y(G = /) can b e expressed
by the relation (Hardin and Drnevich, 1972):
(25)
G =
1+7/*
5
=^[i+aexp(-j8^)],
7r
7r
ax
/ G 0,
'] - ( - 5
cfv)
(26)
Machine foundations
Traffic vibrations
Off-shore gravity structures
Strong earthquakes
'Static' loading tests
io-
"
10"
Shear strain y
Fig. 15. Range of shear strain amplitudes for different types of loading conditions. (After Andrason, 1979)
Fundamentals
28
^{P
failure c o n d i t i o n / * ^
/in
' (
situ c o n d i t i o n ^
\
max \
'
4.3
Stress
dependence
T h e preloading history of the soil has a great influence on the deformation properties. One
of the m o s t important parameters in soil mechanics, which has to be investigated in order
to m a k e possible a correct evaluation of the settlement in loading, is the m a x i m u m past
pressure or, as it is generally termed, the preconsolidation
pressure.
A correct evaluation
Time
dependence.
volume
change.
T h e t i m e - d e p e n d e n c e of v o l u m e changes is very
important in the case of fine-grained soils with low permeability and is d u e to p o r e water
being gradually squeezed out of (or sucked u p into) the voids in the soil, so-called
hydrodynamic
time-lag.
consolidation,
is continuing
29
Fundamentals
Effective pressure ' , in kPa (log. scale)
40 I
0
100
200
400
600
800
Effective pressure \ in kPa (lin. scale)
L_J
1000
Fig. 17. Oedometer curve presented in lin.-log. (full line) and lin.-lin. scales (broken line). Lin.-log.
curve used for determination of the preconsolidation pressure according to Casagrande. Lin.-lin. curve
used for detewrmination fo the preconsolidation pressure according to Sllfors.
d& dt
de
of effective
stress.
In this expression, of course the void ratio e can be replaced by the relative compression .
In classical soil m e c h a n i c s the v o l u m e c h a n g e caused by creep p h e n o m e n a is
considered to begin at the end of primary consolidation, and is termed
consolidation
secondary
( 2 7
Fundamentals
30
of consolidation.
process, pore water is squeezed out (sucked up) in the vertical direction w e speak of the
coefficient of consolidation in vertical pore water flow, cv. Correspondingly, we speak
of the coefficient of consolidation in horizontal pore water flow, ch. T h e consolidation
coefficients are related to the oedometer modulus M and the permeability of the soil
through the relation:
cv =
kvMlyw
(28)
c =
h
where kv mdkh
k M/r
h
directions, respectively,
y w = unit weight of water.
The primary consolidation process is generally defined according to Terzaghi (1923,
1925). T h e basic assumptions behind Terzaghi's theory are:
the soil is water-saturated,
pore water flow is taking part only in the vertical direction (one-dimensional
consolidation),
Darcy's law is valid,
the decrease in excess pore water pressure equals the increase in ef fective stress (Acf = - Au),
there is a unique relationship between void ratio and effective stress.
On these assumptions w e can derive the consolidation equation:
du
M d
=
dt
du
du
~V^V^ vTT
(29)
Ywdz
dz
dz
(30)
=/
(31)
In reality, the permeability is a function of the strain, and consequently by the excess
pore water pressure, and therefore the consolidation equation is more correctly written:
31
Fundamentals
Void ratio e
Slope = Ca
Primary consolidation stage
Secondary
consolidation stage
da'
0
dt
da'
=0
dt
logr
Fig. 18. Decrease in void ratio with time of loading of a clay specimen. Primary and secondary consolidation periods.
du
du
M dk du
- = c y L + _ _ ( )
dt
ywdu
dz
(32)
dz
cfe r
dt
dz
dev
dz
(33)
T h e error introduced by ignoring the last term in this equation is less important than in
the prevous case, the reason being that the c v value does not usually vary as m u c h with
changes in as the k value with changes in u (in a").
(iii) Secondary
consolidation.
period, the relative compression (the change in void ratio) is directly proportional to the
logarithm of time elapsed. We have entered into the period named secondary consolidation.
T h e rate of void ratio decrease during secondary consolidation is usually expressed by
the relation (Fig. 18):
Ae = C a A ( l o g i )
(34)
(35)
Fundamentals
32
w h e r e cxs(l+e0)
Ca.
(iv) Creep due to shear. Shear deformations are the p r i m e cause of changes in shape. T h e
main part of these deformations take place instantaneously w h e n the load is applied.
However, even under working load condition the shear deformations caused by longterm creep are mostly of the same order of m a g n i t u d e as those taking place immediately
in connection with load application. T h e rate of creep deformation Ascr/At
in this case
STRENGTH CONCEPTS
5.1
Strength
(i) Effective
parameters
and total strength.
built up of either of two c o m p o n e n t s or both: one that is only dependent on void ratio but
independent of the effective stress (cohesion) and the other that is dependent on the
effective stress (friction). In practice, the shear strength of soil is either expressed in terms
of effective stress by the so-called effective strength parameters c (the effective cohesion
intercept) and ' (the effective angle of internal friction,) or in terms of total stress by the
so-called total strength parameters c (the total cohesion intercept) and (the total angle
of internal friction). In the first case, the shear strength has been determined in a way that
gives us full k n o w l e d g e of the excess pore pressure developed during the test, whereas
in the second case the pore pressure is u n k n o w n . We also speak of the drained strength
parameters cd and in the sense that the shear strength has been determined in drained
condition with complete excess pore pressure dissipation.
' 3
'
Fig. 19. The Mohr-Coulomb failure criterion is represented by the envelope to the Mhr failure circles.
33
Fundamentals
c,
cr3 = ' 3 + 3
/
= + Au3
T h e strength parameters of the soil constitute the basis for the analysis of the bearing
capacity of the ground. Failure is reached w h e n the shear stresses in the soil reach an
upper limit represented by the envelope to the M h r failure circles (Fig. 19). T h e MohrC o u l o m b failure criterion is given by the relation:
Tf= c' + c/tan ' - (a + c/)tan '
(36)
strength
strength of the soil in undrained condition, i.e. w h e n the water content along the failure
surface remains unchanged. T h e undrained shear strength is particularly important in the
case of cohesive soils. T h e strength parameters in undrained condition are termed cu and
. In water-saturated cohesive soil, the effective stresses b e c o m e unaffected by changes
in total stress (Fig. 20) and h e n c e = 0. T h e failure condition is governed by the relation
(iii) 'True'strength.
failure criterion at constant void ratio. T h e true shear strength can b e expressed as the sum
of two c o m p o n e n t s : o n e I independent of , the other D d e p e n d e n t on ( S c h m e r t m a n n
and Osterberg, 1961). T h e independent c o m p o n e n t Ie can be expressed as ieo'c
and the
quicksand
due to the effect of h u m u s acids m a y give rise to an instable clay skeleton, reminding of
a house of cards, so-called quick clays (Fig. 21).
Fundamentals
34
Fig. 21. Quick clay in undisturbed (left) and remoulded state. The disturbance of the clay structure due
to remoulding has turned it into a liquid. (By courtesy of AAB).
quicksand.
Quicksand phenomena can be expected in sand with rounded grains having a relative density
below 0.5 and a uniformity coefficient below 5. Shear caused by earthquakes may lead to
quicksand phenomena (liquefaction) in loose to medium dense sand (Seed and Idriss, 1971).
5.3
Creep
failure
As pointed out in paragraph 4.4 , creep in cohesive soils induced by deviatoric stresses
of a lower m a g n i t u d e than those conventionally representing the failure condition may
end in failure (Fig. 22). T h e shear stress leading to long-term failure in undrained
condition is defined as creep strength
ccr
T h e ratio of deviatoric stress level leading to long-term failure to the deviatoric stress
level leading to failure in standard testing procedures decreases with increasing content
of organic matter and increasing range of plasticity (cf. Fig. 70).
5.4
Strength
anisotropy
35
Fundamentals
Fig. 22. Observed axial rate of strain vs. logarithm of time in undrained triaxial test under two different
stress conditions. Bckebol clay with natural water content w=80% and consistency limits wL =75% and
wP = 30%. Undrained shear strength Tj= cu = 17 kPa. The creep strength ccr ~ 0.7cu.
structural anisotropy ( m a d e evident for e x a m p l e by the fact that the permeability in the
horizontal direction is generally higher than in the vertical direction). Moreover,
unisotropic strength properties derive from the fact that the overburden pressure in the
vertical and horizontal directions differ from each other. This entails that every soil
element in its natural state is subjected to a shear stress vector (Fig. 23) which adds to the
shear stress vector induced by the i m p o s e d load.
Fundamentals
36
0.4
4
f
1>
<
0.3
cS
I
<! )
& 0.2
<
0.1
'
= 48%; cfCL&Q
DraiTimen clay (/ p = 2 9 % ;
-90
-60
-30
:/
= 1.2 )
=15)
30
60
90
CUC(/<JQ
= (1 - K0)sinacosa+
0.6[de(cos
- ( 1 -K0)
a + AT 0sin
a)
(37)
(f0
Slow test
Quick test
j1
0.2
0.4
0.6
0.8
1.0
1.2
Plate diameter, m
Fig. 25. Volume dependence of undrained shear strength exemplified for Swedish boulder clay. Results
of plate loading tests. (Harden, 1974).
37
Fundamentals
and d=0.29-0.20
Volume
dependence
The yield
surface
F r o m what has been said in paragraph 4 . 2 - 3 w e realise that the soil structure starts
yielding w h e n it enters into a state of shear failure or w h e n the stresses exceed the
preconsolidation pressure G'C. In a case w h e r e the principal normal stresses are acting in
the vertical and horizontal directions, the yield surface will thus b e represented b y the
M o h r - C o u l o m b failure lines on the one hand (the 0'lines, Fig. 26) and the preconsolidation
pressure in the vertical and horizontal directions on the other. Inside the yield surface
border lines, the soil can b e a s s u m e d to b e h a v e m o r e or less as an elastic material.
2o'c
0.4t-
Fig. 26. Example of yield stresses and theoretical yield border lines. Medium plastic Drammen clay ( 0
= 30; K0 = 0.5). Inside the border lines, the clay can be assumed to behave elastically. Test results
according to Berre (1975). After Larsson (1977).
So/7
38
classification
SOIL C L A S S I F I C A T I O N
1. O B J E C T
Soil classification is a very important part of the identification process necessary for the
solution of problems involved in foundation engineering. For classification purposes
routine testing p r o g r a m m e s have been m a d e up, the extent of which depend upon the type
of soil to be classified. In foundation engineering classification with regard to geological
background, to particle size ranges and distribution, to plastic properties and organic
constituents and to strength and deformation properties is of principal interest.
Various classification systems exist depending upon tradition and regional geological
conditions. In this textbook, the main outlines of the classification system proposed by
ISO (the International Standardisation Organisation) and C E N (Comit Europen de
Normalisation) will be presented. Other classification systems of current interest until the
ISO and C E N standards have been finally adopted will be mentioned briefly.
2. G E O L O G I C A L B A C K G R O U N D
As regards the geological origin, we differ between sedentary
and sedimentary
m o d e s of
formation. Sedentary soils have been formed on the spot while sedimentary soils have
been transported in one way or the other to the spot. A m o n g sedentary formations w e find
residual soils (such as weathered rock and weathered gravel) and alluvium. Organic soil
formations created on the spot (such as peat) also belong to this group. A m o n g sedimentary formations we find the water-deposited glaciofluvial and postglacial sediments
(such as clay, silt and sand, esker formations, and sludge) and the wind-deposited
sediments (such as dunes and loess). Till formations which belong to a special type of
glacial sediment are characterised by their wide grain size distribution and the irregular
shape of the grains. Often the grains are characterised by having sharp edges. In regions
subjected to glaciation, till formations usually form a hard cover to bed-rock. T h e hardness depends upon the fact that the till has generally been subjected to a heavy overburden
pressure of inland ice. In consequence, till usually forms an excellent basis for the foundation of structures.
It is often difficult to clearly distinguish the boundary line between rock and soil. From
contractual point of view, soil can be defined as the part of the ground that can be dislodged by excavation while rock cannot.
T h e geological origin has a very great influence on the geomechanical properties of
So/7
39
classification
soil and represents, therefore, a very important part of soil identification and classification.
Profound k n o w l e d g e of the geological features at the building site is also of great value
for the evaluation of which investigations should b e carried out and to what extent. If the
geotechnician has a lack of such k n o w l e d g e he ought to w o r k in close co-operation with
geologists.
3. C L A S S I F I C A T I O N A C C O R D I N G T O C O M P O S I T I O N
3.1 Grain size and grain size
distribution
Every type of mineral soil contains a mixture of grains (particles) of different sizes and
shapes. Grain size and grain size distribution form the fundamental basis for designating
mineral soils using soil fractions to distinguish the major classes of geotechnical behaviour. Grain size, therefore, serves as a basis for soil classification. Unfortunately, as yet
the fractional limits, which are applied w h e n classifying a soil, are not the same in
different countries. This must be paid attention to in cases of soil classifications which
are presented without reference to any specific classification system.
(i) Fractional
limits of mineral soils. As regards grain size, mineral soils are divided into
and sand (coarse fraction) and silt and clay (fine fraction). T h e main fractional groups and
their subdivisions are presented in Table 5.
M o s t of the European countries follow the principles established by I S O / C E N , with
TABLE 5.
Fractional limits according to ISO/CEN. The fractional limits according to the Nordic classification
system, wherever they differ from the ISO/CEN limits, are given in parenthesis.
Main groups
Grain size, mm
Boulders
>200
(> 600)
200-60
(600-60)
Cobbles
Gravel
60-2
Sand
2-0.06
Silt
0.06-0.002
Clay
< 0.002
Sub-groups
Grain size, mm
(Large boulders
> 2000)
(Large cobbles
(Small cobbles
Coarse gravel
Medium gravel
Fine gravel
Coarse sand
Medium sand
Fine sand
Coarse silt
Medium silt
Fine silt
Fine clay
600-200)
200-60)
60-20
20-6
6-2
2-0.6
0.6-0.2
0.2-0.06
0.06-0.02
0.02-0.006
0.006-0.002
< 0.0006
Soil
40
classification
distribution.
and 0.06 m m is determined using a series of sieves with different m e s h widths. T h e same
set of sieves is used in International standards (ISO/R 565) and in G e r m a n standards (DIN
4188), except that International standards do not include sieves with m e s h widths 0.2,
0.63, and 6.3 m m . F r o m a practical point of view, any set of sieves can b e used to give
a good picture of the grain-size distribution.
Guiding values for the division of coarse-grained and
fine-grained
given in Table 6.
TABLE 6.
Guiding values for soil classification on a basis of the contents of various fractions.
Fraction
Content of fraction
in wt% of material
< 60 mm
Gravel
20-40
gravelly
Sand
>40
10-20
sandy
Fines
(silt+clay)
>20
5-15
15-40
>40
Content of clay
in wt% of material
< 0.06 mm
Name of soil
Modifier
Main term
gravel
sand
<20
>20
<20
>20
< 10
10-20
20-40
>40
somewhat silty
somewhat clayey
silty
clayey
clayey
silty
silt
silt
clay
clay
So/7
41
classification
T h e grain-size distribution of material with grain size < 0.06 m m must be determined
by sedimentation analysis, while the size of boulders and cobbles is determined in the
field by sieving through a grating or by direct measurement.
T h e grain size is indicated by the symbol d (Fig. 27). T h e size of those grains which
correspond to 6 0 % , 4 0 % , etc. on the grain-size distribution curve is denoted d60, d40, etc.
T h e inclination of the grain-size distribution curve, also called the grading curve, is
indicated by the so-called uniformity coefficient C0 = d6Q/dl0.
proposal, the soil is called poorly graded if Cv < 6 and well-graded if Cv > 6.
T h e uniformity coefficient is sometimes not representative of the grading. This is the
case w h e n o n e or m o r e intermediate fractions are strongly under-represented. Such soils
2
are termed gap-graded. They are characterised by a coefficient of curvature Cc=d^Q /dQ
c o m m o n l y b e l o w 1 or above 3.
(iii) Designation
purposes.
simplified by n o m o g r a m s which vary according to the country in which they are drawn
up (see Wiegers, 1974). T h e n o m o g r a m drawn up by the Swedish Geotechnical Society
for general classification of mineral soils (Fig. 28) is simple to use and provides an
u n a m b i g u o u s identification of the soil.
(iv) Visual observations.
m a d e of the size of the grains based on the size of m o r e familiar objects. For e x a m p l e ,
cobbles are larger than h o c k e y balls and gravel larger than lead shots. Sand, at its lower
limit, contains particles that are hardly visible to the naked eye.
In the case of silt and clay where the particles are invisible to the naked eye certain
methods can be utilised to help in the identification. Coarse silt particles can be felt between
Boulder and cobble fractions
Coarse fractions
Fines
clay
sand
silt
gravel
I cobbles , boulders
with boulders
0,002
0,006
0,02
Grain size d, mm
0,06
0,2
0,6
20
1.85 mm
60
200
d*60
t
i / 3 0 = 20 mm
600
2000
170 mm CU = 92
C c= 1 . 3
Fig. 27. Examples of grain size distribution curves and how to determine the uniformity coefficient.
42
So/7
classification
Fig. 28. Example of nomogram used for classification of sedimentary soils (Karlsson & Hansbo, 1984).
In accordance with the exemplified fraction percentages the soil should be designated sandy, silty day.
Example 4: Use the nomogram in Fig. 28 for designation of soil whose content of clay is (a) 20% [(b)
4%] of fines (a) 70% [(b) 45%] and of sand (a) 10% [(b) 22%]. All figures are given in wt.% of material
<60 mm.
Solution: The designation of the soil in example (a) is silty clay and in (b) sandy, gravelly silt.
Soil
43
classification
the fingers and give a rough feeling. In the wet state the particles stick together but can
easily b e m o v e d in relation to each other. If the hands are r u b b e d with wet coarse silt, the
material dries rapidly to a p o w d e r which can easily be brushed off. T h e colour is normally
grey.
Medium
silt and fine silt in a dry state h a v e a floury character and feel smooth between
the fingers. In wet state the material is plastic and at high water content, sticky. T h e
material can easily b e w a s h e d off the hands. T h e colour of dry material is normally whitegrey.
Clay in a wet state is plastic and sticky. It cannot easily be w a s h e d off the h a n d s without
a brush. O n drying, the samples turn into hard l u m p s that cannot b e crushed between the
fingers. T h e colour is normally grey, b r o w n - g r e y or red-grey.
3.2
Lime
content.
calcareous (very marly or very calciferous clay or silt) and those with
calcareous
According to the I S O proposal, the carbonate content is assessed on the basis of the
reaction of soil to dilute hydrochloric acid, HCl. T h e soil is t e r m e d non-calcareous if there
is no effervescence, slightly calcareous if there is a slight, but not sustained, effervescence
and highly calcareous if there is a strong and sustained effervescence.
3.3
Organic
content.
Soil
44
classification
organic and mineral soils. T h e colour is dark and b e c o m e s black w h e n the organic content
exceeds about 5 % . Organic soils also have a distinctive odour.
Peat is a soil formed from vegetable remains, deposited in fens ( l o w m o o r peat) or in
raised b o g s (highmoor peat), with an organic content that exceeds 20 w t % of dry matter
(grain size < 2 m m ) , see Table 7. Depending upon the degree of decomposition, peat is
classified as fibrous peat, pseudo-fibrous peat and a m o r p h o u s peat, according to the von
Post classification system (von Post, 1921).
Fibrous peat has a low degree of decomposition, a fibrous structure and an easily
recognisable plant structure (primarily white m o s s ) . W h e n squeezing a sample of fibrous
peat in the p a l m of the hand, only water, and no solid matter, passes between the fingers.
Pseudo-fibrous
plant structure. W h e n squeezed, less than half of the solid matter passes between the
fingers.
Amorphous
mushy consistency. W h e n squeezed, more than half of the solid matter passes between
the fingers without free water being separated.
An exact determination of the content of organic matter is rather difficult to achieve.
T h e most c o m m o n methods are the ignition loss method and the colorimetric method. In
the first case, the soil sample should gradually be brought to 9 5 0 C (550C as recommended
in the A S T M Standards seems not always sufficient) and held until completely ashed (not
less than one hour), while in the second case, the organic matter is determined through
rapid wet combustion, as described by Walkley and Black (1934), followed by a
colorimetric test using a light filter for wavelengths of about 6 2 0 p m .
T h e ignition loss method is preferably used for a determination of the organic content
of highly organic soils, especially peat, provided that the lime content of the soil is not
too high (< 2 0 % ) . T h e lime content must be determined and the ignition loss corrected
TABLE 7
Terms for the designation of organic content
Term
Examples
Slightly organic
2-6
Moderately organic
6-20
Highly organic
>2()
So/7
45
classification
method
normally gives m o r e accurate results ( 1 percentage point) than the ignition loss method.
4.
Classification systems based on the geotechnical properties of the soil are applied in order
to serve as a m e t h o d o l o g y in the appraisal of foundation problems. Normally, the soil is
classified with regard to relative density, strength characteristics, sensitivity, consistency
limits, consolidation characteristics and frost activity.
4.1
Relative
density
(38)
Very loose
Loose
Medium dense
Dense
Very dense
density index
<0.25
0.25-0.50
0.50-0.75
0.75-0.90
0.90-1.0
CPT
? c( M P a )
^30
WST
halftums/0.2m
<2.5
2.5-5.0
5.0-10.0
10.0-20.0
>20
<4
4-10
10-30
30-50
>50
< 10
10-30
30-60
60-100
> 100
SPT
So/7 classification
46
Strength
properties.
friction between the grains and partly from dilation energy. T h e dilation energy which is
c o n s u m e d in the shearing of dense material has a great influence on shear resistance. In
m e d i u m dense material, the influence of dilation energy is negligible. In such a case, the
soil is said to be in a state of critical density. In loose soils w e h a v e to deal with negative
dilation.
Coarse-grained soils are typically non-cohesive. However, cementation between the
grains as well as capillary forces can give rise to apparent cohesion. A l t h o u g h the effect
of apparent cohesion m a y be very useful, it rarely can b e taken into account in the design
of foundations.
(ii) Cohesive
soils. Cohesive soils are characterised by the fact that the shear strength is
due to both friction (between coarser grains and between aggregates that are formed by
clay particles) and cohesion within the material (through the action of sorption forces or
organic c o m p o n e n t s ) .
Clay and organic mineral soils are typically cohesive. Since it is not possible to
determine the relative density of these types of soil, w e c h o o s e instead to group them
according to shear strength properties, determined under undrained condition, the socalled undrained shear strength cu (or Tfu). T h u s , the soil can b e said to b e very soft if cu
< 20 kPa, soft if 2 0 < cu < 4 0 kPa, firm if 4 0 < cu < 75 kPa, stiff if 75 < cu < 150 kPa and
very stiff if cu > 150 kPa.
Cohesive soils are also classified according to their sensitivity, i.e. the ratio between
the undrained shear strength of a specimen in undisturbed and in r e m o u l d e d states.
Sensitivity is very important for the estimation of h o w m u c h the undrained shear strength
may decrease in a case of disturbance, for instance d u e to piling. T h e sensitivity is m o s t
easily determined by m e a n s of the fall-cone test. Soils are termed slightly sensitive w h e n
the sensitivity St < 8, moderately sensitive w h e n 8 < St < 30 and highly sensitive w h e n
St > 30. Clays with St > 50 are called quick clays (cf. Fig. 2 1 , p. 34).
Soil
47
classification
(iii) Intermediary
soils. Silt and composite soils (content of fines 15-40 wt% of total
material < 60 m m ) occupy, from the point of view of strength, an intermediate position
between non-cohesive and cohesive soils. T h u s , the shear strength is due to both
frictional and cohesive resistance.
4.3
Consistency
(i) Consistency
great extent governed by the water content. If, for e x a m p l e , the water content of a clay,
originally in a liquid state, is gradually reduced it passes through a plastic into a firmer,
brittle state in which it easily crumbles. T h e water content limits, within which the soil
has a plastic consistency in the remoulded state, cannot be given exactly since the
transition from liquid/plastic consistency, on the one hand, into plastic/brittle, on the
other, takes place gradually. T h e consistency limits are therefore a matter of definition.
T h e m e t h o d s generally applied for their determination were originally suggested by
Atterberg, and in c o n s e q u e n c e these limits are alternatively called the Atterberg
limits.
Atterberg p r o p o s e d the testing procedures for the determination of three limits: the liquid
limit w L , the plastic limit Wp, and the shrinkage limit ws. A full and detailed description
of the testing procedures applied for determination of the consistency limits, and their
historical back-ground, can be found in a d o c u m e n t presented by the Laboratory
C o m m i t t e e of the Swedish Geotechnical Society (Karlsson, 1977).
The liquid limit, which defines the transition from liquid to plastic state, is usually
determined according to C a s a g r a n d e ' s method, representing a final d e v e l o p m e n t of the
method originally suggested by Atterberg. T h e liquid limit is defined as the water content
at which the soil, in a r e m o u l d e d state, when placed in b o w l - s h a p e d c u p and parted into
two halves b y a V-shaped groove as shown in Fig. 29, flows together about 13 m m at the
b o t t o m of the cup after the cup has been dropped freely 25 times from a height of 10 m m
against a b o t t o m plate of micarta or ebonite.
Fig. 29. Determination of the percussion liquid limit. To the left, the groove has just been formed. To
the right, the groove has flown together 13 mm lengthwise.
So/7
48
classification
The plastic
liquid
liquid
limit.
limit, which defines the transition from plastic to semi-solid state, is the
lower water content limit at which the sample can b e rolled to a thread, 3 m m in thickness,
without crumbling. T h e test is e a r n e d out by hand by rolling the sample in a plastic state
repeatedly on a water-absorbing paper until the given requirement is fulfilled.
The shrinkage
no further shrinking w h e n its water content is reduced. It can also be defined as the
m a x i m u m water content at which the soil transforms from semi-solid to solid state.
Evidently, the soil in a remoulded state has a plastic consistency w h e n its water content
is between wL and wF T h e difference:
Fig. 30. Determination of the fall-cone liquid limit is generally performed by using a 60g/60 cone. To
the left, the cone is adjusted before being dropped. To the right, after the cone has been released and
penetrated into the soil.
Soil
49
classification
TABLE 9
Classification of fine-grained soils on a basis of plasticity
Designation
Non-plastic
Low plasticity
Intermediate plasticity
High plasticity
Very high plasticity
<35
35-50
50-70
70-90
< 1
1-7
7-17
17-35
35-50
>90
>50
(39)
index,
W-
W ~Wp
WL
the liquidity
wP
Wp
(40)
Wp
index,
Wj
- w
(41)
k= = 1 - 4
w
the consistency
index.
A plastic soil has a liquidity index 0 < IL< 1 and a corresponding consistency index 1>
/c>0.
T h e consistency index is sometimes used as an alternative basis for characterisation
of the strength properties of silts and clays. T h u s , according to I S O , soils are characterised
as very soft if Ic<
0.05, soft if 0.05 < Ic< 0.25, firm if 0.25 < Ic<
1.0.
O n the basis of the plasticity characteristics, soils are divided into four groups as shown
in Table 9.
A statistical study of the consistency limit values of m a n - m a d e h o m o g e n e o u s clays
(one and the s a m e for each test series) arrived at in different S w e d i s h geotechnical laboratories (Figs. 3 1 - 3 2 ) s h o w e d a large coefficient of variation (Karlsson, 1977). T h e
most consistent values w e r e obtained with regard to the fall-cone liquid limit (Fig. 31).
(ii) Classification
50
So/7
Low-plasticity clay
10r
classification
High-plasticity clay
(a)
(b)
_ .is
>
MM
15i
(d)
(c)
J3
a
10
20
25
30
35
45
50
55
60
65
Fig. 31. Statistical variation of the consistency limit w 7 determined in different laboratories, (a) and (b)
represent the percussion liquid limit, (c) and (d) the fall-cone liquid limit.
Low-plasticity clay
High-plasticity clay
1
1
Average value: 18%
Standard deviation: 1.3%
10
15
20
25
25
20
Test results, wp
30
35
40
Fig. 32. Statistical variation of the consistency limit wP determined in different laboratories.
So/7
51
classification
Soils with the same geological origin seem to fall within narrow zones, closely parallel
to the A line. However, the A line function itself is not unique but changes with regard
to geological conditions. For Scandinavian fine-grained soils, for example, the upper
boundary of organic clay and silt seems to follow the relation IP = 0 . 9 5 ( w L - 26) while
that of inorganic clay follows the relation IP = 0 . 8 ( w L - 8), where wL = fall-cone liquid
limit in %.
The plasticity index divided by the clay content, the so-called activity
number,
is a
measure of the colloidal activity of clay. The activity number is first of all dependent on
the ion exchange capacity and the specific surface of the clay minerals and on the content
of organic colloids.
4.4
The unified
soil classification
system
In 1952, the US Corps of Engineers and Bureau of Reclamation agreed on a soil designation system, the so-called Unified Soil Classification System (USCS), which has
become the most well-known and most widely adopted soil designation system throughout
the world. According to this system, the soil should be identified by a group name and
a group symbol, based on certain criteria using results of laboratory tests. According to
U S C S , soils are designated coarse-grained if more than 5 0 % by dry weight of material
with grain size < 75 m m is retained on No 200 sieve (mesh width 0.074 mm) and finegrained if the corresponding figure is less than 5 0 % . Thus, just as in other classification
systems, coarse-grained soils represent sand and gravel while fine-grained soils represent clay
and silt. A more detailed picture is given in Table 10.
Fig. 33. The Casagrande plasticity chart. Soils of equal composition lie in zones more or less parallel
with the A line. The soil classification symbols are those of the USCS (cf. Table 10) determined in
different laboratories.
Soil
52
classification
TABLE 10.
USCS. Criteria for assigning group symbols and group names.
Symbol
Name
Gravels,
more than 50%
> 4.76 mm
Clean gravels,
< 5% fines
Gravels with
> 12% fines
GW
GP
GM
GC
Well-graded gravel
Poorly graded gravel
Silty gravel
Clayey gravel
Sands,
50% or more
< 4.76 mm
Clean sands,
< 5% fines
Sands with
>12% fines
SW
SP
SM
SC
Well-graded sand
Poorly graded sand
Silty sand
Clayey sand
Inorganic
wL < 50%
Organic
Inorganic
Organic
PT
Lean clay
Silt
Organic clay/Organic silt
Fat clay
Elastic silt
Organic clay/Organic silt
Peat
The symbol O L means organic clay when IP > 4 and on or above the " A " line in the
/ p / w L - d i a g r a m and organic silt when IP< 4 and below the " A " line, while the symbol OH
means organic clay when IP is on or above the " A " line and organic silt when IP is below
the " A " line.
A full description of U S C S can be found in, for example, the 1986 Annual of A S T M
Standards under Designation: D 2487.
4.5
Frost
activity
Frost penetration into the soil leads to formation of ice crystals which create a pore water
underpressure in the adsorption layers surrounding the particles. As a result of the
underpressure, pore water is sucked up from the environs and concentrated as ice lenses
and/or ice layers in the soil. The inflow of water continues as long as water can penetrate
into the adsorption layers between the particles and the growing ice crystals. The ice
crystals continue to grow as long as the inflow of water can keep pace with the ice
formation. The thickness of the water film between the grains and the ice crystals in which
the water is moving is independent of the grain size. Therefore, the total flow area is much
smaller in frozen coarse-grained material than in frozen fine-grained material, which
reduces the possibility of water flow into the frozen zone. Consequently, coarse-grained
soils are not subjected to frost heave due to suction of water from the environs and
So/7
53
classification
formation of ice lenses or ice layers. Fine-grained soils, on the other hand, and silt in
particular, are very disposed to frost heave.
T h e frost h e a v e to b e expected obviously depends on the permeability of the soil
beneath the frozen z o n e and on the grain size in the frozen z o n e as well as on the rate of
penetration of the frost.
Classification of soils with regard to frost activity can be m a d e as follows:
G r o u p I - Frost-insusceptible
frost-susceptible
frostheave and by severe softening during thawing. R o u g h l y speaking, this group covers
silt soils and silty clays.
A diagram for the determination of frost-susceptibility based on grain size distribution
has been presented in a w o r k report 1 9 8 5 - 1 9 8 9 by the I S S F M E Tecnical C o m m i t t e e on
frost (Fig. 34).
0.002
0.006
0.02
0.06
0.2
0.6
Grain size d, mm
Fig. 34. Determination of frost-susceptibility of a soil on the basis of the grain size curve. Region 1L
corresponds to group II (moderately frost-susceptible soils) and region 1 to group III (strongly frostsusceptible soils). If the grain size curve falls inside either of regions 2-4 the soil belongs to group II
(frost-insusceptible soils). However, if the lower part of the grain size curve permanently passes the the
next region on the finer side, the soil belongs to group II-III. (ISSMFE Technical Committee on Frost).
54
So/7
classification
5. D I S C O N T I N U I T I E S A N D B E D D I N G
5.1
Discontinuities
whichwereformeddur'mg
discontinuities
which
include physical breaks in the soil due to shrinkage and/or ice loading in the Pleistocene
or tectonic disturbance. Fissures, faults and shears are other e x a m p l e s of mechanical
discontinuities.
T h e frequency of discontinuity occurrance is expressed by noting their spacing. T h e
terms r e c o m m e n d e d by ISO are given in Table 11.
TABLE 11
Terms for the designation of discontinuity spacing.
Term
Mean spacing, mm
Very widely
Widely
Medium
Closely
Very closely
Extremely closely
> 2000
2000-600
600-200
200-60
60-20
< 20
5.2
Bedding
Bedding is another factor of great importance for the engineering behaviour of soil. As
was the case with the Furre slide event, undetected fine-grained layers e m b e d d e d in
coarse-grained soil m a y entail unforeseen risks of failure. Coarse-grained layers in finegrained soil represent another e x a m p l e . T h e s e generally provide drainage paths which
govern the rate of settlement induced by loading.
T h e terms r e c o m m e n d e d by ISO for the designation of bedding are given in Table 12.
TABLE 12
Terms for designation of bedding thickness.
Term
Mean thickness, mm
> 2000
2000-600
600-200
200-20
60-6
< 6
Rock
55
classification
ROCK CLASSIFICATION
T h e most important aspects of rock classification and identification in foundation
engineering concerns rock types and the structure of the rock mass. As in the case of soil
classification, a rock classification system is being worked out by the International
Standardisation Organisation, ISO (Price, 1992). Here, a short s u m m a r y of the proposal
put forward to I S O will be presented.
1
ROCK IDENTIFICATION
Bedded rocks
Sedimentary
Foliated
Metamorphic
Coarse
>2
Conglomerate
Gneiss
Granite
Gabbro
(rounded particles
in a finer matrix)
Breccia
(angular particles
in a finer matrix)
Medium
0.06 - 2
Sandstone
Schist
Microgranite
Dolerite
Fine
0.002 - 0.006
Mudstone
Shale
Slate
Rhyolite
Basalt
< 0.002
Flint, chert
Mylonite
Obsidian
Rock
56
classification
Chalk, clastic limestone, crystalline limestone and dolomite, evaporites, coal and
lignite also belong to the group of sedimentary rocks. T h e s e h a v e a grain size in the whole
range shown in Table 13.
Fossils m a y b e found in sedimentary rocks. T h e mineral calcite in calcareous rocks
m a y b e scratched with knife and will react with dilute hydrochloric acid. Quarts scratches
steel. B r o k e n crystals in crystalline rocks reflect light.
T h e r o c k to b e identified is best seen in outcrop or as large fragments showing broken
surfaces.
2
ROCK MASS
2.1
Mass
structure
Metamorphic
Igneous
Bedded
Interbedded
Laminated
Folded
Massive
Graded
Cleaved
Foliated
Schistose
Banded
Lineated
Massive
Flowbanded
2.2
Weathering
weathered.
weathered.
weathered.
and/or
Rock
57
classification
G r a d e V I Residual
and material fabric are destroyed. T h e r e is a large c h a n g e in v o l u m e , but the soil has not
been significantly transported.
2.3
Strength
of rock
material
Very weak
Weak
Moderately weak
Moderately strong
Strong
Very strong
< 1.25
1.25-5
5-12.5
12.5-50
50-100
100-200
Extremely strong
>200
2.4
Discontinuities
(i) Discontinuity
spacing.
discontinuities in the rock mass govern its mechanical behaviour. As in the case of soil
there are two m a i n groups of discontinuities: genetic discontinuities
foliation) and mechanical
discontinuities
Spacing, mm
> 2000
2000-600
600-200
200-60
60-20
20-6
< 6
Rock
58
classification
TABLE 17
Terms to describe mechanical discontinuity spacing,
Term
Spacing, mm
>2000
2000-600
600-200
200-60
60-20
<20
(ii) Aperture.
< 0.1 mm
0.1-0.25 mm
0.25-0.5 mm
Very tight
Tight
Partly open
'Closed' features
0.5-2.5 mm
2 . 5 - 1 0 mm
> 10 mm
Open
Moderately wide
Wide
'Gapped' features
10-100 mm
0.1-1 m
> 1 m
Very wide
Extremely wide
Cavernous
'Open' feature
Rock
59
classification
(iii) Infilling.
its shear strength are important for the j u d g e m e n t of the stability of the rock mass.
(iv) Water seepage.
60
So/7
investigations
SOIL I N V E S T I G A T I O N S
1. OBJECT
T h e object of soil investigation is to establish a reliable picture of the building site
conditions with regard to geological and geotechnical characteristics, necessary as a
basis for design. This purpose, which, of course, is likely to be in the m i n d of all
geotechnicians, is not always fulfilled. Not only do w e h a v e the inevitable pressure of
competitive tendering, which may result in a reduction of the level of site investigations
but also the p r o b l e m that the client, often inexperienced in the value of thorough and
professional site investigations, prefers good e c o n o m y to good engineering, ignorant of
the fact that insufficient information about the subsoil conditions m a y result in very poor
e c o n o m y in the long run. T h e client m a y also feel suspicious of the value of thorough and
expensive soil investigations since there are a great n u m b e r of geotechnicians w h o carry
out soil investigations, the results of which are m o r e or less irrelevant to the problems met
with in foundation design but serving merely as a display of their a c a d e m i c k n o w l e d g e
of soil mechanics.
W h a t kind of information, then, do w e need to be able to arrive at an o p t i m u m design?
First of all w e h a v e to form an opinion of the geological characteristics of the site, for
instance on the basis of geological maps. T h e s e m a p s , and the geological features
observed by inspection, give us a b a s i s for t h e p l a n n i n g of direct soil investigations. These
generally begin with some type of penetration testing, the results of which may give
valuable information about h o w the subsoil conditions vary at the site. M o r e advanced
methods of soil investigation can then b e determined with due regard to soil type,
variation in sounding results and type of project in question.
T h e m o s t vital information about the subsoil required for the design concerns the
strength and deformation properties and their variation at the site and the in situ stress
distribution. Although w e m a y do our best, on the basis of sounding results, to take
representative samples or to m a k e adequate in situ tests for determination of these properties, w e m u s t realise that, after all, it is merely a r a n d o m survey. In consequence,
statistical m e t h o d s h a v e been applied in order to account for the variations normally
occurring.
In the next paragraphs, the most c o m m o n m e t h o d s of investigation, and the relevance
of the results obtained for the solution of foundation engineering p r o b l e m s , will b e briefly
discussed.
So/7
61
investigations
2. P E N E T R A T I O N T E S T S
An extensive d e v e l o p m e n t and mechanisation of different sounding e q u i p m e n t s have
taken place in the last few decades. T h e most advanced sounding m e t h o d today is the
CPT-test (the c o n e penetration test), w h i c h has b e c o m e very popular in Europe. In the
U S A and in m a n y other parts of the world, the S P T (the standard penetration test) is still
extensively used due to long familiarity and experience. Other sounding m e t h o d s , such
as d y n a m i c probing (reminding of S P T ) and weight sounding, are also quite c o m m o n .
T h e a i m behind the innovations in sounding e q u i p m e n t has been to achieve a m a x i m u m
of information about the soil at the lowest possible cost. E r g o n o m i e considerations have
also played an important role.
All the penetration m e t h o d s mentioned a b o v e h a v e been standardised under the
auspices of the International Society for Soil M e c h a n i c s and Foundation Engineering
( I S S M F E ) . T h e r e c o m m e n d e d standard is publicised in Volume 3 of the Proceedings of
the International Conference on S M F E , held in Tokyo in 1977 (pp. 1 0 1 - 1 2 0 ) and in Volu m e 4 of the Proceedings of the International Conference on S M F E , held in S t o c k h o l m
in 1981 (pp. 1 2 0 - 1 2 1 ) . National standards also exist.
In the following, a general description of various penetration m e t h o d s will be given.
For those w h o w a n t a m o r e detailed description, reference is given to the standards
presented in the Proceedings mentioned above.
2.1
Dynamic
penetration
tests
(i) SPT In the standard penetration test, a split-barrel sampler is driven from the b o t t o m
of a prebored h o l e into the soil by m e a n s of a 63.5 kg h a m m e r , d r o p p e d freely from a
height of 0.76 m. T h e sampler (Fig. 36) has a length of 457 m m , an outer diameter of 51
m m and an inner diameter of 35 m m . T h e diameter of the prebored hole varies normally
b e t w e e n 6 0 and 2 0 0 m m . If the h o l e does not stay open by itself, casing or drilling m u d
should b e used. T h e sampler is first driven to a depth of 0.15 m b e l o w the b o t t o m of the
prebored hole, then the n u m b e r of blows required to drive the sampler another 30 c m into
the soil, the so-called N30 count, is recorded. T h e rods used for driving the sampler should
h a v e sufficient stiffness. Normally, w h e n sampling is carried out to depths greater than
around 15 m, 5 4 m m rods are used.
T h e S P T has the advantage over other sounding m e t h o d s in that it provides samples,
certainly disturbed but still m a k i n g a classification of the soil possible and it suits
practically all types of soils. However, it m a y b e t i m e - c o n s u m i n g and e x p e n s i v e to
perform unless c h e a p labour is available.
(ii) Dynamic probing.
62
Soil
investigations
511 mm
coupling
4 vents, 13 mm
diameter, min.
steel ball, 25 mm diam
split tube
driving shoe
1.6 mm
351 mm
As can be seen from Table 19, the main difference between the various methods is the
driving energy. T h e r e c o m m e n d e d driving rate is 30 blows per m i n u t e but up to 60 blows
per minute can b e used in non-cohesive soils. In cohesive soils, however, the rate should
not exceed 30 blows per minute. T h e n u m b e r of blows required for every 0.2 m of
penetration is recorded in the site log.
TABLE 19
Data on various dynamic probing methods.
Method
DPA
DPB
DPL
Point diam., mm
Point length, mm
Point angle, degr.
Rod diam., mm
Insertion
Hammer, kg
Drop height, m
62
62
90
40-45
51
51
90
32
35.7
71.4
90
22
preboring
63.5
0.75
63.5
0.75
45
90
90
32
no preboring
10
63.5
0.50
0.50
So/7
63
investigations
32 m m
320.3 m m
solid or
hollow
45 m m
510.2 m m
fixed or "lost"
90
~f
mm
Fig. 37. Dynamic probing equipment, types DPB (left) and HfA.
tests
(i) WST. Weight sounding is a fairly old-fashioned m e t h o d but is still used, particularly
where there is too little space for other sounding m e t h o d s . In weight sounding, a screwshaped point (Fig. 38) fixed to sounding rods is pushed into the soil by m e a n s of weights,
350.2 m m
22
64
So/7
investigations
pushrod
Seal
friction sleeve
Soil
65
investigations
is that the tip is cone-shaped with an apex angle of 6 0 (or, m o r e seldom, 90) and a diameter
of 35.7 m m (Fig. 39). It is usually provided with stress transducers which register
electrically the tip resistance and the friction against a friction sleeve. Hydraulic and
mechanical m e a s u r i n g devices exist. According to the standard, the penetrometer tip
should h a v e the s a m e diameter as the cone over a length of 1000 m m a b o v e the cone base.
Moreover, the friction sleeve should be placed immediately above the base of the cone
2
and h a v e a surface area of 150 c m . O n e has to remember, however, that there are
divergencies from the standard r e c o m m e n d e d , both with regard to the diameter of the
cone and to the location and surface area of the friction sleeve.
In its most m o d e r n design, the tip is also provided with a pore pressure transducer
which registers the pore water pressure during penetration the so-called piezocone. In
this case, the penetration test is usually referred to as the C P T U test. T h e p i e z o c o n e offers
excellent possibilities of identifying soil type and soil stratification. It is also a promising
method, based on local experience, of finding realistic values of strength parameters and
deformation moduli as well as consolidation characteristics (see, for e x a m p l e , Senneset
etal,
T h e cone tip is connected to a push-rod, with the s a m e diameter as the tip (or slightly
smaller) and is pushed into the soil by m e a n s of a thrust m a c h i n e at the r e c o m m e n d e d rate
of penetration of 20 m m per second (1.2 m per minute) T h e pushing force required for
penetration varies, of course, with the soil type, and a n u m b e r of thrusting rigs of various
powers exist.
Based on the C P T results, a report on soil type, relative density and undrained shear
strength in cohesive soils is often included. H o w well these parameters represent the truth
is as yet an open question.
3.
GEOPHYSICAL METHODS
Seismic
methods
66
Soil
investigations
Source of vibration
S wave J
R wave
Spherical propagation Circular-cylindrical propagation
along ground surface
Fig. 40. The different modes of soil particle movement induced by a shock at the ground surface. The
longitudinal and the transversial S waves spread spherically while the R wave moves in circles along
the ground surface.
s i n a /sin/3 =
(42)
v lv
x
w h e r e = angle of incidence,
= angle of refraction,
v 1 = w a v e velocity in m e d i u m 1,
v 2 = w a v e velocity in m e d i u m 2.
If in a three layer m e d i u m 1-3 (Fig. 41) the w a v e velocities v 3 > v 2 > vh then for
certain limiting angles of incidence
Source of vibration
oc
Geophones
Medium 3
Fig. 41. Principles of the seismic refraction method in a three layer medium with v 3 > v 2 > v t . The fastest
wave propagation lines from the shock point to the respective geophones indicated.
67
investigations
So/7
equal to 90, w h e n c e s i n a l r = 2
will run parallel with the respective interfaces with velocities v 2 and v 3 . A s a result of
particle motions in the planes of refraction, new w a v e s with velocities vx and v 2 respectively, and refraction angles equal to the limiting angles of incidence are created in the
upper m e d i a 1 and 2.
T h e m o d e of p r o c e d u r e is as follows: a shock is p r o d u c e d at the ground surface,
normally by firing an explosive in a shallow hole, and the time it takes for the reflected
and refracted w a v e s to arrive at a n u m b e r of sensors (so-called g e o p h o n e s ) , placed at
different distances from the explosive, is registered. T h e arrival times form the basis of
interpretation.
Evidently, according to the interrelation b e t w e e n velocity and m o d u l u s of elasticity,
the b o u n d a r y b e t w e e n a soft m e d i u m below a stiff m e d i u m (a m e d i u m with low m o d u l u s
of elasticity b e l o w a m e d i u m with high m o d u l u s of elasticity) cannot b e detected since,
in such a case, the angle of refraction will b e smaller than the angle of incidence.
Since the w a v e is a compression w a v e its velocity b e l o w water is m u c h higher than
above water. Therefore, the seismic m e t h o d can also b e used for location of the water
table.
3.2
Electric
resistivity
method
Echo
sounding
E c h o sounding is often used for the determination of the soil layer s e q u e n c e beneath sea
or lake b o t t o m s . It provides not only direct information about the b o t t o m profile but also,
to a certain extent, the transition from one soil layer to another. It m a y also b e possible
to discern the b e d r o c k profile if it is not too deep below the soil surface. T h e depth of
penetration and resolving p o w e r depend upon the frequency applied. With m o d e r n
instruments, and under favourable conditions, changes in soil strata can b e observed to
depths of 2 0 - 3 0 m .
4.
Ever since the evolution of the effective stress concept, p o r e pressure prognostications
and m e a s u r e m e n t s h a v e been considered imperative. Nevertheless, soil investigations
and geotechnical reports h a v e contained very little information about groundwater
68
Soil
investigations
_ One-in. pipe
. Test adaptor
_ Hypodermic needle
- Disc of resilient material
. Filter tip
Soil
69
investigations
a solenoid and attached to a bellows connected with the filter stone. T h e frequency of the
string is registered by m e a n s of sound w a v e s through the sounding rods and thus no
connecting cables are required. T h e results are stored digitally in a solid state memory.
In practice, in most occasions, satisfactory information about the pore pressure
situation is obtained by the use of open standpipes with filter tips. T h e m o r e advanced
piezometers are required w h e n it is important to study q u i c k pore pressure changes.
5. S A M P L I N G
Sampling is normally carried out on the basis of the results of penetration tests
(sounding). Sampling is necessary to ensure a m o r e accurate soil classification than that
which is possible on the basis of sounding alone. Sampling is also d o n e for the purpose
of laboratory investigations of the strength and deformation characteristics. This is
particularly important in the case of cohesive soils w h e r e consolidation tests on
undisturbed samples are the only m e a n s of investigating the characteristics required for
analysis of long-term settlement. In cohesionless soils, undisturbed sampling is not
possible. It is, therefore, questionable if laboratory testing of samples of cohesionless
soils will yield strength and deformation parameters of any practical use.
Details about different samplers and methods of sampling from all over the world, with
special emphasis on their application to undisturbed sampling of soft cohesive soils, are
given in an international m a n u a l , prepared by an I S S M F E s u b - c o m m i t t e e on soil
sampling and published by Tokai University Press, Tokyo, in 1981.
5.1
Undisturbed
(i) Intermittent
sampling.
sampling.
Fig. 43. Principle of undisturbed sampling using the Swedish Standard Piston Sampler I:
(a) Piston locked to the sampling tube while the sampler is inserted into the soil.
(b) Piston rigidly fixed vertically during the sampling operation.
smaller than the inner diameter of the tube k n o w n as inside clearance. Therefore, it is
usually necessary to let the sampler stay in the soil after sampling for about 5 to 10
minutes to give the sample time to swell. (Swelling is d u e to the horizontal p r e s s u r e
release obtained w h e n the sample is punched into the tube). In this w a y e n o u g h adhesion
will b e developed for the sample to be torn off at the tip of the sampler w h e n it is
Soil
71
investigations
edge taper angle
withdrawn. T h e sample can also be sheared off at the tip by turning the sampler before
it is withdrawn. In s o m e cases, however, it m a y be necessary to use s o m e kind of sample
retainer or to h a v e s o m e arrangement for releasing the v a c u u m effect w h e n the sampler
is withdrawn from the soil. Retainers should be avoided if possible, as they always cause
some disturbance.
T h e inner diameter of the sampling tube varies with the type of sampler from 50 m m
to 100 m m and a length of 8 - 1 2 times the inside diameter is c o m m o n . T h e Swedish
Standard Piston Samplers h a v e an inner tube divided into three centrally placed 170 m m
lengths and t w o outer 85 m m lengths, i.e. a total sample length of 6 8 0 m m , or 13.6 times
the sample diameter. However, the soil samples in the two outer tubes, the one nearest
to the piston and the other nearest to the cutting edge, are considered as being disturbed.
T h e tubes are m a d e of reinforced plastic and can be used repeatedly. Using the Swedish
method, cutting of the tube in the laboratory is avoided.
A good sampler should fulfil certain requirements with regard to inside clearance ratio,
(^-),
inside clearance ratio depends on the diameter of the sampling tube and on the type of soil.
In s o m e countries, inside clearance is not considered necessary. T h e r e c o m m e n d e d edge
taper angle depends on the area ratio; the higher the area ratio, the smaller the angle. M o s t
of the samplers h a v e an area ratio of about 1 0 - 2 0 % and an e d g e taper angle of 5 - 1 0
degrees. (The Swedish piston sampler, for e x a m p l e , which is considered to yield highquality samples of soft cohesive soils, has an area ratio of 21 % and an e d g e taper angle
of 5 degrees. T h e inside clearance ratio is 0.4%).
(ii) Continuous
sampling.
which can entail serious disturbance, is the foremost obstacle to the possibility of taking
long, undisturbed samples. Therefore, in this case, friction has to be eliminated in one
way or another.
T h e first sampler, designed to fulfil the zero-friction requirement, w a s the Swedish foil
sampler. Here, the friction b e t w e e n the soil and the inner walls of the sampler is
eliminated b y m e a n s of foil strips of steel that are locked to the piston and unrolled from
a foil m a g a z i n e (placed in the sampler head) w h e n the sampler is a d v a n c e d into the soil
(Fig. 45). Consequently, the only friction obtained during sampling is the friction caused
by tensile strain in the foils. T h e inner diameter of the cutting e d g e of the foil sampler is
So/7
72
investigations
. Fastening of foils
. Foil magazine
lubricant with a density of about 1.7 t / m , placed in the space between the stocking and
the surrounding plastic tube. T h e inner diameter of the B e g e m a n n sampler is either 66
m m or 29 m m . T h e latter is not considered to yield undisturbed samples and is therefore
only used for classification purposes.
Continuous sampling offers a great advantage over intermittent sampling in that it
gives a detailed picture of the soil strata. Such a detailed picture can be of great
importance w h e n erratic soil conditions h a v e to b e dealt with. However, the sample
quality is not as good as in the case of a high-quality piston sampler being used.
5.2
Disturbed
sampling.
If the only purpose of sampling is to classify the soil, undisturbed sampling m a y not b e
required. It is also m o r e or less impossible to carry out such sampling by conventional
m e t h o d s in the case of non-cohesive soils.
Soil
73
investigations
T h e r e are a great n u m b e r of m e t h o d s used for taking disturbed samples, one of the most
c o m m o n m e t h o d s being the standard penetration test. A u g e r boring and core boring are
other c o m m o n methods of obtaining disturbed samples. Representative disturbed samples
can also b e obtained by the use of thick-walled rugged piston samplers. For shallow
sampling of hard soil, digging test pits is often the easiest and m o s t e c o n o m i c solution.
6. D E T E R M I N A T I O N O F D E F O R M A T I O N P R O P E R T I E S
6.1
Introductory
remarks.
T h e loading conditions are very important for the response characteristics of the ground.
Normally, soil investigations are carried out with the a i m of determining the properties
of the soil in static loading conditions. However, the u n d e r g r o u n d r e s p o n s e in d y n a m i c
loading is m o s t probably very different from the response in static loading. T h e underlying causes of deviation between soil behaviour in static and d y n a m i c loadings are
differences in m a g n i t u d e of deformations and time effects. On o n e hand, the m a g n i t u d e
of the amplitude of shear deformation in d y n a m i c loading is generally m u c h smaller than
in static loading and, on the other hand, the loading conditions are quite different (generally cyclic loads of short duration).
In static loading, both strength and deformation properties are governed by consolidation
(hydraulic time lag) and creep p h e n o m e n a . L o n g - t e r m effects on strength and strain in
d y n a m i c loading are mainly due to the n u m b e r of loading cycles in combination with the
m a g n i t u d e of the amplitudes.
T h e soil can b e either strain-softening or strain-hardening, or b e h a v e like a m o r e or less
ideally elasto-plastic material. It can dilate or contract in shear d e p e n d i n g u p o n the density of its structure. Water-saturated granular soils in a loose state easily liquefy in shear.
Liquefaction can also take place in relatively dense soils subjected to long-term d y n a m i c
loading.
Investigations of strength and deformation properties of soil are chosen with due
regard to the the loading condition (static or dynamic), the permeability of the soil and
the possibility of taking undisturbed samples. If the latter is considered possible then
laboratory testing is often chosen, especially in cases w h e r e the soil characteristics are
governed by time-bound p h e n o m e n a (e.g. primary and secondary consolidation). If, on
the other hand, undisturbed sampling is not possible, in situ m e t h o d s are preferable.
T h e determination of the strength parameters is associated with a n u m b e r of uncertain
factors. In the case of n o n - c o h e s i v e soils, there is n o possibility of taking undisturbed
samples. Even if it w e r e possible to re-establish the in situ stress condition and the in situ
void ratio, possible cementation forces w o u l d h a v e been destroyed, the soil skeleton
subjected to rearrangement, ageing effects eliminated, etc. M o r e o v e r , considering the
size of soil v o l u m e tested and all the possible heterogeneities that can b e expected in the
underground, the limitations in the results obtained are obvious.
74
Soil
6.2 Laboratory
(i) Determination
investigations
investigations
ofK. T h e bulk m o d u l u s can be determined by m e a n s of triaxial tests.
of a shear test apparatus of the type shown in Fig. 4 6 . In order to prevent a v o l u m e change
during shearing, the specimen is surrounded by a n u m b e r of interspaced parallel metal
rings (or s o m e other similar arrangement which admits unprevented shearing), and the
vertical distance b e t w e e n the top piston plate and the b o t t o m plate is kept constant (direct,
simple shear test). Normally, the specimen is first consolidated in the apparatus under a
vertical pressure equal to that prevailing in situ before it is subjected to shear. T h e
inclination of the shearing stress vs. shearing strain curve yields the (tangent) shear
m o d u l u s G.
(ii) Determination
determining the elastic (or rather the pseudo-elastic) parameters J5and v, well-known from
Fig. 46. Direct shear test apparatus. Consolidation phase (a) and shearing phase (b). Before shearing,
the oedometer ring is removed and replaced by the rubber membrane and the metal rings (or by a
reinforced rubber membrane). The horizontal load is applied in steps until failure takes place.
So/7
75
investigations
(-or)
(43)
E = ^ ^
a-<y = ^rr
<>
a + bea
w h e r e a and b are inverted moduli w h o s e determination is demonstrated in Fig. 4 8 .
Derivation of Eq. (44) in respect of yields the tangent m o d u l u s :
44
Soil
76
investigations
0.5-10-3
'Tarct m b
- kPa
12
Axial strain ,%
Fig. 48. Determination of the parameters and b in Eq. (44).
where Eq=1/
(\ +
eaE0b)
(45)
Also the value of Poisson's ratio can be determined by the s a m e type of triaxial test
where the cell pressure is kept constant. We have:
-
Aer
(46)
of consolidation
characteristics.
Soil
77
investigations
Loading plate
Porous s t o n e -
Floating ring
Soil specimen
consecutive steps is doubled. T h e M value represented by the inch nation of the oedometer
virgin curve is called the primary
compression
modulus.
AelA(\ogo\),
03
0.
&
'
<
/
M0
= 2/\og2
- 'c
XiL
I
I
I
I
'B
<U
1
)m _
'
^ '
_ ' - '0
1
= 0 + '
&c -
' - &c
=
- +
1
1
50
_
(' - '
ln[l +
ML
WH
<u
C/5
,
if &c < ' < a'L
>r
150
/
Effective vertical pressure ', kPa
200
Fig. 50. Correlation between M and effective overburden pressure "(Sllfors and Larsson, 1981 ; Larsson
and Sllfors, 1986).
78
Soil
investigations
20
1
0.8 L- I
0
20
40
40
1
80
1
160
1
320
1
640
1
1280
1
80
160
Effective pressure & v, in kPa (lin. scale)
320
Fig. 51. Oedometer test result taken from a consultant's report. The consultant evaluated the preconsolidation pressure in the semi-log. plot at 80 kPa (according to Casagrande's method). By rewriting
the oedometer curve in linear scale we realise that there is no visible sign of a preconsolidation pressure.
In the conventional oedometer of the type shown in Fig. 4 9 , the soil layer submitted
to consolidation is drained at both top and bottom. Moreover, it can b e a s s u m e d that the
initial excess pore pressure u0 is constant throughout the specimen. A s s u m i n g that the
specimen has a thickness of 2h, the average consolidation degree {7V
in terms of the time factor Tv-
av
can b e expressed
t/
v > av
= 2/7^7
(47)
w h e n Tv < 0.2,
f/ v av = 1 _ 0.81 l e x p ( - 2 . 4 6 8 T v )
(48)
cv (Fig. 52).
So/7
79
investigations
_dd_\?_
dt
w h e r e dcfldt
(49)
2ub
2ub),
compression
the tail of the o e d o m e t e r curve as shown in Fig. 18, p. 31 (cf. c o m m e n t s by Zeevaert, 1986).
According to Mesri and Godlewski (1977), the secondary c o m p r e s s i o n index Ca is
directly proportional to the primary compression index Cc. T h e most typical values of
Ca/Cc
for inorganic soft clays and organic soft clays are 0.04 0.01 and 0.05 0 . 0 1 , respectively. For sand, Mesri (1990) reports Ca/Cc
Time of consolidation t
= 0.02 0 . 0 1 .
24 h
Load steps
0 - 1 0 kPa
1 0 - 2 0 kPa
2 0 - + 4 0 kPa
4 0 - 80 kPa
- 1 6 0 kPa
Fig. 52. Evaluation of the coefficient of consolidation from the results of oedometer tests with step by
2
80
So/7
6.3
Field
investigations
investigations
Direct in situ determinations of the deformation characteristics of soils can be carried out
in several w a y s , for instance by pressuremeter tests or dilatometer tests, and of course
by full-scale or half-scale loading tests. T h e deformation properties of soil under dynamic
loading conditions are c o m m o n l y determined by seismic m e t h o d s .
(i) The pressuremeter
proven to b e a most reliable tool for the determination of the deformation characteristics
of various types of coarse-grained soil and rock as well as overconsolidated fine-grained
soil. T h e pressuremeter (for details, see Baguelin et al, 1978), which was developed as
a practical tool by the French engineer Louis Mnard, consists of a cylindrical body with
originally three cellsa central measuring cell and two guard cells (Fig. 53). N o w a d a y s ,
pressuremeters with only one cell (a measuring cell), long enough to ensure that the end
effects are negligible, are also utilised.
After the pressuremeter is installed in the soil, the pressure in the cell(s) is increased,
which brings about a state of cylindrical expansion of the soil surrounding the measuring
cell. T h e radial deformation of the outer boundary of the measuring cell, according to the
M n a r d procedure, is obtained directly from the amount of water that is inflated into the
cell. T h e cell pressure is increased in steps and kept constant during each step for 2
Pressure gauge
minutes. Readings are taken after 30, 60, and 120 seconds. T h e readings h a v e to be
corrected with regard to the stiffness of the measuring device itself. T h e corrected
pressure vs. creep deformation from 30 to 120 seconds is plotted together with the
corrected pressure vs. total deformation (after 120 seconds) in a diagram of the type
shown in Fig. 54.
T h e results of pressuremeter investigations are greatly influenced by the installation
technique. To avoid disturbance in the best possible way, the M n a r d pressuremeter is
generally installed in the soil in a carefully prebored hole of the s a m e diameter as the
pressuremeter. However, in difficult soil conditions direct insertion inside a driven
slotted tube m a y b e necessary. A comparison between the pressuremeter moduli obtained
with and without the use of the slotted tube shows quite different results above, but no
significant difference beneath, the groundwater level. In dense and m e d i u m dense sand,
the moduli obtained without the use of the slotted tube h a v e been shown to vary from
about 40 to 7 5 % of the moduli obtained by its use (Hansbo and Pramborg, 1990). In the
literature it is clearly stated that direct insertion inside a slotted tube should be resorted
to only after all other m e t h o d s of installation h a v e failed. This, in m y opinion, is an
overstatement considering the costs involved in each unsuccessful trial. In Sweden, for
example, direct insertion inside a slotted tube has b e c o m e the rule rather than the
exception, and the results obtained have yielded settlement values that h a v e shown
acceptable agreement with those observed in practice.
In the evaluation of the pressuremeter test, the soil is assumed to b e h a v e as an elastic
medium. Since, in such a case, the stresses induced in the soil are of deviatoric character
the pressuremeter test e a r n e d out in a prebored hole yields the shear m o d u l u s :
82
Soil
Gpr =
investigations
(Vc+Vm)Ap/AV
(50)
Av +v )(v + v )
c
(51)
pressuremeter first-load modulus. In the M n a r d interpretation this has been taken into
account by introducing a set of rheological coefficients with regard to soil type and
loading history.
Usually, the shear modulus determined by the pressuremeter test is replaced by the socalled pressuremeter m o d u l u s Epr,
i.e.:
Epr=2Gpr(l+vs)
(52)
&GprB
Camcometer
autofaureur).
These
tools can only be used in relatively fine-grained soils and have, therefore, a restricted field
of application in comparison with the preboring pressuremeters of M n a r d type.
(ii) Dilatometer
Soil
83
investigations
96 m m
84
Soil
investigations
is therefore mainly used in sand and silt. Under difficult conditions it m a y be advanced
by driving, but then the results should be treated with caution.
T h e evaluation of the dilatometer test is based on two limiting pressure values, namely
px representing the pressure required to produce 1.10 m m m o v e m e n t of the m e m b r a n e ,
and p0 being the pressure at zero m o v e m e n t of the m e m b r a n e . A s s u m i n g that the soil
behaves elastically, the dilatometer modulus can be deduced from the relation ( see
discussion by Ekstrm, 1989):
ED =
48A(Pl-Po)
(53)
_P\
iD -
-Po
Po - "o
and u0 = pore water pressure at rest (no excess pore pressure),
Ovo
(54)
Soil
85
investigations
T h e ID value varies from about 0.6 to 1.8 for silt and is above 1.8 for sand. For normally
consolidated soil KD ~ 2.5. If KD > 2.5 it is r e c o m m e n d e d that the ID value b e corrected.
T h e correction to b e m a d e depends upon the depth of investigation.
For depths < 2 m :
IiD(corr) = / D - 0 . 0 7 5 ( t f D - 2 . 5 ) .
For depths > 2 m
D(corr)
= / D- 0 . 0 3 5 ( ^ D- 2 . 5 ) .
T h e correction of the m e a s u r e d ID value in the case of sand and silt is generally of minor
importance.
T h e dilatometer is sometimes utilised as a m e a n s of determining the deformation properties of overconsolidated clays. However, the results are difficult to interpret as they
represent a partly u n d r a i n e d condition.
(iii) Plate
loading
considered the best m e a n s of determining the deformation characteristics of soils, but are
only used in exceptional cases b e c a u s e of the costs involved. T h e best w a y of performing
these tests is to use a testing procedure in w h i c h the load is applied in steps of equal
duration. First the failure load m a y be estimated on a theoretical basis. T h e n the load m a y
be applied in steps of about 5 - 1 0 % of the theoretical failure load. R e a d i n g s of the
settlement m a y b e taken 1, 2, 4, 8, and 16 minutes after the application of each new load
step. B y plotting the creep settlement from 1 to 16 minutes against the load, the critical
load can b e interpreted in a w a y similar to that used in the p r e s s u r e m e t e r test (see Fig. 54,
p. 81). T h e determination of the critical load which leads to excessive creep settlement
is j u s t as important as the determination of the failure load.
T h e settlement s observed at a certain load per unit area q (below the critical load) in
a plate loading test can b e used for the determination of the pseudo-elastic m o d u l u s of
elasticity of the soil according to the relation:
2
K(\-V )qD
E
55
4s
<>
v )qb/s
(56)
86
Soil
investigations
nearest below the plate (to a depth of approximately five times the plate width or plate
diameter, cf. pp. 1 4 1 - 1 4 4 ) .
Plate loading tests using small-scale plates are frequently e a r n e d out. With regard to
normal variations in soil properties, the width of such plates should not be below 0.6 m
(cf. Fig. 2 5 , p. 3 6 ) . T h e so-called field compressometer developed at the N o r w e g i a n
Technical University (Janbu and Senneset, 1973) belongs to a special category of plate
loading tests that can b e carried out at various depths below the ground surface.
(iv) Seismic
investigations.
determining the moduli of deformation to be applied in the case of small strains of the
order of m a g n i t u d e of ~ 1 0
-5
(57)
Input
rg
Oscilloscope
Trigger
Trigger
geophonc
Hammer impulse
Impulse
Horizontal v e l o c i t y
transducers
String and
cable
I m p u l s e rod
S c r e w plate
Receiver geophones
Fig. 57. Down-hole and hole-to-hole methods for determination of the dynamic shear modulus of soil.
So/7
87
investigations
whether the soil is above or below the groundwater level. T h e shear m o d u l u s is obtained
from the relation:
2
(58)
G0 = pvs
where as above.
T h e shear w a v e velocity can be determined directly in boreholes by so-called downhole (alternatively up-hole) or cross-hole techniques (Fig. 57). All these m e t h o d s comprise a shock i m p u l s e generator (anything from an ordinary h a m m e r to an explosive), a
pick-up (for e x a m p l e m o u n t e d in a seismic cone of the type developed at the University
of British C o l u m b i a ) and a registration instrument (normally an oscilloscope). By
generating shock w a v e s with inverted w a v e amplitudes, the shear w a v e velocity can be
determined with great accuracy.
T h e d y n a m i c shear m o d u l u s can also be obtained by measuring the surface w a v e (the
Rayleigh w a v e ) velocity
G0~U5pvR*
(60)
correlations
correlations.
with reference
to deformation
properties
As previously shown (Fig. 17, p. 29), the stress history of the soil
88
So/7
investigations
Consequently, it is important to k n o w both the vertical and the horizontal stress histories
in relation to the actual vertical and horizontal stress levels.
T h e effective octahedral stress level c f o c i has also a certain influence on the magnitude
of the shear m o d u l u s G 0 (see, for example, Hardin and Richart, 1963; Hardin and Black,
1969). Hardin and Richart investigated the shear w a v e velocity
void ratios. For sands with rounded and sharp-edged grains, respectively, the following
correlations w e r e established:
vs
= 160(2.17 -e)()
(61)
O
vs
= 110(2.97 - 0 ( )
1 M
(62)
G ~690 f
(2
(63)
1+e
( 2
^ ~
1+ e
g )
\/&
(64)
&
oa
(65)
//',
0.45wL
(66)
Soil
investigations
/=
(ii) Strength
correlations.
(67)
0.11+0.37/p
10000
100
50
101
-
I
7
I
6
I
5
I
4
io-
Fig. 59. Empirical correlation between G/cu and ^according to Larsson ( 1986) and Larsson & Mulabdic (1991).
90
So/7
G 0 - (208//p + 2 5 0 ) c u
investigations
(68)
or, alternatively,
G0=504c>
(69)
G 0 - 6cu
(iii) Correlation
with sounding
resistance.
+ 500cM
(70)
aspects related to the possibilities of estimating G 0 from sounding resistance are referred
to L o P r e s t i and Lai (1989). T h e empirical studies e a r n e d out on this matter are mainly
based on comparisons between shear w a v e velocities obtained by seismic methods
(cross-hole and seismic cone tests) and penetration resistance by the use of the S P T and
C P T sounding m e t h o d s . T h e results obtained, which mainly depend upon geological
24
a i
\
\ -*
(X
20
16
\
V \
\ \
12
OCR= 10
\ \ \
s
OCR= 1
0
200
300
500
1000
2000
3000
Fig. 60. Correlation between G 0 and cone penetration resistance qc for quartz sand (Jamiolkowski and
Robertson, 1988).
Soil
91
investigations
origin, grain size, cementation effects and stress history of the soil, show great dispersion,
particularly with reference to the S P T penetration resistance.
According to Imai et al. (1982), a rough estimate of the shear m o d u l u s G 0 , based on
comparison b e t w e e n S P T resistance and shear w a v e velocities, can b e obtained from the
relation (Imai et al,
1982):
G 0 14.1(N 3 0)0-68MPa
(71)
G0 = 20Epr
7.
investigations
of effective strength parameters.
there, the s p e c i m e n is loaded in the radial direction by applied cell pressure and in the
axial direction by a combination of cell pressure and an additional external axial load. T h e
total axial stress applied can b e either higher (active test) or lower (passive test) than the
radial stress (Fig. 61).
In order to k n o w the effective stress level at failure, the tests h a v e to b e carried out
either in a drained condition or with pore pressure m e a s u r e m e n t s . Moreover, the specimen has to b e water-saturated. As regards fine-grained soils, the tests are generally performed in an undrained condition with simultaneous observation of the pore pressure
induced due to loading.
92
Soil
investigations
b)
a)
Fig. 61. Active (a) and passive (b) triaxial tests. In the active test = { and Gr - 3 while in the passive
test - 3 and =
test, i.e. when the sample is in axial compression, the axial (major
principle) stress aa is increased either in steps (each step about 1/15 of the estimated
ultimate stress) or increased continuously. T h e rate of loading is chosen so as to obtain
complete pore pressure dissipation (drained test) or, if possible, to admit an equalisation
of the excess pore pressure induced due to loading (undrained test).
In the passive test, i. e. when the sample is in axial extension, the axial (minor principle)
stress is reduced in a corresponding way and the test otherwise e a r n e d out as described
above.
T h e test results are generally presented in a diagram showing the effective stress path
followed during the test. In the diagram shown in Fig. 62, the M o h r - C o u l o m b failure
criterion yields the relation:
c
= ^(
(73)
- 2\).
relation:
q = (p' + a)
in the active test, and:
q = (p' + a)
3 sin '
3 - sin '
3 sin '
3-fsin0
(74)
(75)
- 100 .
Fig. 62. Effective stress paths obtained in active and passive triaxial tests on organic sulphide silt. The
marks on the curves represent axial strains 0 , 0 . 2 , 0 . 5 , 1, 2 and 3%. The 'failure lines' correspond to an
internal angle of friction of '= 35 (a = 0).
sine'
q = (&r + ah
"",
(76)
1 - sin
in the active test, and:
q = fr + a)-
sine'
^
1 + sin 0
(77)
using
special shear apparatuses, of which the C a s a g r a n d e shear box (Fig. 63) is c o m m o n l y used
in the case of coarse-grained soils, and, for example, the G e o n o r or the SGI shear
apparatus (Fig. 46, p. 74) is used in the case of fine-grained soils. In the C a s a g r a n d e shear
box, shear failure takes place along a horizontal plane, while in the G e o n o r and the SGI
shear apparatuses, it takes place in simple shear of the w h o l e specimen. In both cases,
94
Soil
investigations
deformations at right angles to the direction of shear failure are prevented and consequently
w e have to deal with a plain strain condition which is often in better a g r e e m e n t with real
conditions in practice.
T h e testing principle is as follows: T h r e e specimens are selected and consolidated
under three different normal (vertical) stresses and then sheared to failure at a rate of
strain that has to b e adjusted with regard to the permeability of the soil (the rate of excess
pore pressure dissipation).
During the test, the principal stresses are undergoing a c h a n g e in both direction and
magnitude which m a k e s the interpretation of the test extremely difficult. T h u s , the i n t e r pretation of the results in respect of angle of internal friction will depend on the quotient
= \\
the angle a, the failure condition can b e expressed by the relation (Hansen, 1961):
sin0 ' s i n 2 a
a + <fv
where
1 + sin ' c o s 2
(78)
is the horizontal shear stress at failure and a'v is the normal vertical stress.
Expressed in terms of the quotient K, the friction angle is obtained from the relation:
sin0' =
a +
(79)
If in the simple shear test the occurrence of failure is not clear, failure can be assumed
to take place at a relative displacement between the piston plate and the b o t t o m plate of
about 5 % of the height of the specimen.
As mentioned above, the rate of shear deformation must be adjusted with regard to the
permeability of the soil, so in the case of clays it has to be kept very low. For e x a m p l e ,
for a high-plasticity clay specimen with a height of 20 m m and a diameter of 5 0 m m , the
Soil
95
investigations
'
Fig. 64. Determination of failure criterion on the basis of the Mhr stress circles.
rate of relative displacement between the bottom plate and the top piston plate should not
exceed
mm/min.
(ii) Determination
that the effective stresses cannot b e determined then the results can only b e used as a
m e a s u r e of the so-called total strength parameters c and . This is, for e x a m p l e , the case
when the soil is not fully water-saturated, because if the soil contains pore gas the
effective stresses will be affected by surface tension forces of u n k n o w n m a g n i t u d e in the
menisci of the p o r e water. T h e total strength parameters are of practical interest when
failure can be assumed to take place under undrained condition. The most common
laboratory m e t h o d s of determination are the triaxial test, the unconfined compression
test, and the fall-cone test. By using either of the unconsolidated undrained triaxial test,
the unconfined compression test or the fall-cone test, only the undrained shear strength
parameter cu (or Tj-U), valid for the prevailing consolidation pressure of the specimen, can
be determined. If, for s o m e reason, the undrained shear strength at various consolidation
pressures is needed, then a n u m b e r of specimens first have to be consolidated in the
triaxial test cells at different pressures and then loaded to failure u n d e r undrained
conditions in the triaxial apparatus.
In the active unconsolidated
undrained
failure under undrained conditions with a cell pressure preferably equal to in situ
horizontal pressure. In the passive
undrained conditions with an axial pressure equal to in situ vertical pressure. Since the
effective stress path is u n k n o w n , the results are generally used as a basis for drawing the
M h r circles at failure (Fig. 64). T h e inclination of the failure envelope represents the
apparent angle of friction , and the intercept on the deviatoric stress axis represents the
apparent cohesion c. For water-saturated soil, w e find = 0 and in this case c is usually
denoted cu.
In the unconfined
compression
twice its diameter, is placed between a top and a b o t t o m plate and loaded axially to failure.
96
Soil
investigations
h =2d
///////////,
T h e test is merely a special case of the triaxial test with the radial pressure or equal to zero
and can evidently only b e used in the case of cohesive soils, preferably clay. It can either
be e a r n e d out with a constant rate of axial deformation or by continuously increasing
axial loading. T h e rate of axial strain ought to b e approximately 2.5%/min. In order to
prevent dessication of the specimen, it m a y be necessary to h a v e it enclosed by a substance such as paraffin oil.
T h e axial failure load is normally obtained from the m a x i m u m value of the load/
deformation curve. If there is no such m a x i m u m value, failure can b e assumed to have
taken place at an axial strain of 10%. T h e undrained shear strength, as given by the M h r
circle, is cu = 12.
T h e most simple and also the quickest testing m e t h o d is the fall-cone
Soil
97
investigations
Fig. 66. The Swedish fall-cone test. A 100 g/30 cone placed in position to be dropped.
contact with the p l a n e surface of the specimen (Fig. 66). T h e c o n e is then dropped freely
into the clay and the depth of penetration measured.
T h e undrained shear strength of clay can be obtained by the following relation
(Hansbo, 1957 and 1962):
0.08 0.1
0.2
0.3 0.4 0.5
Parameter
Km g
lL
1.0
Fig. 67. Relation between the parameter and the apex angle of the fall-cone.
(80)
98
So/7
investigations
w h e r e m - m a s s of the cone,
g = acceleration of gravity,
i - depth of penetration (indentation),
h - distance between cone apex and soil surface when the cone is dropped (normally h - 0)
= function of the apex angle of the cone.
If calibrated against the field vane test, for samples taken by m e a n s of the Swedish
standard piston sampler can be taken from Fig. 67.
T h e fall-cone test is frequently used as a standard procedure in the determination of the
sensitivity of cohesive soils. T h e results are generally in good agreement with the results
obtained by the field vane test.
7.2
Field
investigations
As mentioned in the introductory remarks to this paragraph, in situ testing has many
advantages over laboratory testing and has b e c o m e increasingly popular in applied
geotechnical engineering. In situ testing can either be used for determination of the
strength parameters to be used as input data in bearing capacity formulae, as was the case
in laboratory testing, or for direct determination of bearing capacity. Tests which allow
the latter type of interpretation are usually preferable in practical applications. A m o n g the
most c o m m o n in situ m e t h o d s w e find the field vane test, used to determine the undrained
shear strength of cohesive soils, the pressuremeter
both for determining strength parameters and the bearing capacity of foundations, and the
Fig. 68. Field vane test apparatus, type Nilcon (left) and type SGI with protective housing for vane
during installation. Standard vane, 130 mm in height and 65 mm in width.
Soil in vestigations
dilatometer toi which can be used both for soil classification purposes and for determining
strength parameters.
(i) Determination
of effective
strength parameters.
parameters (normally the angle of internal friction) in situ is usually carried out by m e a n s
of the pressuremeter or the dilatometer. However, there is seldom a need for translating
the results of these tests into effective strength parameters. In foundation engineering this
is an unnecessarily r o u n d a b o u t method. T h e results can b e used directly for calculation
of the bearing capacity of foundations (see p p . 1 3 5 - 1 3 9 and 1 9 0 - 1 9 2 ).
(ii) Determination
of undrained
strength
parameters.
generally determined by m e a n s of the field vane test (Cadling and Odenstad, 1950).
In the field vane test, a fourbladed vane attached to a sounding rod (Fig. 68) is pushed
into the soil to the depth of strength determination. T h e assembly is then rotated until
failure is reached. T h e relation between torque and angular rotation of the vane is
generally registered on a diagram. Investigations h a v e shown that the soil fails along a
cylindrical surface with the same diameter as the width of the v a n e and along plane
horizontal circular surfaces at the top and b o t t o m of the vane.
T h e standard vane is 130 m m in height and 65 m m in width. Vanes with heights of 80
m m and 170 m m and with width of 40 m m and 80 m m , respectively, are used in cases
where the standard v a n e is not applicable. According to the standard procedure, the vane
is left in the soil o n e minute before rotating it to failure, which should occur a minute later.
T h e undrained shear strength, according to the field v a n e test, can b e calculated from the
observed m a x i m u m torque, M m a x, according to the relation:
2M
u
nd h(\+d/3h)
(82)
If the shear strength is different in the vertical (= cuv) and in the horizontal directions
(= cuh ), w e have:
j
Mmai
(iii) Correction
= -nd h(cuv
+cuhd/3h)
(83)
the shear strength mobilised in clay in an undrained condition is dependent on the strain
rate the shorter the time to failure, the higher the mobilised strength (Fig. 69). This is
( 8 1
100
Soil
10
20
30
50
investigations
70
particularly the case with clays of high plasticity. Moreover, the strength m a y vary with
the direction of shear deformation because of anisotropic properties of the clay. T h e
undrained shear strength will therefore have to be corrected with regard both to time and
to anisotropy. For soils of extremely high plasticity, the undrained shear strength will only
have to b e corrected with regard to time (cf. Eq. 84). T h e lower limit of the shear strength
Plasticity index Ip (%)
0
20
40
40
60
80
120
Liquid limit wL
80
100
160
200
(%)
Fig. 70. Correction of undrained shear strength with regard to liquid limit (Andrasson, 1974)
continuous lineand plasticity index (Bjerrum, 1973)dashed line.
Soil
101
investigations
prevalent in long-term loading is defined as the creep strength ccr (cf. p. 34). T h e correction
factor ^ i s generally related to the plasticity index or, alternatively, to the liquid limit (Fig.
70). T h e value presented as a function of the plasticity index by Bjerrum was based
on results of test e m b a n k m e n t s loaded to failure and on case records. T h e liquid limit has
served as a basis for correction in S w e d e n since the 1950s.
T h e undrained shear strength determined by the field vane test can be corrected with
regard to both time and anisotropy according to the relation (Larsson, 1977):
c
c o s 0 + ( 0 . 1 7 + 0.7vv L) s i n 0
:
^,corr =
0.45wL
(84)
w h e r e 0 i s the inclination of the main principal stress to the vertical [ 0 i s 0in the active
test, 90 in the passive test and 60 in the direct shear test (' assumed equal to 30)].
7.3 Empirical
correlations
(i) Correlations
with preconsolidation
with reference
to strength
properties
pressure.
the undrained shear strength of clay and the preconsolidation pressure have been
suggested. For instance, in a case w h e r e the preconsolidation pressure of the clay is
known, the undrained shear strength can be estimated on the basis of Eqs. ( 6 6 - 6 7 ) .
(ii) Correlations
with penetration
resistance.
shear strength parameters by laboratory testing will yield unreliable results, partly due
to sample disturbance (breakdown of original structure) and partly to the effect of
changes in stress history in comparison with in situ conditions. Field investigations adapted
for direct m e a s u r e m e n t s of the shear strength m a y be considered too expensive or, for
s o m e reason or other, be impossible to use. In such cases, local experience can be used
ol
o.i
I2
I3
I4
102
So/7
investigations
to establish fairly reliable correlations between the results of penetration tests and the
strength parameters.
A m o n g the various penetration methods, the cone penetration test with a friction
sleeve and pore pressure transducers mounted at the tip (the C P T U penetration test)
seems to b e quite a useful tool for determination of the effective strength parameters
(Sandven et , 1988; Senneset et al, 1989). A pore pressure parameter Bq is introduced,
determined from the measured excess pore water pressure AuT and net c o n e resistance
qn (= qT - ' 0) according to the relation:
(85)
Bq=AuT/qn
and a cone resistance parameter Nm, determined from the effective overburden pressure
' ^ and the net cone resistance according to the relation:
Nm = qnK^v0
+ a)
(86)
w h e r e a = attraction.
A s s u m i n g that the failure zone beneath the cone tip is in a g r e e m e n t with the Prandtl
solution for a shallow footing, the cone resistance parameter can b e expressed by the
relation (Sandven et ai, 1988):
Nm = (Nq-l)/(l+NuBq)
(87)
w h e r e Nq = tan (45+072)exp(7Ctan0O,
N M 6 t a n 0 ' ( l + tan0OIntroducing the values of Bq and Nm into Eq. (87), the angle of internal friction ' can
be determined (Fig. 72).
If the soil is h o m o g e n e o u s , the attraction a can be obtained directly from the penetration
resistance curve b y extrapolating it to zero overburden pressure. For i n h o m o g e n e o u s
soils, the following values of a (in kPa) m a y b e applied (Sandven et ai, 1988):
5 - 1 0 for soft clay, 1 0 - 2 0 for m e d i u m clay and 2 0 - 5 0 for stiff clay,
0 - 5 for soft silt, 5 - 1 5 for m e d i u m silt and 1 5 - 3 0 for stiff silt,
0 for loose sand, 1 0 - 2 0 for m e d i u m sand and 2 0 - 5 0 for dense sand.
8.
PRESENTATION
Soil
103
investigations
10
20
50
100 150
classification, shear strength values, consistency limits, natural water content, etc.) as
exemplified in Fig. 7 3 .
Presentation of laboratory test results should include all information needed for the
appreciation of the testing routines applied and of the deformation and strength parameters
stated in the report. It should always b e possible to check the results presented. This is
a vital part of a quality assurance system.
An interpretation of the results of the geotechnical investigation should b e presented
in a special geotechnical report aimed at forming a basis for solving the p r o b l e m s which
arise in the foundation project in question.
In order to avoid misinterpretations of the information given in the report, the final
solution to the foundation problems should b e w o r k e d out in close cooperation with the
structural engineers involved in the project as well as other representatives of the client.
T h e situation previously experienced, w h e r e the geotechnical engineer, after having
delivered his report, had no connection at all with the structural engineer or the contractor, or the client for that matter, until something w e n t w r o n g with preparatory foundation works or d a m a g e caused by excessive settlement took place, has hopefully c o m e
to an end.
104
So/7
investigations
Fig. 73. Example of a borehole section and corresponding inteipretation of subsoil stratification. By
courtesy of J&W. Abbreviations for names of soil according to Karlsson and Hansbo (1981).
Spread
105
foundation
SPREAD FOUNDATIONS
1.
INTRODUCTION
Spread foundations comprise footings placed at shallow depth for the support of
individual structural columns and walls. A footing that supports a single c o l u m n is called
an individual footing whilst one that supports a group of columns is a c o m b i n e d footing
and one that supports a wall or a row of columns is a strip footing. In soft soil conditions,
and also in s o m e other special cases w h e r e there is a need to reduce the m a x i m u m
foundation pressure, it is advantageous to replace foundation on footings by a c o m m o n
raft designed to carry the total load of the building. In both types of foundation, the load
of the building will h a v e to be carried by direct contact stresses at the footing/soil
interface which places strong requirements on the geotechnical characteristics and
behaviour of the subsoil.
1.1 Depth of
(i) Regional
foundation
requirements.
and deformation characteristics of the soil but also on the climatological conditions and
the response to these conditions exerted by the soil. In s o m e places w e encounter soils
that can give rise to serious foundation difficulties, for e x a m p l e soils susceptible to frost
action, soils disposed to swelling, collapsible soils, chemically unstable soils, etc.
In places with frost-susceptible soils, the footings have to be placed below the depth
of frost penetration, or in regions with permafrost below the depth of thawing.
Different m e t h o d s of reducing frost penetration and the depth of thawing can be used,
mostly by insulation or by feeding heat or cold into the soil.
In places with swelling soils, such as smectite, the footings must be located at a depth
that is not influenced by drying. S o m e t i m e s it is necessary to anchor the foundation in
deeper layers to prevent the building from being d a m a g e d when water is sucked up by
the soil during rain seasons. Efficient drainage around the building and protection agai nst
water infiltration is needed.
Residual soils, such as latrites, are often covered by a collapsible top layer. This layer
is ordinarily firm and stable. However, when it b e c o m e s saturated, the soil skeleton
undergoes a complete collapse. Therefore, the footings h a v e to b e placed b e l o w the collapsible top layer or the top layer has to be compacted.
Buildings with shallow foundations on clay frequently suffer d a m a g e by differential
Spread
106
foundation
settlements caused by water-suction from nearby trees, such as poplars, oaks, birches,
elms and linden trees. In order to avoid d a m a g e , trees should not be planted closer to a
building than its expected full-grown height.
Building activities often affect the infiltration of rain water and m a y cause a lowering
of the groundwater table. In consequence, settlements can be induced both by dry crust
formation and by long-term increase of effective vertical stresses throughout the soil
profile. T h e dry crust formation is mainly governed by vegetation and other sources of
evaporation. Withering of vegetation starts at a water binding pressure of around 1.5
MPa. This entails that the shrinkage limit is decisive for the m a x i m u m v o l u m e decrease
entailed by dry crust formation.
(ii) Depth of foundation
requirements.
T h e choice between
shallow and deep foundations of a structure is last of all depending upon the depth to
bearing strata and/or the preloading history of the subsoil. In a n u m b e r of cases, a deep
foundation is c h o s e n a l t h o u g h not n e e d e d b e c a u s e of the designer's belief that,
otherwise, settlement would exceed the permissible value. A m o r e detailed or advanced
soil investigation than the one at hand m a y often result in a m o r e cost-effective solution
with shallow foundation, or with a combination of shallow and deep foundations.
K n o w l e d g e of the deformation characteristics and of the loading history of the subsoil
is of p a r a m o u n t importance in the design.
Lack of suitable ground for building purposes, especially in densely populated areas,
makes it necessary to optimise the design of the building by increasing the n u m b e r of
basement floors. In consequence, the weight of the soil that has to be excavated for the
basement often exceeds the weight of the building itself. We then h a v e to deal with a socalled compensated
foundation.
Spread
107
foundation
due to the uplift caused by the combined effect of effective pressure acting at the soil/raft
interface and pore water pressure. Obviously, in this case a raft foundation is always
possible, although in s o m e situations, for instance under heavily loaded c o l u m n s , the raft
may h a v e to b e supported by piles.
2.
2.1
Foundation
on
footings
T h e distribution of contact pressure at the footing/soil interface
depends on the mechanical characteristics of the soil and the bending rigidity of the
footing. Regarding individual and continuous footings, the bending rigidity is generally
very large in comparison with the deformability of the soil. T h e analysis of the contact
pressure distribution can therefore be m a d e on the assumption that the footing is infinitely
rigid. However, confined rigidity may h a v e to be given certain considerations in the case
of c o m b i n e d footings.
A characteristic of an infinitely rigid footing is that there is equal vertical deformation
at the soil/footing interface. A s s u m i n g the soil to b e h a v e as an elastic m e d i u m , the contact
pressure distribution beneath a strip footing is governed by the relation (Fig. 75):
(88)
0.5b
P/b
Fig. 75. Contact pressure distribution beneath a rigid strip footing on an ideally elastic medium
Spread
108
foundation
Fig. 76. Observed contact pressure distribution at various loads in stiff clay. Rigid footing, 0.3 m in
diameter, placed on ground surface. (Faber, 1933)
Spread
109
foundation
Fig.77. Observed contact pressure distribution at various loads in sand. Rigid footing, 0.3 m in diameter,
2
placed on ground surface without and with a surrounding surcharge of 7 k N / m . (Faber, 1933).
pressure along the e d g e of the footing also close to zero. T h e pressure distribution will
follow the trend s h o w n in Fig. 77.
Usually, b e c a u s e of the footings being placed at a certain foundation depth, whether
this is related to the ground surface or to the b a s e m e n t floor, a certain overburden pressure
is always acting at the foundation level outside the footings. This allows for a higher edge
pressure also for footings on cohesionless soil (Fig. 77, right).
T h e contact pressure distribution for footings on silt is in b e t w e e n the distributions
given for clay and sand (Fig. 78).
B e c a u s e of the various factors that influence the contact pressure distribution, not least
the unforeseen variations in soil properties, effect of capillary forces, cementation, etc.,
it is extremely difficult to predict the real contact pressure distribution induced b y the
load. In practice, however, the structural design of a footing can generally b e e a r n e d out
on the assumption of evenly distributed contact pressure.
(ii) Pressure distribution
of the pressure caused by the load applied on the footing is usually calculated on the basis
of the theory of elasticity. Practical experience has s h o w n that this can be done without
serious errors. Solutions also exist w h i c h are based on statistics w h e r e the soil is
considered as a particulate m e d i u m (see p. 115).
E v e n though the contact pressure distribution under a rigid footing is subjected to large
variations, the difference in stress condition u n d e r n e a t h a rigid footing on the one hand
and a flexible footing on the other is m o r e or less negligible at depths e x c e e d i n g the width
of the footing (Table 20). Therefore, the stress increase occurring at greater depths can
be calculated on the assumption that the contact pressure is equivalent to an evenly
distributed flexible load. O n this basis, the analysis of the stress increase is generally
performed according to Steinbrenner (1936) or N e w m a r k (1942).
110
Spread
foundation
Fig. 78. Observed contact pressure distribution at various foundation loads in silt. Rigid footings, 0.31
m in diameter, placed on ground surface and at different depths df. (Helenelund, 1965).
The principal stresses induced by a flexible strip load can be found by the relation (Fig. 79):
q
= - ( y / + s i n )
(90)
T A B L E 20
Stress distribution at depth underneath rigid and flexible strip footings with width b according to the
theory of elasticity (Szechy, 1965).
zJb
xlb = 0
rigid
flexible
xlb =0.5
rigid
flexible
xlb = 1
rigid
flexible
xlb = 2
rigid
flexible
0
0.25
0.5
1.0
1.5
2.0
3.0
0.637
0.683
0.676
0.513
0.383
0.300
0.206
oo
0.710
0.535
0.407
0.329
0.271
0.196
0
0.104
0.186
0.215
0.209
0.170
0
0.006
0.031
0.061
0.085
0.103
1.000
0.960
0.818
0.550
0.396
0.306
0.208
0.500
0.493
0.480
0.409
0.334
0.275
0.198
0
0.084
0.185
0.211
0.205
0.170
0
0.005
0.029
0.059
0.083
0.103
Spread
111
foundation
Fig. 79. Geometric determination of the principal stress at a given point below a flexible strip load.
3 = ~(-
sin )
2q
2 -
(91)
(92)
= ( - 3 ) / 2 = - sin
(93)
an arbitrary point underneath, or outside, a rectangular load (or a load that can be divided
into several rectangular part areas). This method is based on the following reasoning.
Assuming a rectangular load of length 21 and width 2b, this can b e divided up into four
congruent part rectangles / b. T h e stress increase underneath the centre of the load can
now, according to the principle of superposition, b e calculated as the sum of the stress
increase obtained below the corner of each o n e of the four part rectangles (in the centre
of the load). T h e stress increase under a corner of a rectangular load of length / and width
b is thus equal to one fourth of the stress increase under the centre of a rectangular load
of length 21 and width 2b.
Accordingly, w e find the following expression for the stress increase at depth below
the corner of a rectangular load with length / and width b\
9
q
mn(\+m
+2n )
= [
2
2
2
2 (i + n )(l + n ) / l + w
m
+arcsin-7=
1
2
2
2
+ n
/(m + n )(l + )
J
/ rx w
(94)
w h e r e m = lib,
- zlb.
N o w , considering a loaded area that can be divided up into a n u m b e r of part rectangles,
Eq. (94) m a k e s it possible to calculate the vertical stress induced b y the load for any point
and for any depth.
T h e pressure distribution under a load area of arbitrary shape can be obtained by the
Spread
foundation
113
Fig. 81. Newmark's influence diagram for determination of vertical stresses under a load area of
arbitrary shape.
114
Spread
foundation
Fig. 82. Newmark's influence diagram for determination of horizontal stresses under a load area of
arbitrary shape.
Example 5: Calculate the vertical stress at 2 m depth below points A, and C due to an evenly distributed
load shown on top of p. 115.
Solution: The stress below point A is the sum of the stresses below corner A of the rectangles GHIA (m
=1.5; = 0.5), IJDA (m = 1.0; =0.5) and DEFA (m = 1.0; =0.5) whence:
azA/q = 0.238 + 2-0.232 = 0.702
The stress below point is the sum of the stresses below corner of the rectangles RGLB (m = 4.0;
= 1.0) plus LHVB (m = 4.0; = 1.0) plus QFKB (m = 3.0; = 1.0) plus PEQB (m = 3.0; = 1.0) plus
VJPB (m = 1.0; = 1.0) minus RAKB (m = 1.0; = 1.0), the effect of which would otherwise be included
twice. Hence:
/ = 0.204-2 + 0.203-2 + 0.175 - 0.175 = 0.814
Spread
" ^~
, HI
115
foundation
'
J1
////////
Q
6m
2 m
U - -^
4m
The stress below point C is the sum of the stresses below corner C of the rectangles SGMC (m = 2.2;
= 0.4) plus TFUC (m = 1.8; = 0.4) minus SAUC (m = 1.0; = 0.4) minus TE NC (m = 9.0; - 2.0)
minus OHMC (m = 11 ; - 2.0) plus OJNC (in - 1.0; = 2.0) which would otherwise be deducted twice.
ozClq = 0.244 + 0.244 - 0.240 - 0.137 - 0.137+ 0.084 = 0.058
The stress below point is the sum of the stresses below corner of the rectangles RGLB (in = 4.0;
= 1.0) plus LHVB (m = 4.0; = 1.0) plus QFKB (m = 3.0; = 1.0) plus PEQB (in = 3.0; = 1.0) plus
VJPB (m = 1.0; = 1.0) minus RAKB (m = 1.0;= 1.0), the effect of which would otherwise be included
twice.
azB/q = 0.204-2 + 0.203-2 + 0.175 - 0.175 = 0.814
The stress below point C is the sum of the stresses below corner C of the rectangles SGMC (m = 2.2;
= 0.4) plus TFUC (m = 1.8; = 0.4) minus SAUC (m = 1.0; = 0.4) minus TENC (m = 9.0; = 2.0)
minus OHMC (m = 11 ; = 2.0) plus OJNC (m = 1.0; = 2.0) which would otherwise be deducted twice.
ozClq = 0.244 + 0.244 - 0.240 - 0.137 - 0.137+ 0.084 = 0.058
(iii) Probabilistic
approach.
ered as a particulate m e d i u m . In such a soil, is zero at any given point located in a void.
T h e real value of in relation to the expected value E(az) can b e calculated b y the
probabitity relation:
p[
n(-lnn)^
()
/ E ( c
[/()]\
(95)
(96)
w h e r e () represents the area below the normal distribution curve from 0 to . They/(x)
value can b e found in standard mathematical tables.
T h e expected vertical stress at any point below a load area of arbitrary shape that can
be divided up into a n u m b e r of rectangular areas can n o w b e calculated by superposition
according to Steinbrenner's m e t h o d as described above.
Spread
116
foundation
Example 6: Calculate for the load according to Example 5 the vertical stress to be expected at 2 m depth
below points A and C in a particulate medium.
Solution: The stress below point A is the sum of the stresses below corner A of the rectangles GHIA (//
= 3; biz =2) plus IJDA (l/z = 2; biz =2) plus DEFA (llz = 2; biz =2) whence:
2
(97)
(1 + zll)(\
+zlb)
(l+z/b)
(98)
b +z
Fig. 83. Stress pyramid used for approximate calculation of vertical and horizontal stress increase under
load area.
Spread
111
foundation
r
h
iJb
/A
y)
M/
ff
II
II
II
Fig. 84. Vertical stress distribution with depth below the centre of square (left) and strip loads. Full line:
2:1 method. Dash line: Rigid footing, theory of elasticity. Dash-dot line: Weak load, theory of elasticity.
A comparison of the stress increase below the centre of a square and a strip footing
according to the empirical method to that obtained according to the theory of elasticity
is m a d e in Figs. 8 4 - 8 5 .
T h e empirical m e t h o d of stress analysis gives results in close a g r e e m e n t with the stress
increase b e l o w the critical point (see p. 142) of a footing according to the theory of elasticity, Fig. 86.
2.2
Raft
(i)
Contact pressure.
foundation
In the case of raft foundations, the limited bending rigidity of the
3
Fig. 85. Horizontal stress distribution with depth below the centre of square (left) and strip load areas
according to approximate and strict calculation methods.
Spread
118
foundation
Fig. 86. Vertical stress distribution with depth according to theory of elasticity (full lines)below the
characteristic point of a square (left) and a strip footing. Broken lines represents vertical stress
distribution according to the empirical 2:1 method.
m o m e n t s and forces in the superstructure (as is the case with the influence of differential
settlements of individual footings).
M a n y attempts have been m a d e to solve, in a m o r e or less exact way, the pressure
distribution at the soil/raft interface. T h e problem is quite difficult for several reasons:
the pressure distribution is a function not only of the bending rigidity of the raft but
also of the rigidity of the superstructure,
the soil m a y be heterogeneous with varying deformation properties below different
parts of the raft,
the rigidity of the structure is subjected to gradual changes during the time of
construction,
stress changes in the subsoil during the time of construction has a strong influence
on the resulting, final stress distribution.
In respect of all the difficulties involved, we are reduced to finding approximate
solutions to the problem. T h e oldest approach is based on the use of a subgrade coeffi cient
while the m o d e r n approach is based on the utilisation of finite element m e t h o d s and
computer-based solutions.
(i)
The subgrade
coefficient
that the soil can be replaced by a bed of vertical, elastic springs without horizontal
coupling. In a physical sense, this assumption means that the load does not entail shear
stresses in the soil and that the contact stresses are directly proportional to the vertical
settlements. The contact stress p a t a given point is thus given by the relation:
Spread
119
foundation
(99)
P = KS
as an elastic medium.
are obtained by the aid of the finite element m e t h o d and computer-aided calculations. T h e
soil at the foundation level is often divided into finite surface elements which are coupled
to the connecting finite raft elements and to the superstructure. Every raft element is acted
upon by a uniform raft load and the point load exerted by the superstructure. T h e s e loads
are uniformly distributed over the element. T h e elements are first di sconnected from each
other and a settlement analysis is performed. T h e settlement of the element is obtained
by s u m m a t i o n of the vertical deformation of the different layers in the subsoil with due
consideration to the settlement contribution obtained from nearby elements. In the
following step of the analysis, enforced forces and m o m e n t s are applied to the elements
in order to maintain the continuity of the raft in the nodes between the elements. T h e
additional settlements thus obtained are considered in the next step of the analysis, and
so on in an iterative process.
T h e superstructure and the raft and the subsoil can also b e treated as a continuous
system divided into substructures that are analysed b y themselves and then connected
with each other to fulfil the the continuity requirements. T h e soil is then divided into a
n u m b e r of rectangular, trapezoidal or triangular elements in a two-dimensional analysis,
or in prismatic or tetrahedral elements in a three-dimensional analysis. Now, if the stress/
deformation characteristics can b e correctly described, a correct solution to the problem
is theoretically possible. In practice, however, these calculations require a very extensive
computer m e m o r y which is not usually available. In consideration of the difficulties
previously m e n t i o n e d (time effects, plastic yield, etc.), this type of analysis is as yet only
of interest in p a r a m e t e r studies and research.
120
Spread
foundation
Fig. 87. Contact pressure distribution caused by point and line loads.
(iii) Approximate
evaluation.
solved by approximate m e t h o d s . Beigler (1976) s h o w e d that the contact pressure distribution under a point and a line load can b e approximated to a cone and a w e d g e ,
respectively, with a base width ry given by the relation (Fig. 87):
(100)
r=\3Kr
w h e r e Kr =
(,/5),
Po =
7ir
:z
(101)
=P/r
Pl
For a Une load acting at distance a from the e d g e of the raft that is below the value of
r (r < bl2), the contact pressure along the edge, according to Beigler (1976), can be
obtained from the relation (Fig. 88a):
r-a
(103)
Pe=P\
a
If the value pe thus obtained exceeds the plastic yield pressure of the soil pcn
m a y occur:
P/pcr
< 2a
two outcomes
Spread
121
foundation
Fig. 88. Contact pressure distribution caused by a line load near the edge of a raft, (a) Edge pressurep e
< pcr (b) Edge pressure pe = pcr and < 2apcr (c) Edge pressure pe = pcr and > 2apcr.
P-a{pcr
x0 =
Pcr
a + 0.5r> Plpcr>
(104)
"Pi
2a
(a + r-x0)
2pcrr
Pi
2Pr
JC0 =
(105)
Pi
c
4
<r * .-a
' **'.<>. '
- L i :
>-V
-V,
11
Px
II
r
i
{
III
Pm
P,
i l l . .
b/2 - r
4^
Fig. 89. Contact pressure distribution under a raft acted upon by evenly distributed load q.
122
Spread
foundation
Px^Pm+iPe-PmK-^)"
r-x
where =
pm
2.SpCfJq,
q(\-mx)(\+m2),
- ra2(l - mx)]/r
<pcr,
mx = 0 . 3 6 3 / [ l - 0 . 0 5 f e / r + 0 . 4 ( f e / r ) 3 ] ,
m 2 = 0.826qmx/pcr
if x0>
0,
m2 = 0 if x0 < 0,
XQ
b{n + \)(q-pm)
r
- - > 0 .
2n(Pcr-Pm)
(If r > fe/2 then put r = b/2, and if x 0 < 0 then put JC 0 = 0 ) .
T h e e d g e p r e s s u r e p e (at = r - x0) is limited by the plastic yield pressure
pcr.
1 0 0 - 0 . 5 ( 1 6 6 + 16.7)
= 0.06 m
166-16.7
Influence of central line load:
We find px = 200/6 = 33.3 kPa
2
Influence of uniform load consisting of surface load 10 k N / m and self-weight of the raft 0.5-24 = 12
2
2
kN/m , i.e. in total 22 kN/m :
We find:
77 = 2.8-166.5/22 = 8.4
Spread
123
foundation
3
mx = 0.363/[l - 0 . 0 5 - 1 5 / 6 + 0 . 4 ( 1 5 / 6 ) ] = 0.051
m 2 = 0.826-22-0.051/166 = 0.0056 (see below)
pm = 22(1 - 0.051)-(1 + 0.0056) = 21.0
15-45-(22-21)
- < 0
2-44.(166-21)
8.4
which yields ra2 = 0.
Thus pm = 22(1 - 0.051) = 20.9 kPa
pe = 20.9 + 22-9.4-0.051-15/12 = 34.1 kPa
Considering the effect of the line load near the edge (which caused yield) the uniform load will extend
the width of the zone of yield to a distance from the edge of JC0 = 0.06 + (0.5 - 0.06)-34.1/(166 - 16.7)
= 0.16 m
Superposition yields:
Contact pressure at raft center:
pm = 20.9 + 33.3 - 54 kPa
57
Contact pressure 0.5 m from e d g e p x ( 55 m) = 16.7 + 20.9 + (34.1 - 20.9)-[5.5/(6 - 0 . 1 6 ) ] - 47 kPa
57
Contact pressure 0.3 m from e d g e p x (5 7 m) = 20.9 + (34.1 - 20.9)[5.7/(6 - 0 . 1 6 ) ] + (0.24/0.44)(166
-16.7) + 1 6 . 7 - 1 3 1 kPa
Contact pressure 0.16 m from edge p^o.iom) = Per=166
kPa
The contact pressure distribution across the raft is shown below.
XQ =
100 KN/M
0.06
200KN/M
0.16 M
-JFTO-i
m, = 7.85 G P a and = 0.2. T h e load on the floor is 2.5 k N / m and the load from the
walls 5 kN/m.
124
Spread
foundation
Fig.90. Observed contact stress distribution in two cross-sections of a concrete raft foundation, 400 mm
in thickness. Column spacing lengthwise of the buildings 7.2 m. In shelter section, the column load 490
kN (to the left) is replaced by a wall load of 52 kN/m. The internal wall load of the shelter = 42 kN/m.
Building completed 2/7 -74 and in use 17/7 -75.
Spread
125
foundation
3.
BEARING CAPACITY
3.1
Individual
footings
on failure
surface.
dimensional case and that the force vector p, acting on a unit length of the failure surface
at a certain point, is k n o w n as well as the inclination of the failure surface at the point
in question (Fig. 92). T h e position of the point considered is determined by the arc length
s, set out from a fixed point. T h e parameters p, v, s and $ ' a r e positive w h e n they have
the directions shown in Fig. 92. Consider an element A B C D in the failure surface. Draw
the lines D C and B C parallel to A B and A D . D u e to the rotation of the sides B C and D C
the force vectors acting against D C ' and B C will change in size as shown in Fig. 92.
Now, from the equilibrium conditions for soil element A B C D an expression for the
stress vector can b e derived, the so-called Kotier 's equations.
(107)
-e 2 v t a n 0
\_^^' + > 1
(
)( 1 0 8
(109)
Fig. 92. Failure surface, affected by the effective stress vectorp, inclined </>'to the normal of the failure
surface (left) and soil element ABCD in the failure surface. AB C D represents a parallelogram.
Spread
126
foundation
p = jp oe x p [ 2 ( v o- v ) t a n 0 ]
(110)
angle of 9 0 '. T h e intersection angle b e t w e e n the failure surfaces and the major
principal stress is 4 5 - '/2 and between the failure surfaces and the minor principal stress
45+ 072.
(ii) The bearing capacity factors. Consider a strip footing of width b resting on the ground
surface and surrounded by an evenly distributed load q0 (without internal friction). T h e
strip footing is carrying a centrical vertical load Q per length unit. T h e effective unit weight
of the soil is / .
Now, if Q is increased up to soil failure and the soil is weightless, it is obvious that the
effective stress equals the contact pressure. T h e unit weight of the soil has no influence
on the cohesion intercept c\ only on the frictional resistance. Considering only the
influence of the weight of the soil beneath the footing, the effective stresses increase from
a zero value along the edges of the footing to a m a x i m u m value below the centre of the
footing (Fig. 93). In consequence of this reasoning, w e can express the bearing capacity
of the footing as the s u m of three bearing capacity factors:
q0, Nc related to the cohesion intercept ( / a n d N r r e l a t e d to the unit weight / o f the soil
beneath the footing. On the basis of the stress distribution at failure, shown in Fig. 9 3 , w e
have:
qf= Qj/b = 0.5bYNY+
qo q
+ c7V c
(HI)
Nq+c Nc
assumed
Spread
127
foundation
Fig. 94. Rankine and Prandtl failure zones developed below the frictionless base of a strip footing.
P l
qf= 4 0 t a n ( 4 5 + 072)exp(7itan0O + c c o t 0 [ t a n ( 4 5 +
072)exp(7rtan0O -
1]
This yields:
2
Nq = t a n ( 4 5 + 072)exp(7rtan0O
Nc = (Nq-l)cot<l>'
(112)
(113)
In the special case of ' = 0 (i.e. w h e n failure is governed by the undrained shear
strength cu) w e find ^ 0 = 1 and
= 2+.
T h e s e solutions for Nq and Nc, originally presented b y Prandtl and Reissner (Fig. 95),
are most c o m m o n l y applied in the analysis of the bearing capacity of strip footings.
While there is a general acceptance regarding Nq and Nc, there is quite a disagreement
128
Spread
about the evaluation of Ny. According to the German code DIN 4017, Ny
foundation
is expressed
NY=2{Nq-\)\smQ'
which can b e approximated to ('
degrees):
# y= 0 . 0 8 e x p ( 0 . 1 8 f )
(115)
(116)
T h e bearing capacity factors given above are only relevant for strip footings placed
either on the ground surface or at a depth w h e r e the material above the foundation level
has neither friction nor cohesion. For individual pad footings, for footings buried in the
soil and for certain special loading conditions, the bearing capacity factors will have to
10001
500
10
20
30
40
SO-
Fig. 95. The bearing capacity factors Nc and vs. angle of internal friction.
Spread
129
foundation
be corrected. This is done by multiplying the bearing capacity factors with coefficients
w h o s e m a g n i t u d e depend on geometry of the footings, depth of foundation, loading
conditions, and so forth.
(iii) Shape and depth coefficients.
of determining the bearing capacity of individual pad footings placed at a certain depth
below the ground surface or the b a s e m e n t floor. Failure will then take place in the
direction towards the lowest adjoining ground surface or floor. Consequently, the depth
coefficients dq, dc and dy should b e based on the foundation depth
regard to failure. Both the depth coefficients and the shape coefficients sq, sc and sy
r e c o m m e n d e d in the literature are determined on an empirical basis.
T h e shape and depth coefficients r e c o m m e n d e d in the G e r m a n code D I N 4017 are:
1 + ysin0'
sq=
s = l-03y
Y
dq
(117)
(118)
(119)
(120)
dc = dY=l
(121)
scQ = l+02-
T h e variation of Nc0 with regard to shape and depth of the footing can b e taken from
Fig. 96.
Results of large-scale plate model tests on sand (Du Thinh, 1984) indicate a better
agreement with the shape and depth factors r e c o m m e n d e d by M e y e r h o f (1961; 1963)
than with those r e c o m m e n d e d in D I N 4017. According to M e y e r h o f w e have:
b
(122)
(123)
Spread
130
10
foundation
1
b/I=\
/3// = 0.5
8
a?
/?// = 0
t-l
dq = dy = 1 + 0.1 ^ tan(45 + 0 7 2 )
(124)
4 = 1 + 0 . 2 ^ tan(45 + 0 7 2 )
(125)
M e y e r h o f (1961) points out that the angle of internal friction to b e used in his
correction coefficients should be determined by direct shear tests (plain strain t e s t s ) ' I n
case the angle of friction is determined by triaxial tests, M e y e r h o f suggests that the value
thus obtained should b e replaced by:
^(U-O.lyWiri.
(126)
( M e y e r h o f ' s correction of the triaxial value is based on results of true triaxial testing
under plain strain condition. T h u s , the angle of friction d e t e r m i n e d under plain strain
condition has been shown to be around 1 0 % higher than the angle of friction determined
in the conventional active triaxial test).
M e y e r h o f ' s correction coefficients given in Eqs. (122) through (126) cannot be applied for values of '< 10.
T h e influence of shape and depth of foundation on the bearing capacity of a cohesive
soil in undrained condition can be obtained from the d i a g r a m s h o w n in Fig. 96.
In case the groundwater level is situated below the foundation level, / i s replaced by
/ a b o v e the groundwater level and / b e l o w the groundwater level, from the foundation
level d o w n to a depth below the foundation level equal to the width of the footing.
Spread
131
foundation
Example 8: Calculate the bearing capacity according to DIN 4017 on one hand and according to Meyerhof on the other for a centrically loaded square footing with width 2.5 m founded at a depth of 1 m below
the ground surface. The soil consists of sand with an internal friction angle ''= 35. The groundwater
3
level is 1 m below the foundation level. The density of the sand is 1.8 t/m above the groundwater level
3
and 1.1 t/m below the groundwater level. The influence of capillary forces and surface tension on
effective stresses can be ignored.
1.0 m
GW
2.5 m
1.0m
Solution: The bearing capacity factors according to DIN are Nq = 33 and Ny = 45 and according to
Meyerhof Nq = 33 and Ny= 37. According to DIN we have sq = 1.574 and sy= 0.7 and dq = dy= 1 while
2
according to Meyerhof sq=sy -1.369 and dq-dy-\
.077. Furthermore, q0 = 18 k N / m and / a v = ( 1 .() 18
3
+1.5-11)72.5 = 13.8 kN/m Thus, according to DIN:
2
Vf= ft2
q
= (0.5-2.5-13.8-45-0.7 + 18-33-1.574)-2.5 = (543 + 935)-2.5 = 9240 kN
and according to Meyerhof:
2
(iv) Eccentric
loading.
load resultant.
b e shallower than w h e n the load resultant is vertical. In consequence, this loading case
b e c o m e s m o r e critical the higher the ratio of the horizontal c o m p o n e n t H to the vertical
c o m p o n e n t V(Fig. 98). In a case w h e r e the horizontal c o m p o n e n t is
to the length direction
achngperpendicular
Fig. 97. Fictive footings to replace real footings under eccentric loading conditions.
Spread
132
OJH
^
" v
Wc'cot^
3
)
^ - T ^ i
i
r
'
foundation
( -TT-77^
V + blc cot
<
1 2 7
>
( 1 2 8 )
1 2 9
(130)
(131)
w h e r e = arctan(///V)
In the case of '= 0, Brinch H a n s e n (1967) p r o p o s e s the correction coefficient:
-W'-^
<132)
Example 9: Calculate according to DIN 4017 and Meyerhof the bearing capacity of the square footing
in Ex. 8 if the load resultant goes through the centre of the base of the footing and has an inclination of
3(vertical): 1 (horizontal).
Spread
133
foundation
..xf&\
GW
52m
1m 0
I 1.0 m
qfb JW/3
Example 10: A square footing under a vertical column load of 200 kN is subjected to a horizontal load
/ / k N and a rotational moment M = 2 / / k N m . The footing has a width of 2.5 m and is placed at 1 m depth.
The groundwater level is 0.5 m below the foundation level. Determine according to DIN 4017 the value
of//, leading to foundation failure at pj- 500 kPa, for a soil with an effective friction angle 0 ' = 32 and
3
3
a density above the groundwater level of = 1 . 8 t/m and below the groundwater level of p ' = 1.1 t/m .
IH *- iti
= 2 0 0 kN
3?
1.0m
0.5 m
GW
2.5 m
Solution: The eccentricity of the load is eb = 2///200 = 0 . 0 1 / / which gives a reduced footing width of
b' = 2.5 - 0 . 0 2 / / m. The correction factors become:
2.5-0.02//
sin*/)'
^ = 1+
2.5
2.5-0.02//
7= 1 - 0 . 3
2.5
0.7//
200
7
200
The bearing capacity factors for '- 32 become Nq - 23.2 and = 27.7 and the average unit weight
=
=
3
7av
Pav
12.2 k N / m . The relation becomes:
500 = 0.5-(2.5 - 0.02//) 1 2 . 2 - 2 7 . 7 - 5 ^ + 1 8 - 2 3 . 2 - ^
which yields / / = 37.5 kN.
to its length
(133)
134
Spread
foundation
(vi) Footings
next to slope. For a footing that is placed near a slope with inclination
to the horizontal, Brinch Hansen (1967) suggested that the bearing capacity factors in the
case of a cohesionless soil be corrected by the coefficients:
2
Si
(1 - s i n < / ) ' ) c o s
e
Zr =
,o
*>s
1 - sin sin(2u + )
- 0
2
^
- 2 + 20) tan 0 ']
(134)
w h e r e u represents the angle of inclination of the failure surface to the down-slope (Fig.
99) which by conditions of equilibrium (cf. p. 2 7 3 - 2 7 4 ) is obtained from the relation:
2u - arccos(sin/sin</0 + - '
(135)
;i36)
T h e approximate expression generally yields results on the safe side but when the
angle of inclination of the slope tends to the angle of internal friction the difference in
results can be up to 4 0 % on the unsafe side.
In the case of an unloaded slope in cohesive soil with
ScO =
2
2+
(137)
Example 11: Determine according to Brinch Hansen how much the bearing capacity of the square
footing in Ex. 8 will be reduced if the footing is placed near aslope with inclination 1 (vertical):4(horizontal).
Spread foundation
Solution: We have = arctan(0.25) = 14 and 2u = arccos(sinl4/sin35) + 1 4 - 35 = 44
Thus:
( l - s i n 3 5 < ) c o s 2 14o
=
6 q
76
l-sin35sin79
180
(vii) Inclined
failure surface change in accordance with the exponential function of 2 t a n $ ' times the
angular change v. Consequently, the correction factor bq is given by:
bq = e x p ( - 2 v t a n 0 O
(138)
The weight of the soil can be considered by the correction factor (Brinch Hansen, 1967):
by-
exp(-2.7vtan0O
(139)
By comparison with the earth pressure against a vertical, smooth retaining wall, Brinch
Hansen proposed that the correction factor in the case of inclined loading be approximated
to (notations given in Fig. 100):
=[1-(1-
300
blc'cot0'
(140)
bc0 = 1
(viii) The pressuremeter
2v
2+
(141)
Spread
136
foundation
+ (& S q u a re
,
0
+ /
(142)
kstIl^)b/l,
Pi~Po
n et
liroit pressure,
/>, = 6 MPA
2 MPA
1 MPA
0.5 MPA
-pt =
6 MPA
2 MPA
0 . 4 MPA
3
'Pi = 3 MPA
1 MPA
0 . 5 MPA
pt = 3 MPA
1 MPA
0 . 5 MPA
dfJb
Spread foundation
Un
Pie = ('Pn'Pi2-'PB-'PiJ
(143)
In this case the real depth of foundation d^is also replaced by an equivalent depth of
f o u n d a t i o n ^ determined by the relation:
dfe = ll(AzrPil/yie)
(144)
where ; ; = dj.
T h e pressuremeter method can also b e used to determine the so-called creep load, i.e.
the load that leads to excessive creep deformations. This is done by using the creep
pressure pcr in the bearing capacity formula instead of the limit pressure/?/ (for the definition
of the creep and limit pressures, see p . 81).
For eccentric
loading, a fictive footing with reduced length and width according to the
principles shown in Fig. 97 is used, i.e. with length / - 2e{ and width b - 2eh w h e r e e{ and
eb are the load eccentricities in the length and width directions, respectively.
For inclined
loading,
w h e r e 0 = arctan(///V),
Xd=
Xd =
\-d/bfoi'0<df/b<\,
0fovd/b>\,
Spread
138
For footings
foundation
near a slope, the reduction factor gs is obtained from Eq. (145) with re-
the slope, the reduction factor is is calculated by using Eq. (145) after replacing by + '.
If the inclined load is directed away from the slope, failure m a y take place either towards
the slope (replace 0 b y 0 + ') or away from the slope (keep 0). T h e t w o cases will have
to be considered separately.
For eccentric
inclined
loading,
inclined in the direction away from the centre of the footing or towards the centre of the
footing. In the former case, the footing is assumed to h a v e the width b + 2eb while, in the
latter, the footing is assumed to h a v e the width b -
2eb.
Solution: The bearing capacity of the footing under a vertical centric load is obtained from:
Pj~~ 1.4*1.2 1.68 MPa
We have:
= 1 - 1 / 2 = 0.5
= 1 - 1.0/1.4 = 0.286
= 0 . 5 0 . 2 8 6 = 0.143
The angle = arctan(0.2) = 11.3 and the angle '= arctan[2/(l+3-3)] = 11.3 whence + '= 22.6.
This yields:
2
is = (1 - 22.6/90) (l - 0.143) + (1 - 22.6/20)*0.143 = 0.46
Thus the reduced bearing capacity is Vj-= 0.46*1.68*2 = 1.5 MN/m
Example 13: Calculate the bearing capacity of a square footing with 2.5 m by 2.5 m base area, placed
at 1.4 m depth 2.5 m away from a slope with inclination 1 (vertical):3(horizontal) in a soil with pt = 1.5
MPa. A horizontal load, equal to one third of the vertical load, is applied 0.9 m above the base of the
footing in the direction towards the slope.
Spread
139
foundation
Solution: The inclination of the resultant R is = arctan(l/3) = 18.4 and the eccentricity eb = 0.9/3 =
0.3 m. The reduced width of the footing is 2.5 - 2-0.3 = 1.9 m. This yields '= arctanfl .9/(2.5+3.3- 1.5)]
= 14.3 from which + '= 32.7. Further we have m=\J.e. Xm = 0 and, consequently, = 0. The reduction factor becomes:
2
i 5 = ( l - 3 2 . 7 / 9 0 ) = 0.40
The bearing capacity factor for d/b = 1.4/1.9 = 0.74 and bll =1.9/2.5 = 0.76 is k = 1.9. The reduced
2
base area is 1.9-2.5 = 4.75 m .
The reduced bearing capacity of the footing is thus
0.4-1.9-1.5-4.75 = 5.3 M N
3.2
Raft
foundations
SETTLEMENTS
4.1
Introductory
remarks
Settlement calculations are generally based on the presumption that plastic yield in the
soil is not taking place or has a negligible effect on the settlement. T h e reliability of
conventional settlement analyses decreases successively the nearer the load is to failure.
Reliable settlement calculations can, of course, also b e carried out in such cases, for
instance b y the u s e of data technique, but these mostly require very a d v a n c e d computer
capacities and are, therefore, mainly utilised in connection with research or for parameter
studies. Before conventional settlement calculations are carried out it is certainly
necessary to m a k e sure that the factor of safety against failure is satisfactory.
The course of settlement is very m u c h dependent on the hy drauli c conductivity and the
creep properties of the soil. Settlement will not terminate until the soil skeleton has
Spread
140
foundation
readjusted itself and b e c o m e strong enough to carry the effective stress increase induced
in the soil by the load of the footing. In water saturated soils with low permeability, such
as clays, the load applied is originally, wholly or partially, e a r n e d by excess pore water
pressure. T h e load is then gradually transferred to the soil skeleton with a corresponding,
simultaneous excess pore water pressure dissipation. For footings on clay, the pore
pressure m a y certainly dissipate rather quickly due to three-dimensional or twodimensional consolidation. However, spread loading of the ground in the vicinity of the
foundation, for e x a m p l e by placement of fill, m a y cause a very drawn-out settlement
process. Even after complete dissipation of excess pore pressure dissipation, long-term
settlement can be expected due to creep p h e n o m e n a .
In coarse-grained material pore pressure dissipation takes place simultaneously with
the application of the load. In coarse-grained soil, settlement d u e to creep often
amounts to 3 0 - 4 0 % of the 'initial' settlement.
T h e settlement of footings on rock are governed by open fissures and other weaknesses
which lead to inelastic behaviour. W h e n the fissures have b e c o m e closed by the ground
pressure the rock behaves more or less as an elastic m e d i u m (Fig. 103). T h e bearing
capacity of footings on rock can be estimated from pressuremeter tests or from the
strength characteristics presented in Table 15.
T h e r e are two main causes of settlement that have to b e considered: ( 1 ) settlement due
to isotropic stress changes (mainly leading to a change in v o l u m e and, consequently, to
pore water being squeezed out of the soil) and (2) settlement due to deviatoric stress
changes (mainly leading to a change in shape). T h e volume change is governed by the
bulk m o d u l u s and the change in shape by the shear m o d u l u s G of the soil. As regards
settlement of footings, settlements due to deviatoric stress changes dominate over
settlements due to isotropic stress changes.
10
20
30
40
50
60
D = 1.0 m
2
D = 0.5 m
D = 0.7 m
Fig. 103. Results of plate loading test on fissured gneiss rock of bad quality (After Pusch et <-//., 1974)
Spread
141
foundation
Settlement
if) Analysis
of
footings
in comparison with the rigidity of the soil, the settlement of the footing can b e considered
equal at all points. According to the theory of elasticity, there are certain so-called
characteristic points at which the surface settlement of an elastic half-space is independent
of the bending rigidity of the footing. In order to find the settlement of a rigid footing, w e
then h a v e to calculate the settlement at the characteristic point. According to Steinbrenner
(1936) this can b e done by superposition of the settlements obtained b e l o w corner I of
the four part rectangles E A F I y F B G I , G C H I and H D E I , corner I being situated at the
characteristic point of load area A B C D (Fig. 104).
Now, the settlement of a corner sc of a rectangular or square area with evenly di stributed
load q, acting on the surface of an elastic m e d i u m with limited thickness d, can be calculated
by the relation (Steinbrenner, 1936):
sjb
(146)
where
/l
(i
| n [
AJVq
(l/b)(l
,n[ ( * + y q ) / Q
llb +
yjQ+Cd-\
lib
--arctan
2nb
Q =
c J- 2
+ yjCi + Q - 1 )
1 d
f2
(dlb)yj
l + ( | )
a n d Q =
Q+Q-X
l + A
1/3.
Spread
142
foundation
0.42D
Fig. 104. The position of the characteristic point. The settlement at the characteristic point I is obtained
by superposition of the corner settlement of the rectangles GCHI, HDEI, EAFI and FBGI.
For calculation of the initial settlement of a footing on water saturated clay, which is due
only to shear deformations, w e have to a s s u m e = 0.5.
T h e settlements obtained by superposition of the sc values according to Eq. (146 ) can
be expressed by the relation:
reqb/E
s =
(147)
0.1
0.2
0.5
<
>
PC
\ \ \ \
\\ \
\\y
10
20
0.5
V \
1.5
2.5
Spread
143
foundation
s/b =
w h e r e Fei and Fej_x
ql (r -r _ )/E
i
e9i
EJ
(148)
(depth d )
t
Example 14: Determine the settlement at midpoint of the long side of a rectangular area with length 10
2
m and width 2.5 m acted upon by a uniformly distributed, flexible load q - 20 k N / m . The soil consists
of sand underlain by bedrock at 4 m depth. The sand is assumed to behave as an elastic medium with
Es = 20 MPa and = 0.3.
Solution: The area is cut into two equal halves, each one with length 5m and width 2.5 m. For these we
have C/ = 5 and Cd = 3.56. Thus:
/ , = l { 2 . 1 n t
( 1 +
^
^ ^
]
2(1 -h V5 + 3 . 5 6 - 1 )
ln[
(2 +
: , )
l i
/
- ^ : ] } =( . 2 2 9
2 + V5 + 3 . 5 6 - 1
0.1
0.2
0.5
\\\
\ \\\YV
10
20'
0.5
1 \ \
1
1.5
Spread
144
foundation
u.i
= 1.6 arctan[
] = 0.109
1.6 V5 + 3 . 5 6 - 1
The settlement becomes equal to:
2
2
3
j = 2-2.5-20[(l - 0 . 3 ) 0 . 2 2 9 + (1 - 0.3 - 20.3 )0.109]/20000 = 1.3-l()- m
2
Example 15: Determine the settlement of a square footing, 2.5 m by 2.5 m, founded at 1 m depth in sand
3
and subjected to a column load of 700 kN. The sand is assumed to have a unit weight of 18 kN/m , a
modulus of elasticity of 20 MPa to a depth of 5 m and below that 30 MPa.
Solution: The settlement analysis is carried out in two steps. For the layer with Es = 20MPa we have
dej-i - 0 and dei = 4 m from which we find Te -{_x = 0 and Te t = 0.52. For the underlying sand we have
fe%i_x = 0 . 5 2 and = 0.77.
The settlement, solved by Eq. (148), becomes equal to:
0
7 7
5 2
, ^ - 1 1 8 ) 2 . 5 ( - = = , + - ) = 0.008
2
3
3
2.5
20 10
30 10
method.
Spread
145
foundation
Fig. 106. Assumed stress distribution around a rigid half-sphere embedded in soil.
footing, enclosed in a half-sphere with the s a m e diameter as the footing, is rigid. As was
shown by D e Josselin de Jong (1957), the settlement of a rigid half-sphere with diameter
D, subjected to a load q per unit area, is equal to:
sd =
12G
(149)
= qsin co (Fig. 106). B y integration over the surface area of the half-sphere w e find an
average value of 2q/3. A s s u m i n g further that 2q/3 is an isotropic stress acting against
the half-sphere w e find the vertical deformation (the settlement s,-) equal to:
qD
s , = -
(150)
Spread
146
foundation
TABLE 21
The value of = El to be applied in different soil and rock conditions.
Gravel
Sand
Silt
Clay
Soil type
EpSPi
Epr/'Pl
EprfPl
Eprhpt
Heavily overconsolidated
Normally
consolidated
Weathered and/or
remoulded
>16
>14
2/3
>12
1/2
>1()
1/3
9 - 16
2/3
8 - 14
1/2
7 - 12
1/3
6-10
1/4
7-9
1/2
Rock type
Extremely
fractured
Other
cc= 1/3
= 1/2
1/2
1/4
1/3
Slightly fractured or
extremely weathered
=2/3
TABLE 22
Shape coefficients A.
1
lib
(1
A,
circle
square
1
1
1.12
1.10
20
1.53
1.20
1.78
1.30
2.14
1.40
2.65
1.50
s = qD(
H
1
12G
1
D
+ ) = ( 2 + )
9K
(151)
9E
bt
b t
^=^(1.2-0.2-f)[-^(-^)^ + - ^ - l
prd
pri
(152)
If b < 0.6 m (which is rarely the case in practice), then b0 in Eq. (152) should be put
equal to b.
T h e rheological coefficient to be applied in different soil conditions is given in Table
21 and the shape coefficient in Table 22.
T h e practical u s e of Eq. (152) for settlement analysis described in the following is in
agreement with the routine method presented by Baguelin et al. (1978).
For the determination of the pressuremeter moduli to b e applied in the deviatoric and
the isotropic terms, the subsoil is divided into five layers as s h o w n in Fig. 107 (Baguelin
etaL,
1978).
E3,-En,
/) =
= HM(El/E2/E3I-
(I)
(153)
is taken as the
1.5/7
_0.5fc
(ID 1.5/7
(3)
1.5/7
(4)
1.5/?
(5)
4/7
Fig. 107. Division of subsoil into layers for determination of pressuremeter moduli to be applied in
deviatoric and isotropic terms of Eq. (152).
Spread
148
4 Epri
-L =V ! _
Eptd
foundation
- L
0.85Epr2
_ ! ! _ ,
Epr3
2.5Epr4
( | 5 4 )
2.5Epr5
EprZ
than in the surrounding soil, the settlement is first calculated on the assumption that the
layer has a fictitious pressuremeter m o d u l u s Epnn
1
(155)
10
20
30
49
Spread foundation
Solution: The ultimate load is governed by the equivalent pressure limit and the bearing capacity factor
k. In our case there is no pressuremeter value above the foundation level. The equivalent value of pj is
thus a function only of the values obtained to a depth of 1.5/?, including four pressuremeter values. We
1 /4
find *ple = (2.30.80.80.6) = 0.97 MPa. The equivalent depth in this case becomes equal to the
foundation depth, 1 m. We find the ratio djb = 1/2.7 = 0.37. From Fig. 101 the bearing capacity factor
k can be estimated at 1.2.
2
3
The ultimate load Qf = ( 18 1 + 1.2970)2.7 = 8.6 10 kN
For the determination of the settlement of the footing we have to calculate the harmonic means of the
pressuremeter moduli in the different layers 1 -4. These become E { = HM( 16.5/5.2) = 7.9 MPa; E 2 = 3.8
MPa; E 3 = HM(4.3/7.3/13.8) = 6.8 MPa and E 4 = HM( 16.2/12.8) = 14.3 MPa. The value of E 5 is assumed
equal to E 4 . We have also to consider the loose layer at depths 7 and 8 m with z = HM(4.0/2.1) =
2.8 MPa.
The moduli to be introduced in the deviatoric and isotropic settlement terms are equal to:
1
+ +
= 5.5 MPa
= f-( +
4 7.9 0 . 8 5 - 3 . 8
6.8
14.3
E p n = E { = 7.9 MPa
For determination of the additional settlement due to the loose layer, E p r m is chosen as the mean value
of 14.3 and 13.8 MPa, i.e. E p n n = 14.0 MPa.
The shape coefficients for a square foundation are Xd = 1.12 and Xt - 1.1. In our case, the ratio EptJp/ is
in the range of 4 to 10. The rheological coefficients can thus be estimated at 1/3. The factor ( 1.2 - 0 2d jib)
= 1.126
The settlement is obtained from the relation:
H ^ ^ 2
3 - 1.5
_L_J_
0 0 30
m
2
2
9-2.7
5.5
0.6
3-7.9
3(2.7 + 7 . 5 ) 2.8
14.0
Entd
^
(iii) Empirical
methods.
which the settlement of footings can be be roughly estimated. Most of these m e t h o d s are
connected with S P T t h e standard penetration testor C P T t h e c o n e penetration test.
The interpretation of the penetration resistance in terms of settlement is very m u c h
dependent on the loading history of the soil. Thus, experiments have shown that the ratio
of the m o d u l u s of pseudoelasticity to the penetration resistance of C P T is m a n y times
higher ( 5 - 1 0 times) in overconsolidated, than in normally consolidated sand (Jamiolkowski t al, 1985). Therefore, if the relations between penetration resistance and settlement are not connected with the preconsolidation pressure, the results can be strongly
misleading. In fact, almost all existing empirical relations do not take into account the
influence of possible overconsolidation. Nevertheless, empirical relations can be useful
for a rough, and generally conservative, estimate of the settlements to be expected.
On the basis of the results of 48 case records, Schultze and Sherif (1973) developed
an empirical correlation
between
penetration
resistance
at the
characteristic point of a load area with length / and width b. T h e settlement is obtained
by the relation:
s=q
f
(N^f^il
OAdf/b)
(156)
150
Spread
foundation
(157)
= 4 1 . 6 + 1.09 N 3 0 M P a
(158)
E=
(159)
Example 18: Determine the settlement of a rectangular footing with a bottom area of 2 m by 3 m
subjected to a column load of 3 MN if the footing is founded at 1.5 m depth in sand with an average
3
standard penetration resistance of 7V 30 = 35.The density of the sand is 1.8 t/m . The groundwater level
is below the foundation depth.
Solution: The settlement will be evaluated on the basis of the empirical relations given above.
Spread foundation
Schultze-Sherif: We have dj/b = 0.75 and lib = 1.5. This y i e l d s / 0.57 m/MPa. The net foundation
pressure is q = 0.5 - 1.5-0.018 = 0.47 MPa. The settlement is obtained as:
0: 57
' = 0.47 8 7
= 0.009 m
35 (1 +0.4-0.75)
D,Appolonia et al. : Assuming that the soil is normally consolidated we have = 19.6 + 0.79-35 =
47.3 MPa. The Te value (Fig. 105c), taking into account a soil layer with a thickness of 8 m (= 4b) is
0.77. The settlement is obtained as:
s = 0.47-2-0.77/47.3 = 0.015 m
Assuming the soil to be overconsolidated we Find = 41.6 + 1.09-35 = 79.7 MPa which yields:
s = 0.47-2.0.77/79.7 = 0.009 m
Parry: The modulus of elasticity = 5-35 = 175 MPa. The settlement becomes:
s = 0.47-2-0.77/175 = 0.004 m
between
is
(160)
s = ClC2qZ-^-Az
Mz
w h e r e q = net load increase at the foundation level
Mz = settlement m o d u l u s varying from 2.5qcz
for a
strip footing (for heavily overconsolidated soil, these values should m o s t probably be
replaced by about \0qc
and I3qc,
respectively).
\-0.5(qlq-\)
C 2 = 1 + 0.21og(10i)
t = loading time in years
T h e p e a k value of Iz is obtained from the relation:
(161)
where ' = vertical stress increase at depth = 0.5b for a square footing and =b
for a strip footing
Spread
152
0.2
0.4
0.6
0.8
foundation
1.0
0.5b
b
y
2b
y
4b
y'
Fig. 109. Influence function Iz for square and strip footings. Function Iz for a strip footing can be used when
/ > 20b. When b <l<
20b the influence value can be obtained by interpolation between Iz for square
Example 19: A static cone penetration test in silty sand with a unit weight of 19 k N / m has given the
result shown below. The groundwater level is at 4 m depth. Determine by Schmertmann's method the
10 years settlement of a strip footing, 0.8 m in width, subjected to a line load = 130 kN/m. The footing
is founded at a depth of 1.0 m.
1.0 m
3.2 m
Solution: The average cone resistance qc varies from 1.2 MPa at the foundation level to 1.5 MPa at a
depth of 3.2 m (4b) below the foundation level. Thus, the settlement modulus Mean be assumed to vary
from 3.5-0.12 = 4.20 MPa to 3.5-0.15 = 5.25 MPa. At 0.8 m below the foundation level we have M =
4.20 + (0.8/3.2)(5.25 - 4 . 2 0 ) = 4.46 MPa. Furthermore, we have ' - ( 1.0 + 0.8) 19 = 34 kPa at a depth
Spread
153
foundation
of 0.8 m ) below the foundtaion level and q = 130/0.8 - 1.0 19 = 143.5 kPa. These values of & z and
1 /2
q yield7 Z m ax = 0.5 + 0 . 1 ( 1 4 3 . 5 / 3 4 ) = 0.705.
The parameter CX = \ - 0.5(162.5/143.5) = 0.434 and the parameter C 2 = 1 + log(10-10) = 3.0.
Dividing the subsoil into 0.4 m thick layers, we find for the respective layers - 4.21, 4.40, 4.53,
4 . 6 6 , 4 . 7 9 , 4 . 9 2 , 5.05 and 5.18 MPa and the corresponding Iz values 0 . 3 2 6 , 0 . 5 7 9 , 0 . 6 4 6 , 0 . 5 2 9 , 0 . 4 1 1 ,
0.294, 0.176 and 0.059. The settlement of the footing becomes:
s = 0.4343.00.14350.4[(0.326/4.27 + 0.579/4.40 + 0.646/4.53 + 0.529/4.66 + 0.411/4.79 +
0.294/4.92 + 0.176/5.05 + 0.059/5.18) = 0.05 m.
The settlement can also be calculated in an approximate way as:
s = 0.434-3.0-0.1435[(0.2/4.2 + 0.705/4.46)0.8/2 + (0.705/4.46)-(3.2 - 0.8)/2] = 0.05 m
4.3
Settlement
of rafts
T h e settlement analysis in the case of raft foundations is an intricate matter and is very
much dependent on the location of columns and load-carrying walls and the stiffness of
the raft and superstructure. Therefore, settlement calculations are generally carried out
by the aid of computers (see Bowles, 1988).
Mostly, calculations can be performed in a fairly simple manner. T h u s , for raft
foundations on soft, compressible soil (such as normally consolidated, or lightly
overconsolidated clay) w h e r e settlement is mainly a matter of primary and secondary
consolidation, the overall average settlement can b e calculated in the s a m e analytical way
as for individual footings, for e x a m p l e by using M values determined by the oedometer
test and the 2:1 m e t h o d for calculation of the stress increase in the subsoil. T h e settlement
distribution underneath the building will then have to be estimated with due consideration
to the gradual build-up of the rigidity of the raft and superstructure during construction
of the building.
Example 20: A building with a bottom area of 15 m by 20 m is founded at 2m depth in normally
consolidated clay using a raft foundation with high bending rigidity. The load of the building is 50 kN/
2
3
m on the average. The clay is underlain by sand at 17 m depth. It has a density of 1.6 t/m and a compression
modulus ML = 0.17 MPa, prevailing in the stress range G*L - o'c - 20 kPa {cf. Fig. 50). The coefficient
2
of consolidation cv = 0.5 m /year. The groundwater level is situated at 1 m depth below the ground surface. Determine the average settlement of the building after 10 years.
Relative compression, %
0
is
5
1
10
15
1 0
Spread
154
2
foundation
Solution: The load release due to excavation is equal to 2 16 = 32 kN/m . Thus, the load increase under
2
the weight of the building is 50 - 32 = 18 kN/m . The stress increase in the soil is below 20kPa and,
therefore, the ML value governs the compression throughout the clay layer. The determination of the
primary compression is exemplified for 5, 10 and 15 m depths.
3
2
5 m depth: = 18/[(1+5/15)(1+5/20)0.1710 ] = 6.4-10'
3
10 m depth: = 18/[(1+10/15)(1+10/20)0.1710 ] = 4.2 K H
3
2
15 m depth: = 18/[(1+15/15)(1+15/20)0.1710 ] = 3.0-10"
The total primary settlement to be expected, which is the area of the primary compression graph, is
sp = 0.85 m. (This settlement corresponds to a CR value of 0.35, see p. 77).
The remaining primary settlement (cf. p. 31)after 10 years is determined by Helenelund's method (p.
2
323). Choosing a layer thickness Az = 3m we have At=(Az) /(4cv)
= 2.25/0.5=4.5 years. The construction
is carried out as shown on the previous page.
After a loading time of 2At (9 years) we have a remaining primary settlement sr = 3(0.026 + 0.056
+ 0.052 +0.038 +0.014) = 0.56 m and after a loading time of 3At (13.5 years) sr = 3(0.021 + 0.048 + 0.049
+0.035 +0.012) = 0.50 m. The remaining primary settlement after 10 years can be estimated by
interpolation which yields sr = 0.56 - (0.56 - 0.50)/4.5 = 0.55 m.
Thus, the settlement after 10 years can be estimated at 0.85 - 0.55 = 0.30 m.
Deep
155
foundations
DEEP FOUNDATIONS
1.
INTRODUCTION
PILES
2.1
Common
pile
types
(i) Timber piles. Timber piles represent the earliest type of piles and w e r e m o r e or less
the only ones used until the end of the nineteenth century. Old buildings on piles,
therefore, p r e s u m a b l y rest on timber piles, often via a mattress of horizontal planks, laid
out in order to achieve an evened-out stress distribution underneath the footings.
N o w a d a y s , timber piles are mainly used in places with deep cohesive soil layers and a
high groundwater level.
T h e m o s t obvious p r o b l e m with timber piles is the deterioration caused by fungi
attacks above and in the vicinity of a fluctuating groundwater level. Rotting of the part
of timber piles that sticks out of the groundwater is very frequent and m a y cause serious
problems. T h u s , as the degree of rotting underneath the different footings is generally
subjected to large variations, rotting usually leads to large differential
settlements
Deep
156
foundations
piles.
Deep
foundations
157
Fig. 111. Pile shoe of steel for protection of pile tip from being damaged during pile driving.
a great influence on the geotechnical properties to be used in the analysis of the performance of the pile when loaded.
The concrete to be used for the production of precast piles has to be of very high quality
in order to withstand the severe treatment of pile driving. In particular, the toughness of
the concrete is important as the number of blows of the pile hammer required to reach the
expected bearing capacity can be very large and lead to destruction of the pile. Moreover,
in order to protect the tip of the pile from being damaged during installation, the pile is
often provided with a pile shoe of steel (Fig. 111).
The width, or diameter, of precast piles is usually between 0.25 and 0.35 m and the
length limited to 1214 m in order to facilitate transport of the piles from the pile factory
to the building site as well as installation. Special splices have been constructed which
have the capacity of taking up equally large tension forces and bending moments as the
pile itself (Fig. 112). Moreover, the splice connecting two pile segments should be
constructed so as to have the same rigidity as the pile segments themselves.
Piles cast in place exist in many different forms among which the most common ones
are bored and auger piles formed by removing the soil. These are also referred to as nondisplacement piles in contrast to displacement piles which represent all the precast and
cast in place piles which are driven into the ground.
A m o n g the driven cast-in-place piles, the Raymond and the Franki piles are probably
the most well-known.
In some places, concrete can be subjected to considerable corrosion due to aggressive
constituents in the groundwater. Certain molluscs exist in sea water which burrow in
concrete or secrete a substance that has a solvent action even on rock.
Deep
158
foundations
(iii) Steel piles. Steel piles represent an alternative to precast concrete piles in places
w h e r e the subsoil conditions entail large risks of the concrete piles being d a m a g e d during
pile driving. Rolled steel shapes, for e x a m p l e H piles, which have high bending rigidity
are c o m m o n l y utilised, but rolled bar irons of circular, square or X-shaped cross-section
Concrete pile
with fixed
steel case
--+.
Deep
159
foundations
are also in use. Steel piles are generally driven into firm, bearing strata often bedrock
or hard till.
Steel piles mostly require some kind of protection against corrosion.
(iv) Composite
piles.
Fig. 114. Micro-piles, in this case so-called steel-plastic piles, being installed in room with low ceiling
height. The steel-plastic piles consists of a steel tube covered by 1.8 mm of polyethylene plastic. The
pile segments are spliced by means of galvanised steel tube sleeves which are pressed upon the pile after
that the plastic cover has been softened by heating. The diameter varies from 76 to 102 mm.
Deep
160
foundations
piles consist of small-diameter steel tubes filled with cement and provided with some
kind of corrosion protection, for example a plastic cover (Fig. 110).
2.2
Displacement
piles
A displacement pile is defined as a solid pile, or a hollow pile driven with its tip closed,
which displaces an equivalent soil volume by compaction or by lateral or vertical
displacement of the soil. Most of the displacement piles are precast concrete piles, steel
piles, or timber piles which by necessity have to be driven into the soil. However, driven
concrete piles can also be cast in place. In this case a closed-ended casing is driven into
the soil whereupon the pile is cast inside the casing. The casing can either be left in the
soil, which is the case with the Raymond pile, or withdrawn from the soil, as is the case
with the Franki pile. The Raymond pile is tapered which improves its bearing capacity.
The Franki pile can be installed in several ways (Fig. 115). There is, however, a common
feature. The casing is filled before installation with 'dry' concrete to a height of 1 to 2 m.
The concrete plug, thus formed in the casing, serves as a water-tight pile tip during
driving. The pile is then driven into the soil by means of 4 - 5 m long hammer dropped
inside the casing. When the pile tip has reached its design level, the casing is fixed to the
ground surface and the concrete plug is driven out of the casing, thereby
causing
compaction of the soil around the pile tip. In this way the Franki pile b e c o m e s
dynamically preloaded.
(i) Driving equipment.
gravity hammers but also by the use of steam/air hammers, diesel hammers of different
kinds or vibratory drivers. The gravity hammer is simply a weight (generally
t) which
is lifted a certain distance with a hoist line and then released to fall and strike a drive cap
(Fig. 116).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Fig. 115. Schematic picture of hammer and driving tube and installation methods for three different
types of Franki piles: a+b+c = standard Franki pile, a+b+d+e = vibrated Franki pile and a+b+f+g =
composite Franki pile.
Deep
foundations
161
T h e steam/air h a m m e r s in use are of type single-acting, double-acting, differentialacting and c o m p o u n d h a m m e r s . T h e single-acting steam/air h a m m e r is similar to the
gravity h a m m e r except that it has a cylinder and a piston to lift the r a m weight instead
of a hoist line. T h e lift of the h a m m e r is produced by compressed air or steam (motive
fluid). T h e double-acting steam/air h a m m e r and the c o m p o u n d steam/air h a m m e r differ
from the single-acting h a m m e r in that the m o t i v e fluid is also introduced over the piston
to accelerate the r a m in its d o w n w a r d stroke. T h e differential-acting steam/air h a m m e r
is another type of double-acting hammer. M o t i v e fluid is introduced b e t w e e n large and
small piston heads to lift the r a m to the top of its stroke. T h e n m o t i v e fluid is introduced
over the large piston head to accelerate the r a m in its d o w n w a r d stroke.
Diesel h a m m e r s are of two different kinds: open-end (single-acting) and closed-end
(double-acting) h a m m e r s . Both these diesel h a m m e r s are one-cylinder diesel engines in
which the potential energy of diesel fuel combustion is converted through the m o v i n g
piston to kinetic i m p a c t energy delivered at the anvil block. T h e difference between the
closed-end and the open-end diesel h a m m e r s is that the gravity-controlled flight of the
r a m of the open-end h a m m e r is shortened in the closed-end h a m m e r . This is produced by
decelerating the r a m ' s u p w a r d stroke by trapping air to form an air spring. This m a k e s
the closed-end diesel h a m m e r operate at about twice the b l o w rate of an o p e n - e n d diesel
h a m m e r of c o m p a r a b l e size.
Vibratory drivers get their driving capacity by the centrifugal force produced by
Deep
162
foundations
hammer
--
pile
Fig. 117. Stresses acting on pile element due to the impact of the pile hammer.
eccentrics in the driver which can be rotated at a steady-state frequency w h e n loaded with
an oscillating pile. Their driving efficiency depends on amplitude, frequency, centrifugal
force, vibrating weight and non-vibrating weight. European vibratory drivers usually
operate at a frequency of 1 0 - 3 0 H z while, for instance, the A m e r i c a n B o d i n e Resonant
Driver operates at a frequency of 5 0 - 1 5 0 Hz.
(ii) Influence
the r a m is travelling d o w n w a r d s the pile until it reaches the pile tip w h e r e it is reflected,
travelling u p w a r d until it reaches the pile head, reflected again and travelling d o w n w a r d s ,
and so on. Stress w a v e analysis is used as a basis for determination of the bearing capacity
of driven piles (see p. 175).
T h e effect of the stress w a v e on the pile can be studied by simply assuming that
damping of the stress w a v e by the surrounding soil can b e disregarded.Considering the
forces acting u p o n a pile element (Fig. 117) under the assumption of no influence
surrounding
of the
Ap
-Appp
(162)
Deep
163
foundations
whence:
2
dw
dt
jd w
(163)
- ^
dz
w(z,0
et)
+f2(z
et)
(164)
-cf\
(z-ct)
+ cf'2(z
+ ct)
(165)
N o w , for the initial (primary) stress w a v e (the first term in Eq. 164) w e find the
following correlation:
dw
-E (z,t)
p
-EJ^z-ct)
az
dw
^ ( z , t )
c m
(166)
Hence:
Oz
where
(cf.
dw
= {z,t)y/Eppp
at
= (Z
dw
IA )(z,t)
p
at
(167)
EpAp/lp).
In the case of the gravity h a m m e r , the velocity enforced on the pile w h e n it hits the pile
h e a d is:
= a/gh
dt
(168)
Deep
164
1
l +
J^~
foundations
\+z /z
p
=
If E - > w
^/ / = aZy/lghlAp
(169)
find a = 1.
If the moduli of elasticity and the cross-sectional areas and the densities of the pile and
the pile h a m m e r are the same, i.e. Eh-
Ep\ Ah-
case the stress w a v e b e c o m e s rectangular with a length equal to double the length of the
pile hammer.
T h e shape of the stress w a v e can be adjusted by choosing a suitable cushion block
placed in the drive cap under the hammer. T h e length of the w a v e is independent of the
drop height.
F r o m Eq. (169) w e find that the normal stress induced in the pile by the impact of the
h a m m e r is dependent on the drop height but independent of the weight of the hammer.
In case the pile tip is driven d o w n to rigid material (big blocks or bedrock), the reflected
stress w a v e will be a compressive stress w a v e with the same intensity as the stress wave
travelling d o w n w a r d s . T h u s , for a short m o m e n t the stresses at the pile tip will
superimpose which often leads to crushing of the pile tip (Fig. 118).
In the case of rectangular stress waves, the m a x i m u m compressive stress at the pile tip
will be:
Fig. 118.Crushing of pile tip due to overlapping of the compressive stress wave caused by pile driving.
Deep
foundations
165
Fig. 119. Tensile cracks in concrete pile caused by the tensile stress wave obtained in pile driving.
(170)
Normally, however, the m a x i m u m compressive stress reached at pile refusal does not
exceed 1.5-1.8 times the initial stress.
If, on the other hand, the soil resistance at the pile tip is negligible, the stress w a v e will
stretch the pile tip and the reflected w a v e travelling u p w a r d s will be a tensile stress w a v e
of equal intensity as the compressive stress w a v e reaching the tip, /. e.\
(171)
In this case, tensile cracks in the pile m a y be obtained (Fig. 119). If the pile is under
water, water will be sucked into the cracks and then forced out of the cracks by the
following compressive stress w a v e , a jetting process that m a y cause erosion of concrete
around the opening of the cracks.
T h e stress w a v e velocity in h o m o g e n e o u s (uncracked) piles is 3 2 0 0 to 3 9 0 0 m/s,
depending upon the quality of the concrete, and in steel about 5 1 0 0 to 5 2 0 0 m/s.
The driving of the piles b e c o m e s most efficient w h e n the stress w a v e has a rectangular
shape. Therefore, the design of the gravity h a m m e r and the drive cap has been carried out
with the attempt of achieving a stress w a v e that b e c o m e s as closely rectangular as
Deep
166
foundations
possible. For e x a m p l e , a cup spring drive cap (Fig. 120) has been developed w h i c h limits
the peak value of the stress w a v e and, consequently, m a k e s it independent of the height
of fall of the hammer. An increase of the height of fall only increases the length (the
duration) of the stress w a v e . By using the cup spring drive cap, crushing of piles driven
by m e a n s of steam/air h a m m e r s can also be avoided.
(iii) Influence
on soil properties
of pile driving.
taking place during pile driving, the properties of the soil are subjected to changes,
sometimes for the better, sometimes for the worse. T h e vibrations induced in the soil and
the enforced soil displacements during pile driving m a y give rise to large and destructive
settlements of adjacent buildings (Fig. 121). This is certainly to be expected w h e n piles
are driven into loose granular soil. In fine-grained loose soils, pile driving m a y induce
high residual excess pore water pressures. As the n u m b e r of piles increases, w e m a y end
up with a state of liquefaction, or the shear strength of the soil m a y decrease to such an
extent that the stability of the place is put at stake. In particular, this peril has to be taken
into account in sloping clay regions with silt and sand layers. Two such cases of failure
due to piling w e r e reported in the beginning of the 1970s (see H a n s b o , 1987). In both
cases, the sites had fairly similar geological conditions: smooth clay slopes with a very
low inclination. In one case, after driving of only 5 piles, a 2 0 - 3 0 c m w i d e crack, 5 5 0
m in length, developed causing severe d a m a g e to three houses. In the other c a s e , pile
driving for s o m e terrace houses induced a 5 - 1 0 c m w i d e crack, 2 5 0 m in length, causing
s o m e d a m a g e to streets and houses and breaking two water conduits.
Deep
foundations
167
Fig. 121. Damage to building caused by pile driving for-foundation of adjacent building.
M o s t probably, piling also triggered the slide disaster at Surte in 1950 comprising an
area of around 25 h a (Fig. 122). In the slide, one person was killed and two badly hurt
while 31 houses were m o r e or less destroyed.
In dense, saturated granular soil, pile driving by m e a n s of a gravity h a m m e r or a steam/
air h a m m e r m a y b e very time-consuming and troublesome and entail considerable risk
of concrete piles fatigue. However, the rate of penetration of the piles can b e increased
considerably by the use of vibratory drivers. In that case, the bearing capacity and the
settlement behaviour of the pile will b e dependent on the frequency of the driver during
installation.
So-called false refusal, i.e. w h e n the resistance observed during pile driving will be
strongly reduced after termination of driving, represents a p r o b l e m often encountered,
particularly in fine-grained cohesionless soils. This p h e n o m e n o n is m o s t probably due
to an increase in soil strength by pore water underpressure (suction) i n d u c e d during pile
driving.
T h e compaction of loose granular soil caused by pile driving m a y obstruct the
possibility of driving the piles to equal depth in a pile group. T h u s , the piles first driven
reach m u c h deeper before the required resistance is obtained than the piles driven at the
168
Deep
foundations
Fig. 122. The slide at Surte north of Gothenburg on Sept. 29, 1950, is assumed to have been triggered
by pile driving in the area.
end. This entails bad e c o n o m y since the piles not driven to full depth will h a v e to b e cut.
Moreover, the load/settlement characteristics of the piles first driven m a y b e quite
different from those last driven. This p r o b l e m can be avoided by providing the piles with
Deep
169
foundations
Fig. 124. Coring operation in soft clay. To the left: clay core just withdrawn from the soil. To the right:
clay forced out of the tube by the aid of compressed air.
central tubes and special arrangements for the application of water jetting (Fig. 123).
W h e n the piles h a v e reached the intended depth, the pile installation can then be
finalised b y driving the pile to greater depth with a pile h a m m e r .
Pile driving in soft clays usually entails a heave and a reduction (sometimes considerable,
for e x a m p l e in highly sensitive clays) of the undrained shear strength w h i c h has to be
taken into account in stability analyses. In order to reduce the disturbance effects, clay
cores are taken w h e r e the piles are to b e driven. T h e coring operation is carried out by
m e a n s of a hollow tube with a vent at the top w h i c h is kept open w h e n the tube is driven
into the soft clay to release trapped air. T h e tube is driven to a depth of 6 to 8 m. T h e vent
is closed and c o m p r e s s e d air is injected into the tube at its b o t t o m to eliminate v a c u u m
during withdrawal. After withdrawal, the clay is forced out of the tube b y m e a n s of
compressed air (Fig. 124).
2.3
Non-displacement
piles
methods.
170
Deep
foundations
Fig. 125. Examples of installt of borec' ^iles: To the le. use of auger boring. To the right: use of
rotary boring (by courtecy of Bachy).
reinforcement, in the form of a prefabricated cage, is usually placed in the hole prior to
concreting. In the case of continuous auger piles, the reinforcement cage is placed
immediately after the auger has been r e m o v e d (Fig. 126).
Deep
171
foundations
In many cases the borehole for the pile can be formed in stable ground condition where
no support of the sides of the hole is required. In other cases, however, the employment
of casings or, more commonly, bentonite slurry may be required to ensure the stability
of the borehole.
Casings are generally used only as a temporary measure to provide stability during
boring and concreting. Permanent casings may be necessary in weak soils which cannot
sustain the lateral pressure of the fresh concrete.
If the borehole is stabilised by means of bentonite suspension a short collar casing is
almost always used. The use of bentonite does not seem to have any negative influence
on the quality or integrity of the pile. On the contrary, it seems safer to use bentonite for
borehole stabilisation than casing (Sliwinski & Fleming, 1984)
Bored piles with a diameter ranging from 0.3 to 0.6 m are generally referred to as small
diameter piles and those with diameters above 0.6 m as large diameter piles or sometimes
as caissons. With modern methods, bored piles can be installed with a m a x i m u m rake of
4:1 (vertical:horizontal). The diameter is in the range of 0.45 to 3.5 m. By means of a
special tool, the piles can be underreamed to a tip diameter of up to 5.4 m (Figs. 127-128).
The diameter of continuous flight auger piles is in the range of 0.35 to 1.5 m.
(ii) Installation
problems.
contains a great number of boulders, blasting may have to be carried out below the tip of
the casing. The shock waves produced by blasting may induce excess pore water pressures and cause compaction of the soil in a similar way as during pile driving. Careless
excavation in sand and silt soils below the groundwater level involves a great risk of
bottom erosion (piping) which can badly affect the ground conditions in the vicinity ofthe
172
Deep
foundations
Fig. 128. Underreamed cave under inspection. Underreaming can only be used in the case of stiff or very
stiff clays.
excavation. In this case, the borehole for the pile has to be stabilised by m e a n s of bentonite
suspension or by filling the borehole with water to a higher level than that of the sur-
Deep
173
foundations
\fs
(negligible)
m
qt
(negligible)
rounding groundwater. As in the case of driven piles, the installation of bored piles may
lead to severe d a m a g e to nearby buildings (Fig. 129).
Before the excecution of holes for piles in soft clay, the shear strength of the clay has
to b e k n o w n . T h e existence of possible sand or silt layers is a reason for taking special
precautions. T h e release of overburden pressure during excavation can lead to a drastic
decrease in the shear strength below the b o t t o m of the hole and adventure stability even
though the original shear strength might h a v e seemed satisfactory.
T h e integrity of bored piles can be jeopardised by segregation of concrete, inclusion
of soil or slurry, cavities in the concrete, displacement of the reinforcement cage, etc.
Therefore, possible sources of defects m u s t be anticipated and measures taken to ensure
the integrity of the piles.
2.4
Load transfer
pile/soil
T h e load carried by the pile is transferred to the soil b y frictional resistance along the pile
shaft and/or b y pile tip resistance (Fig. 130). If the major resistance to load is derived
merely by side friction (or by adhesion in the pile/soil interface), w e speak offriction
piles
(or, less commonly, floating piles). On the other hand, if the major resistance to load is
exerted by the pile tip, w e speak of point bearing piles or, alternatively, end bearing
piles.
A friction/end bearing pile is a pile that carries the load by both frictional resistance and
point resistance.
T h e bearing capacity of end bearing piles is of course dependent on the characteristics
of the soil on which the pile tip is resting and on tip dimensions. In h o m o g e n e o u s soil,
the end bearing capacity increases with depth until the pile tip reaches the so-called
'critical depth' below which no further increase can be noticed. Dynamic preloading
Deep
174
foundations
improves the bearing capacity of the soil. Therefore, driven piles can usually c a n y a
higher load than non-displacement piles with the same tip area.
T h e bearing capacity of friction piles depends on the angle of internal friction in the
soil, the coefficient of friction between soil and pile and the normal pressure against the
sides of the pile. A considerable n u m b e r of pile loading tests h a v e s h o w n that the bearing
capacity of friction piles in cohesionless soils generally increases with time after pile
installation (stedt et , 1990). This is most probably due to postdensification effects,
similar to those observed in connection with soil i m p r o v e m e n t by blasting or vibratory
compaction (Mesri et , 1990).
In clay soils, the bearing capacity is governed by the long-term shear strength of the
clay. T h e roughness of the sides of the pile and pile material h a v e certainly a considerable
influence on the frictional resistance in granular soils but s e e m negligible in cohesive
soils. W h e n , in the latter case, the pile is driven, the soil around the pile will be remoulded
within a zone that extends to at least one pile diameter from the pile surface. T h e resulting
excess pore water pressure around the pile m a y reach and even locally exceed the total
overburden pressure. Reconsolidation of the r e m o u l d e d clay leads to an increase in the
undrained shear strength of the clay. T h e reconsolidated, r e m o u l d e d clay adheres to the
pile surface and rupture takes place in the clay with lower shear strength outside the pile
surface (cf. B r o m s & H a n s b o , 1981).
T h e shape of the pile is important for the bearing capacity of friction piles. Conic piles,
for instance, such as timber piles with a pile tip diameter considerably smaller than the
diameter of the pile head, are very advantageous from the point of view of frictional
resistance.
3.
Pile loading
tests
In the case of piled foundations, full-scale loading tests are often prescribed as a check
on the bearing capacity predicted according to the m e t h o d of analysis applied. Full-scale
loading test h a v e the advantage of providing information not only of the ultimate pile load
but also of the w h o l e load vs. settlement relationship.
T h e ultimate pile load by definition is the load that leads to failure. T h e r e are, however,
Deep
175
foundations
Fig. 131. Determination of the ultimate load by the Polish method (Mazurkiewicz, 1972), (left), and by
Brinch Hansen's 80% criterion (Brinch Hansen, 1963).
cases w h e r e the ultimate load according to this definition is difficult to e s t e e m from the
shape of the load/settlement curve. Therefore, rules h a v e been w o r k e d out to ease the
determination of the ultimate load irrespective of the shape of this curve. Unfortunately,
however, such rules exist in great n u m b e r s and they do not give us o n e and the same
answer (This has been thoroughly discussed in Sellgren, 1981).
Also the w a y the pile loading test is carried out affects the result obtained. Certainly,
the best w a y of performing the test is to use the loading procedure described previously
in the case of plate loading tests (see p. 79), a so-called " M L T " (maintained load test), in
which the load is increased in defined increments and each load level is held at equal
length of time. This gives us not only the ultimate load but also the critical pile load, also
called the creep load, which represents the load w h i c h causes a sudden increase in the rate
of creep settlement of the pile (in this case equal to the creep that takes place in each load
step).
T h e ultimate load can preferably b e interpreted on the basis of the so-called Polish
m e t h o d or by Brinch H a n s e n ' s 8 0 % criterion (Fig. 131).
3.2
Methods
of
analysis
(i) Pile driving energy. In this approach it is a s s u m e d that the pile and the h a m m e r are both
totally stiff (cf. stress w a v e analysis).
Let us a s s u m e that the pile is driven by m e a n s of a gravity h a m m e r . Let us further m a k e
the following assumptions:
Deep
176
foundations
Fig. 132. Assumed correlation between pile load and pile movement. The shadowed area represents
consumed energy by the impact of the hammer.
m a s s of gravity h a m m e r
mh
mass of pile
mp
velocities of
hammer
pile
at the impac
vh
immediately after
Vh
v'p
e.
T h e correlation between the velocities of the gravity h a m m e r and the pile then
becomes:
(172)
v'u = v'p-evh
T h e law of m o m e n t u m yields:
mhvh
= mhvh
= mhv'h
+ m/p
+ mp(v
(173)
+ evh)
whence
m
v'h
* -
(174)
mh + nip
and
v'p = v
(175)
- ^ - ( \ + e )
mh 4- nip
= -(mhvh
I
+m pv p ) =
2
m h + mn
(176)
Deep
111
foundations
2
or, since vh
gravity,
2
mh +
e mp
^
W = gmhh
(177)
mh+mp
This energy is transferred into pile m o v e m e n t . A s s u m i n g that the elastic r e b o u n d of the
pile head is sel and that the remaining, plastic (irrecoverable) m o v e m e n t is spl (the set),
w e have (Fig. 132):
1
(178)
Qf(spl+-sel)
W=
Spl
2d
m
h
e m
w h e r e is the h a m m e r efficiency.
T h e m a g n i t u d e of the coefficient of restitution depends on whether or not a drive cap
is used and, in such a case, on the properties of the drive cap unit (see, for e x a m p l e ,
Chellis, 1951).
A s regards the h a m m e r efficiency w e can a s s u m e = 1 for free-fall h a m m e r s and
7] = 0.75 for single hoist line h a m m e r s .
In the S w e d i s h pile code, a modified H i l e y ' s formula, especially useful w h e n the pile
is driven b y the aid of a mandrel, has been accepted which takes into account results of
stress w a v e m e a s u r e m e n t s :
*
Spi +sel/2
Qf=
where
A E
Pp p P
EA
ff
mh>3t,
Ep = elastic m o d u l u s of pile,
Ap = cross-sectional area of pile,
Pp = density of pile,
If = length of follower,
Ef-
elastic m o d u l u s of follower,
0.8(1-0.1-^)
m h
(180)
178
Deep
foundations
Example 21: Determine the ultimate pile load according to Hiley's pile formula, on the one hand, and
according to the modifed pile formula of the Swedish pile code, on the other, if the permanent set spl
= 3 mm per blow using a 3 1 single hoist line hammer and a drop height of 0.3 m. The rebound sd is estimated
at 10 mm. The pile is 12 m in length and has a cross-sectional area of 0.25 m by 0.25 m and a modulus
3
of elasticity of 3 GPa. The density of the pile is 2.4 t/m . A 2 m long steel mandrel with the same crosssectional area as the pile is utilised for the pile driving. The elastic modulus of the steel mandrel is 20
GPa. The coefficient of restitution e = 0.5.
2
Solution: The mass of the pile mh = 120.25 2.4 = 1.8 t. For a single hoist line hammer the hammer
efficiency = 0.75.
Hiley's pile formula yields:
-6 +
) = 10.74- 10" m
-4A
2
6
2 . 4 0 . 2 5 3 0 .
0 . 2 5 2 0 0
0 . 7 5 . 9 . 8 1 . 3 . 0.5
1 0
^
0.003 + 0 . 0 1 0 7 4 / 2
i.e. the same value as that obtained by Hiley's formula.
Hiley's pile formula can be applied also for single-acting steam/air h a m m e r s . In the
case of double-acting steam/air h a m m e r s or diesel h a m m e r s being used, the impact
velocity of the r a m vr is obtained from the relation:
vr
= 27] Wlmr
(181)
w h e r e W i s the impact energy, is the h a m m e r efficiency and mr is the mass of the ram.
This yields the pile formula:
Of =
Spl + 2 el
(182)
Wr + Mp
Deep
179
foundations
Qf=
E+gmvrn
P
,
rp +fsL
083)
duced by the impact of the h a m m e r (cf. p. 162) can be used as a m e a n s to determine the
bearing capacity of the pile. By the aid of a stress transducer and an accelerometer, placed
on the pile j u s t b e l o w the pile head, the force and the particle velocity at the measuring
point is registered as a function of time. T h e data thus obtained are analysed in different
w a y s , m o s t c o m m o n l y by the so-called Case and C A P W A P m e t h o d s .
According to Eq. (167), the initial compressive force induced by the impact of the
h a m m e r can b e expressed by the relation:
= ^ ( , 0 =
dt
w h e r e = EpAplc
(184)
is the pile i m p e d a n c e ,
= particle velocity.
T h e deduction of Eq. (167) w a s based on the assumption that the pile w a s unaffected
by the surrounding soil during pile driving. In reality the soil will offer d y n a m i c resistance
against the pile motion. T h e influence of this resistance must b e taken into consideration
when predicting, b y m e a n s of stress w a v e analysis, the load/settlement characteristics of
the pile.
In the Case analysis, the d y n a m i c resistance is represented by the so-called d a m p i n g
factor Jc. It is a s s u m e d that Jc is a linear function of the particle velocity of the pile. By
the aid of the stress transducer and the accelerometer, the force Q and the particle
velocity
are m e a s u r e d j u s t b e l o w the pile head. T h e total pile force is the s u m of the force
travelling d o w n w a r d s and the force travelling u p w a r d s , /. e.\
Q = IQ + Q
(185)
Deep
180
foundations
T h e particle velocity is a function of d y n a m i c force and can be expressed in a corresponding w a y by the relation (velocity in d o w n w a r d s direction positive):
(186)
v = dw/dt = lQ/Z-Q/Z
v(t)Z]/2
= [(*+2Z/c) -
v(f+2//c)Z]/2
Q(t + 2l/c)-Zv(t
2l/c)
(188)
(189)
or
r r
(r)+Zv(Q
dyn=4l
( /+ 2 / / c ) - Z v ( ; + 2 / / c ) n
U^U;
+ Zv(i)-
(191)
t o
m ) + ()
- t
o t
(192)
Deep
181
foundations
sand
silty sand
sandy silt
0.05-0.20
0.15-0.20
0.20-0.30
silt
silty clay
clay
0.20-0.45
0.40-0.70
0.60-1.10
Solution: In this case, 7 c c a n be assumed equal to 0.3. The stress wave velocity is obtained by the relation:
9
ax
= 1610kN) we find
In the CAPWAP analysis (Rausche et , 1972) the pile and the h a m m e r are represented
by finite rigid e l e m e n t s (Fig. 133) j o i n e d together by springs representing the stiffness
of the pile and the h a m m e r while the soil resistance is represented by elastic springs and
rigid plastic s p elements in combination with viscous dashpots. A s s u m i n g that the total
dynamic and static soil resistance per unit length of the pile is Rd w e h a v e the stress w a v e
equation (cf. Eq. 163):
Deep
182
foundations
viscous dashpot
Fig. 133. Pile/soil model used in the CAPWAP analysis. Elements 1 and 2 represent the hammer and
the pile cap, elements 3-11 the pile itself. The resistance of the surrounding soil is represented by springs
in parallel with viscous dashpots.
D2
dw
2 W
= c
w h e r e pp-
dz
(193)
A
Pp p
CR)v
(194)
Deep
183
foundations
Force, kN
2000
Measured
1500
Calculated
10
15
20
2 5 ^ " 30
35
40
50
Time, ms
Fig. 134. Result of computerised CAPWAP analysis. Good agreement has been obtained between the
measured and calculated force vs. time function by successively readjusting the spring and damping
coefficients of the soil.
viscous damping,
CH=
hysteretic damping,
CR = radiation damping,
= dw/dt = particle velocity of the pile element.
Regarding the shaft resistance
ks = nGAl
9 =-
^
4G
5/(1-)
P
Dp
I+ >
2
(196)
CH
TiDpMDyfp~G
(197)
184
Deep
foundations
(198)
CR =nDpAl/pG
w h e r e D = d a m p i n g ratio,
pp = density of the pile,
= soil density.
T h e d a m p i n g ratio D can be obtained by the the relation:
= 7 ^ 1+7/*
(199)
y
by
w h e r e yh = [1 + a e x p ( - ) ] .
Yr
7r
T h e parameters a, b and D m
ax
TABLE 23
Parameters and b and maximum damping ratio D m
ax
Soil type
0.6yV- -l
1 /6
0.54/V- - 0.9
b
1/6
1_/12
0.65(1
3 3 - 1.5-logW )
-N~ ) 2 8 - 1.5-logW
m2
^ = the number of loading cycles (in this case the blow counts for the last m of penetration)
k^lGD^l-v)
(200)
(201)
Deep
185
foundations
0.85Z)/
CR =
(203)
1-v
G = G 0 ( l + 7/>)-
l(h ,
with and a c c o r d i n g to Table 4, p. 27 (/Vin Table 4 represents in this case the b l o w count
for the last m e t r e of penetration).
T h e shear strain a m p l i t u d e can be taken as the ratio of the a v e r a g e particle velocity
of the pile and the shear w a v e velocity vs of the soil, i.e.:
(204)
As the shear strain a m p l i t u d e s , p r o d u c e d by pile driving, are quite large, both shear
m o d u l u s and d a m p i n g factors m u s t b e d e t e r m i n e d as functions of the i n d u c e d strain in
Example 23: Determine the input values of shear modulus, soil stiffness, soil rigidity and damping
factors to be applied in the SVIDYN analysis for a pile shaft element at 10 m depth in saturated sand
3
with density = 1.8 t/m , angle of internal friction
32 and Poisson's ratio = 0 . 4 5 . The shear wave
3
velocity in the sand is 200 m/s. The steel pile has a diameter Dp = 0.5 m and a density pp = 7.78 t/m .
The observed number of blow counts per m of penetration = 500. The average particle velocity can
be assumed equal to 1 m/s.
Solution: The shear modulus can be determined from Eq. (25). According to Table 4 we have a - 3
0.2-log500 = - 0.540 and = 0.16. From the shear wave and particle velocities we find y= 1/200 - 5 10
2
3
and G 0 = 1.8-200 =72 10 kPa. The r m ax value depends on the depth in the soil. At a depth of 10 m, c\
2
can be estimated at 110 kPa. Choosing K0 = 0.5 in Eq. (26) we then find T m xa = [(0.75 1 1 0 s i n 3 2 ) 2 1 /2
4
( 0 . 2 5 - 1 1 0 ) ] = 34 kPa. This yields yr = 0.034/72 = 4.7 10" and ylyr = 10.6 whence yh = (10.6)[1 3
3
0.540-exp(-0.16 10.6)] = 9.6 and G - 7 2 - 1 0 / 1 0 . 6 = 6.8 1 0 kPa. Maximum damping ratio, according
to Hardin & Drnevich, becomes D m ax = (28 - 1.5-log500)/100 = 0.24. With regard to damping we have
yh = (10.6)11 - 0.708-exp(- 0.263-10.6)] = 10.1. Thus, D = 0.24-10.1/11.5 = 0.21. This leads to the
following results for a pile shaft element at 10 m depth:
soil stiffness ks = nG = 21.3 MPa.
soil resistance T m xa = 34 kPa.
3 1 /2
hysteretic damping CH = -0.50.21(7,78-6.8-10 ) = 76 kNs/m
3 1 /2
radiation damping CR = 0 . 5 ( 1 . 8 6 . 8 1 0 ) = 173 kNs/m
Deep
186
foundations
analysis.
the strength parameters of the soil. In the case of pile foundations this type of analysis,
however, is connected with several difficulties. T h u s , as previously mentioned, pile
installation in itself m a y h a v e an unpredictable influence on the soil properties and it is
therefore hard to know which properties should be used to get the right answer. T h e basis
for a successful application of geostatic analysis obviously has to b e carefully considered
from case to case.
As previously mentioned, the bearing capacity of the pile is built u p of shaft resistance
and tip resistance, the respective contribution of which depends on the soil conditions and
the pile length in the soil. T h e bearing capacity Qjof a single pile in cohesionless soils
is generally expressed in the form:
1
Pp
20
40 45
Fig. 135. Bearing capacity factors Nq and Nc and critical depth for different lplDp ratios. lp = pile length
in soil; Dp = pile diameter. (Meyerhof, 1976).
Deep
187
foundations
(205)
Qf = ptfAt+fsfAs
w h e r e ptf=
c'0tNq<pcr,
fsf= s'osten<fcr,
o'0s = average effective overburden pressure along the shaft,
Ks = earth pressure coefficient (average value),
= angle of friction at the soil/shaft interface,
As - shaft area.
If the soil consists of layers with different characteristics the total shaft resistance is
obtained b y s u m m a t i o n of the contributions given by each separate layer.
B e l o w a certain depth dcr (Fig. 135), the tip resistance does not increase with increasing
depth. T h e limits reached at the critical depth is denoted with pcr
a n d / c r.
T h e diagrams given in Fig. 135 show that Nq reaches its m a x i m u m value at about half
the critical depth. O n the other hand, the product Nqo'0
is reached.
T h e choice of the special parameters required for the determination of the shaft resistance ( S a n d Ks) is quite delicate and m o r e difficult than the choice of '. T h e parameters
are very m u c h dependent on the relative density, initial state of stress, shape and diameter
of the pile and the m e t h o d of pile installation. For e x a m p l e , in a n o n - c o h e s i v e soil with
a given value of the angle of internal friction, the bearing capacity of a b o r e d pile is
normally only o n e third to o n e half the bearing capacity of a driven pile.
If the piles are installed in normally consolidated or lightly overconsolidated cohesive
soils, then the tip resistance is usually negligible in comparison with the shaft resistance.
T h e tip resistance will not b e mobilised until the shaft resistance has passed its m a x i m u m
value (Fig. 136) and, furthermore, its value will b e comparatively very small. Therefore,
piles in soft c o h e s i v e soils can b e considered to b e h a v e as friction piles (since the term
friction piles seems to m a k e reference to cohesionless soils, piles in soft cohesive soils
are also referred to as cohesion piles or floating piles).
Also in the case of b o r e d u n d e r r e a m e d piles in stiff clay the shaft resistance will be
mobilised long before the base resistance. T h u s the b a s e resistance develops very slowly
and is s e l d o m fully mobilised until the settlement reaches 1 0 - 2 0 % of the b a s e diameter
(Burland, 1986). O n the other h a n d the shaft resistance is fully mobilised w h e n the
settlement reaches only about 0 . 5 % of the shaft diameter (or about 5 m m ) and then
remains m o r e or less constant with increasing settlement.
T h e total bearing capacity of a floating pile can b e determined on the basis of the
undrained shear strength of the soil according to the relation:
Deep
188
foundations
Fig. 136. The maximum shaft resistance in clay is reached at very small relative displacement between
clay and pile, long before tip resistance is mobilised. The influence on shaft resistance of pile material
is negligible. (Torstensson, 1973).
Qf =<xcusAs
+ cut Nc0At
(206)
w h e r e oc= empirical factor w h o s e value depends on soil characteristics, shape of pile, pile
diameter, type of pile (driven or bored), installation method, time to failure, time after pile
installation, etc.
cus = undrained shear strength along the shaft (average value),
cut = undrained shear strength at the pile tip,
= the bearing capacity factor,
Deep
189
foundations
As = shaft area,
At = tip area.
piles in soft clays can be put equal to 1 for short-term loading and 0.7 for long-term
loading ( a = 0.7 can b e considered representative of the 'creep failure' of the pile, i.e.
the load that leads to excessive creep settlement). For piles with constant cross-section
driven into in soft clays (such as concrete piles), has to be taken s o m e w h a t lower ( 0 . 8 0.9 of the values given above). For bored straight-shafted piles and bored u n d e r r e a m e d
piles in stiff clays with cu determined by unconsolidated, undrained triaxial tests on 38
m m diameter samples, a c a n b e assumed equal to 0.45 ( S k e m p t o n , 1 9 5 9 ; B u r l a n d , 1986).
In layered soil with varying characteristics, the total shaft resistance is obtained by
summation of the contributions given by the various layers.
In the case of a pile w h o s e diameter varies linearly with depth and which is installed
in clay with an undrained shear strength increasing linearly with depth (cu = cu0 + kz), the
bearing capacity can be determined by the relation:
1
Qf = ccnlp[-(Dph
1
+ Dpt )cu0 + - (Dph + 2Dpt )klp]
6
(207)
Qf = As&0s(l
where
(208)
190
Deep
foundations
Example 24: A 15 m long timber pile with a tip diameter of 150 mm is driven into a clay deposit with
an undrained shear strength c M= 10 + 1. 8z kPa, where z, in m, denotes the depth below the ground surface.
3
The clay is normally consolidated and has a density of 1.6 t/m . The groundwater level is 2 m below the
ground surface and the pore pressure distribution with depth is hydrostatic.
Determine the ultimate load of the pile if the diameter of the pile increases by 8 mm per m from the
pile tip upwards. Use both undrained and drained analysis. In the latter case the angle of internal friction
can be taken as '' - 20. The friction angle in the pile/soil interface <5can be assumed equal to '.
Solution: The pile head diameter Dph = 150 + 8-15 = 270 mm.
Undrained analysis: Qf= 15[(0.27 + 0 . 1 5 ) 4 0 / 2 +(0.27 + 2 0 . 1 5 ) 1.8 15/6] = 220 kN. For longterm loading can be put equal to 0.7 which yields / = 154 kN
Drained analysis: The effective stress distribution increases linearly from about 20 kPa (= 2-1-9.81)
at the ground surface to 31 kPa (= 1.6-2-9.81) at 2 m depth and from there linearly to 108 kPa (= 31 +
/
13-0.6-9.81) at 15 m depth. Since the clay is normally consolidated o {j&c - G'^G'V = 1.
We find:
Qf= [2(0.27 + 0.254)(20 + 31)/4 + 13-(0.254 + 0.15)(31 + 108)/4](1 - sin 20)-tan 20 = 147 kN.
method.
(b)
Depth of e m b e d m e n t
10
d/Dp
Fig. 137. values vs. pile embedment for driven piles (Bustamente et al, 1981).
(a) Very dense sand and gravel.
(b) Medium dense sand, firm silt and firm clay.
(c) Loose silt, clayey sand and soft clay.
Deep
191
foundations
Ptf=
where '0t
(209)
Pie ^yJpixPllPn
'"Pin
(210)
T h e correlation between shaft resistance/ v ^and the net limit pressure *p{ can be obtained
from Fig. 138. T h e shaft resistance depends not only on the net limit pressure but also on
the m e t h o d of pile installation and the pile material.
150
( a ) _
120
*
*
Ou,
Jt
/
90
'
i
60
/ _ - v u/
/
(d)
/
30
0.5
1
1.5
2
Net limit pressure *pt, MPa
2.5
Deep
192
foundations
Example 25: Determine the bearing capacity of a concrete pile with a cross-sectional area of 0.25 m
by 0.25 m driven to a depth of 8 m in a sand deposit whose properties, determined by pressuremeter tests,
are given in the figure shown below. The groundwater level is at 1 m depth. The unit weight of the sand
3
3
is 18 k N / m above and 11 k N / m below the groundwater table.
Pressuremeter modulus
0
10
Ept
L i m i t p r e s s u r e pt, M P a
20
16.5
\
\
_ _
/
4 1)
1
r j
16
Solution: The point resistance is governed by the bearing capacity factor K~ 2.7 and the pressure limit
ple - 0.5 MPa. The shaft resistance is governed by the average of pt from the pile head to the tip which
is equal to (2.3 + 0.8 + 0.8 +0.6 + 1.2 + 1.8 +1.0 + 0.5)/8 = 1 . 1 MPa. From Fig. 133, curve (b), we find
the average v a l u e ~ 80 kPa, and from curve ( d ) , / ^ ~ 110 kPa.
2
U s i n g / 9 /= 80 kPa, the bearing capacity of the pile becomes Qf= 0.25 ( 18 + 7 11 +2.7-500) + 804-0.25-8
= 730 kN, while, using fSf= 110 kPa, the bearing capacity becomes Qj - 970 kN.
+ K*Pi)At+f5fAs
(211)
methods.
/ =
4 0 0 ^ ^ + 2 ^ 3 ^
(212)
w h e r e 3 0 represents the /V 3 0 count at the pile tip level and N30s the average / V 3 0 count
along the shaft of the pile.
Deep
193
foundations
At and As in m ) :
Qf=KN30tAt
+ 1 0 ( t f 3 0 / 3 +1)A,
(213)
w h e r e = 2 0 0 for clayey silt, 300 for sandy silt and 4 0 0 for sand.
T h e tip resistance should b e limited to a m a x i m u m of about 10 M P a .
B u s t a m a n t e and Gianeselli (1982) p r o p o s e the following relation based on the results
of C P T investigations:
(214)
Qf=qcpkcAt+fcAs
w h e r e qcp = average point resistance in C P T from 1.5D t above the pile tip to 1.5D, below
the pile tip,
kc = parameter w h o s e size depends on pile type and soil characteristics,
fc =
qcs/cx,
<7 c(MPa)
/cm
k P
( a)
IA
IIA
<5
>5
0.40
0.45
0.50
0.55
60
60
35
1
35 ( 8 0 )
5 - 12
0.40
0.50
100
80(120)
> 12
0.30
0.40
150
120 (150)
<5
>5
0.20
0.20
0.30
0.40
100
150
35
120 (150)
The values in parenthesese refer to very careful pile installation, involving minimum disturbance of soil.
194
Deep
foundations
Example 26: Pile loading tests were carried out on concrete piles with square cross-section and a side
length of0.235 m. Three piles, driven to depths of 15 and 19 m, were loaded to failure. Results of piezocone soundings showed that the soil consisted of clay underlain by sand at 5 m depth. A typical result
of the piezo-cone sounding and the pile tip level of the three tested piles are presented below. Calculate
the bearing capacity of the piles on the basis of the penetration resistance according to Bustamente &
Gianeselli.
Point resistance
MPa
0
10
Pore pressure
MPa
S l e e v e friction
MPa
Solution: The CPT point resistance at the pile tip qc ~ 8 MPa for all the piles tested. As regards the pile
shaft we find qc 0.6 MPa to a depth of 5 m, qc 6 MPa, on the average, from 5 m to 15 m depth and
qc ~ 7.5 MPa from 15 m to 19 m depth. The piles belong to Category IIA. Since there is no value for
clay, the contribution to the bearing capacity of the clay layer will be based upon the value for silt.
The bearing capacity of the different piles can now be calculated.
2
Pile PI: Qf= 0.5-8-0.235 + ( 5 0 . 6 / 6 0 + 10-6/100)40.235 = 0.88 MN
Piles P2 and P3: Qf = 0.88 + 4 ( 7 . 5 / 1 0 0 ) 4 0 . 2 3 5 = 1.16 MN
According to the loading tests we have:
P i l e PI: / = 0 . 6 8 MN
P i l e P2: Qf = 1.07 M N
P i l e P3: Qf= 1.16 M N
The ratio of calculated to observed ultimate pile loads varies from 1.0 to 1.3. The agreement between
theory and reality in this case is quite satisfactory.
Deep
195
foundations
25
30
40
50
Design parameter Ns
5
10
Relative depth of penetration
60
15
lxIDp
Fig. 139. Top: Relation between sounding resistance and design parameter Ns. Range of validity for
different penetration methods shaded. Bottom : Critical tip resistance vs. pile penetration depth for different
values of Ns (After Berggren, 1981).
be a s s u m e d to increase linearly with increasing load (Fig. 139). In the top diagram, a
design parameter Ns is determined on the basis of the results of different penetration tests.
Then knowing Ns, the critical tip r e s i s t a n c e p t c is obtained from the b o t t o m diagram and
the corresponding pile tip settlement from Fig. 140.
196
Deep
foundations
60
201
I
I
I
I
I
I
I
I
I I
0.05
0.10
0.15
Ratio of elastic settlement limit to pile diameter
Fig. 140. Pile tip settlement at critical point resistance as function ofNs and uniformity coefficient
Cv.
Fig. 141. Equipment used for dynamic preloading of bored piles, (a) 3.8 t hammer, (b) Guide casing,
(c) Measuring cable, (d) Casing for work, (e) Damper, (f) Measuring cell, (g) Pile point of steel.
Deep
197
foundations
increase the bearing capacity of the piles, thereby reducing the required length of the piles
and in c o n s e q u e n c e leading to cost-effective solutions.
T h e r e c o m m e n d e d procedure of d y n a m i c preloading is as follows (Berggren &
Bengtsson, 1985). A special set-up for d y n a m i c preloading is utilised, consisting of a 3.8
t h a m m e r , a guide casing and a 'pile point' equipped with accelerometer, strain gauges
and displacement transducers, Fig. 1 4 1 . T h e diameter of the 'pile point' should not be
below 0.8 times the pile diameter. D y n a m i c preloading is carried out in several steps by
gradually increasing the drop height of the hammer, e.g. 0.2 m, 0.5 m, 1.0 m, 2.0 m and
4.0 m, and so that the ratio of remaining plastic settlement spi to m a x i m u m settlement s m a x
caused by the impact of the h a m m e r decreases in each step. T h e increase in drop height
can take place when the ratio spi/smx<
Example 27: Determine the critical point load of a bored pile, 0.9 m in diameter, by the aid of the graphs
shown in Fig. 139. The pile tip is at 1 m depth in gravel with a cone resistance qc - 8 MPa. The gravel
is overlain by a soft clay deposit, 10 m in thickness. The groundwater level is situated at a depth of 2
3
3
m. The unit weight of the clay is 16 k N / m and of water-saturated gravel 21 kN/m .
Solution: From Fig. 139 we find the design value Ns - 35. The depth of penetration into the gravel is
1 m which yields l\IDp = 1 . 1 . From the bottom diagram we find ppj~ 22'0 = 22 (2-16 + 8-6 + 1-11)
3
2
= 2-10 kPa. Thus, the critical point load is equal to 20.9 /4 = 1.3 MN. The elastic settlement limit
according to Fig. 140 is around 0.09 to 0.14 m depending on theuniformity coefficient of the gravel.
4.
W h i l e the bearing capacity is decisive of the ultimate state, the s e t t l e m e n t a b o v e all the
differential settlementis decisive of the serviceability state. T h e possibility of predicting
pile settlement with satisfactory accuracy is limited. T h e r e are several reasons for this.
Pile installation generally alters the stress/strain conditions in the soil. In soft, sensitive
clays disturbance effects m a y give rise to a decrease of the preconsolidation pressure
followed by consolidation. In loose cohesionless soil pile driving causes compaction
which will i m p r o v e the settlement characteristics. Driven piles will b e subjected to
residual stresses which m a y
factors not mentioned here can increase the difficulties of settlement prediction. Therefore,
in practice the permissible pile load is mostly coupled with a certain factor of safety
against pile failure.
A s a general rule it can b e stated that in-situ methods for determination of the deformation
characteristics are preferable in the case of cohesionless soils w h e r e a s laboratory methods are generally adaptable in the case of cohesive soils. Stress-wave m e a s u r e m e n t s and
analyses can b e used not only for determination of the bearing capacity of piles but also
for prediction of pile settlement under working load condition.
T h e column and wall loads in a building are mostly too large to b e carried by single
piles and, therefore, pile groups h a v e to b e installed. E v e n if the settlement of a single pile
Deep
198
foundations
can be predicted with satisfactory accuracy, the settlement of a pile group can b e entirely
different. This is important and has to b e recognised.
T h e settlements occurring in the serviceability state are limited which, in the case of
friction piles, m e a n s that load will be carried m o r e or less merely by shaft resistance. T h e
tip resistance will either not be mobilised at all or will b e of negligible m a g n i t u d e . On the
other hand, piles driven to refusal, or with the pile tip in layers of high bearing capacity,
such as hard till, will carry the service load almost entirely by tip resistance. D y n a m i c
preloading of the pile tip, as is sometimes utilised with bored piles, represents another
case w h e r e the service load is carried mainly by tip resistance.
(i) The pressuremeter
w h e n the shaft resistance is not yet fully mobilised, will mainly b e governed by the shear
m o d u l u s of the soil. T h e foremost aim of the pressuremeter is to determine the shear
m o d u l u s of the soil and the results of pressuremeter tests therefore serve as a reliable basi s
for pile settlement calculations. A strict analysis based on the results obtained by the
M n a r d pressuremeter, taking into account both shaft and tip resistance, was carried out
by Cassan (1966). In C a s s a n ' s analysis it is a s s u m e d that the shaft resistance fs = Bs(z)
and the tip r e s i s t a n c e p p = Sp/Dp where s(z) and sp represent pile settlement at, respecti vely,
depth and pile point level and and represent soil resistance. C a s s a n ' s analysis, later
modified by Sellgren (1981), yields the following relation:
where
\ + [/()]1<()
nDp
+ )
Q_
\-QIQf
Q
\-QIQf
(215)
EpDp
with Lq = 0.3 m for driven piles and L q = 0.9 m for bored piles,
= GP/L$
average of GPR along the pile shaft and GPRT = GPR at the pile tip),
EP = m o d u l u s of elasticity of pile,
Dp = pile diameter.
Using instead of GPR the pressuremeter m o d u l u s EPR = 2Gpr(l+vs)
1
i r r and
=4.5EPR
should b e e x c h a n g e d for bp w h e r e
Deep
199
foundations
TABLE 25
Examples of values for bored and driven piles, assuming Ep = 30 GPa.
lp (m)
Epr (MPa)
(m m/M )
bored piles
Dp=03m
Dp =0.6 m
Dp=()3
12.2
8.6
6.8
5.8
3.8
2.7
3.3
2.9
2.8
1.4
1.1
5
10
20
10
34.2
22.0
14.6
16.8
10.5
6.5
5
10
20
50
7.7
5.8
5.0
3.6
2.5
1.9
driven piles
m Dp - 0 . 6 m
1.0
D0
9APEPR
MPa
L i m i t p r e s s u r e Pj, M P a
200
Deep
foundations
Solution: The average value of the pressuremeter moduli along the pile shaft is Epr = 6.7 MPa. At the
pile point, the pressuremeter value is taken as the harmonic mean of 2.1 and 16.2 MPa, i.e. Epr = 3.7
MPa. This yields = 1.25-6.7 = 8.4 MPa/m and = 13.5-3.7 = 50.2 MPa. The equivalent pile diameter
3
1 /2
1
Dp = 40.25/ = 0.318 m, whence = 2-[8.4/(30-10 0.318)1 = 0.0593 m" .
The value according to Eq. (214) becomes:
1
a- 0.25
1+-
50.2
0.0593-30-103-0.318
tanh(0.0593- 8)
: 0.0139
ra/MN,
from which:
s = 0.0139(0.73/3)/(1 - 1/3) = 0.005 m
(ii) Empirical
methods.
s=fs/Ks
w h e r e fs = a v e r a g e frictional resistance along the shaft,
K = a v e r a g e pile d i s p l a c e m e n t m o d u l u s .
10
2*10
Fig. 142. Diagram for determination of settlement of friction pile (Hansbo & Bengtsson, 1979). Legend:
Ep= Elastic modulus of pile; Dp= pile diameter; lp= pile length in soil; G= shear modulus of soil; K =
pile displacement modulus.
Deep
201
foundations
for cohesionless
soils.
In c o h e s i v e soils, the settlement at 8 5 % of the u l t i m a t e load and at failure can be
estimated at respectively 2.4 and 4 times the the settlement at half the u l t i m a t e load
(Torstensson, 1973).
In c o h e s i o n l e s s soil, the settlement at 7 5 % of the u l t i m a t e load and at failure can be
estimated at respectively 2 a n d 5 times the settlement at half the u l t i m a t e load (Sellgren,
1985).
Example 29: In order to investigate the load vs. settlement behaviour of piles in soft clay, test loading
was carried out on two piles, one with a diameter of 0.33 m and the other with a diameter of 0.5 m, both
driven to a depth of 25 m through a top layer of peat, 3 m in thickness, underlain of a 10 m thick layer
of volcanic clay on alluvial, silty clay (Simonini & Soranzo, 1988). A finite element analysis was carried
out, based on elastic properties determined by means of unconfined compression tests at half failure
load. The undrained shear strength was determined by means of field vane tests. The mean value of the
undrained shear strength was 13 kPa in the volcanic clay and 26 kPa in the alluvial clay. The result of
the test loading and the finite element analysis is shown below.
Determine the load vs. settlement behaviour of the piles by means of Eq. (217) and Fig. 142.
Pile load, kN
Pile load, kN
Solution: The failure loads of the two piles is determined according to Eq. (207). The contribution to
the bearing capacity of the peat layer is neglected since the load/deformation characteristics of peat are
quite different from those of the clay.
For the pile, 0.33 m in diameter, we have (a assumed equal to unity):
2
Qf= 0.33(10 13 + 12-26) + 9260.33 /4 = 478 kN
and for the pile, 0.5 m in diameter
2
Qf= 0.5(10 13 + 12-26) + 9260.5 /4 = 740 kN
The mean value of the shear modulus along the shaft of the piles can be estimated at:
Gs = 1 5 0 ( 1 0 13 + 12-26)/22 = 3000 kPa
4
Assuming that the elastic modulus of the pile is 30 GPa we find Ep/Gs = 1 0 . For the pile, 0.33 m in
diameter , we have lplDp = 22/0.33 = 67 which, according to Fig. 142, yields KsDpIGs = 0.33 and thus
Ks = 3.0 MPa/m
The settlement at half the failure load for the pile, 0.33 m in diameter, ( g = 239 kN) becomes:
3
3
s = 239/(310 0.3322) = 3.5-10~ m
and at 85% of the failure load ( g = 406 kN)
3
3
s = 2.4-3.5-10- = 8 . 4 - 1 0 - m
202
Deep
foundations
For the pile, 0.5 m in diameter, w e have lplDp = 44 whence KsDp/Gs = 0.41. In this case we find Ks
= 2.5 MPa/m. Thus the settlement at half the failure load( = 370 kN) is equal to:
3
3
s = 370/(2.5 10 0.522) = 4 . 3 1 0 - m
and at 85% of the failure load ( = 629 kN)
3
3
s = 2.4-4.3-10- = 10.3-10~ m
Assuming a safety factor of 3, the calculated and measured results are as follows:
Pile diameter, m
Working load, kN
Approximate
0.33
0.50
160
245
2.3
2.9
Settlement, mm
FEM
2.5
2.4
Measured
1.4
1.5
Example 30: Determine by the approximate method, used in Example 30, the settlement of the pile
according to Example 28.
Solution: We have IJD? = 8/0.318 = 25. The Gs value can be put equal to Gpr which yields Gs = 6.7/2.6 =
4
2.6 MPa. Thus EJGS - 30-103/2.6 =1.15 10 which gives ^=0.46.
Hence we find
Ks=0.46-2.6/0.318
= 3.8 MPa. The settlement for a load of one third of the failure load 800 kN becomes:
s = 0.8/(3-3.8) = 0.007 m
which is 14% above the value obtained in Example 28.
(218)
5.1
Ultimate
resistance
Deep
203
foundations
Fig. 143. Pile movement and ultimate earth pressure distribution against horizontally loaded short pile
in cohesionless soil. Restrained pile.
lateral limiting capacity will b e reached either w h e n the earth pressure reaches its upper
limit or w h e n the pile is broken in bending.
(i) Non-cohesive
(Barton, 1982), indicate that the limiting earth pressure against the pile, below a depth
of about 1.5 pile diameters, follows the relation K^Dp\,
pressure coefficient [Kp= t a n ( 4 5 + 072)], ' is the effective overburden pressure and
Dp is the pile diameter. Nearest to the ground surface, the limiting pressure was found to
b e about KpDpa'v
Kj-o'v
Let us a s s u m e that the point of rotation of a short pile subjected to a horizontal force
at height e a b o v e the ground surface is at depth lp-2a
that the limiting earth pressure against the rotating pile is distributed according to Brinch
Hansen (1953) which can b e simplified according to Fig. 144c (cf. Fig. 205).
Fig. 144. Pile movement and ultimate earth pressure distribution against horizontally loaded short pile
in cohesionless soil. Unrestrained pile, (b) Probable earth pressure distribution, (c) Assumed earth
pressure distribution.
Deep
204
foundations
Fig. 145. Nomogram for determination of lateral capacity of short piles in cohesionless soil.
conditions of equilibrium:
(219)
Hf = Kp YD/-^--a(lp--2)]
Hfe = [(
a j2 (L-P
- - ) -
3a)
3
(220)
Hf - KpjDp-^
while, if it is yielding (one hinge with a yielding m o m e n t Mpf,
(221)
Eq. (221) takes the form:
Deep
205
foundations
Fig. 146. Assumed limiting earth pressure distribution against long piles in cohesionless soil.
M pf
(L - 3) 2
a 2?
Kp YDp[^-^-a(lp--) ]
(222)
Fig. 147. Nomogram for determination of the lateral capacity of long piles in cohesionless soil.
Deep
206
foundations
Example 3 1 : In a test carried out by Koskinen (1991), the lateral bearing capacity of a steel pile, 273
mm in diameter, embedded in sand was investigated. The sand had the following characteristics:
3
internal angle of friction '= 37 and unit weight / = 9 k N / m . The length of embedment of the pile
was lp = 4 m and the lever arm of the horizontal load e - 0.8 m. The yield moment Mpj of the pile was
assumed equal to 92.7 kNm. Determine the lateral capacity on the basis of Eqs. (219-220) and (223-224).
Solution: We have e ID = 0.8/0.273 = 2.93 and lpID =4/0.273 = 14.65. The earth pressure coefficient
2
2
4
) = 114.7. From the diagram, Fig. 147 (the pile
Kp = tan (45 + 3 7 7 2 ) = 4.02. The value Mpfl(Kp YD
2
3
is assumed to be broken in bending), we find H^k/yD^)
- 17 whence Hf= 174.02 90.273 = 50
kN (an exact solution yields Hj= 49.9 kN). From the diagram, Fig. 145 (the pile is not broken), we find
2
3
Hfl(Kp YDp )
20 whence Hf= 59 kN (an exact solution yields Hf= 58.6 kN). In reality the pile was
loaded up to 60 kN which can be assumed to correspond to the failure load (the occurrence of failure
is not distinct). The horizontal displacement of the top of the pile at = 54 kN was 0.13 m.
2
f
Hf - KpjDp
(223)
(224)
If the pile is restrained (two hinges), Eq. (223) will be replaced by:
(225)
Mpf=Kp YDp -
Hf,which
soils. In the case of cohesive soils the lateral earth pressure against a short,
restrained pile varies normally as shown in Fig. 148 (left). It can be a s s u m e d (Fig. 148,
right) that the lateral resistance, below a depth of 1.5 pile diameters, is equal to 9 times
the undrained shear strength of the soil multiplied by the pile diameter (Broms, 1964).
If the undrained shear strength cu is constant with depth the conditions of equilibrium
of a short, unrestrained
Hfie
zo) = 9 c
d /
- [
^ -
(226)
(227)
Deep
207
foundations
9c,pn
up
%cj}p-\2cj>r
Fig. 148. Limiting earth pressure distribution against a short restrained pile in cohesive soil (left) and
assumed limiting earth pressure against unrestrained, short pile (Broms, 1964).
Hf = 9cuDp(lp-\5Dp)
(228)
If the pile is yielding (one h i n g e with a yielding m o m e n t Mpj, Eq. (227) takes the form:
(229)
Fig. 149. Assumed limiting earth pressure distribution against a long pile in cohesive soil.
(230)
208
Deep
Fig. 150. Nomogram for determination of lateral capacity of short piles in cohesive soil.
Fig. 151. Nomogram for determination of lateral capacity of long piles in cohesive soil.
foundations
Deep
209
foundations
Mpf
(f-\.5D)
^
= Hf(e +f)-9cuD/
2
pJ
(231)
Mpf = 9cuDp
f -{\.5DD)
(232)
Deflection
dy
dz
= ^
psDD
J L
(233)
Eplp
w h e r e Kls =
y =
(234)
Deep
210
Transverse force, kN
-5
Bending moment, k N m
10
foundations
Displacement, mm
6
10
20
Fig. 152. Internal forces and deflection of a concrete pile subjected to a horizontal load H= 10 kN at
a height of 0.2 m above ground surface. Pile with square cross section, 0.35m in width. Modulus of
lateral subgrade reaction kls - 40 MPa. Elastic modulus of pile Ep - 30 GPa. (= 5.7).
_
k
4 / KisDp
~ \ a e
_ 4 /
~ \ a e
k[s
p
M - [eK(cosKz
+ sinKz)
+ SINK*z] e x p ( - K 2 )
(235)
[COSKZ
(2 + l ) s i n r c ] e x p ( - x z )
(236)
Deep
211
foundations
value of kis can b e a s s u m e d equal to 8 0 c u . Of course, the kls value thus obtained can only
be applied w h e n the lateral soil r e a c t i o n p s = kls y is b e l o w the pressure leading to failure.
In cohesionless soil the coefficient of subgrade reaction according to Terzaghi (1955)
can be a s s u m e d to increase linearly with depth according to the relation kls = nhz, w h e r e
nh can b e estimated at 7 5 / t o 2 2 5 / f o r loose sand and at 7 5 0 / t o 1 5 0 0 / f o r dense sand
w h e r e / i s the effective unit weight of the soil.
In practice, a possible way of determining the m o d u l u s of lateral subgrade reaction is
by using the results of pressuremeter tests. According to the correlations between
settlement and pressuremeter m o d u l u s Epr previously given, Eq. (152), p. 146, w e find:
9F
k ls
if Dp < 0.6 m,
kls
(237)
2 ( 2 . 6 5 ) + 1.5
9E
^
a
2(DQ/Dp)(2.65Dp/D0) +
(238)
15a
i f D p > 0 . 6 m,
w h e r e is the rheological coefficient at depth (Table 2 1 , p. 139),
D0 = 0.6 m.
T h e kls value determined accordingly can only be applied w h e n the lateral soil reaction
Ps
kisy
c r
pressure, the lateral subgrade reaction b e t w e e n creep pressure and failure can b e assumed
equal to half the values determined by Eqs. ( 2 3 7 - 2 3 8 ) .
N e a r the ground surface, the lateral subgrade reaction is less than at depth because of
ground h e a v e . T h u s Eqs. ( 2 3 7 - 2 3 8 ) are only valid below a critical depth zc w h i c h varies
from about 2Dp for cohesive soils to about 4Dp for granular soils. A b o v e the critical depth,
the kh value should b e replaced b y kls where:
. \+zlz
(239)
where
and EprR
respectively.
(240)
+ EprR
pressuremeter
moduli,
Deep
212
foundations
(241)
Example 32: Determine the horizontal displacement at the ground surface and at 1 m depth below the
ground surface of a steel pile with square cross-sectional area, 0,10 m in width, installed in clay with
constant shear strength cu - 30 kPa, if the pile is subjected to a horizontal load of 5 kN at a height of 0.5
m above the ground level.
4
Solu non : For a steel pile we have Ep=210 GPa. The moment of inertia islp=0.1 /12
6
6
1 /4
1
that kls = 20cu we find = [600/(4-210-10 8.31- )] = 0.542 m" .
The lateral movement becomes:
= 0 (ground level)
y = (250.542/600)(0.50.542 + 1) = 0.011 m
= 1 m
= 8.3 l O ^ m . Assuming
6.
BUCKLING OF PILES
Fully embedded
piles
(i) Straight pile. T h e buckling load of a straight pile e m b e d d e d in soil can be analysed
in the same w a y as in the case of a column e m b e d d e d in an elastic m e d i u m (Timoschenko,
1936). A s s u m i n g that the pile is hinged at both ends we have:
is
w h e r e kls,lp,Ep
m , pip
(242)
and Ip as above
Deep foundations
213
1
+
* - * w
* >
>
= I (-
+ -)
= 0
TCEpIp
(244)
X = X = K(EpIp/kls)U*
from which:
(246)
Qb,min ~
Introducing = Qb/Qb,mm
a dn
k[sEpIp
In practice, Qb
m in
(247)
I. Then,
if m is not an integer, the integer values mx and m2 are chosen that are closest to m (mx
m2). This yields t w o values of i:
<m<
I] = { -m
lmx
T h e value ix or i2 that gives the lowest value yields the buckling load (Mascardi,
1970):
(248)
Qb ~
Qb, min
If the pile is free to m o v e laterally at either of its ends the buckling load will b e only
half the value obtained for the hinged pile.
Example 33: Determine the buckling load of a straight, 5 m long steel pipe pile (outer diameter = 50
mm; wall thickness = 4 mm) installed in clay with an undrained shear strength of 10 kPa.
4
Solution: We have kls = 200 kPa and Ip = ( 0 . 0 5 - 0.042 )/64 = 1.54 10" m . Furthermore, Ep = 210
GPa. Thus:
6
7 1/2
(2 /? = 2 ( 2 0 0 2 1 0 1 0 1 . 5 4 1 0 - ) = 161 kN
6
7
1 /4
A = (210 10 1.54-10~ /200) = 1.99
m = 5/1.99 = 2.51 and, consequently, m{ = 2 and m2 = 3 from which:
1 1 = 2.51/2 = 1 . 2 5
1 ^ = 1.10
1 2 = 2.51/3 = 0.84
v2= 1.06
( 2 4 3
Deep
214
foundations
Fig. 153. Illustrative example of pile 'wandering'. A 37 m long steel -pile installed by means of the
pile driver shown to the left (top picture) is being curved during installation, then moving in a wide bow
below the ground surface, finally penetrating the ground surface 20 m away and hitting the parked
Honda which is being lifted 1.5 m before discovery. (By courtesy of Herkules Grundlggning AB).
The buckling load is equal to 1.06-161 = 171 kN
The steel stress in the pile under the buckling load is
= 1 7 1 4 / [ ( 0 . 0 5 2 - 0 . 0 4 2 2 ) ] = 2 9 6 1 0 3 kPa
(ii) Initially bent pile. B y experience w e k n o w that it is difficult to maintain the straightness
of driven piles (Fig. 1 5 3 ) . T h e danger of the pile being curved during installation is
particularly important in the case of small-diameter piles, so-called micro-piles. T h e
bearing capacity of such initially curved piles is governed by elasto-plastic buckling
(buckling in combination with plastic yield). A pile may, of course, b e curved along its
w h o l e length and fail in bending because the lateral soil resistance is insufficient. In the
case of buckling, however, the pile is a s s u m e d to b e initially curved between the
inflection points determined by the sinusoidal half w a v e s appearing in buckling (Fig. 1 5 4 ) .
A s s u m i n g that the initial deflection of a pile, hinged at both e n d s , is 0 and the total
deflection caused by the application of the load is ^ + y w e h a v e (cf. E q . 2 4 3 ) :
Qb = 21(-
(249)
+ -*-)->
2 , 2Xk,s
2(-
A3
**Epl/y
y
+ 0
(250)
Deep
215
foundations
line of thrust
Y -
^ _ \{ I + I 2 I
e of thrust
Fig. 154. Buckling of initially bent pile. Radius of curvature = p. Initial deflection from line of thrust
= 0. For a pile curved along its whole length (right picture), the thrust line can be assumed to coincide
with a circle drawn through A, and C. In this case <50 can be taken as the average of the deflections
and b\ (Bernander & Svensk, 1970).
QbMm
2 / k i s E p I Pp
(251)
J + <So
T h e stresses in the pile are obtained from the relation (Bernander & Svensk, 1970):
Q
.Mfn
(252)
x 2
y y ^ p h
QbMmj,
lim
-7?,max
0 +
(253)
Deep
216
foundations
( p, yield ~ jp) An
(254)
a,iim
P
V1 + yv ++ Q
g f t, A
2 W
y +<5Q _
S0y/kl5EpIp
(255)
2y/khEpIp-Qbyim
Inserting the expression for (y + <50)/2 into Eq. (254), w e finally have:
a,Hm
\[Qi
+ aod
-^y:I
ftoo + ^ ) ] - 4 a
2
(256)
w h e r e Qx = ( " ^ 1 ( 1- ) ,
and
0.001
0.002
0.003
0.004
0.005
Deep foundations
T h e value of can b e found from the correlation ( p =radius of pile curvature):
Eplp
8<50
whence:
'^ /
Aff =
(257)
F r o m the results of Eqs. (246) and (256) w e find that the limiting load of initially bent
piles is generally governed by yield rather than by elastic instability (Fig. 155).
6.2
Partially
embedded
pile
For a pile that is not fully e m b e d d e d in soil, it can b e a s s u m e d that the pile is restrained
(rigidly fixed) at a depth below the ground surface equal to (Davidson et al,
1965):
ls=\A{Eplplklsy
w h e r e EpIp
(258)
(259)
T h e buckling length depends on the degree of restraint at the pile head. For a pile that
is rigidly fixedat the pile head it can b e a s s u m e d that = 0.5(ls + / 0 ) w h e r e l0 is the free
length of the pile. T h e length of the e m b e d d e d part of the pile should b e > 2.5ls.
Instability is reached for the Euler buckling load:
Qb = n*EpIpM
(260)
Example 34: Determine the buckling load of a steel -pile, HEB 200, installed in sand to a depth of
10 m and with a free length of 5 m. The sand has a pressuremeter modulus Epr = 6 MPa.
Solution: The coefficient of subgrade reaction, determined on the basis of the pressuremeter tests (Eq.
238), becomes equal to:
9-6
kh =
7T,
= 16.5 MPa
1 / 3
22.65 +1.5/3
4
In the weak direction of the beam we have Ip = 28.43 1 ( H m . Moreover Ep = 210 GPa.
Deep
218
foundations
This yields:
3
6
1 /4
/, = 1.4(21010 28.4310- /16.5) = 1.1 m
= ( 1 . 1 +5)/2 = 3.05m
The buckling load is equal to:
2
3
6
2
Qb = 21010 28.43 1 0 / 3 . 0 5 = 6.3 M N
Since the cross-sectional area of the -pile is 9.104-10
approximately 690 MPa (in most cases above the yield stress).
7.
7 . 1 Bearing
capacity
in load/settlement behaviour between the individual piles in the group. This is particularly
the case with friction piles. If the soil were still h o m o g e n e o u s after pile installation, the
frictional forces along the pile shafts would cause a settlement bowl underneath the pile
footing with a tendency towards increasing load share a m o n g the outer piles and a
decreasing load share a m o n g the inner piles in the group. In granular soil, however, pile
installation, particularly in the case of displacement-type piles, generally causes changes
in the deformation characteristics of the soil inside the pile group in such a way that the
load distribution a m o n g the piles will change in another direction than that mentioned.
T h u s , the piles in the centre of the pile group will generally be subjected to a higher load
than the outer piles.
T h e ultimate bearing capacity of a pile group with friction piles is generally different
from the s u m of the ultimate loads of the individual piles in the group.
According to Kishida & Meyerhof (1965) the total bearing capacity of a piled
foundation can be estimated as the bearing capacity of the foundation and its surcharge
effect on the point resistance of the piles in the g r o u p e i t h e r by considering the bearing
capacity of the pile cap as a whole and the bearing capacity of the individual piles in the
group (individual pile failure) or by considering only the contribution to the bearing
capacity of the outer rim of the cap, outside the pile group (the pier), and the bearing
capacity of the pile group as a whole (pier failure), Fig. 156.
In an extensive test series on bored pile/cap/soil interaction effects in sand, comprising
51 pile groups and 23 single piles, Liu etal. (1985) investigated the group effect on both
pile groups with free space between cap and soil and pile groups with pile cap in direct
contact with soil. T h e pile groups consisted of 2 - 1 5 piles, 1 2 5 - 3 3 0 m m in diameter, and
with lengths of 8 - 2 3 times the pile diameter. Pile spacings were 2 - 6 times the pile
diameter. F r o m the results obtained they found no evidence of block failure and therefore
propose that the analysis of pile group failure be based upon the bearing capacity of the
cap as a w h o l e in combination with individual pile failure.
T h e ultimate load Qgrf of a pile group is generally expressed as:
Deep
219
foundations
tr
1^--
\
N\
\
1
Leap failure
\
base failure zone J
(b)
(a)
Fig. 156. Assumed failure zones at piled foundations: (a) pier failure, (b) individual pile failure. (Kishida
& Meyerhof, 1965)
2^=77/7*0,/
(261)
= ultimate load of a single pile under the s a m e soil conditions as for the pile
group.
(262)
Deep
220
foundations
(263)
Fig. 157. Average lateral earth pressure coefficient with reference to the centre pile in pile groups with
pile spacing equal to 4b, driven into sand with relative density ID ~ 4 7 % . The letter L represents results
after loading test. Arrows indicate change in lateral earth pressure with time, in days. (Ekstrm, 1989).
Deep
221
foundations
TABLE 26.
Example of ultimate total pile loads and ultimate shaft loads observed by Ekstrm ( 1989). Free-standing
square pile groups. Individual pile tests. Pile loads in kN.
Pile spacing
Initial ID %
3bp
47
4bp
47
6.5bp
60
3bp
47
4b
47
Total
Single pile
6.5bp
60
Shaft
8-10
11
24
3-6
10
Centre
Corner
18
11-18
21
33
25-26
13
5-8
14
20
8-11
22
14
27
25
38
23
15
6
18
12
22
9
35
25
36
35
19
10
21
17
37
25
33
25
19
17
7
TABLE 27
Group efficiencies for bored piles according to Liu etal. (1985). Dp = 0.25 m. Loose silty sand. 77^ and
77, include pile/soil interaction effects while % also includes action of pile cap in contact with soil.
SJDp
ID
PP
Group
Shaft
Tip
Total
Tis
3
3
3
3
3
8
13
18
18
23
3x3
3x3
3x3
3x3
3x3
0.36
1.09
1.16
1.42
1.16
1.44
1.51
1.49
0.91
1.12
1.64
1.69
1.51
1.36
1.15
2
4
6
18
18
18
3x3
3x3
3x3
0.98
1.11
0.82
0.70
0.93
1.06
1.21
1.46
2.23
3
3
3
3
18
18
18
18
1x4
2x4
4x4
2x2
1.11
0.88
1.03
1.20
1.10
1.51
1.45
1.22
1.49
1.40
1.19
1.60
222
Deep
foundations
T A B L E 28.
Efficiency factors at pile failure obtained by Phung (1993). Pile cap action included.
Tip ,
Slbn
ID(%)
Shaft 5
Total
Free-standing group:
38
67
62
2.6
3.2
2.0
2.0
0.8
1.0
2.4
1.1
1.2
3.2
4.4
4.4
3.0
0.7
1.4
3.1
1.3
2.0
0.5 m
-50 L
0
50
100
150
Cap load, kN
200
250
150
0.5 m
100
0.75 m
50
1.25 m
M
-50
rTi
1.75 m
20
40
Settlement, mm
60
Fig. 158. Influence on lateral earth pressure against the shaft of the centre pile in the group due to the
surcharge induced by the pile cap (After Phung, 1993).
Deep
foundations
223
Fig. 159. Comparison of load vs. settlement relationship of single pile to those of cap (shallow
foundation), free-standing pile group and pile group with cap in direct contact with the soil. Pile spacing
= 4b. Sand with In = 38%.
Deep
224
foundations
pile failure at large pile spacing, or, w h e n the pile spacing is small, by the shear strength
along the perimeter and nearest below the b o t t o m of the pile group, so-called block
failure. A group efficiency factor of 0.7 is r e c o m m e n d e d for pile spacings in the range
of2.5Dp-4Dp
(ii) End bearing pile groups. T h e ultimate bearing capacity is generally calculated as the
sum of the bearing capacity of a single pile.
7.2
Settlement
Different proposals have been presented about how to define, under equal soil conditions,
the settlement of pile groups in relation to the settlement of an individual free-standing
pile the so-called settlement ratio . A m o n g these, it seems preferable to use either one
of the following definitions:
is the ratio of pile group settlement to single pile settlement at equal pile loads
Load, kN
0
10
Load, kN
20
100
200
300
400
Fig. 160. Comparison of load vs. settlement relationship of single pile to that of cap (shallow foundation),
free-standing pile group and pile group with cap in direct contact with the soil. Pile spacing = 8/?. Sand
w i t h / D = 62%.
Deep
225
foundations
i s the ratio of the initial slope of the average pile load vs. settlement curve of the pile
group to the initial slope of the load vs. settlement curve of the single pile.
(i) Pile groups in granular
soil. A fairly large number of tests have been carried out to find
the value but the results obtained are often difficult to analyse both because of the
settlement ratios being related to different factors of safety against pile failure and the
settlement ratios not being defined. A s settlement is very much dependent on the factor
of safety applied, the results presented by different authors show great scattering and are
often contradictory. Roughly speaking, results of loading tests on free-standing pile
groups indicate that >\ in dense sand while <\ for driven piles in loose to medium
dense sand. Thus, in the latter case h a s been found to vary from about 0.2 at BgrIDp
3 (where Bgr = the width of the pile group) to about 0.7 at Bgr IDp ~ 10. A s for bored piles
in loose sand, h a s been found to vary from about 0.6 at BgfJDp
~ 3 to about 2 at
Bgr/Dp
-5.
Based on the results of full-scale investigations, Vesic (1969) suggests that the Rvalue
o.iof-
CS
_ o Outer piles
Centre pile
'S
.05
0.25
0.50
0.75
1.00
226
Deep
foundations
the soil behaves as an elastic m e d i u m with P o i s s o n ' s ratio v = 0.5. T h e settlement thus
obtained is also valid for the long-term settlement of pile groups in overconsolidated
clays w h e r e the in situ pressure induced by the pile group does not exceed the
preconsolidation pressure of the clay.
In clay soils, the m o d u l u s of elasticity Es to be applied in the analysis can b e chosen
on an empirical basis, for e x a m p l e in relation to the undrained shear strength cu of the clay.
TABLE 30
Rvalues for free-standing pile groups with lp/Dp= 25 in an elastic medium with v = 0.5. Rigid cap. Pile
spacing = S. Depth of the medium = d.
d/lp =
CO
2
3
5
10
2.91
2.59
2.19
1.70
2.80
2.46
2.08
1.63
2
3
5
10
5.38
4.64
3.74
2.73
2
3
5
10
8.34
6.96
5.34
4.43
2.5
1.5
1.2
S/Dp
2.46
2.10
1.69
1.29
2.01
1.70
1.39
1.16
5.00
4.22
3.27
2.20
Group 2x2
2.76
2.41
2.00
1.54
Group 3x3
4.88
4.06
3.05
1.98
4.10
3.25
2.30
1.48
3.09
2.39
1.75
1.27
7.56
6.12
4.43
2.66
Group 4x4
7.29
5.77
4.00
2.29
6.02
4.01
2.82
1.60
4.18
3.05
2.05
1.33
With regard to the pile length, the values given in Table 30 can b e adjusted by the
factors given in Table 3 1 .
TABLE 31
Adjustment factors for
lp/Dp.
S'Dp
lptDp=
2.5
5
0.82
0.77
0.74
10
10
lp/Dp=
1.00
1.00
1.00
100
25
1.20
1.30
1.45
Deep
227
foundations
^TmiltkTIIIIIIIIIIIIIIIII
(Bgr+zHLgr+z)
Thus, for normally consolidated clay Es ~ 150c w while for heavily overconsolidated clay
Es - 500c M.
T h e load distribution a m o n g the piles in working load condition is in good agreement
with the theory of elasticity (Fig. 161).
A c o m p r e h e n s i v e theoretical study of the relation between individual pile settlement
and pile group settlement in an elastic m e d i u m w a s presented by Poulus (1968). H e
assumes the soil to behave as an elastic medium and the pile cap to be either rigid or
perfectly flexible. In the rigid cap case, their analysis yields the settlement ratios given
in Table 30.
T h e Rvalues given in Tables 30 and 31 can b e considered representative for settlements
in undrained condition (immediate settlements). However, according to Poulos the
immediate settlements of pile groups represent the p r e d o m i n a n t part of the total, final
settlements.
T h e Rvalues obtained on a theoretical basis (Table 30) are supported by results reported by, for example, Berezantzev et (1961), Sowers et (1961) and H a n n a (1963).
(iii) Equivalent
applied on the pile group is assumed to act at the lower third of the pile length (Fig. 162).
T h e settlement of the pile group is calculated as the s u m of the equivalent raft settlement
and the compression of the upper t w o thirds of the piles. According to Poulos (1993), a
good correlation can b e expected between the results obtained b y this m e t h o d of analysis
and computer-based analysis of settlement, based on his o w n data p r o g r a m D E F P I G which
takes into account pile/soil interaction in an elastic m e d i u m (Poulos, 1990).
Regarding pile groups in low-permeable, cohesive soils, w h e r e the load induces in situ
stresses in excess of the preconsolidation pressure, the long-term consolidation settlement
228
Deep
foundations
can be analysed in a similar way. For pile spacings presumably less than 8 times the pile
diameter Dp,
surrounding soil seems m o r e or less negligible (Hansbo, 1973). T h e soil above the lower
third of the pile length behaves as an 'incompressible' layer (compression m o d u l u s M
tending to infinity). In consequence, the course of settlement can be analysed on the
assumption of full drainage (cv = kMlyw tending to large values) at the fictitious foundation
depth 2 / ^ / 3 .
7.3
Design
As shown in Section 2.8, the behaviour of individual piles in a pile group can b e quite
different from their behaviour as single piles. This fact is generally neglected in
conventional design.
By tradition, the pile group design is carried out on the basis of very simplified
assumptions:
T h e piles in the group are assumed to function as axially loaded c o l u m n s hinged at
pile head and pile tip. (Computer p r o g r a m m e s exist which take into consideration possible restraint of the pile head as well as lateral soil resistance).
Every pile is a s s u m e d to have the same axial stiffness, i.e. they are a s s u m e d to have
equal lengths and cross-sectional areas and to be supported by an unyielding m e d i u m .
T h e pile cap is considered rigid and is not assumed to contribute in carrying the load
applied.
Forces applied at the centre of gravity of the pile group are assumed to cause pure
translation. M o m e n t s applied at the centre of gravity of the pile group are a s s u m e d to
cause pure rotation.
In c o n s e q u e n c e of this model of pile group analysis, horizontal forces acting on the pile
group h a v e to be taken by raker piles. Moreover, in order to reduce the m o m e n t of
rotation, the pile group has to be arranged in such a w a y that its centre of gravity is as close
as possible to the line of action of the external force resultant.
In reality, the piles in a pile group m a y deviate considerably from the position given
in the design. Therefore, the real position of the piles has to b e checked and the pile forces
recalculated after pile installation is terminated.
According to conventional design, the load acting on the pile group, irrespective of
whether the piles are in cohesionless or cohesive material, is assumed to be carried by the
piles alone with a certain factor of safety against failure. This approach is rational w h e n
the piles are end bearing or mainly end bearing or w h e n w e h a v e to deal with footings on
normally consolidated clay. However, it is used even in cases w h e r e the bearing capacity
of the footing itself would b e satisfactory, the reason being that the settlements without
piles are felt to b e too large. In situations where the piles installed are friction piles for
w h i c h the load/settlement relationship does not show a m a r k e d decrease after peak, this
approach is quite conservative and unnecessarily expensive. A m o r e cost-effective
Deep
229
foundations
approach is first to investigate h o w m u c h of the load can be carried by the pile cap without
causing excessive settlement and then design the pile group to carry the r e m a i n i n g part
of the load. T h e intricate p r o b l e m of analysing the influence of pile/soil/cap interaction
on the ultimate load of the piles in the pile g r o u p and on the settlement can b e totally
disregarded. T h u s , on the basis of the investigation carried out b y P h u n g (1993), the
settlement obtained under the load taken by contact pressure at the cap/soil interface in
a piled footing is very nearly equal to the settlement obtained under an equally large load
taken by a corresponding unpiled cap (spread footing). This fact simplifies the design
procedure. T h e n u m b e r of piles required to limit settlement can b e d e t e r m i n e d on the
basis of the ultimate load of the single pile and the settlement can b e calculated as if the
pile cap w e r e a shallow footing carrying the load not taken by the piles.
T h e principle of pile group design can thus b e s u m m a r i s e d as follows:
D e t e r m i n e the load (Qx) that can b e placed on the unpiled footing without causing
unacceptable settlements.
T h e r e m a i n d e r of the load (Q - Qx) should b e carried by settlement reducing piles.
As the permissible settlement will be large e n o u g h for shaft resistance to b e fully
mobilised, the piles can be designed as friction piles in a state of failure (in clay, in a state
of creep failure).
T h e settlement of the piled footing can b e estimated at about the s a m e value as the
settlement of the unpiled footing under load Qx. This leads to a conservative design. For
e x a m p l e , for footings on sand the design b e c o m e s m o r e conservative the looser the sand.
This approach has b e e n used with great success to b o r e d large-diameter piles in stiff
L o n d o n clay (Burland, 1986).
Example 35: Determine the load that can be carried by the piled cap shown in Fig. 160, following the
principle of design proposed above. The soil characteristics, determined by pressuremeter tests, are
given below (Phung, 1993):
Depth, m
Epr MPa
'Pi MPa
p c rM P a
0.5
1.0
1.5
2.0
2.5
4.74
2.55
5.89
8.23
6.33
0.33
0.22
0.47
0.53
0.52
0.33
0.15
0.32
0.38
0.37
Solution: The ultimate load of the single pile determined on the basis of the pressuremeter tests is given
by Eq. (211), p. 191. The mean value of the limit pressure along the pile shaft is 0.39 MPa and of the
creep pressure 0.30 MPa. According to Fig. 138, case (d), the shaft resistance can be estimated at 15 kPa.
This yields a total shaft resistance of 1 5 0 . 0 6 4 - 2 . 1 = 7.6 kN (observed value = 1.8 kN). For the
determination of the base resistance, Eq. (209), we have a value of 2.6. The net limit pressure is 0.49
2
MPa. This yields a tip resistance of 2.64900.06 /4 =3.6 kN (observed value = 8.2 kN). Thus, the total
bearing capacity is 7.6 + 3.6 ~ 11 kN (observed value = 10 kN). Choosing instead of *p{ the net creep
pressure *pcn the creep failure load can be estimated at 5.7 + 2 . 6 - 8 kN.
230
Deep
foundations
Fig. 163. Piled raft with equally distributed wooden piles, 18 m in length. Soft highly plastic clay
reaching to a depth below foundation level of about 45 to 85 m.
The ultimate load / c f m e unpiled cap is determined according to Eq. (141), p. 136. We have ple
= (0.33-0.22) 1 72 = 0.27 MPa and bearing capacity factor k = 0.8. This yields Qfc = 0.8-270-0.8 2 = 138
kN (observed value 200 kN). The critical load (the 'creep' load) becomes 120 kN. Assuming a factor
of safety of minimum 1.5 we find Qx =90 kN (which is below the creep load), i.e. qx = 0.144 MPa.
The settlement is determined according to Eq. ( 151 ), p. 146. We have the shape coefficients
(1=2
and := 1.10 and the rheological coefficients ad = a J; = 1/3. The pressuremeter moduli to be applied in
the analysis are determined according to Eqs. (152-153). We find = 4.7 MPa and Eprd = 4/[ 1/4.74
+ 1/(0.85-4.74) + 1/4.39 + 2/(2.5-7.15)] = 5.0 MPa. Thus, the settlement is given by the relation:
.v =
1.2. 0 . 1 4 4 . 2 0.6 1 . 1 3 - 0 . 8 1 3/
1.10-0.8.
. n A c/ :
[
(
),M+
] = 0.0065 m
9
5.0
0.6
3-4.7
The total load that can be carried by the piled cap without exceeding a settlement of 6 - 7 mm is the
sum of the unpiled cap load and the pile loads at failure, i.e.:
Q = 1 + m - s /= 9 0 + 5 11 = 145 kN
The observed settlement under this load, according to Fig. 160, is 4 mm. Taking into account that
settlement, determined on the basis of pressuremeter tests, refers to settlement after a loading time of
10 years, the deviation between the calculated and observed values will most probably be strongly
reduced with loading time. However, in spite of the fact that the factor of safety against failure for the
unpiled footing is as low as 1.5, the design of the piled footing turns out to be conservative.
8.
PILED RAFTS
Piled rafts (Fig. 163) are used instead of piled footings in poor soil conditions, particularly
w h e r e the subsoil consists of soft, normally consolidated clay. T h e traditional approach
in this case is the s a m e as for piled footings the piles h a v e been designed to carry the
Deep
foundations
231
Fig. 164. Settlement contours in mm for two adjacent residential buildings in Gothenburg. Equal soil
characteristics (normally consolidated, highly plastic clay to great depth) and equal building loads (
2
60 kN/m ). Top: Traditional foundation design (factor of safety against pile failure equal to 3). Centre:
Raft foundation with settlement reducing creep piles. Bottom: Average settlement vs. time for the two
buildings, (cf. Hansbo, 1984, and Jendeby, 1986).
w h o l e load with a certain factor of safety against failure. A design principle, similar to
that applied to piled footings, has been used in S w e d e n since long ago with great success
(Hansbo & Kllstrm, 1983; H a n s b o , 1984; Jendeby, 1986; Svensson, 1 9 9 1 ; Randolph,
1993).
T h e approach, suitable for piled rafts on normally or lightly overconsolidated clays,
is as follows:
Deep
232
foundations
determine the effective overburden pressure and the preconsolidation pressure a ' c
at different depths in the clay,
determine the decrease in overburden pressure due to excavation and the increase
caused by the construction of the building,
decide h o w m u c h of the building load can be carried by contact stresses at the soil/
raft interface without exceeding the preconsolidation pressure in the soil,
the remainder of the building load shall b e carried by piles in a state of creep failure
(pile load equal to creep load); the piles should be distributed in such a way that the
preconsolidation pressure is nowhere exceeded and so that the differential settlement is
minimised.
Besides the savings in foundation costs, this approach has the advantage that the pile
forces acting against the raft from below are k n o w n in size. C o m p u t e r p r o g r a m m e s have
been developed for the design of the raft, taking into consideration the stiffening effect
of internal walls in the b a s e m e n t (Svensson, 1991).
An e x a m p l e of the settlement distribution for two adjacent residential buildings, one
of which is designed according to the 'creep pile' approach, the other according to the
traditional approach, is given in Fig. 164.
9.
DOWNDRAG
skin friction,
design of end bearing piles. T h e piles will carry not only the applied load but also part
of the weight of the surrounding soil.
In old days, negative skin friction was seldom, if ever, considered and all the same the
buildings rarely suffered d a m a g e . However, in certain circumstances, the consequences
of ignoring negative skin friction can be serious. T h u s , in practice it has h a p p e n e d that
piles h a v e been pulled out of the foundation due to negative skin friction which has
entailed serious d a m a g e to the buildings.
T h e negative skin friction ca can b e estimated from the equation:
ca = tan0' fl G'V
(265)
Deep
233
foundations
ca - 0.6c t t
(266)
Negative skin friction affects the individual piles only d o w n to the neutral point, w h e r e
the relative m o v e m e n t between pile and soil in the pile/soil interface is zero. B e l o w the
neutral point, the skin friction is positive. The position of the neutral point d e p e n d s on
the length of the piles and the bearing stratum at the pile point. For piles driven through
deep layers of soft clay to bedrock of high bearing capacity, it can be a s s u m e d that the
whole pile is subjected to negative skin friction. However, for piles driven into a sand or
gravel or to bedrock with an ultimate bearing capacity less than that of the pile section,
the position of the neutral point can be taken at the level w h e r e the settlement of the
surrounding soil is 5 m m (Norwegian Pile C o m m i s s i o n , 1973).
For pile groups with large pile spacing, each individual pile can be a s s u m e d to be
subjected to negative skin friction according to Eqs. ( 2 6 5 - 2 6 6 ) . At small pile spacing, the
d o w n - d r a g forces on the pile group can be assumed to comprise the weight of the fill
above the pile group plus the shear resistance of the soil on the perimeter area of the pile
group d o w n to the neutral point. In consequence, the d o w n - d r a g will be larger for the
peripheral piles than for the central piles.
10.
10.1 Introductory
remarks
Deep
234
foundations
Fig. 165. Secant pile wall prepared for intermediate floor slab. By courtesy of Bachy.
bored piles that intersect to form a solid wall, resembling the d i a p h r a g m wall, and walls
consisting of piles with interspace w h e r e the soil is retained by arching (Figs. 1 6 5 - 1 6 7 ) .
T h e construction of piers is carried out in very m u c h the s a m e w a y as of large-diameter
bored piles. A hole is excavated or drilled into the soil d o w n to the foundation level and
the pier is built inside. T h e sides of the hole usually have to be stabilised, for instance by
m e a n s of bentonite slurry or sheet pile walls. Another m e t h o d for construction of a pier
is the use of caissons, generally provided with a cutting edge to facilitate the caissons
being lowered to the depth of foundation.
10.2 Bearing
capacity
T h e analysis of the bearing capacity of piers, caissons and underground walls can b e
carried out in the same w a y as for piles with due consideration to shape and depth/width
relations. T h e main contribution to the bearing capacity is obtained from base resistance
(cf. Brandl, 1993). In the case of granular soils, the most reliable m e t h o d of analysis seems
to b e the one based upon the results of pressuremeter tests. Calculations based on the
shear strength parameters c ' a n d (//of granular soils are carried out on the assumption of
Fig. 166. Secant wall with unreinforced 'female' piles and intersecting, reinforced 'male' piles.
Fig. 167. Assumed zone of failure for a deep wall foundation. Two cases may occur: the failure zone
does not reach the ground surface (left) and the failure zone intersects the ground surface (right).
'global' failure which does not usually agree with reality. For the sake of completeness,
this type of analysis will all the same be included.
(i) Geostatical
failure load as the load/settlement curve is generally quite flat r e m i n d i n g of that obtained
for strain hardening soil. This has its explanation in local shear failure taking place before
total failure of the foundation. O n e w a y of taking this into consideration is to apply a
reduced value of the shear strength of the soil or to define failure in relation to a certain
relative settlement sib, w h e r e b is the width of the foundation. A c o m m o n assumption is
that failure takes place at sib = 0.10.
T h e analysis of the bearing capacity of piers and underground walls can be carried out
in principally the s a m e way as for shallow foundations, see p. 126. Let us first consider
the contribution to the bearing capacity represented by the bearing capacity factors Nq and
Nc. A s s u m i n g that w e h a v e a case of ' g l o b a l ' shear failure, comprising a combined
Prandtl-Rankine failure zone, t w o cases m a y occur: (1) the failure z o n e reaches the
ground surface, (2) the failure zone does not reach the ground surface (Fig. 167).
Assuming that the failure zone reaches the ground surface as shown to the right in Fig.
Deep
236
foundations
167 and that w e h a v e a case of plain strain condition (strip foundation), the solution to
the p r o b l e m can b e derived as follows.
In the soil w e d g e A C D , the relative lengths A C / C D / D A , according to the law of sines,
can b e expressed as
COS(t] +
directions yield:
T5cos0'= [cos(r+</Ocosj - sin?] sin(rj+0O + (fx
aoCos0
[COS(t] +
^ - sin?7 C O S ( t j
")
/ = =
+ c / . t a n ^ ^ ^
( 2 6 ?
COS0
7
c + \ tan
,
,
- - -
[sin(2i] 4- ') - sin ']
cos '
= cfx
By this equation system, and
,./m
(268)
to = r ( c + CTotan0O
2 6 9
(270)
T h e equilibrium condition for the R a n k i n e failure zone below the base of the wall
(wedge A B E ) yields:
qf=
(271)
Substituting Eq. (268) for ' 0 and Eq. (270) for r 2 and expressing the bearing capacity
under the conventional form:
q^ibllWNy+NyG'v
+ c'N,
_ (1 + s i n 0 ) e x p ( 2 0 t a n 0 )
9
"
l - s i n 0 ' sin(2rj+0')
( 2 7 2
Deep
237
foundations
1000
J3 = 90
//
100
A
10
V
VA
// /
/3 = 30
/
//
j
Y // ////
//','///
//
<///.
r -= 0
10
/3 = 60
20
30
r=1
40
yVc = ( i V , - l ) c o t f
where = 3/4 +--
(273)
072,
= the angle of inclination to the horizontal of the upper boundary surface of the
failure zone (Fig. 167).
The value is a function of 0 ' a n d r and is governed by the relation:
cos(277
+ 0 )
v
1
r =
{1
COS0'
tan<*
[sin(27] + 0 - sin ']}
COS0'
(274)
, can be assumed
to vary between active earth pressure and unity. The ' value should be chosen as an
average along the boundary between the failure zone and the sides of the foundation with
a maximum value governed by the critical depth according to Fig. 135 where the ratio dcr IDp
is replaced by
dcrlb.
Deep
238
foundations
= 90
10
20
30
40
In the analysis of the bearing capacity factor 7Vr it is assumed that the failure zone has
a shape similar to the one prevailing in the former case treated above. T h e m i n i m u m
values of A^are found by trial and error. T h e R v a l u e s , according to Meyerhof (1951),
are given for = 30, 60 and 90 in Fig. 169.
For values of djb > 5, local shear failure seems to govern the b a s e bearing resistance.
M e y e r h o f (1951) r e c o m m e n d s in this case that the shear strength of the soil be reduced
to 8 5 % of the value determined.This is s o m e w h a t higher than 2/3 of the value which was
suggested by Terzaghi (1943).
TABLE 32
Shape factors based on Meyerhof s proposal.
Friction angle '
Shape factor
30
35
>1
1
>5
1
2
5
> 10
1.0
1.2
1.0
1.9
1.5
1.3
1.0
40
Deep
239
foundations
= 1
(275)
2
(276)
sc=\+0A5b/l
Example 36: Determine the bearing capacity of a diaphragm wall with a thickness of 0.5 m, founded
at 10 m depth in a homogeneous sand layer with an internal angle of friction of 35. The groundwater
3
3
level is at 2 m depth. The effective density of the soil is 1.8 t/m above and 1.1 t/m below the groundwater
level.
Solution: First find out whether or not the zone of failure reaches the ground surface. With the notations
given in Fig. 167 we have AB = 0.25/cos(45 + 35/2) = 0.54 m. The radius vector of the logarithmic
spiral, representing the Prandtl failure zone, if extended to the sides of the diaphragm wall, is equal to
0.54-exp{7U-[(180 + 45 - 3572)/180]-tan35} - 6.8 m < 10 m. Thus the bearing capacity factors are
governed by = 90.
Since djb > 5, the angle of internal soil friction to be applied in the determination of Nq and ^ s h o u l d
be reduced to 0 ' = arctan(0.85-tan35) 31. According to Figs. 168-169 this yields Nq ~ 140 and
~ 160. Since = 90, we have cr^ = <fs. The magnitude of <fs is uncertain and has to be estimated. In
our case we assume Ks = 0.5. The & s value is taken as the average along the sides of the wall inside the
failure zone, i.e. from the base of the wall to a height of 6.8 m above the base.The critical depth ratio
for '= 31 according to Fig. 135 is about 10 which yields dcr = 5 m. The maximum value of ' to be
applied is thus 2-18 + 3-11 = 69 kPa. At 6.8 m above the base we have ' = 2 18 + 1.2 11 - 49 kPa.
The bearing capacity becomes:
^=0.25-11.160+
240
Deep
foundations
TABLE 33.
Maximum values for strip foundations and corresponding, minimum embedment ratios. (Baguelin
et ai, 1978)
Category
( d / b ) mm
Rock
1
2
5
10
1
1
1
7
1.9
2.2
2.6
3.0
0.4
2
6
0.1
0.5
1
3
6
9
12
3
4
5
6
1.9
3.1
4.4
1.2
1.6
1.9
2.2
0.1
1
4
3
4.5
5
1.3
1.8
2.2
Silt
Clay
max
Na
= (1 + 0.2y ){5.14 + - A s
/
3 b
- iV }
b
(278)
jf-O^ + i ^ - o W - ^ -
(279)
Deep
241
foundations
Example 37: Determine, by the pressuremeter method, the bearing capacity of a diaphragm wall, 0.8
m in thickness, with its base at 5 m depth in a sand deposit. The pressure limits pt in the sand, observed
at depth intervals of 1 m, are 2 . 5 , 2 . 1 , 1 . 9 , 2 . 2 , 1 . 8 , and 2.3 MPa, starting from 1 m depth below ground
surface downwards. The groundwater level is at 2 m depth and the porosity of the sand = 27%.
Solution: The density of the sand below and above groundwater is obtained by Eq. (5). Assuming Sr
3
3
= 100% below groundwater and pg = 2.65 t/m we obtain = 2.65(1 - 0.27) + 0.27-1.0 = 2.2 t/m ,
3
and assuming Sr = 0 above groundwater we find = 2.65(1 - 0.27) = 1.9 t/m . The effective overburden
pressure at the foundation level is then ' = 9.81(2-1.9 + 3-1.2) 73 kPa.
1 /3
The equivalent limit pressure with regard to end bearing capacity is ple = (2.2-1.8-2.3) = 2.1 MPa
from which the net limit pressure pte ~ 2.0 MPa.
1 /2
For dlb = 5/0.8 = 6.25 and (d/b)min = 9 we find = 0.8 + (3.1 - 0 . 8 ) ( 6 . 2 5 / 9 ) = 2.7.
The pressure limit with regard to shaft resistance can be taken as the average of the observations down
to 5 m depth, Le. pt pt = (2.5 +2.1 +1.9 +2.2 +1.8)/5 = 2.1 MPa
The shaft resistance, taken from Fig. 138, is t h e n / ^ ~ 40 kPa
The bearing capacity is now obtained from Eq. (211). We find:
qf= 0.073 + 2.7-2 + 2-5-0.04/0.8 = 6.0 MPa
10.3 S e t t l e m e n t
S e t t l e m e n t s of piers and u n d e r g r o u n d walls can b e analysed by g e o m e c h a n i c a l m e t h o d s
b a s e d on deformation
IM
9-0.8
2_0_6 2 , 6 5 , 0 8
22.6
0.6
+ 1
5018
3-20
242
Dynamically
loaded
foundations
INTRODUCTION
Periodic disturbances.
disturbances.
with vibrations
of foundations
by a lumped
representing
it is generally
mass-spring-dashpot
the effective
representing
system
assumed
whith
that the
subsoil
the mass
the elastic
of the vibrating
of the
response
system.
Here,
2.1
Free
(i) Undamped
vibrations
free vibrations.
springs with a spring constant equal to k (Fig. 170). In a state of rest the shortening of the
springs required to balance the body is <50. We thus have:
Dynamically
loaded
243
foundations
Amplitude
Fig. 170. Undamped free vibrations with one degree of freedom can be illustrated by a body resting on
a perfectly elastic spring system. The vibration becomes a sinusoidal function of time.
(280)
0 = mglk
By pushing the b o d y d o w n w a r d s a distance z 0 from its position of rest and then letting
it free, the b o d y will start swinging around the position of rest. T h e oscillatory motion of
the b o d y thus achieved can be demonstrated graphically as s h o w n in Fig. 170. T h e
velocity of the b o d y is given by the tangent to the zlt curve. Since the spring is perfectly
elastic w e h a v e according to N e w t o n ' s second law:
d\
mg - (mg +kz)
= m
(281)
that is:
m'z+kz
= 0
(282)
z=Zo
fk
cos(\/ t)
Ym
= zo
cos(<*v)
(283)
w h e r e is an integer.
(284)
Dynamically loaded
244
foundations
mg + kz
mg
Fig. 171. Damped, free vibrations with one degree of freedom can be illustrated by a body resting on
a spring-daspot system.
Introducing k - mg/S0
T h e oscillation amplitude of the b o d y will decrease with time and finally the body will
turn into a position of rest. D a m p i n g counteracts the m o v e m e n t of the body. Assuming
that w e h a v e to do with viscous damping which is directly proportional to the vibration
velocity (Fig. 171), N e w t o n ' s second law of motion takes the form:
mg - c i - (mg + kz) = m
(287)
(288)
T h e solution to this differential equation is of type = exp(Ar). Substituting this into Eq.
(288) w e find:
Dynamically loaded
245
foundations
2 - - {c
2m
yjc -4mk)
= i^expUjf) + 5 i e x p ( ^ i )
(289)
= - c/m, w h e n c e :
= (A2t + B2)exp(-
ctllm)
(290)
Fig. 172. Viscous damping can give rise to three different time dependencies: (a) Overdamping, (b)
Critical damping, (c) Underdamping.
Dynamically loaded
246
z = z
( ^ + l)exp(-^-)
Zm
Zm
foundations
(291)
=2y/mk
Ccr
(293)
~
= conD
Zm
2
A 1 2 = -nD
1
2
2
y/c -c
2m
(294)
= n{-D\y/\-D )
= [A 3 sm(conty/1
- D ) + B3 cos(conty/1
-D )]
exp(-CunDt)
(295)
= z0[
w h e r e =
ny/l-D
y 1
D
2
-D
w(dt)
+ c o s ( ) d0 ] ( - )
(296)
Eq. (296) shows that oscillation is continuing with a gradually decreasing amplitude
and with a d a m p e d circular frequency
(d - cnyl
2.2
damping
Forced
vibrations
with viscous
-D
(Fig. 172c).
Dynamically loaded
247
foundations
kz + mg
mg-cz-{mg
+kz)
sin()f) = m f
whence:
mz + cz
= <2 sin(&)0
0
(297)
After a certain period of time w h o s e length depends on the d a m p i n g ratio the vibrating
system will keep p a c e with the vibrations enforced upon it. T h e system is subjected to
continuous, h a r m o n i c vibrations with a phase shift between the oscillating force and
the oscillating system.
T h e simplest w a y of solving the resulting effect upon the system due to the enforced
vibrations is by vectorial studies (Fig. 174). During oscillation, the velocity vector zcois
perpendicular to to the displacement vector and the acceleration vector zo is perpendicular to the velocity vector. Furthermore, the m a x i m u m amplitude z 0 is reached for
sincot = 1. Consequently, the following relation can be established:
kzo
(
Fig. 174. Vectorial presentation of displacement, speed and acceleration (left) and of forces acting at
maximum amplitude.
Dynamically loaded
248
foundations
(298)
T h e conditions of equilibrium in the vertical and horizontal directions (Fig. 174, right)
yield:
kz0+
Q0cos(p = mzoco
(299)
0 sin<p + cz0co = 0
(300)
^(k-mco )
tan -
F r e q u e n c y ratio
+ (cco)
ceo
2
k - m
flfn
Fig. 175. Dynamic factor as a function of the frequency ratio f/fn and the damping ratio D.
(301)
(302)
Dynamically
loaded
249
foundations
///>///////////
Fig. 176. Vertical impact. A body with weight mx is dropped from height h against a foundation block
with mass m supported by a spring system.
Now, if o represented a static load (it has to b e realised that the spring constant k which
represents the soil refers to the d y n a m i c r e s p o n s e of the soil even w h e n determining the
(50 value) w e would h a v e the displacement <50 = Q$lk.
T h e ratio of the m a x i m u m amplitude z 0 to the 'static' displacement <50 is defined as the
dynamic factor
. Substituting c/2m
for
co
2
n
= ?
So
yj[ 1 - ( I ) ] + (2 I )
2
(303)
2Dco/
tan =
(304)
\-( )
Impact
m
t
m +m
(1 + g)
(305)
Dynamically loaded
250
foundations
1
= -kz
2
IF
(306)
2 k
(307)
vJmk
s t a t = mg
k0
which yields:
(308)
T h e total load transmitted to the subgrade is the sum of the d y n a m i c and static loads,
that is:
^tot^statO+)
(309)
= 1 t
1.8
m a
2m3m
m3
= 50t
2.5 m 3.5 m
2.4 m
Dynamically loaded
251
foundations
Determine the the maximum contact pressure and the maximum amplitudes for the upper and lower
concrete blocks due to the impact of the tilt hammer if the impact energy is 30 kNm. The oak bed has
2
a thickness of 0.38 m and a modulus of elasticity = 1000 MPa. The base area of the oak bed is 2.3 m
and its coefficient of restitution e = 0.6.
Solution: The calculation is carried out in two steps, firstly with respect to the oak bed and secondly with
respect to the subsoil.
Step 1. The contact pressure between the upper concrete block and the oak bed is:
qx = 9 . 8 1 - 2 6 / 2 . 3 = 111 kPa
The corresponding deformation of the oak planks is:
3
3
(5 01 = 0 . 3 8 - 0 . 1 1 1 / 1 0 = 0 . 0 4 0 - m
The velocity v 1 of the tilt hammer when it hits the concrete block is found by the relation:
2
= 30 kNm
mxvx /2
whence
2
V l
'
3 0
nn<
^ =
V -
VgSi
8 1
6
0
' -
=22.7
0 4
'
1 0 3-
50-9.81
= 56 kPa
2.5-3.5
From the result of the seismic investigation we find the dynamic shear modulus:
2
3
G 0 = 1.15-1.9-375 = 307.3-10 kPa
Assuming that Poisson's ratio of the soil is 0.3, we have
6
0 = 0.80-10 kPa
From Eq. (146) and Fig. 100c, yielding Te = 0.75, we find:
6
= 0.75-56-2.5/(0.8-10 ) = 0.13-10~ m
Moreover:
62
vu = vi
(1 + 0 . 6 ) = 0 . 2 5 m / s
26 + 50
Thus, the dynamic factor becomes:
025
V 9 . 8 - 0 . 1 3 - ICH
-7.0
252
Dynamically
loaded
foundations
Fig. 177. General modes of vibration with six degrees of freedom. Translation movement designated
as ,
etc., rotational movement with ^, ^ etc.
3.
3.1
Introduction
T h e linear vibration theory which is applicable for vibrating systems with one degree of
freedom is sufficient for the solution of m a n y vibration p r o b l e m s . However, vibrating
machines often give rise to non-linear vibrations. Considering the general case, the
vibrating b o d y has six degrees of freedom: displacements in the directions of the three
coordinate axes and rotation around each one of the three axes (Fig. 177). T h e vibrations
m a y include every kind of combination of these motions.
A so-called rocking and sliding m o d e of oscillation represents a type of non-linear
vibrations w h i c h is c o m m o n l y occurring and, therefore, quite important for dynamically
loaded m a c h i n e foundations. Torsional oscillations represent another type of non-linear
vibrations that is also c o m m o n .
3.2
Rocking
vibration
In the case of m a c h i n e foundations, the horizontal, principal elastic axis does not go
through the centre of gravity of the foundation block but normally through the contact
surface b e t w e e n the foundation and the subsoil. A horizontal, d y n a m i c force gives rise
to a horizontal translation of the foundation b l o c k only if it is applied at the elastic centre
(the intersection point between the horizontal and vertical principal axes) of the
foundation (Fig. 178) and has the s a m e direction as the horizontal principal axis. A
rotational m o m e n t acting in the plane formed b y the vertical and horizontal principal axes
only causes rotation around O. In practice, a horizontal force acting on the foundation
s e l d o m passes through the elastic centre and therefore the vibrations induced are
neither purely horizontal, nor purely rotational. Instead t w o rotational vibrations will take
place around t w o fixed points situated on the vertical principal axis. This t y p e of
vibrations is defined as rocking-sliding.
Dynamically
loaded
253
foundations
Fig. 178. Foundation block subjected to a horizontal, dynamic load outside of its elastic centre.
Let us a s s u m e that point (Fig. 178) represents one of these fixed points and that the
foundation block is turned at the angle y/from its position of rest. T h e centre of gravity
of the foundation C is then displaced a distance = by. In order to induce this m o v e m e n t
of the foundation, it has to b e subjected to a turning m o m e n t around and a horizontal
force through O. T h e m o m e n t and the horizontal force in combination can be replaced
by a horizontal force through point A at a distance from the centre of gravity.
Introducing = \/kx and y/y = l/k^
=Px
+ P(a + s)y/}<>
(311)
y/=/b
= P(a + s)y/y
(312)
whence:
+s+s
= ob + bs
(313)
Dynamically loaded
254
foundations
Fig. 180. Forces belonging to poles A and in connection with rocking movement. The radius of inertia
/ is the geometrical mean of the distances a{ and a2.
(314)
m'S+ko = 0
(315)
+=0
where lis the mass moment of inertia of the foundation with respect to the horizontal axis
of gravity perpendicular to the plane of the figure.
112
But = /b and = k. Hence, Ilm = ab. Introducing the radius of inertia / = (Ilm) ,
we
find:
ab = i
(316)
Thus, the radius of inertia iis the geometric mean of the distances a and b. Consequently,
if the distance / is set off from the centre of gravity C and if its end point is connected
with the fixed points A and B, then the angle AEB will be 90 (Fig. 180).
Substituting for b the radius of inertia according to Eq.(316), a can be solved from the
2nd degree equation:
2 , 1 , ^ , 2
+ - ( + s
.2,
-)-
.2
riM\
(J1/)
whence:
a
= -a0
y/afi + i
(318)
Dynamically loaded
255
foundations
Fig. 181. Principle for dividing up an arbitrary force into components acting in the directions of the three
principal modes of vibration.
where 0 - ( + s - i ) .
2s y/y
As expected w e get two values of a, of which the positive value ax corresponds to a in
Fig. 178 and the negative value - a2 corresponds to b. (In order not to mistake the term
b for width of foundation, the terms ax and a2, with positive
following). The force Px, causing rotation around pole goes through pole A, and the
force P2, causing rotation around pole A, goes through pole (Fig. 180). The foundation
body with mass m can be replaced by two bodies, one at pole A with mass mx and the other
at pole with mass ra2, where:
m i
= - ^
ax +a2
(319)
m2
(320)
ax
+a2
These mass 'points' are dynamically equivalent to the total mass of the foundation, i.e.
the resulting horizontal movement can be obtained by studying the horizontal movements
of the two mass points. If, for example, mass point mx is subjected to a horizontal impact,
the mass point m2in
(321)
Dynamically loaded
256
fa
= J =
foundations
( 3 2 2 )
2y/2
where and 2 represent the horizontal displacements (in m) of the mass points mx and
m2 due to horizontal loads of magnitude gml and g m 2 , acting at the respective mass point.
The two displacement values and 2 are calculated on the basis of and in the
following way: the horizontal force gm x is replaced by a horizontal force gm x, acti ng through
the centre of gravity C, and a turning moment gmx(a
\ =gmx[x
+ (ax + j ) V y
82^gm2[x
+ (a2-s) yy
(323)
By analogy w e have:
2
(324)
where , =
1
and P2 = 2Q
( 3 2 5 )
\fn\
r ) - f / I
f2 n
and
'
Vn2-f \
Impact:
--gmx - ! m j
P2 = v2m2y/J7o 2
(326)
( 3 2 7 )
Now, by the aid of equivalent static forces, w e can easily solve the problems caused by
dynamic forces acting in arbitrary directions on the foundation. For example, the
dynamic force R (Fig. 181) can be divided up into three components, one vertical V and
two horizontal Hx and H2. Each component is then studied separately. It has to be noticed
that the dynamic factor will differ between the three components.
Dynamically loaded
3.3
Torsional
257
foundations
vibrations
To be able to study study the influence of any type of d y n a m i c forces, there remains to
analyse the influence of a torsional m o m e n t acting around the vertical axis through the
centre of gravity. A s this axis represents o n e of the three principal axes, w e simply h a v e
to do with p u r e torsional vibrations. In this case, N e w t o n ' s law of motion yields:
'+^
= 0
(328)
77
(329)
= QtM
(330)
w h e r e Qt =
f
fnt
5 - /
4.
T H E SUBSOIL AS D A M P E D SPRING S Y S T E M
In the elementary vibration theory it w a s p r e s u m e d that the m a s s of the spring system was
negligible. In reality, the subsoil affected by the oscillating b o d y has a considerable mass
which seemingly w o u l d invalidate a direct application of the results of the mass-less
spring response. Test experience, however, gives full e v i d e n c e that the elementary
vibration theory can b e applied provided that the subsoil is treated both as a mass-less
spring system and a d a m p e r with the d a m p i n g ratio D.
In Fig. 182, typical rheological m o d e l s are presented w h i c h are utilised for the analysis
of foundations subjected to vertical, horizontal and torsional d y n a m i c loads
4.1
Vertical linear
vibrations
Dynamically loaded
258
foundations
Rigid block
with equivalent mass
Section
f// - ///
5
Rigid block with equivalent mass and
mass moment of inertia around horizontal axis
Section
JAW
Equivalent system
Real system
Fig. 182. Examples of rheological models used for the analysis of dynamically loaded foundations
<50 under the w e i g h t mg of the foundation with regard to the a m p l i t u d e in question. Since
the vibration amplitudes are generally small, <50 can b e calculated on the basis of the
theory of elasticity with a m o d u l u s of elasticity determined by d y n a m i c m e t h o d s , for
e x a m p l e from the results of Rayleigh w a v e or shear w a v e velocity m e a s u r e m e n t s . T h u s ,
in the case of a shallow foundation with rectangular b o t t o m area, <50 can b e determined
according to Eq. (147) by substituting for EQ = 2 G 0 ( 1 + v) and q for mg which yields:
0=
8 0 )
2 G 0( 1 + ) /
(3313
Dynamically
'
loaded
259
foundations
'
Ui
C3
CU
0.1
0.2
0.4
0.6 0.8 1
_ I
L_ __L
8 10
According to Eq. (280), the spring constant can n o w b e expressed by the relation:
kz = mg/<5b
(332)
1-
(333)
260
Dynamically loaded
foundations
I I I I I I I I I I I I I
Fig. 184. Influence of embedment upon the vertical spring constant for a circular foundation with radius
R (After Kaldjian, 1969). Top curve represents rigid foundation, bottom curve weak foundation.
4.2
Rocking
mode of
vibration
Horizontal forces in the y direction that are not acting through the elastic centre of the
foundation (which, therefore, give rise to a rocking m o d e of vibration around the y axis)
produce a triangular stress distribution under the bottom area of the foundation.
According to Rausch (1959), this stress distribution can be replaced by a uniform
pressure as shown in Fig. 185. For rocking m o d e of formation, the spring constant k^ can
be chosen according to Gorbunov-Possadov & Serebrajanyi (1961):
b
I
Fig. 185. Theoretical stress distribution due to rocking mode of vibration. However, in practice only one half
of the foundation will be loaded in turns. Replacement distribution suggested by Rausch (1959).
Dynamically
loaded
261
foundations
1-
where
vyAl
(335)
Unlike the state of vertical vibrations, a lengthening of the base of the foundation in the
direction of the acting forces increases the spring constant considerably and, consequently,
also the natural frequency of the rocking foundation.
L o o k i n g n o w at the effect of horizontal forces acting through the elastic centre of the
foundation, these are resisted by shear stresses in the foundation/subsoil interface. T h e
horizontal displacements induced by the horizontal forces in this case are thus due to
shear deformations. For a shallow foundation with base area A acted u p o n b y horizontal
forces through the elastic centre in the direction, Barkan (1962) proposes a spring
constant equal to:
(336)
w h e r e according to Fig. 183.
4.3
Torsional
vibrations
T h e spring constant for a shallow circular foundation can be chosen according to Reissner
& Sagoci (1944) from the relation:
(337)
Soil damping
ratio
damping,
damping.
D =
As
(338)
262
Dynamically loaded
foundations
damping.
Translation:
Re =
Rocking:
Re = [ / / ( 3 ) ]
Torsional:
Re = [A(l +
(/)
2
4.5 Instructions
for practical
1 /4
b )/(6n)]
1/4
analysis
= 0
The contact stresses under the foundation base caused by the design load must not
exceed the allowable stresses.
Determine the maximum amplitudes obtained under the influence of the static load
o-
Dynamically loaded
263
foundations
TABLE 34.
Internal damping ratio for different soils
Soil type
Damping ratio D
0.01 - 0.03
0.03-0.07
0.03-0.10
0.02 - 0.05
0.4
03
00
G
C3
10
TABLE 35.
Geometrical damping ratio D for various modes of vibration.
Mode of vibration
Vertical
Horizontal
7-8v
~ 32{1-)PR*
Damping ratio
^
m
^
=
y[x
8 ~ ~ pR*
Torsion
= ~
0.288
d+B^y/B
0 . 5 0
Dynamically
264
loaded
foundations
Example 40: A rotating mass oscillator is placed on a rectangular block foundation with a rectangular
base area 2 by 3 m. The total weight of the mass oscillator and the foundation is 251. The rotating masses,
0.5 t each, have a lever arm of 0.6 m and produce a vertical oscillation with a frequency of 10 Hz. The
foundation is placed with an embedment of 1 m on a 10 m thick sand layer on bedrock. Seismic refraction
measurements have given a Rayleigh wave velocity in the sand equal to 210 m/s. The total density of the
3
3
sand above the groundwater level (at 2 m depth) is 1.8 t/m and below the groundwater level 2.1 t/m .
500 kg
W^- ft =
///SW
/ = 1 0 Hz
3 / // //"/ HV/7
3 in 2 m
Sand with = 1 . 8 t / m
Solution:
Subsoil condition:
The initial (dynamic) shear modulus G 0 of the sand is calculated from the R wave velocity. Assuming
that the average value of the density is 2 t/m3 we have (see Eq. 59):
2
fn =
263
= 0.14.10-
= 42 Hz.
1.38 m;
Bz =
4
0.7 25
3 =
:0.83;
3
p/?
4-2.0-1.38
= ^ = 0 . 4 7 .
B7
Internal damping ratio can be assumed equal to 0.03, whence total damping ratio:
D
1
/
[ 1 - ( 1 0 / 4 2 ) 2 ] 2 + (20.50 10/42)
= 1.03
Dynamically loaded
265
foundations
ax
=
AEo
_ 3i n
n , o n1 . 0 3 - 2 3 6 9 - 2
=0.804
= 2 . 5 - 10
m.
3
6 - 2 6 3 - 10
Example 41: A machine foundation of concrete with a base area of 8 m by 3 m and a height of 3 m is
affected by two oscillating masses, 10 t each. The oscillations take place without phase shift in the
horizontal plane, 3.8 m above the base area, with a circular frequency of 60 cycles/s and a maximum
centrifugal force of 25 kN. The foundation is resting on deep layers of sand with a total density of 1.9
2
t/m and a R wave velocity of 250 m/s. Calculate the maximum foundation pressure and the amplitude
at point D in the xlz plane.
' 50sin)f k N
VU
25cos)/kN 25cosft)/kN
/// = /v-^///
S a n d w i t h = 1 . 9 t/iiv^
Solution:
Oscillating force:
F r e q u e n c y / = 2 = 60/2 = 9.55 Hz
Maximum force amplitude 2Q0 = 50 kN
Subsoil:
The dynamic shear modulus of the sand is obtained from the shear wave velocity:
2
3
G 0 = 1.151.9250 = 137 1(> kPa
whence, for = 0.3, E0 = 355 MPa
Foundation block:
Mass:
m =8-3-3-2.4 = 173 t
Centre of gravity:
5(173 + 2 0 ) = 173-1.5 + 20-3.8
s = 1.74 m
Mass moment of inertia (r designates the polar distance of an infinitesimal mass dm from the centre
of gravity):
2
/ = jV dm
For rocking in the xlz plane we have to calculate the mass moment of inertia
2
Iy = l(x
+ z )dm
Iyx+Iyz
We have :
2
2
Iyx= 173-8 /12 = 923 t m
2
2
2
2
Iyz = 173(3 /12 + 0 . 2 4 ) +20(1.26 + 0 . 8 ) = 225 t m
whence:
2
Iy = 923 + 225 = 1148 t m
The radius of inertia / = / I l m
iy = 2.44 m
Dynamically loaded
266
foundations
Spring constants:
The maximum displacement amplitude is obtained when the maximum force amplitude is directed
perpendicular to the longer side of the foundation, i.e. in the y direction.
Remains to determine and .
For determination of we have to know kx. According to Eq. (336) we have kx = </
. For
the side ratio lib = 8/3 (Fig. 183), = 1.0 whence kx = 355/S^3
- 1740 M N / m . For determination
of y/y we have to know k^. According to Eq. (335) we have k^ = [GQ/(1 - v^yyAL For the side ratio
8/3 (Fig. 183),
= 0.65, whence k^ = 136-0.65-24-8/0.7 - 24250 MNm/rad.
3
6
Thus, = Ukx = 0.575 1 0 - rn/MN and = Ukw = 41.2 1 0 - rad/MNm.
Rotational centres of rocking movement:
Now, ax 2 can be calculated according to Eq. (318). First we calculate a0.
2
a0 =
2s
= ^ ( + 1 . 7 4 - 2 . 4 4 ) = 3.17 m
2- 1.74^41.2
+s -L )
whence
2
a h2 = -3.17/3.17 +2.44
Thus, the two rotational centres of the rocking movement (poles 1 and 2) are defined by ax = 0.83 m
and \a2 I = 7.17 m.
Pole displacements:
The pole displacements ( of the upper pole 1 and 2 of the lower pole 2) are given by Eqs. (323
324 ):
3
3
6
3
{ =9.81193(7.17/8.0)10- [0.57510- + (0.83 + 1.74)^41.2-10 ] = 1.44 10" m
3
3
2
6
3
2 = 9.81 193(0.83/8.0) 10- [0.574 -f (1.74 - 7.17) 41.2 1 0 ] = 0 . 3 5 - 1 0 m
Natural frequencies:
3
fi = ( 2 / 1 . 4 4 1 0 - ) " = 13 H z
3 -1
fn2 = ( 2 / 0 . 3 5 " )
Total damping:
Geometrical damping:
Bw
pRe
= 27 H z
4 8
- !2
V
45 /4= 0.272
8 1.9 (24 8 / 3 )
whence:
Vy =
Q , 1
i
=0.23
1.272/0.272
1
2
/[1-(9.55/13) ] + (20.269.55/13)
= 1.7
2 = -^
= =2 1 . 1
2 2
/ [ 1 - ( 9 . 5 5 / 2 7 ) ] + (2 0.26 9 . 5 5 / 2 7 )
Equivalent external force system:
The external force system = 2)0 and M of around the elastic centre is replaced by an
equivalent force system distributed between poles 1 and 2. We find:
P =
P{+P2
M =
Pl(al+s)-P2(a2-s)
whence:
Dynamically loaded
M +
foundations
267
P{g2-s)
\ + a2
P(ai+s)-M
r2 =
a\ +a2
We have = 1.7-50 = 85 kN,
M= 1.7-50-3.8 = 323 kNm,
ax = 0.83 m; a2 = 7.17 m and s = 1.74 m.
Introducing these values, w e find:
= 9 8 kN
P2 = - 13 kN
The moment around the elastic centre produced by Px: Mx = 98(0.83 + 1.74) = 252 kNm
Maximum contact stresses:
193#
3-252
a m ax =
= 103 kPa
2
8-3
3 8 /6
W = 3 - 8 5 / 2 4 = 11 kPa
Maximum amplitude at point D:
In direction:
zD =
Mx
kyyl
252-l(h3.
= 0 . 0 4 - 10
24250-2
In the direction:
Pi
Mxh
85-10-3
252-10-3.3
xD = + =
+
= 0.08 10
z
0 9 1 0 3- m
kx
kyy
Total amplitude
= V D + 1740
* D = - ' 24250= 0.09 mm.
Retaining
268
structures
RETAINING S T R U C T U R E S
1. I N T R O D U C T I O N
O n e of the first analysed problems in foundation engineering, also considered as one of
the major ones, is h o w to evaluate the earth pressure distribution on earth retaining
structures and structural elements e m b e d d e d in soil. A theoretical analysis of the earth
pressure distribution in the upper and lower limit states was published by C o u l o m b
already in the beginning of the 18th century. With increasing urbanisation and consequential
need of utilising expensive land property, buildings are provided with an increasing
number of underground floors. D a m a g e s to underdimensioned retaining structures often
lead to heavy additional foundation costs. Therefore, a correct evaluation of the earth
pressure is no doubt imperative in foundation engineering.
Earth pressure is defined as the force or the stress acting at the boundary between a
structural element and the soil. T h e magnitude, distribution and direction of the earth
pressure depend on the relative m o v e m e n t between the structure and the soil, on dynamic
actions, frost action, etc. If the soil displacements b e c o m e large e n o u g h for soil failure
to occur, certain limits will be reached depending upon the shearing resistance of the soil,
the roughness of the structure and, even m o r e important, the size and direction of the
relative m o v e m e n t s between structure and soil. Obviously, in order to find the most
probable earth pressure distribution the m o v e m e n t of the retaining structure has to be
clarified.
2. E A R T H P R E S S U R E A G A I N S T R E T A I N I N G W A L L S
2.1 Introductory
remarks
An extensive analysis of the various failure patterns that can occur depending upon the
m o v e m e n t of a retaining wall has been presented by Brinch H a n s e n (1953), s o m e of
which are presented in Fig. 187. As can b e seen, t w o types of failure patterns exist: one
characterised by the development of two groups of failure surfaces intersecting each
other by the angle 9 0 - 0 ' ( z o n e failure) and another by the d e v e l o p m e n t of a single failure
surface (line failure). A combination of zone failure and line failure is c o m m o n .
In the design of retaining walls, the p r o b l e m of earth pressure evaluation is generally
simplified to three cases:
earth pressure at rest (no m o v e m e n t between soil and wall),
active earth pressure (the wall is m o v i n g a w a y from the soil until failure takes place),
Retaining
269
structures
> 1.26
> 1.21
1.21 >
1.26 >
> 0.52
0.52 >
> 0.49
> 0.67
I
0.67>>.33
0.49 > 7] > 0
n=
z/h
= ZJh
0 >
> -
0 >
> -
Fig. 187. Examples of different types of failure depending upon the situation of the rotational centre of
the wall and the direction of rotation.
Passive earth pressure (the wall is m o v i n g against the soil until failure takes place).
T h e m e c h a n i c a l b a c k g r o u n d to these three cases will b e elucidated in brief.
(i)
Earth pressure
subjected to an evenly distributed surface loadit is evident that the horizontal normal
stresses \ acting in an imaginary vertical plane through the soil is i n d e p e n d e n t of the
direction of the plane. Moreover, due to reasons of symmetry, shear stresses cannot exist
in the plane. Vertical and horizontal stresses are thus principal stresses.
Now, if the imaginary plane is e x c h a n g e d for a rigid, unyielding wall, the stress situation in the soil m a s s obviously does not change. On the assumption that no vertical
displacement b e t w e e n the wall and the soil is taking place, no shear stresses along the wall
will appear. Consequently, in such a case, the roughness of the wall has no influence on
the stress situation.
T h e horizontal stresses acting at the soil/wall interface u n d e r these p r e m i s e s represent
the earth pressure at rest.
Retaining
270
structures
Dense state
Earth pressure
Critical density
Critical density
From the soil
Wall movement
Fig. 188. Influence of wall movement on the development of earth pressure exemplified for dense and
loose cohesionless soil
position of rest, away from the soil mass. T h e earth pressure is then gradually decreasing
as shown in Fig. 188. For cohesionless soil, two borderline cases m a y occur depending
upon whether or not the soil is in a dense or in a loose state. In the case of dense soil, the
earth pressure, first of all, rapidly reaches a m i n i m u m value, but, by additional m o v e m e n t
of the wall, it increases again until it b e c o m e s constant, independent of further wall
m o v e m e n t . On the other hand, if the soil is in a loose state, the earth pressure gradually
decreases until it b e c o m e s constant, independent of further m o v e m e n t . In both cases,
when the earth pressure has b e c o m e independent of the wall m o v e m e n t , the soil has
turned into a state of critical void
ratio.
decrease
in the active earth pressure. However, the opposite case m a y occur w h e r e the wall is
m o v i n g d o w n w a r d s in relation to the soil. T h e n the shear stresses will contribute to the
d e v e l o p m e n t of failure and, consequently, cause an increase in the active earth pressure.
(iii) Passive
earth pressure.
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271
structures
soil. For dense, cohesionless soil the m a x i m u m value will b e reached at a fairly small wall
m o v e m e n t , m u c h larger though than in the case of active earth pressure. At increasing
wall m o v e m e n t , the earth pressure will decrease until it reaches a constant value
independent of further wall m o v e m e n t s . For loose, cohesionless soil the earth pressure
will gradually increase with increasing wall m o v e m e n t until it reaches a constant value.
In both cases the constant values represent the earth pressure at critical void ratio.
In the case of cohesive soil, the earth pressure reaches a m a x i m u m determined by the
undrained shear strength of the soil. Further wall m o v e m e n t will cause a decrease of the
earth pressure until it reaches a m i n i m u m value determined by the residual shear strength
of the soil. If the wall is arrested after the earth pressure has reached its m a x i m u m it will
decrease with time due to creep effects.
W h e n the wall is m o v i n g inwards, towards the soil, the latter will travel upwards
against the gravitational forces. T h e shear stresses thus generated at the soil/wall interface will prevent soil failure and consequently cause an increase in the earth pressure. On
the other hand, if the wall is m o v i n g u p w a r d s in relation to the soil the shear stresses at
the wall/soil interface will contribute to soil failure and consequently cause a
decrease
Influence
at rest
distributed
surface
Po= Ko'v
w h e r e K0 = the coefficient of earth pressure at rest,
' = the effective overburden pressure.
T h e earth pressure coefficient K0 can be calculated on the basis of the theory of elasticity. Since, by definition, no horizontal movement of the wall is taking place, i.e. eh = 0, we
have for an isotropic elastic m e d i u m :
/
(340)
K0 = T ^ 1-
(341)
\-
+ ) = 0
w h e r e = P o i s s o n ' s ratio,
\ = the earth pressure p0.
In other w o r d s :
T h e evaluation of Poisson's ratio for natural soil is quite a difficult matter. Therefore,
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272
structures
= l -
sin0'
(342)
= 0.15 + 0.7 wL
(343)
wL).
where K0NC
(344)
L8
If the ground surface is inclined at angle to the horizontal (Fig. 189), K0 can be found
by multiplying the K0 values according to the a b o v e by the factor (1 + ) w h e r e is in
radians and \ is determined on the assumption of horizontal ground surface.
(ii) Influence
z
p0 = c o s 0 s i n
(346)
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273
structures
Fig. 189. Ground surface inclination . The depth coordinate is measured from an assumed horizontal
ground surface.
w h e r e = the angle between the vertical and the line connecting the point of action of
the line load as shown in Fig. 190,
= the depth b e l o w the ground surface.
(iii) Influence
principles as in the previous case, i.e. p0 is the resulting horizontal stress in the plane of
symmetry b e t w e e n two point loads Q (Fig. 190):
z
l-2v
Q3z p
Po = - [ ~ - ~
(347)
a(a + z)
w h e r e a = distance from the point of action of the point load to the wall at depth z,
= radius vector from the point load to the wall.
Again w e h a v e difficulties in evaluating the value of Poisson's coefficient v. For sand
can be assumed equal to 0.3, while for water-saturated clays in the undrained state
is equal to 0.5.
P(Q)
'/< . o//a~ tit = _ / / / w
/
>/
Fig. 190. The earth pressure at rest due to the action of a line load or a point load Q is calculated on
the assumption of an identical action on the opposite side of the wall.
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274
structures
Fig. 191. Directions of active and passive failure lines in a zone of failure and their angles of inclination
to the minor principal stress ' 3 and the major principal stress \.
in the state of
failure
As previously mentioned, w e differ between line failure (failure along a single failure
surface), z o n e failure and combined failure (partly zone failure and partly line failure,
Fig. 187). Z o n e failure is characterised by two sets of failure surfaces, the intersection
angle between the two being 90 ' (Fig. 191). T h e intersection angle between the
failure surfaces and the major principal stress is 4 5 - 0 7 2 and b e t w e e n the failure surfaces
and the minor principal stress 45 + 072.
(i) Inclination
to the horizontal
assume that the ground surface has an angle of inclination to the horizontal equal to .
A s s u m i n g further that the ground surface is subjected to a vertical overload q per unit area
of the inclined ground surface, the equilibrium condition for the triangular soil element
A B C , shown in Fig. 192, yields:
c' sin / 3 s i n ( 2 v 0 + ' - ) cos ' + (q sin ' +
+ c ' c o s c o s 0 ' ) c o s ( 2 v o + 0 ' - / ? ) + qsin=
(348)
(349)
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275
structures
Fig. 192. Intersection between two sets of failure surfaces and ground surface. Passive failure.
and if also = 0:
v0 = 4 5 - 072
If 0 ' = 0, i.e. if w e h a v e an undrained case (a clay with c ^ c j w e
(350)
find:
v 0 = - arccos(- sin) +
2
cu
(351)
and if also - 0:
v 0 = 45
(352)
Example 42: Determine the intersection angle between the failure surface in a non-cohesive soil and
the ground surface under passive failure condition if the inclination of the ground surface = 10 and
the angle of internal friction of the soil 0'= 36.
Solution: Eq.(351) yields:
2 v 0 = arccos(- sinl07sin36) - 36 + 10 = 81
Thus, the intersection angles become v 0 - = 30.5 and 90 - 0 ' - v 0 + = 23.5
Example 43: Determine the corresponding value under active failure condition.
/
Solution: In this case, the angle 0 shall be inserted with a negative sign. This yields:
2 v 0 = arccosf- s i n l 0 7 s i n ( - 36)] + 36 + 10 = 153
The intersection angles become v 0 - = 66.5 and 90 - '- v 0 + = 59.5
Example 44: Determine the intersection angle between the failure surface in a cohesive soil under
undrained passive failure condition, if the ground surface has an inclination of 20 and the undrained
2
shear strength of the soil cu = 20 kPa. (a) The surface load q = 10 k N / m . (b) q = 0.
Fig. 193. Intersection between two sets of failure surfaces and wall.
v 0 = 65
(ii) Inclination
to the horizontal
the wall has a positive angle of inclination to the vertical equal to a as s h o w n in Fig. 193.
A s s u m i n g further that the angle of friction b e t w e e n wall and soil is <5 and the adhesion
cw,
the equilibrium condition for the triangular soil element A B C (Fig. 193) yields:
,/
~
c o s ( 2 v ! + ' + -2a)
w h e r e = c'+ c r ^ t a n ^ '
s i n [ T ! - c c o s 0 + ( c v v/ 2 ) c o t s i n ( 2 0 ) ]
=
(
3
sin
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277
structures
'
Fig. 194. Failure condition when cjc
' - tan5/tan0 .
0, w e find:
2vx = arccos(sin<5/sin0O + 2 - - 0 '
(354)
In the undrained case where c''= c w and both 0 ' = 0 and <5 = 0, w e have:
2vx = a r c c o s ^ / c j + 2 a
(355)
Example 46: Determine the intersection angle between the wall and the failure surface in a noncohesive soil under passive failure condition if the inclination of the wall a = 20 and the internal angle
of friction in the soil '- 36 on the assumption that (a) - 24; (b) =().
Solution: According to Eq. (354) we have:
(a)
2vx = arccos(sin247sin36) + 2-20 - 24 - 36 = 26.2
The angles of intersection with the wall become 90 - v{ +cc = 96.9 and v{ + '- a - 29. 1
(b)
2vx = arccos 0 +2-20 - 36 = 94
The angles of intersection with the wall become 90 -vx+a
= 63 and vx + '- a = 63
Example 47: Determine the intersection angles between the wall and the failure surface for the case
given in Example 46 under active failure condition.
Solution: In this case both <//and shall be inserted with negative signs. We find:
(a)
2vx = arccos[sin(-24)/sin(-36)l + 2-20 + 24 + 36 = 146.2
The angles of intersection with the wall become 90 -vx+a=
36.9 and vx - '- a - 17. 1
(b)
2vx = arccos 0 + 2-20 + 36 = 166
The angles of intersection with the wall become 90 - vx +a - 27 and vx - '- a - 21
Example 48: Determine the intersection angles between the wall and the failure surface for the case
given in Example 46, case (a), under passive failure condition, if the relative movement between wall
and soil is opposite to the general case, i.e. the wall is moving upwards in relation to the soil.
Solution: In this case shall be inserted with negative sign and 0 ' w i t h positive sign. We have:
(a)
2vx = arccos[sin(-24)/sin36] + 2-20 + 24 - 36 = 1 6 . 8
The angles of intersection with the wall become 90 -vx
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278
structures
Fig. 195. Mhr circle at failure, showing the relation between effective overburden pressure <fz and passive
and active earth pressures p a and p p . Lines a and represent the inclination of the active and passive
failure surfaces.
Example 49: Determine the intersection angle between the wall and the failure surface under active
failure condition in a cohesive soil with an undrained shear strength cu = 25 kPa if the inclination of the
wall a- 20 and the adhesion between wall and soil cw = 20 kPa. The relative movement between wall
and soil is opposite to the general case, i.e. the wall is moving downwards in relation to the soil.
Solution: The shear strength cu shall be inserted with negative and the adhesion cw with positive value.
We have:
2vj = arccos[20/(-25)] + 20 = 163. 1
The angles of intersection with the wall become 90 - Vj + a = 38.4 and vx - a = 51.6
due to zone
failure
ground surface: smooth vertical wall. In the case of a smooth vertical wall,
no shear stresses will appear at the wall/soil interface. Vertical and horizontal stresses in
such a condition are principal stresses. In the active case, when the wall is translated away
from the soil, the horizontal stress b e c o m e s the minor principal stress corresponding
to active earth pressure while in the passive case, when the wall is translated against
the soil, the horizontal stress b e c o m e s the major principal stress corresponding to
passive earth pressure. T h e magnitude of active and passive earth pressures can be
obtained geometrically from the M h r stress circles (Fig. 195).
By geometrical and trigonometrical considerations w e find:
' =
' ~*
;=
' ~ ':
(356)
Expressing the active and passive earth pressures p'a and p'p with the customary forms
p'a = '
andp'p
= '
w e find {Mohr-Coulomb's
failure
criterion):
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279
structures
2c'
(357)
&JNA
(358)
= +
w h e r e N* =
<
1 -sin0'
= tan (45
4 - 0 7 2 )
If the effective cohesion intercept c ' i s e x c h a n g e d for attraction (Fig. 195) w e find:
2c'J
sin0
(359)
and
2c'
= 2a-
sin0 '
(360)
1 +sin0 '
2 a ).
1 - sin 0 '
As soil cannot take long-term tensile stresses, this will eventually lead to fissuring.
T h e active and passive earth pressure h a v e been expressed in terms of effective stress
conditions. T h e total earth pressure is obtained by adding the g r o u n d w a t e r pressure uw
(which is positive b e l o w but negative a b o v e the g r o u n d w a t e r level in the capillary
zone). Consequently, the total active and passive earth pressures are obtained by the
respective relations:
pa = Kao'z+uw=Kaaz
Pp = '+
(361)
+ uw(l-Ka)
K
w = pz
- w( p
(362)
90-'
90 + </>'
Fig. 196. Failure surfaces and major and minor principle stresses in the active and passive states of
failure.
280
Retaining
structures
(363)
= az-2cu
Pp = <*z +
2 c
(364)
In the case of active earth pressure presented above, the major principal stress is in the
vertical direction and is equal to the effective overburden pressure. Consequently, the
shape and inclination of the failure surfaces in the zone of failure are known since they
always form an angle with the major principal stress direction of 45 - 072. Then the
failure surfaces b e c o m e plane and have an inclination of 45 + 072 to the horizontal. In
the passive case, the major principal stress is in the horizontal direction and is equal to
p'p.
T h e failure surfaces become plane and have an inclination of 45 - 072 to the horizontal
(Fig. 197).
T h e stress vectors acting against one group of parallel failure surfaces b e c o m e parallel
with the other group of intersecting failure surfaces.
In the deduction of the active and passive earth pressures it was presumed that the wall
is either subjected to a translation or a rotation around an axis situated below the foot of
the wall or at the foot level {cf. Fig. 187). Moreover, the wall m o v e m e n t w a s presumed
to be large enough. (Generally, the wall m o v e m e n t required for the development of active
earth pressure can be assumed to be 0.1 % to 0.5% and for passive earth pressure 1 % to 5%).
(ii) Horizontal
ground surface:
deal with a retaining wall with a rough surface which entails full or partial friction (and
adhesion) between the wall and the soil. Consider a case where the wall is translated in
parallel against the soil and where the angle of friction between the soil and the wall is
equal to the angle of internal friction of the soil (<5= 00- T h e angles of inclination v 0 and
vj (Figs. 192-193) according to Eqs. (349) and (354) become v 0 = 45 - 072 and vx = - '. Thus,
the angles of intersection between the failure surfaces and the wall become 90 + 0'and 0
respectively. B e t w e e n the wall and the failure zone, close to the ground surface with plane
failure surfaces forming an angle with the ground surface of ( 4 5 - 072) the "Rankine
Fig. 197. Prandtl and Rankine failure zones in cohesionless soil. The Prandtl zone forms a transition
zone between the wall and the Rankine zone. Arrows indicate the major principal stress directions.
Retaining
281
structures
Fig. 198. Passive earth pressure resultant and failure zone in non-cohesive soil with an internal angle
of friction of 30.
='.
zone", a zone of transition is formed with one group of plane failure surfaces, starting
radially from the line of intersection between the wall and the ground surface and another
group of logarithmic failure surfaces, intersecting the radial failure surfaces by an angle
of 90 0 t h e "Prandtl zone"(Fig. 197).
In the normal passive case, the earth behind the wall m o v e s in the u p w a r d direction
relative to the wall. Consequently, the frictional forces acting along the wall h a v e a downward direction (Fig. 198).
A s s u m i n g instead that the frictional forces h a v e the opposite direction, /.e. --
\ w e
find = 90 and the angles of intersection with the wall 0 and 90 - ', respectively. T h e
size of the failure z o n e in this case b e c o m e s strongly reduced (Fig. 199) and so is the case
with the passive earth pressure also.
T h e failure zones obtained for = 0.5 ' and for = 0 are shown in Fig. 200.
The earth pressure coefficients for various values of wall friction are presented in Table 36.
For cohesive soils in undrained condition, w e find v 0 = 45 and vx = 0 in the normal
passive case, when cw = cu (cw acting downwards on the soil), and = 90 when cw - - cu.
T h e failure zones obtained for cw = 0.5c w and cw = 0 are shown in Fig. 2 0 1 .
Fig. 199. Passive earth pressure resultant and failure zone in non-cohesive soil with an internal angle
of friction of 30. = -'.
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282
structures
Fig. 200. Passive failure zones in cohesionless soil for (1) = - 0 . 5 0 ' (2) (5 = 0 and (3) =0.5'. Angle
of internal friction '- 30. Centre of logarithmic boundary of Prandtl failure zone for
-0.50'shown.
Introducing r = cw/cu,
the normal
component
+2r/3
(365)
Pan = - 2 c M \ / 1 + 2 r / 3
(366)
Ppn = z +
id
Fig. 201. Passive failure zones in clay for (1) cw = - 0.5c M, (2) cw - 0 and (3) cw = 0.5c u. Centre of
circular boundary of Prandtl failure zone for cw = - 0.5cu shown.
Retaining
283
structures
-1
'
'=
10
15
20
25
30
35
40
45
1.060
1.057
1.042
1.017
0.981
0.937
0.880
0.820
(iii) Leaning
-2/3
0.814
0.724
0.636
0.554
0.476
0.403
0.336
0.274
2/3
0.704
0.589
0.490
0.406
0.333
0.271
0.217
0.172
0.656
0.537
0.442
0.364
0.300
0.247
0.202
0.163
1
0.649
0.531
0.440
0.367
0.308
0.260
0.219
0.185
1/2
-1/2
-2/3
1.66
2.20
3.1
4.4
6.5
10.5
18
35
1.55
1.97
2.55
3.40
4.60
6.50
9.60
15.0
1.42
1.70
2.05
2.45
3.00
3.69
4.60
5.83
1.25
1.38
1.53
1.65
1.80
1.93
2.08
2.20
1.17
1.26
1.33
1.40
1.46
1.51
1.54
1.55
surfaces and the ground surface on one h a n d and the wall on the other are given by
Eqs.(349) and (354). This m a k e s possible a graphical determination of the earth pressure
as will b e shown in the following chapter. Quite complete tables have been worked out
by Caquot etal (1973) and some of the values given there are summarised in Figs. 202-205.
An analytical expression for the earth pressure in the case of weightless soil can be
obtained by K t t e r ' s equations and the equilibrium conditions governing the respective
intersection angles between ground surface/failure surface and wall/failure surface.
Thus, for a cohesionless soil, the normal earth pressure coefficient from a vertical load
q per unit area of the ground surface is obtained by the relation:
K
(367)
Pnq = nQ
where:
2
Kn
cos (v1-a) + s i n ( 0 + v 1- a )
e x p [ 2 ( v 0 - vx) tan0 J
Kq = Kn/cosS
For a soil with cohesion, the cohesion term can b e expressed as:
/ : c = ( / i : n- l ) c o t a n 0
(369)
A conservative value of the earth pressure coefficient due to the weight of the soil is
obtained b y the approximate relation:
Retaining
284
-30
-20
-10
10
20
30
-30
-20
-10
10
20
30
structures
Fig. 202. Earth pressure coefficient Ka and Kp as functions of wall inclination and ground surface
inclination . Angle of internal friction ' = 35 and = 0. (In Figs. 202-205 = \ p p = Kp/Q
Ky~Kncos(-a)
(370)
(371)
F o r ' - 0 w e h a v e :
Kq = cos
Kc = 2 ( v 0 -
) + s i n [ 2 ( v 0 - j3)] + sin[2(v! - a)
V l
^ - c o s a + sinjScosiv^aysinCvo-jS)
(372)
(373)
Retaining
0.06
-30
285
structures
-20
-10
1
10
1
20
I
30
Fig. 203. Earth pressure coefficient^ as function of wall inclination aand ground surface inclination Angle
of internal friction '= 30 (top) and '= 35. = 2073. Broken lines refer to negative values of .
2.5
Graphical
determination
of earth
pressure
If the wall and the ground surface are irregularly shaped, the analytical solution to the
earth pressure problem becomes very intricate and graphical solutions are then a helpful
means of solving the problem. In the majority of cases, although it is true only when the
wall is smooth, it can be assumed that the failure surfaces are plane. It is true that the
resulting error is on the unsafe side but in the case of active earth pressure the error is
generally negligible. However, in most practical cases, the use of combined plane and
curved failure surfaces is a relatively simple procedure and ought to be applied, at least
in the case of passive earth pressure.
Retaining
286
-30
-20
-10
10
20
structures
30
Fig. 204. Earth pressure coefficient Kp as function of wall inclination and ground surface inclination
. Angle of internal friction 0' = 30 and = 072. Broken lines refer to negative values of .
surfaces.
pressure based on plane failure surfaces h a v e been presented, generally to determine the
respective earth pressures obtained at a n u m b e r of a s s u m e d inclinations of the outer limit
of the failure zone. In this way, the inclination that leads to the m o s t critical earth pressure
value can b e established.
A m o n g the m e t h o d s introduced, Engesser's
to use.
T h e use of E n g e s s e r ' s method is demonstrated in Fig. 2 0 6 . T h r e e plane failure
surfaces, ( 1 ), (2) and (3), h a v e been selected to represent the outer limit of the failure zone.
For each one, the resultant of the soil weight inside the failure z o n e and possible overload
is k n o w n in size and line of action as well as the lines of action of the earth pressure
resultants against the wall and the plane failure surface in question. T h e earth p r e s s u r e
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287
structures
Fig. 205. Earth pressure coefficient Kp as function of wall inclination and ground surface inclination
. Angle of internal friction '= 35 and = 072. Broken lines refer to negative values of 5.
vector
reaching the envelope to the earth pressure resultants against the failure
surfaces in the force vector polygon, represents the true active earth pressure.
(ii) Curved failure
surface.
Retaining
288
structures
First determine the angles v 0 and vx according to Eqs. (349) and (354). T h e angle v 0
defines one of the two sets of failure surfaces in the R a n k i n e z o n e A B C while the other
set of failure surfaces is fixed by the intersection angles 90 '. T h e angle vx defines the
intersection angle b e t w e e n the failure surface and the horizontal nearest to the wall in the
Prandtl failure z o n e A C D .
T h e centre of the logarithmic spiral, forming the Prandtl failure zone, is situated
on the boundary line between the Rankine and the Prandtl zones (or on its extension). It
is also situated on the failure line extended from the b o t t o m corner D of the wall and
forming an angle with the logarithmic failure surface of 90 - '. T h e intersection point
between these two b o u n d a r y lines forms the centre of the logarithmic failure surfaces
with the centre angle determined by the relation:
- lv 0 v xl
T h e distance [OC] is determined in relation to the k n o w n distance [OD] by the
correlation:
[OC] = [OD]exp(->tan ')
w h e r e in radians and:
[OD] = [ A D ]
S i n ( V
o
^ sin(| v 0 - v x \ )
Retaining
289
structures
Fig.207. Example of graphical determination of the passive earth pressure Pp on the assumption of a
combined Rankine/Prandtl failure zone. ('= 35; =- 0.50'; = 20; = 10).
the centre of the logarithmic spiral and by the intersection point b e t w e e n Pp (the earth
p r e s s u r e resultant acting against the wall) and the resultant R.
H a v i n g established the line of action of Q, the force p o l y g o n can finally b e closed
w h i c h yields the value of the earth p r e s s u r e
Pp.
Example 50: Determine the passive earth pressure against a retaining wall with height 5 m and an angle
of inclination = arctan(2/5) = 21.8 if the ground behind the wall consists of sand with a surface
3
inclination of 1:1.5 [ = arctan (1/1.5) = 33.7]. The sand has a unit weight of 18 k N / m and an angle
of internal friction '= 35. The friction angle in the soil/wall interface is = 20. The groundwater level
is below the foot of the wall.
Solution: The intersection angles v 0 and v 1 are:
v 0 = 0.5[33.7 - 35 + arccos(- sin33.7/sin35)] = 82.0
V! = 0.5[2-21.8 - 35 - 20 + arccos(sin207sin35)] = 2 . 0
The intersecton angle between failure surface and ground surface is v 0 - = 82.0 - 33.7 = 48.3
The opening angle of the Prandtl zone = v 0 - v 1 = 6 1 (1.064 rad.)
The length of the radii vector [OD] and [OC] of the logarithmic spiral are:
[OD] = y/29 -sin(82 + 3 5 - 21.8)/sin(82 - 21) = 6.13 m
[OC] = 6.13exp(1.064-tan35) = 12.92 m
This gives us the outer boundary BC of the Rankine zone.
The weight of the soil belonging to the Rankine zone can now be calculated. We find Gx = 102 kN.
Retaining
290
structures
The weight of the soil belonging to the Rankine zone can be calculated as the weight of the
logarithmic spiral sector area [OCD] minus the weight of the triangular area [ADO]. The area [OCD]
is found by the relation:
[OCD] = -i^^-[exp(2u)tan0 )- 1]
4tan0
/
Retaining
structures
291
the line of action of Q which makes it possible to close the force polygon (which is the requirement of
equilibrium). W e find Pp = 3000 kN which yields the passive earth pressure coefficient = 3000/
(29-18/2) = 11.5, corresponding to = 11.5-cos5 = 10.8.
This value can be compared with the approximate value obtained by Eq. (367)which yields Kpn = 8.5
and, consequently, = 8 . 5 - c o s ( - ) = 8.3 (Eq. 370).
Example 5 1 : Determine graphically the passive earth pressure in the soil conditions presented in
example 50 if the ground surface has an inclination of 1:3 and =- 20 (the soil behind the wall is settling
in relation to the wall).
292
Retaining
structures
Retaining
293
structures
surface at D. The centre angle of the Prandtl zone = v 0 - vx = 9.4 = 0.164 radians. The lengths of the
vector radii [OD] and [OC] are:
[OD] = / 2 6 -sin(57.0 - 35 + 1 l.3)lsin(9.4) = 17.0 m
[OC] = 17.0exp[0.164-tan(- 35)] = 15.2 m.
Furthermore, we have 0)Q = vx - '= 12.6.
The logarithmic spiral area[ODC] and the distance from its centre of gravity to the vertical through
can now be calculated by the relations given in Example 50. Inserting the values of r 0, 0 and , and
2
' = - 35, we find [ODC] = 21.27 m and = 3.14 m.
2
The triangular area [OAD] = 17.0 / 2 6 -sin(^ + )/2 = 17.43 m and the distance from the vertical
through to its centre of gravity is 2.8 m.
2
The area of the Rankine zone is [ODC] - [OAC] = 3.84 m and the distance from the vertical through
to its centre of gravity is (21.27-3.14 - 17.43-2.8)/3.84 = 4.7 m.
Proceeding along the lines given in Examples 5 0 - 5 1 , we find PA = 46 kN which corresponds to the
active earth pressure coefficient = 46/(18-26/2) = 0.20.
The approximate value obtained by Eq. (367) is Kn = 0.23 which yields Kay= 0.208/cos 11 .3 = 0.21.
Example 53. Determine the passive pressure against a 4 m high wall supporting a cohesive soil with
3
an undrained shear strength cu = 20 kPa and a unit weight of 16 k N / m . The ground surface is inclined
1:5 and the wall is leaning backwards by the angle arctan( 1/4) = 0.245. The adhesion between wall and
soil cw= 10 kPa.
Solution: We have cjcu = 0.5, = arctan(0.2) = 0.197 and a = - 0.245. The angles v 0 and vx thus become:
v 0 = 0.5-arccos 0 + 0.197 = 0.982 (56.3)
V! = 0.5-arccos 0.5 - 0.245 = 0.279 (16.0)
The boundary failure line between the Rankine and the Prandtl failure zones has an inclination to the
D
0
100
200 kN/m
294
Retaining
structures
horizontal of /2 - 0.982 = 0.589 (33.7). The centre of the Prandtl zone is situated at the intersection
between this boundary failure line and a line perpendicular to the tangent of the Prandtl zone at D. As
the boundary of the Prandtl zone is a circle with ;ts centre at O, the outer boundary of the Rankine zone
is also given. The radius [OD] is found equal to 5.97 m and the opening angle co= 0.704.
The forces acting against the boundary of the Rankine zone represent the shear forces and normal
forces along BC and AC. The shear forces are given by the shear strength times the lengths of BC and
AC while the size of the normal forces are obtained from the force polygon. The resultant of the forces
acting at the interface between the Rankine and the Prandtl zone R and its line of action is obtained from
the force polygon. Regarding the forces acting against the Prandtl zone we only know the size and lines
of action of the shear forces C 2 and Cw, but not the size, nor the lines of action, of the normal forces Pp
and N2. The line of action of depends on the magnitude of Cw, and the line of action of N2 (passing
through O) depends on the line of action of Pp. However, it can be approximately assumed that N2
coincides with the bisector to the angle . By this assumption the force polygon can be closed and we
find Pp = 450 kN/m.
This can be compared with the result obtained by Eqs. (372-373). Accordingly, we have:
Kc = 2(0.982 - 0.279) + sin[2(0.982 - 0.197)] + sin[2(0.279 + 0.245)] = 3.27
Ky= cos (-0.425) + sin(0.197)-cos(0.279 + 0.245)/sin(0.982 - 0.197) = 1.21
This yields:
Pp = 4.12(1.21-4.12 16/2 + 3.27-20) = 434 kN/m
2.6
Earth pressure
distribution
at enforced
wall
rotation
As s h o w n in Fig. 180, the level of the rotational axis of the wall is vitally important to the
d e v e l o p m e n t of failure and earth pressure distribution. T h e zone failure normally used
as a basis for earth pressure evaluation tales place only w h e n the rotational axis of the wall
is b e l o w or at the level of the foot of the wall. A combination of z o n e failure and line
failure takes place w h e n the rotational axis is below the mid-height of the wall and line
failure w h e n the rotational axis is above the mid-height of the wall. In consideration of
the situation of the rotational axis, the normal component
can b e
component
= K/z
+ Kqq
(374)
+ Kcc
by the relation:
/= /<
+ (Kqq
+ Kcc')tenq
+ cw
(375)
7]
= /
Fig. 209. Graph for determination of the value for smooth wall ( = 0; top figure) and rough wall (<5= ').
Retaining
296
structures
0.05
Fig. 210. Kyx a n d Kyy for smooth w a l l (<5= 0; top figure) a n d rough w a l l ( = 0 0
P'xn = + qx<!
P'yn = /
+ qy
<l +
Kc
cx '
(376)
&cf'
(377)
A
+ c
fx = ,^+
fy = Kyyfztan5Y+
(Kqy q + KyCtanq + c
(378)
(379)
Retaining
structures
297
Example 54: Determine according to Brinch Hansen's method the earth pressure distribution against
a retaining wall, 5 m in height, if the wall rotates clockwise around a horizontal axis 1 m below the top
of the wall. The soil consists of sand with an internal friction angle '= 30. The friction angle in the
2
soil/wall interface = 30. The ground behind the wall is loaded with q = 20 k N / m .
Solution: W e have = ZJh = 4/5 = 0.8 and a positive rotation. According to Fig. 208 we obtain = 0.87
which yields Zj = 0.87-5 = 4.35 m. From Figs. 209 and 210 we find
= 5.6,
= 0.22, Kqx = 1.8 and
Kqy = 0 . 1 8 . The friction along the wall, according to Fig. 205, is t a n 5 7 = t a n ^ = - 0.58.
The earth pressure distribution follows the relation:
above a depth of 0.65 m ( < 0.65 m) / / n = 5.6-18-z + 1.8-20 = 101z + 36 kPa
below a depth of 0.65 m (z > 0.65 m) p'n = 0.22-18-z + 0.18-20 = 4.0z + 3.6 kPa
The earth pressure vector is inclined 30 downwards and is equal to:
2
2
/ / = / / n ( l + 0 . 5 8 ) " = 1.156/7;
Example 55: Compare the earth pressure distribution according to Brinch Hansen's theory with the
passive earth pressure distribution according to the classical Coulomb theory if the rotational axis in the
case given in Example 54 is situated at the foot of the wall.
Retaining
298
structures
ZJh
earth pressure vector is inclined 30 upwards and the earth pressure coefficient becomes Kp = Kx( 1 +
2
1 /2
2.7
Design
of gravity
6.5.
walls
Retaining
299
structures
40-^
30
-20^
- u J
-40-
- 20
1 LI
T T
.40--"
30^
\\
-20
OP-H
;40_
_
-4 -1
1 .,
-0.1 0
W 0
-L
0.5
1.4 2
r\ = ZJh
Retaining
300
structures
7\ = ZJh
The vertical component V= (4.35-19 + 0.65-24)-5.2 + 5 0 . 8 - 2 4 = 607 kN/m, acting 2.5 m inside the
toe of the wall.
The inclination of the resultant is arctan( 199/607) 18 which, ignoring the restraint from the earth
pressure at the passive side of the wall, is also the friction angle required to prevent sliding.
The line of action of the resultant intersects the base of the wall at a distance form the toe of b'= 2.5
- 2 . 2 5 - 1 9 9 / 6 0 7 - 1.76 m.
Retaining
"- -
<
20
30 -
40^
30- %
20
301
structures
>
40^
0
40 - I
--
>
""
40- i
"
20
30
40
^ ( ^ \
ono
T T "
-20'
-40
40^
/
I
- 4 -1
-0.1 0
ft
\ ^
0.5
1 1
1.4
= ZJh
cjc'
For a friction angle of 35 we have Nq = 33 (Eq. 112) and = 45 (Eq. 114). According to Eqs. ( 127)
and (129) we find:
3
iq = (\ - 0.7 199/607) = 0.46
3
i y = ( l - 199/607) = 0.30
The bearing capacity of the retaining wall is thus:
Vf= 1.5 19-330.46 + 1.76-19-45 0.3 - 884 kN/m (> 607 kN/m)
This result is unsatisfactory in practice. Thus the design value of the friction angle, for example
according to Eurocode 7, is chosen as arctan (tan07l .25) which in our case would lead to a design value
of '= 29. This gives K0 = 0.52, Nq ~ 17 and ~ 18. The increase in the K0 value yields a horizontal
force = (0.52/0.43) 199 = 241 kN/m. Now the line of action of the resultant intersects the base of the
wall at a distance from the toe of b'= 2.5 - 2.25-241/607 = 1.61 m. The inclination factors will be changed
to ^ = 0.38 and i r = 0.22.
To satisfy the demands of Eurocode, the bottom plate must be corbelled out along the front of the
wall. Assuming a corbelling width of x m we find the correlation:
Retaining
302
I I I Iii I I I ,
Ph
A ydz
Phtan6
structures
+ dz
3.
3.1
Silo
pressure
p / ?tan<5 w h e r e 5 i s the angle of friction between the silo walls and the material inside. T h e
unit weight of the material is .
Equilibrium in the vertical direction is satisfied by the relation:
dz.
-)
+ Cph tan = 0
(380)
(381)
Retaining
303
structures
Ph = [1 - e x p ( ) ]
ji
R
(382)
Originally, the value was determined with regard to the earth pressure obtained
during filling of the silo or in at rest condition. Later investigations (Bergau, 1959;
Turitzin, 1963), initiated by a number of silo damages, showed that the most critical
condition occurred during emptying of the silo where its pressure was found to be
approximately twice as high as it had been during filling. The reason behind this fact is
that of arching phenomena. Special arrangements can be made in order to create an even
outflow of the fill material and thus prevent arching. The discharge gates should be placed
centrically. In the case of eccentric discharge gates, the silo pressure according to
Janssen-Knen will have to be corrected. The German code D I N 1055:6 recommends to
consider the eccentricity by assuming an idealised silo, with its centre coinciding with the
eccentric discharge gate (Fig. 217).
In a supplement of 1977 to D I N 1055:6 it is recommended to use the correction factor:
c = 1 +0.20t + e/1.5)
(383)
Retaining
304
-1
-2/3
-1/3
'
1/3
2/3
structures
/'
Fig. 218. Theoretical silo pressure coefficients in the active and passive state.
the silo pressure coefficient during filling, should not be b e l o w the earth pressure
coefficient at rest:
Xf=K0=l
-sin0'
(384)
Example 57: A concrete silo with an inner diameter of 8.5 m and a height of 20 m shall be designed
for the storage of iron ore. The silo has two rectangular discharge gates (2.8m by 5.0 m) located 2.1 m
from the centre of the silo. The iron ore to be stored has an iron content of 55% by volume and shall be
filled into the silo from the top by means of a conveyor. The porosity of the ore material filled into the
silo is estimated at 45% and the water content w = 10%. The angle of internal friction is '= 40 and
the angle of wall friction 5 = 0.750'. Determine the design silo pressure.
Retaining
305
structures
Solution: The grain density pg of the ore material is obtained by the relation:
- 0.55-(p fe - p s t) ]
P8 = PftPj\Pk
3
where p f e = specific weight of pure iron = 7.85 t/m
3
p s t = specific weight of stone material = 2.75 t/m
3
This yields pg = 4.2 t/m
The bulk density of the material becomes equal to (Eq. 12):
3
= pg{\ - n){\ +w) = 4.2-0.55-1.1 = 2 . 5 4 t/m
Due to the eccentricity of the discharge gate, the hydraulic radius becomes equal to:
2
R = (8.5 /4 + 22.18.5)/(8.5 + 2-2.1) = 3.0 m
The correction factor with regard to eccentric discharge gates:
c = 1+0.20-2.1-3.0/1.5 = 1.84
With the values of 0'and 8given, we find (Fig. 211) a discharge pressure coefficient Xe - 1.4 while
= tan(0.75-40) = 0.577. Thus the design silo pressure varies with depth below the top surface of the
material filled into the silo according to the relation:
OA
Ph = 1 . 8 4
2 . 5 4 - 9 . 8 1 - 3 . 0 ri
0.577 1.4
1
[l-exp(
)] = 2 3 8 [ l - e x p ( - 0 . 2 6 9 z ) ]
2.54-9.81-3.0M
, 0.577 0.5 -
_1
, _ n Q. ^
[l-exp(
)] = 1 3 0 [ l - e x p ( - 0 . 0 9 6 z ) J
0.577
3.0
The maximum lateral design pressure just above the bottom of the silo (z = 20 m) is obtained during
emptying of the silo (completely filled) and becomes 238 kPa.
The maximum pressure during filling to full height becomes 111 kPa, i.e. 0.47 times the pressure
during emptying.
Using instead ^= (1 - sin^O = 0.36 and = 2^= 0.72, the maximum values of the silo pressure
become 223 kPa during emptying and 97 kPa during filling.
3.2
Earth pressure
against
underground
pipelines.
In the design of buried pipes, one m u s t distinguish b e t w e e n rigid pipes and flexible pipes.
A pipe can be considered as rigid if the pipe stiffness Sp fulfils the condition:
Sp = -Ep(b)
3
A + tp
(385)
306
Retaining
structures
measured at the crown level of the pipe, can b e assumed equal to 4 times the outer diameter
of the pipe.
A great n u m b e r of design m e t h o d s h a v e been developed with reference to different
types of pipe installation. Extensive information regarding types of pipe installation and
relevant design m e t h o d s is presented by Liedberg (1991). In this connection, only the
most c o m m o n semi-empirical design principle, the so-called M a r s t o n / S p a n g l e r theory,
will b e presented.
Retaining structures
In the case of trench pipelines, the back-fill is a s s u m e d to settle in relation to the natural
soil at the sides of the trench. Consequently, shear stresses at the interface between backfill and natural soil are acting on the back-fill sides in an u p w a r d direction (Fig. 219).
Now, the vertical earth pressure at the crown level, exerted by the weight of the soil,
can b e deduced according to the s a m e principles as applied in the silo case. With the
symbols used in Fig. 2 1 9 , the condition of equilibrium for the soil element inside the
trench yields:
' = 0
^b-Yb+2Kcfzt<m(l)
dz.
(386)
2Ku
(387)
2Kph
Yb
= ^[l-exp(-^-)]
2Kji
(388)
chosen equal to 0.5 for loose backfill and equal to 1.0 for c o m p a c t e d backfill.
In the case of embankment pipe lines (Fig. 220) placed on a less compressible soil than
that of the e m b a n k m e n t , the settlement at a certain depth below the e m b a n k m e n t surface
will b e larger on either side of the pipe than directly above it. T h e condition of equlibrium
for the soil element with height dz is now obtained by the relation:
Retaining
308
d&7
--2<\'
dz
structures
= 0
(389)
(390)
at depth
- hs, w h e n c e :
2
YD
2Kphs
,
2Kphs
Q* = ^ [ e x p ( - ^ ) - 1] 4 - / D ( A - ^ e x p ( p )
2Kji
D
D
T h e depth h-hsto
(391)
depends on the rigidity of the pipe and the compressibility characteristics of the natural
ground on the one h a n d and the e m b a n k m e n t on the other. For the determination of
hs,
a reduction factor rsd is introduced with reference to the relative vertical settlements at
the crown level of the pipe, above and outside of the pipe, according to the relation:
j . + - f r + M
above.
A b o v e the p i p e w e have:
rhS/i
s = L(h-h
s)Y
0
exp(2Kpz/D)
V P
;
, K
J
-dz=
(h-hsWD
[exp(2^/D)-l]
2
(393)
Retaining
309
structures
s
(h-h)Y
J
M
(394)
(hs + rsdoD)
2Kuhs
= 1 + 28
(395)
If rsd has a n e g a t i v e value, the frictional forces along the vertical section lines h a v e the
opposite direction to that s h o w n in Fig. 2 1 3 . This implies that will h a v e an u p w a r d
instead of a d o w n w a r d direction, /. e. will h a v e to b e inserted into E q s . (391 ) and (395)
with a negative sign. W e thus find:
YD
25
,
28
-exp(p)] +/D(A-AJ)exp(^)
D
D
e x p ( ^ )+
(396)
(397)
= 1 - 23(1
s= f '[
J
- (' z
dz
t-aoD
] + [ / f t - ( < 4 - Yh)
)
2jD
(398)
2jD
'hs -
or z
hs (
at
qz = YQi - hs +z).
Equating this expression and Eq. (393), w e find:
(2/)-1
2
1
(
h
H
hs
rsda1
1+2/
h h,
= -(DD
+rsda)-
d s
^(2^/)
) +
1+2/
1
(- - )
2D D
Ku
h, h,
hs
. / ( - ) =
D
(399)
T h e hs values according to M a r s t o n (Eq. 397) and Spangler (Eq. 399) for rsd a = 0.5
and 0.8, respectively, and - 0.3 are given in Fig. 2 2 1 .
Retaining
310
1.5
Marston , rsda
= 0.8
0.5
Marston , rsda=
Spangle r,
il
hJD
0.5
structures
rsda = 0.%
1
I
1
1
1
1
10
HID
Fig. 221. The level of equal settlement according to Marston and Spangler for rS(fi - 0.5 and 0.8,
respectively, and - 0.3. In Spangler's expression j is assumed equal to 1.
T h e design values of the settlement ratio rsd according to the A m e r i c a n Concrete Pipe
Association
(1988)
220
37.
(1950)
(1978).
distribution of the design earth pressure a ' n against the top part of the pipe and a \
The
against
n = <7cr [
cos
(400)
2(l-ab/2n)
= 4 i n Vc o s ( )
a
(401)
in
TABLE 37.
1
Design values of the settlement ratio rsd according to ACPA (1988)
Foundation condition
Usual range
Design value
1.0
0.5 - 0.8
0.0 - 0.5
1.0
0.7
0.3
For ordinary soil the rsd value depends on the degree of compaction of the fill adjacent to the sides of
the pipe. With construction methods resulting in proper compaction of bedding and side-fill materials,
it is recommended to use the design value rsd = 0.5.
Retaining
311
structures
Fig. 222. Theoretical stress distribution around rigid pipe according to Smith (1978).
cylindrical walls can b e neglected, the pipe will deform under the earth pressure until the
line of thrust b e c o m e s m o r e or less coincident with the centre line of the pipe walls. T h u s ,
the ring force and the consequential risk of buckling will generally determine the pipe
design. However, traffic load or other sources of load variation m a y lead to pipe
deformations w h i c h do not recover.
T h e buckling stress ob in the pipe is determined by the relation ( T i m o s c h e n k o & Gere,
1961):
Retaining
312
structures
Fig. 223. Observed stress distribution around two reinforced concrete pipes, 0.6 m in diameter, one
placed directly on the trench bottom (open circles) and the other on a soft inclusion, 0.10 m in thickness
(filled circles). (Liedberg, 1991)
t +
2yklsD/Gf
w h e r e kls = m o d u l u s of s u b g r a d e reaction,
t = t h i c k n e s s of the p i p e walls,
Of- the failure stress of the p i p e m a t e r i a l in c o m p r e s s i o n .
(402)
Embankments
313
INTRODUCTION
Fig. 224. Subsidence due to placement of fill around a building founded on piles.
Embankments
314
LOADING BERMS
Constant
undrained
shear
strength
As previously shown (p. 120), the bearing capacity qyof cohesive soil with an undrained
shear strength equal to cu can be determined by the relation:
9/=<
where
(403)
= 2 + .
(404)
= cu'2cxa lxfia
or:
cu - Q(a -
e)sm oc/2oca
(405)
Embankments
315
Embankments
316
Fig. 226. Centre of critical failure surface with regard to loading condition.
i.e.
^ = 5.52
T h e situation of the critical failure surface is determined by the condition that the
driving m o m e n t of external load around its centre shall be m a x i m u m . Considering only
vertical, external load w e h a v e (Fig. 226):
= * (-)<*
+
(412)
T h e condition of m a x i m u m m o m e n t yields:
| = ^
- f V = 0
(413)
/?
4 1 4
>
qi
q2<5.52cu2
w h e r e cul
(415)
(416)
317
Embankments
e m b a n k m e n t and cu2 the undrained shear strength nearest below the outer boarder of the
embankment.
T h e centre of the failure surface is given b y the area relation = (Fig. 226), i.e.:
(qx-q2)c
(417)
= q2(b2-c)
whence:
(418)
c = b2q2/qx
+ qxx and the line of action of the resultant is obtained by the relation:
(q2b2
+ qxx)e
= q2b2(b2
(419)
/2 + x) + qxx /2
Q\
<7i
^= M - + - +- - ( - ) ] / +
b2
2q2
b2
< )
42
q2b2
Furthermore w e have:
a = c+x
= b2
<7l
(1+ )
(421)
<72^2
2cua a
- V z"
sin a
= (92^2 +qxx){a-e)
Fig. 227. Failure surface intersecting ground surface inside embankment. Depth restricted.
(422)
Embankments
318
w h e r e = 2arctan(D/a)
and e according to Eqs. ( 4 2 0 - 4 2 1 )
Introducing , and e into Eq. (422), x/b2 can be solved by the relation:
> 2 2
2 c w(
v 2
Q\
<7l
X ) arctan(
q\
<?i
b2
) =
b2q2X
2q2b$
b2q2X
where X = 1 +
qib2
Eq. (423) is valid only \fx<bx.
q\lq2 the value of that yields the m a x i m u m shear strength has to be found. T h e width
has to b e chosen in order that the safety requirements are satisfied.
In order to facilitate the design of the loading berm, J a c o b s o n and Odenstad have
presented the diagram shown in Fig. 228.
Fig. 228. Diagram for determination of width of loading berm when the failure surface touches firm
bottom and intersects ground surface inside or at the opposite side of the embankment.
319
Embankments
Example 58: Determine the loading berm required to achieve a factor of safety of 1.5 for a road embankment, 12 m in width and 3 m in height, placed on a 10 m thick clay deposit with an average undrained
shear strength cu = 10 kPa. The clay is underlain by bedrock. The fill material in the embankment and
3
in the loading berm has a unit weight 7 = 18 kN/m .
2
Solution: The load qx - 3-18 = 54 kN/m . The hight h of the loading berm is obtained by the relation:
5 4 - 18/ = 5.5240/1.5
2
whence h = 1 m and q2 = 18 kN/m .
The design value of the undrained shear strength is cud = 10/1.5 = 6.7 kPa.
We have qx/q2 - 3 and (cu/qx)/Fs = 0.123. From Fig. 228 we find b2/D 2.5 which yields b2 = 25 m.
Moreover, we find xlD 1 . 7 . Since > bx the critical failure surface will intersect the ground surface
at = 12 m. Inserting the value = bx = 12 m and the given values of qx and q2 into Eq. (423) the solution
yields b2 = 22.2 m which is very nearly the same value as that obtained from Fig. 228.
(ii) The failure
surface
b e c o m e s equal to:
Q = q\b\ + 9 2 * 2
Furthermore:
qxb\l2
L LJ
e = -
+
qxbx+q2b2
(425)
w h i c h yields:
2
(qxbx
UM[qlb L/2
q2(bx+b2l2)
-
+q2b2)
+
q2b2(b1+b2/2)]
Example 59: Determine the loading berm required to achieve a factor of safety of 1.5 for a road
embankment, 12 m in width and 3 m in height, placed on a deep clay deposit with an average undrained
shear strength cu - 10 kPa. The fill material in the embankment and in the loading berm has a unit weight
3
7 = 18 kN/m .
Embankments
320
2
Solution: As in example 58, the embankment load qx = 54 k N / m and the hight of the loading berm h
= 1m
The width of the berm is determined according to Eq. (426):
(54 1 2 + 1 8 - f r 2 )
JLO
b2/2)]
whence b2 = 33.7 m ~ 34 m.
2.2
Shear strength
increasing
linearly
with
depth
Assuming that the undrained shear strength increases with depth according to the relation
cu = cu0 + kz and that the horizontal projection 2a (Fig. 229) is given, the stabilising
m o m e n t is obtained by the relation:
CO0
Ms = \
+ 2a
(cll
C)
r)0
+2a
+kz)R dco
= \ [cu
n
2a
m
+ kR (sin -sin
0)]R dco
(427)
whence:
2a cu0[a
- acota)]
+ (ka/cu0)(l
(428)
sin
T h e m i n i m u m value of Ms is determined by the condition dMJda
ka
CHQ
a sin 2a - s i n
a + sin2a(acota-
- 0, w h e n c e :
(429)
1.5)
T h e opening angle of the failure surface is governed by the ratio of D/a according to
Eq.(409), i.e. cx = 2arctan(D/a).
2
is presented in Fig. 2 3 1 .
In this case the loading b e r m can be inclined as shown in Fig. 232 with a m a x i m u m load
step between embankment and berm at failure of 5.52c w 0. T h e m a x i m u m rotational m o m e n t
is obtained when the centre of the failure surface is placed so that the area A is equal to
u0 +
Fig. 230. Failure surface when the undrained shear strength increases linearly with depth.
321
Embankments
0.5
0.4
0.3
Dia
0.2
0.1
20
40
60
80
100
120
0
140
k a / c
u0
ka/cu0
= 54 1.5/10 = 8.1. From Fig. 220 we find k/a= 0.355 kPa/m which gives
2.3
Influence
of weak
layers
T h e shapeof the failure surface m a y b e governed by the existence of layers in the soil with
lower shear strength than than that of the soil as a w h o l e (cf Slope stability, p. 345). T h e
m e t h o d of analysis in the case of a w e a k horizontal layer will b e demonstrated for a case
w h e r e a w e a k layer exists at a depth equal to D.
Now, let us a s s u m e that the undrained shear strength of the soil a b o v e the w e a k layer
is cui and in the w e a k layer cu2. Let us further a s s u m e that the width of the loading b e r m
is b2 and that the load of the e m b a n k m e n t is qx and of the loading b e r m q2 (Fig. 233). Then
stability requires that the resultant to the active earth pressure on the vertical section at
Embankments
322
Fig. 232. Diagram for determination of width of loading berm in the case of shearing strength increasing
linearly with depth. The width of the loading berm b = {q- 5.52c
)/a.
u0
the r i m of the e m b a n k m e n t and the passive pressures on the vertical section just outside
the b e r m can b e counteracted b y the shear resistance exerted in the w e a k layer along the
loading berm.
T h e stability condition can thus b e written:
qxD - DNcQcul/2
= DN^cJl
+ b2cu2
(430)
from which:
b2 = D(q1-Nd)cul)/cu2
(431)
w h e r e Nc0 = 2 + n = 5.14
A s in the previous case qx-q2<
Nc0cul.
If
323
Embankments
I I IM
yKJIl
<?2
sDNc0cuJnj
2
yp l2
j j T i D - D N c 0c M /l 2
2
y / > / 2
~-Wcak layer
Fig. 233. Assumed failure condition in the case of a weak horizontal layer.
bx > Dyfl
Kah/2)-Nc0cul]/cu2
(432)
w h e r e Ka can b e taken as the active earth pressure coefficient for /' = 1 (see Table 36,
p. 283).
Example 6 1 : Determine the loading berm required to achieve a factor of safety of 1.5 for a road
embankment, 12 m in width and 3 m in height, placed on a clay deposit with an average undrained shear
strength cu = 10 kPa. At a depth of 3 m there is a continuous, weak horizontal clay layer with an undrained
shear strength of 7 kPa. The fill material in the embankment and in the loading berm has a unit weight
3
7 = 18 k N / m . The angle of friction in the fill material can be assumed equal to 35.
Solution: According to Table 36, the active earth pressure coefficient in the embankment material is Ka
= 0.26. Utilising the relation given by Eq. (432), we find:
b2 = [18-3(3 + 0.26-1.5) - 5.14 10/1.5] 1.5/7 = 33.6 m - 34 m.
3.
SETTLEMENT
3.1
Vertical
settlement
Embankments
324
decrease due to a decrease in void ratio and v o l u m e change due to shear. T h e instantaneous
settlement taking place when the e m b a n k m e n t is placed on water saturated, lowp e r m e a b l e soil is completely due to shear while, in the case of non-saturated soil, a
v o l u m e decrease takes place by compression of gas bubbles.
Instantaneous settlement taking place in connection with filling operations is generally
insignificant. However, if i m m e d i a t e settlement should b e of s o m e concern, it can be
calculated on the basis of the theory of elasticity on the assumption that the elastic
m o d u l u s Eis in the range of 150c M for water saturated, normally consolidated clay and
5 0 0 c w for overconsolidated clay.
Doubtless, the dominating part of settlement, taking place under the load of embankments
and fill on soft clay and organic soils, is due to primary and secondary consolidation. T h e
analysis of primary consolidation settlement is generally carried out according to
Terzaghi's consolidation theory. Refined computer p r o g r a m s exist which take into
account c o m b i n e d primary and secondary consolidation, self-induced excess pore water
pressure and c h a n g e s in the consolidation characteristics (primarily c h a n g e s in
permeability) during consolidation. However, from a practical viewpoint the information
available regarding soil parameters does not ordinarily justify advanced calculations by
the aid of computer p r o g r a m s .
(i) One-dimensional
primary
consolidation.
process generally includes the determination of the final primary settlement, based on the
results of o e d o m e t e r tests, and the average degree of consolidation utilising diagrams of
the kind shown in Fig. 234.
T h e average degree of consolidation, however, is not a satisfactory m e a s u r e of the
consolidation process. It is quite important to get an idea of h o w the consolidation process
is advancing throughout the soil layer. For the determination of the degree of consolidation
Uv,
gives a quick and quite accurate answer. A s s u m i n g that the soil profile is divided into
layers of equal thickness Az, Terzaghi's consolidation equation (Eq. 29) can be written:
'+1~
At
where ut_
l9
ut and ui+
Az
Az
Ut_ ]
Az
Az
(433)
ui+l-2u$
(434)
325
Embankments
Sand
0.01
0.05
0.1
0.5
10
Time factor Tv
2
Now, if CyAt/Az is constant, the c h a n g e in excess pore water pressure with time can be
calculated stepwise. A simple graphical solution is made possible by assuming ATV = 1/4,
whence:
1 II; + U:
u, + , = "(^
U: + Uj+
+ ~ -J )
!
(435)
i.e. the excess p o r e water pressure in the middle of layer i at time t + A n s the arithmetic
m e a n of the excess pore water pressures existing at time t in the interfaces between layer
i and the t w o surrounding layers (Fig. 235).
T h e assumption of ATV = 1/4 entails that the clay layer is divided into part layers of
thickness:
Az = lyfc^At
(436)
T h e time increment At can now be chosen in order that at least 3-4 isochronal lines of
pore pressure distribution are obtained u p to the consolidation time in question. T h e
practical application of the m e t h o d is illustrated in Fig. 236.
Embankments
326
' / 2 | _
"_
4f
/2
/2
Fig. 235. Graphical determination of the change in excess pore water pressure from time t to time
t + At.
If the soil is not h o m o g e n e o u s the thickness of the respective layers will vary in
accordance with the variation of the coefficient of consolidation. Moreover, the condition
of continuity of flow between the different layers has to b e satisfied. This can also be done
numerically in a fairly simple way. T h u s , assuming validity of D a r c y ' s law the condition
of continuity of flow yields:
du
ki()i
dz.
= ki
du
+ l
()
dz
+l
(437)
- tana
- tan/3. With the aid of the auxiliary lines and the construction shown w e have
kx t a n a = k 2 tan/3. H e n c e it follows that the boundary flow condition between layers 1 and
2 is satisfied. T h e graphical m e t h o d indicated for layers 1 and 2 is applied to all the layers
of the soil profile.
(ii) Two-dimensional
consolidation.
settlement of road e m b a n k m e n t s , the width of the load is often limited in comparison with
the thickness of the soft soil deposit. In such a case the consolidation process is shortened
by two-dimensional consolidation taking place. This can also b e solved numerically by
H e l e n e l u n d ' s m e t h o d in combination with Carillo's equation (Carillo, 1942):
U=Uv+Uh-
UvUh
(438)
321
Embankments
Sand .
V * Sand '
/ ' . -
- Sand
Impermeable base
Fig. 236. Graphical determination of the excess pore pressure distribution at times At, 2At and 3At after
the initial stage (t = 0). Homogeneous clay deposit. (In reality, the isochronal pore pressure lines form
a curve through the values obtaining in the middle of the respective layer). In the case of impermeable
base, the graphical construction is determined by the fact that the hydraulic gradient AuIAz must be zero
in the interface between the clay deposit and the impermeable base.
f/
'."* /
. .
I"- :
. .Sand
Sand
/,
;: v
Fig. 237. Graphical construction of the pore pressure distribution at times At, 2At and 3 At after the initial
stage (t = 0). Inhomogeneous deposit. (In reality, the isochronal pore pressure lines are curves passing
through the pore pressure values obtaining in the middle of the respective layer and fulfilling the
condition of flow continuity in the interface between the layers).
328
Embankments
(439)
u0
Example 62: Determine the degree of consolidation attained after 4 years at 2.5 m depth below the
centre of a 4 m wide embankment, placed on a 6 m thick, water saturated, normally consolidated clay
underlain by sand. Determine also the excess pore water pressure after 4 years at 2.5 m depth, 5 m from
the centre of the embankment. The clay deposit is fully drained at top and bottom. The coefficient of
2
consolidation is cv = 0.25 m /year. The permeability in the horizontal direction is 4 times that in the
vertical direction. Skempton's pore pressure coefficients are = 1.3 and = \ .
Solution: Utilise the Carillo equation, i.e. divide the consolidation process into two phases: one with
pore water escape only in the vertical direction and one with pore water escape only in the horizontal
direction. The stress increase induced by loading is assumed to obey the theory of elasticity, Eqs. ( 9 0 91). The initial pore pressure distribution with depth under the centre of the strip loading is found by
Eq. ( 18). Its distribution at 2.5 m depth is symmetrical around the plane of symmetry of the strip loading
and is also given by Eqs. (90-91) and Eq. (18). Thus, for example, below the centre of the strip loading
we find Au/q = 1.11 at 1 m depth, Au/q = 0.93 at 2.5 m and Aujq - 0.51 at 6 m depth. At 2.5 m depth,
4 m away from the plane of symmetry, we find Au^Jq = 0.41, and 6 m away from the plane of symmetry,
Auo/q = 0A0.
1 /2
l /2
Choosing the time interval At = 1 year we find Az = 2(1 - 0 . 2 5 ) = 1 m and Ax = 2 ( M ) = 2 m. The
construction of the pore pressure decrease with time can now be carried out as shown below.
We find Uv = 0.11/0.93 = 0 . 1 2 and Uh = 0.25/0.93 = 0.27.
According to Carillo's equation, the total degree of consolidation at 2.5 m depth is:
U = 0.12 + 0.27 - 0.12-0.27 = 0.35 (35%)
At a distance of 5 m from the centre line we find uv = 024q and uh = 0.37q, whence ulq = 0.24-0.37/0.29
= 0.31.
329
Embankments
Example 63: In the test field at Sk-Edeby, established by the Swedish Geotechnical Institute in 1957,
3
a gravel fill, 1.5 m in thickness and with a unit weight of 18 k N / m , was placed on an undrained test area
with a diameter of 35 m. The soil at the site consists of water saturated clay underlain of sand at 12 m
depth. The clay is normally consolidated and has a dry crust of 1 m thickness. The coefficient of
2
consolidation can be estimated at 0.4 m /year. Determine the excess pore pressure at 5 m depth in the
soil under the centre and the periphery of the test area, and also at the distances 10 m and 20 m from the
centre. Skempton's pore pressure coefficients were estimated a t = 1 and = 0.75. Determine also the
excess pore pressure distribution below the centre of the area after 30 years.
Solution: Taking into account the slope at the boundary of the fill, the nominal diameter is 35 - 2.3 =
32.7 m. The pIR values in question are thus 0, 0.61, 1.0 and 1.22. The pore pressure distribution is
determined on the basis of Eq.(18) and Fig. 77. The stresses induced by the load for the different values
of are:
Aoxlq
p = 0
0.98
0.58
0
0.98
10 m
0.90
0.48
0.14
0.94
16.35m
0.45
0.37
0.29
0.70
20 m
0.12
0.29
0.16
0.39
Aa3/q
0.58
0.44
0.12
0.02
Aojq
where 13
, 3 = ^ ( + ) ^ - ) ] +
Introducing these values into Eq. ( 18) and q = 27 kPa, the excess pore pressure becomes equal to AuQ
= 24 kPa for = 0, Au0 = 22 kPa for = 10 m, Au0 = 15 kPa for = 16.35 m and Au0 = 8 kPa for =
20 m. These values are almost identical with those observed: Au0 = 23 kPa for = 0, Au0 = 22 kPa (max.
25 kPa; min. 20 kPa) for = 10 m and Au0 = 6 kPa for = 20 m.
27 kPa
Embankments
330
The excess pore pressure distribution with depth below the centre of the area becomes equal to AuQ
= 27 kPa for = 0, Au0 = 25 kPa for = 2.5 m, Au0 = 24 kPa for = 5 m, Aw0 = 22 kPa for = 1.5 m and
Au0 - 18 kPa for = 12 m. In this case, the influence of horizontal pore water flow can be disregarded.
The pore pressure dissipation can now be determined according to Helenelund's method. Assuming the
top dry crust to be drained because of the existence of frequent fissures and dividing the remaining 11
2
m of clay into 4 part layers, we have = 2.75 m. This yields = (2.75/2) /0.4 = 4.73 years. Thus 30
years correspond to 30/4.73 = 6.3.
The construction of the pore pressure dissipation is shown at bottom of p. 328.
The maximum excess pore water pressure after 30 years, about 20 kPa, is found at 6-7 m depth.
(iii) Secondary
consolidation.
be formulated as:
= as\og(t/tp)
(440)
= as(l
(441)
+ e0)\og(t/tp)
for inorganic
soft clay and for organic soft clays are 0.04 0.01 and 0.05 0 . 0 1 , respectively. The
correlation between as and Ca is given by the expression as = CJ(\
Horizontal
displacements
Long-term settlement due to shear is very much dependent on the shear stress level in
relation to the shear strength. Often, the factor of safety against failure is between 1.3 and
1.5. With a factor of safety of 1.3 the horizontal displacement in the soil along the rim of
the embankment can be quite important (Fig. 238).
331
Embankments
4.
EMBANKMENT PILES
Horizontal displacement, mm
100
50
50
100
150
15
Fig. 238. Horizontal displacement underneath a road embankment on soft clay deposits at Ska- Edeby.
Shear strength cu at the time of construction around 7 kPa. Height of embankment 1.5 m (load q - 27
2
kN/m ). After Larsson, 1986.
Embankments
332
10
resisting m o m e n t around the centre of the potential failure surface induced by shaft
friction. If the piles m o v e with the sliding b o d y (as, for e x a m p l e , pile 1 in Fig. 240), the
frictional forces along the part of the piles that are outside of the sliding b o d y should be
taken into account. If, on the other hand, the piles do not m o v e with the sliding body (as,
Fig. 240. Analysis of stability of embankment placed on floating embankment piles. The part of the piles
contributing to stability are indicated by arrows. The lever arms for the piles with respect to the centre
of the potential failure surface exemplified for piles 1 and 2.
333
Embankments
for e x a m p l e , pile 2 in Fig. 240), the frictional forces along the part of the piles that are
inside the sliding b o d y should b e considered.
Example 66: A 4m high embankment shall be built on a deep clay layer with an undrained shear strength
increasing with depth below the ground surface according to the relation cu = 10 + 1.5 kPa, where
3
in m. The density of the embankment material is 1.8 t/m . Determine the maximum pile spacing if the
pile caps are 0.8 m by 0.8 m and the required pile length. The safety against pile failure is taken as
minimum 1.3.
Determine also the number of batter piles using a pile inclination of 5:1
2 2
Solution: W e have hi = 4/0.8 = 5 whence S /a = 4.8. This yields 5 = 1.75 m.
The pile working load is = 4.8-0.644 18 - 220 kN
Using spliced timber piles with a total length of 25 m, the lower pile segment consisting of a 17 m
long pile with a tip diameter of 0.13 m and a head diameter of 0.30 m and the upper of an 8 m long pile
with a diameter varying from 0.30 m at the splice to 0.22 m at the top, the bearing capacity of the piles
according to Eq. (207) becomes:
h=4m
Embankments
334
Solution: The value (Fig. 220) of the slide surface is 29-sin(2.33/2) = 26.6 m. The driving moment of
the embankment is thus:
Md = 41926.6 2/2 =26887 kNm/m
The stabilising moment is the sum of the shear resistance along the failure surface and the resistance
exerted by the piles. The pile lengths contributing to the stability are l{ = 8.0 m, l2 - 7.8 m, / 3 = 8.5 m,
/ 4 = 11.2 m, / 5 = 11.5 m, l6 = 6.6 m and / 7 = 2.0 m. The lever arms are rx = 3.2 m, r2 = 7.3 m, r3= 11.4
m, r 4 = 13.0 m, r5 - 17.0 m, r 6 = 18.3 m and r 7 = 22.3 m. This yields:
Ms = 15[2.33-29 2 + (8.0-3.2 + 7.8-7.3 + 8.5-11.4 +11.2-13.0 + 11.5-17.0 +6.6-18.3 + 2.0-22.3)/2]
= 34537 kNm/m
The factor of safety Fs = 34537/26877 = 1.3
5.
In cases w h e r e settlement requirements are less strict, the soft soil can b e displaced by
overloading the soil to p r o d u c e ground failure. In this way the load of the fill will cause
an outward m u d w a v e and fill will replace the soft soil thus displaced. If the e m b a n k m e n t
fill is not displacing the soft soil in a satisfactory way, the overload m a y either be
increased or masses in front of the fill b e excavated.
Alternatively, a combination of overloading and blasting can be used in order to obtain
a m o r e complete removal of the soil underneath the fill (Fig. 241). T h e explosives are
usually placed in plastic tubes, 7 5 - 1 0 0 m m in diameter, which are inserted in the soil to
the required depth in a square grid with 3 - 4 m spacing. T h e best result is obtained when
the explosives are evenly distributed along the lower third of the tubes and a delayed
blasting technique is used. A b o u t 100 g of explosives per m 3 is generally chosen.
335
Embankments
the corre-
Slope
336
stability
S L O P E STABILITY
1.
INTRODUCTION
Before building activities close to river valleys or other natural slopes are begun, the
overall stability condition at the building site has to b e checked. By e x p e r i e n c e w e know
that m a n y disastrous and unforeseen slides with consequential fatal casualties and heavy
e c o n o m i c losses h a v e occurred due to unsatisfactory stability conditions (Fig. 242).
In the following an outline of the analytical m e t h o d s utilised to solve the slope stability
problems will b e presented. For m o r e detailed studies, reference is m a d e to text books and
papers specially devoted to the problem.
2.
2.1
Undrained
analysis
In the case of cohesive soil, the traditional analytical approach is based on the assumption'
of circular-cylindrical failure surfaces acted u p o n by the undrained shear strength cu of
Fig. 242. Air view of the slide at Tuve north of Gothenburg that took place on Nov. 3 0 , 1 9 7 7 . In the slide
65 dwelling-houses were carried away and 9 people were killed. The slide, which was initially of limited
extent but developed into a retrogressive type of slide, took place after a long period of raining.
Slope
337
stability
the soil (Fellenius, 1 9 1 8 , 1 9 2 6 , 1 9 3 6 ) . Strength anisotropy (see pp. 3 4 and 1 0 1 ) with regard
to the inclination of the failure surface and time effects are generally taken into account. T h e
critical failure surface with regard to the loading condition is found by trial and error.
T h e rotational driving m o m e n t of the dead weight of the soil comprised by the sliding
surface, external load included, is easily established. T h e counteracting stabilising
m o m e n t is obtained by s u m m a t i o n of the shear strength contributions exerted along the
a s s u m e d failure surface times their lever a r m / ? . Failure occurs w h e n the driving m o m e n t
exceeds, or equals, the stabilising m o m e n t thus obtained.
In the case of a simple, elementary slope w h e r e the total unit w e i g h t of the soil is
constant, the driving m o m e n t Md due to the weight of the soil enclosed by the circularcylindrical surface can b e obtained by the relation (Fig. 2 4 3 ) :
Md = ^ [ R - x - y
+ H(y-")
+ B(x-
")]
(442)
If the slope is subjected to water pressure the influence of the water pressure against
the slope has to b e added.
In h o m o g e n e o u s soil conditions the critical failure surface can b e determined once and
for all according to J a n b u ' s direct m e t h o d (Janbu, 1 9 5 4 ) . In this case, the safety against
failure in undrained condition can be expressed as:
' = ^ ^ ^
<) 4 4 3
w h e r e the stability factor jV 0 is given in Fig. 2 4 4 and the correction coefficients pq with
Fig. 243. Notations used for the analysis of rotational moment due to weight of soil comprised by
circular-cylindrical slide surface along slope.
338
Slope
stability
Example 67: The risk of failure of a slope with 5 m height and an inclination of 1:1.5 shall be
investigated. The water depth outside the slope is 3 m. Calculate the rotational moment around the centre
of a circular-cylindrical failure surface assuming x = 2m,y = 1 0 m and R = 12 m. The unit weight
3
of the clay = 18 kN/m .
7.5 m
Solution: We have H = 5 m and = 1.5-5 = 7.5 m. The water pressure exerted on the slope is "equal to
5.4-3-9.81/2 = 79.6 kN/m and its lever arm with reference to is 5.41 m.
The rotational moment becomes:
2
2
2
Mr = (18/2)-5[12 - 2 - 1 0 + 5(10 - 5/3) + 7.5(2 - 7.5/3)] - 79.6-5.41 = 3076 kNm/m
regard to external load q,
Hjof
w i t h regard to depth
90 80
60
40
20
Slope angle
Fig. 244. Stability factor N0 under homogeneous soil and undrained failure conditions
Slope
339
stability
n i l
//
DIH =oo
/3=0
1.0
.60
0.9
90
> ) / / / ) > > r ;
0.5
HJH
D/H = o o
^7
0.1
0.2
0.3
0.4
0.5
0.1
0.2
0.3
0.4
0.5
+ W 2) s i n a
(444)
w h e r e P0a is the rotational m o m e n t due to an external horizontal force (e. g. water pressure
or earth pressure in fissures),
Wi denotes full weight of the soil in the lamella a b o v e the water level outside the
slope,
W2 denotes s u b m e r g e d weight of the soil in the lamella b e l o w the water level
outside the slope,
u s i n a i s the lever a r m with respect to the centre of the circular-cylindrical failure
surface.
T h e stabilising m o m e n t Ms is obtained by the relation:
Ms = / ? I c MA /
(445)
w h e r e R is the radius of the failure surface and cu is the undrained shear strength along
the part Al of the failure surface belonging to the lamella in question..
Example 68: Express in general terms the stabilising moment for a circular-cylindrical failure surface
in a clay slope if the undrained shear strength of the clay increases linearly with depth according to the
relation: cu = cu0(l+kz)
kPa, where is the depth, in m, below the top of the slope.
Solution: With the notations given in the figure we have:
Slope
340
o +
MS=R
(Q+a
cuRda>= R
CQ
whenceM s = Rcu0{Bs
2.2
Effective
stability
cu[l+kR(s'm-s'mo)]d
CQ
stress
analysis
T h e first attempt to determine the slope stability for a soil with an internal angle of friction
was publicised in connection with the investigation of the cause for the slide (Fig. 246)
which t o o k p l a c e on M a r c h 5 , 1 9 1 6 , at the Stigberg quay (Pettersson, 1916). T h e analysis,
which was carried out by the Gothenburg Harbour engineer S. Hultin (Hultin, 1916), was
based on the assumption of a circular-cylindrical failure surface and is generally referred
to as the friction circle m e t h o d (Fig. 247). T h e m e t h o d is very laborious and requires trial
and error to find the friction angle which satisfies the equilibrium condition of the sliding
b o d y investigated.
Fig. 246. The slide at the Stigberg quay in the harbour of Gothenburg on March 5, 1916,' initiated the
development of the slope stability analysis based on the assumption of circular-cylindrical failure
surfaces. Notice that the quai front wall is fully intact and hardly affected by the slide while the ground
behind with the railway tracks has sunk below water. This indicates that the centre of the circularcylindrical slip surface is situated nearly straight above the quai front wall.
Slope
stability
341
Fig. 247. The analysis of the slide at the Stigberg quai in 1916 was based on the assumption of clay
behaving as a friction material. The method of analysis, named 'the friction circle method', based on
the assumption of a circular-cylindrical failure surface, was applied for the first time in history by Sven
Hultin (1916).
Slope
342
stability
Pi0
Fig. 248. The slice method applied in effective stress analysis of a circular-cylindrical failure surface.
+ W2)x = 0 + RliW^
+ W 2) s i n a
(446)
w h e r e Wx denotes full weight of the soil a b o v e the water level outside the slope,
W2 submerged weight of the soil below the water level outside the slope.
T h e stabilising m o m e n t is now given b y the expression:
(447)
T h e & value is obtained from the equilibrium condition of the slice. A vertical projection
equation yields:
Wx + W2 +AT-l(uecosa+
smoc/F^)
c o s a + sinatan0 //^0
(448)
w h e r e ue = ywz = the pore water pressure at the b o t t o m of the slice expressed as an excess
over the hydrostatic pressure corresponding to the water level outside
the slope. (If no part of the slice is submerged, ue = u).
Slope
343
stability
Substituting / for b/cosoc and inserting ' into Eq. (447) w e have:
/
Ms =
R X [ c + ( W 1 + W2 + AT-ueb)
Fc<t>
c o s a ( l + tan tan 0
tan ']
(449)
7F^)
w h e r e the s u m is taken over all slices with their respective width b and soil characteristics.
T h e factor of safety can thus b e expressed by the relation:
Rl[c'b
cd>
+ (Wl + W2+AT-ueb)ton<l>
sin a
0 + Rl{Wx+W2)
w h e r e ma = cosoc (1 + t a n a tan
']lmc
(450)
'/F^).
Rl[c'b
+ (WX + W 2 - n g f r ) t a n 0
P0a-^Rl(Wl
']lma
(451)
W2)s'ma
20 m
344
Slope
3
stability
groundwater level, is 1.8 t/m . The density of the silt is 1.7 t/m on the average above groundwater and
3
1.75 t/m below groundwater. The pore pressure increase with depth is hydrostatic.
Solution: The body comprised by the potential failure surface is sliced up, as shown, into 10 lamellae.
For each slice, the values and the total and effective vertical pressures, acting on the potential failure
surface, are calculated. As a first attempt, put F^= 1.05. Then the result, ignoring the A7term, and
denoting Wx + W2 = W, is as follows:
Slice
b m
1
2
3
4
5
6
7
8
9
10
3.5
3
3.5
3.5
5
"
6
8
WkN/m
57.5
51.5
45.0
39.7
33.0
26.3
18.8
12.8
5.0
-4.0
173
390
588
627
825
665
585
543
380
180
ueb
kN/m
0
25
123
158
210
125
60
30
0
0
'
37
35
"
AMJR kN/m
108.7
221.2
265.4
261.6
341.2
302.1
301.4
304.7
240.4
126.2
2472.9
Thus assuming Fc(p - 1.05, the stabilising moment around the centre of the circular-cylindrical surface
becomes 111280 kNm/m.
The driving moment around the centre of the circular-cylindrical surface becomes 105335 kNm/m.
The factor of safety becomes Fc(p = 1.06.
3.
Slope stability
345
failure in a cohesionless soil is then limited to the problem of determining the friction
angle ' for w h i c h the body, under the weight of the soil and the influence of possible
external forces, is in equilibrium. T h e factor of safety is generally expressed as the ratio
of the tangent of the existing value of ' to the tangent of the ' value required for equilibrium. T h e logarithmic spiral for which the lowest factor of safety is obtained is found
by trial and error.
If the cohesion intercept c' 0 the stabilising m o m e n t due to c' is obtained b y the
relation (Fig. 249):
COS0'
Ms = -^(R-R)c'
2tan0
4.
(452)
= (T+dT+
T)dx/2
(453)
Dividing by dx and omitting terms of m i n o r significance, the shear force Tii+l at the
interface b e t w e e n slices / and i+1 is obtained by the relation:
Fig. 250. The slice method applied for the analysis of a composite failure surface.
Slope
346
Ti,i+ = (
dP
dx
i(tana,)M+
stability
(454)
where at is the angle of inclination of the internal pressure line at the interface between
slices i and
i+l.
The exact position ht of the earth pressure resultant, acting in the interface between two
slices, cannot be determined. However, the influence of its position on the safety factor
is relatively small and, therefore, from a practical viewpoint it can be assumed that it is
acting at the lower third of the sides of the slices.
The value of AP is obtained from a tangential projection equation along the base of the
slice:
(Wx + W2 + AT) una - AP c o s a =
(455)
b/cosa
w e find:
=
- -
(456)
(457)
tanO"
tana
The value of dPIdx in the interface between two slices i and i+l can be obtained by
the approximate relation (Fig. 251):
Slope
347
stability
AP + APi+ 1
(458)
k +h
S u m m i n g u p over all slices, the condition of force equilibrium yields:
1AP
= -P
1B-(FCO)-^1A
(459)
=0
(460)
+ 1(WX
+ W2 + AT) tan a]
(461)
w h e r e na = c o s a (1 + t a n a t a n ^ ' / F ^ ) .
Eq. (461) is referred to as J a n b u ' s general procedure of slices (GPS).
T h e iteration procedure takes the following course:
a s s u m e a potential slide surface and cut the sliding body into slices w h o s e widths are
governed by the strength properties of the soil,
calculate for each slice its weight (W=Wl
determine for each slice the value of t a n a and tan0 (and cO,
a s s u m e a value of Fc(p (for instance F^ = 1.5),
calculate the values of A (= A0) and (= B0) for AT= 0 and s u m up the values thus
obtained to get a corrected factor of safety Fc(p = 0/(0
+ 0) ,
calculate AP = B0- AJF^ for the respective slices and determine the values at each
interface b e t w e e n the slices by summation of ,
determine the values of dPIdx at the interface between the slices according to Eq.
(458),
draw the thrust line through the lower third of the height of the elements at the
respective interfaces and determine its inclination (tanc^) at each interface between the
slices,
calculate the value of T a t each interface (which in turn yields A T o v e r the respective
slice),
check that = - P0 and that = 0,
the values of A T t h u s obtained yield new values of A(=A{)
factor of safety F^ = ^ I( +
\),
Fig. 252. Correction coefficient/^ with regard to the effect of internal forces acting between lamellae.
1956) by
which the safety factor, calculated without consideration of the internal forces, should be
multiplied (Fig. 252). Accordingly:
_^ ?Xbc' +
~ J o
7^
^7777
{Wx+W2-bue)\suL$']
777
~
(462)
Example 70: The river embankment in example 69 is to be built upon with terrace houses and the factor
of safety required is F C (^> 1.5. For the construction of the buildings, the embankment is excavated and
2
the buildings placed as shown below. The weight of the buildings is 20 kN/ m , except for the uppermost
2
building whose weight is 26 k N / m . In order to guarantee against erosion by the river, a rock fill is placed
at the toe of the slope with an inclination of 1:3. The soil characteristics are those given in example 2.
3
The rock fill has = 2.0 t / m and '= 45. Determine the factor of safety for the composite failure surface
indicated.
Slope
349
stability
Solution: The body comprised by the slide surface is sliced up, as shown, into lamellae 1 to 12. For each
of the lamellae the value and the total and effective normal stresses along the slide surface are
calculated. The calculation is carried out step by step by successive approximation. In the expression
for a it is assumed, as a first assumption, that FCQ = 1.5. It is further assumed, as a first approximation,
that = 0. The values of ( = B 0 ) and (= A 0 ) thus obtained yield a corrected factor of safety and are
applied in a second iteration for the determination of and which in turn yield new values of A ( = A { )
and (= j), etc. The analysis is carried out as shown in the arrangement shown below.
Slice
1
2
3
4
5
6
7
8
9
10
11
12
WkN/m
bue kN/m
8
5.5
42
37
34
26
20
12
8
6
3
0
-2
-13
540
855
1190
1185
1000
1030
995
640
525
370
480
310
0
129
358
410
275
224
163
100
60
0
0
0
"
7
6.5
5
"
7.5
8
<t>' 0 ( k N / m ) A (kN/m)
37
35
"
45
486.2
644.3
802.7
578.0
364.0
218.9
139.8
67.3
27.5
0
-16.8
-71.6
3240.3
507.3
589.6
644.6
547.2
491.4
536.6
557.5
364.4
318.7
259.1
342.1
385.9
5544.5
The factor of safety F C (^ = Aq/ B 0 = 1.71, i.e. higher than the assumed value 1.5.
The values of A 0 and B 0 are now applied for calculating with a factor of safety of Fc(p = 1.7.
Introducing dP/dx - (, + AP t + ) / ( + bi+l)
and the ht and tana, values at the interface between the
respective slices, the and values can be calculated as shown below. Inserting the AT values thus
obtained, new values of A ( = A { ) and ( = B{) are found as shown below. These values yield a revised
factor of safety.
Slope
350
Slice
B0
^0
1 486.2
507.3
5
6
7
ht
644.3
802.7
589.6
644.6
578.0
364.0
218.9
139.8
547.2
491.4
536.6
557.5
189.7
36.3
0.625
2.5
-28
489.4
66.0
0.578
3.0
-85
915.4
62.2
0.532
3.5
-269
1173.6
30.5
0.466
3.5
-440
1250,4
-1.4
0.325
3.5
-411
1155.8
-20.8
0.176
3.5
-276
969.7
-28.8
0.105
3.8
-211
824.1
-30.5
0.140
3.7
-228
665.3
-31.0
0.176
3.0
-210
513.9
-29.4
0.140
2.3
-140
297.2
-33.2
1.5
-50
299.7
426.0
258.2
76.8
-94.7
-186.0
8
67.3
9
27.5
10
0
364.4
318.7
259.1
-145.7
-158.8
-151.4
11 - 1 6 . 8
12 - 7 1 . 6
342.1
385.9
-216.7
-297.2
0
<p
AT
*1
-28
461.0
500.2
-57
601.3
561.2
-184
678.6
517.3
-171
494.6
436.4
29
374.5
520.3
135
247.6
633.5
65
149.0
605.6
-17
65.5
355.0
18
28.5
332.0
70
308.1
90
-19.9
405.4
50
-83.1
2997.5
438.4
5613.4
*1
tana.
189.7
dPIdx
stability
= 1/
0
1
= 5613/2997 = 1.87
In a third iteration process the factor of safety found in the second iteration process is applied. The
result is as follows.
Slope
stability
Slice B
351
tana,
ht
1
2
3
4
5
6
7
8
9
10
11
dPIdx
461.0
601.3
678.6
494.6
374.5
247.6
149.0
65.5
28.5
0
-19.9
500.2
561.2
517.3
436.4
520.3
633.5
605.6
355.0
332.0
308.1
405.4
438.4
193.9
36.7
0.625
2.5
-29
495.5
64.0
0.578
3.0
-94
897.9
60.3
0.532
3.5
-266
1159.4
32.6
0.466
3.5
-426
1256.1
0.5
0.325
3.5
-406
1165.4
-19.6
0.176
3.5
-274
991.0
-26.0
0.105
3.8
-203
866.9
-27.3
0.140
3.7
-222
718.2
-31.3
0.176
3.0
-220
553.6
-32.1
0.140
2.3
-151
317.2
-35.7
1.5
-53
301.6
402.4
261.5
96.7
-90.7
-174.4
-124.1
-148.8
-164.5
-236.4
-317.2
0
<p
AT
B2
A2
193.9
12 - 8 3 . 1
-
-29
460.1
511.6
-65
595.3
666.1
-172
686.6
536.8
-160
499.9
450.7
20
371.2
519.9
132
247.0
635.9
71
149.8
612.5
-19
65.3
354.8
27.6
321.6
69
307.4
98
-20.2
410.5
53
-83.8
2998.9
436.2
5764.3
= 5764/2999 = 1.92.
A fourth iteration yields 3 = 5678 kN/m and A 3 = 2999 kN/m whence F c 0 = 1.89.
Using instead the correction factor/ 0 to be applied on the safety factor calculated without consideration
to the internal forces between the slices we find the ratio Lid = 11/78 = 0.14, whence f0 = 1.08.
In this case the factor of safety becomes Fc<p = 1.08-1.71 = 1.85 which is about 2% below the value
obtained by the iteration procedure.
Example 71 : Investigate how the internal forces acting in the interfaces of the slices will affect the factor
of safety with regard to moment equilibrium of the circular-cylindrical slide surface analysed in
Example 69.
Solution: For this purpose we have to establish the equilibrium condition for each slice and for the sum
of slices according to Eqs. (454-460). Starting the iteration procedure on the assumption that 7 = 0
and that F c 0 = 1.05, as was previously found in Example 69, we get the following result (forces in kN/
m and lengths in m):
Slope
352
3.5
3
3.5
3.5
5
57.5
51.5
45.0
39.7
33.0
26.3
18.8
12.8
5.0
-4.0
173
390
588
627
825
665
585
543
380
180
Slice
1
2
3
4
5
6
7
8
9
10
"
"
"
6
8
ueb
0
25
123
158
210
125
60
30
0
0
/0
stability
^0
37
212.3
490.3
588.0
520.5
535.8
328.7
199.2
123.4
33.2
-12.6
3077
35
"
271.6
373.1
390.7
357.1
427.2
353.8
334.3
328.0
253.3
132.8
3163
The values of A 0 and B0 are now applied for calculating AP with a factor of safety of Fc<^ = /^Introducing dPIdx = (, + APt + x)l(bi + bi + )l and the ht and tana, values at the interface between the
respective slices, the and AT values can be calculated as shown below. Inserting the AT values thus
obtained, new values of (= A t ) and (= ) are found.
Slice
AP
B0
8
9
tan<7,
ht
1
2
3
4
5
6
7
dPIck
212.3
490.3
588.0
520.5
535.8
328.7
199.2
123.4
33.2
271.6
373.1
390.7
357.1
427.2
353.8
334.3
328.0
253.3
10-12.6
132.8
3077
3163
65.0
29.6
0.93
1.9
-4
192.2
51.5
0.84
3.0
-7
400.0
54.4
0.78
3.4
-127
573.0
34.5
0.66
3.5
-257
693.0
10.4
0.60
3.5
-379
677.4
-14.2
0.41
3.2
-323
551.3
-32.1
0.31
3.6
-286
355.5
-37.2
0.25
3.2
-207
142.2
-25.4
0.19
1.9
127.2
207.8
173.0
120.0
-15.6
-126.1
-195.8
-213.3
-4
265.3
205.3
-3
486.5
362.6
-120
468.0
287.6
-130
412.6
256.3
-122
456.5
340.5
56
356.3
388.7
37
211.7
356.4
79
141.3
377.6
133
44.9
341.6
75
-17.8
188.6
2825
3105
-75
-142.2
*1
65.0
AT
Slope
353
stability
Slice Bx
AP
* 1
265.3
205.3
486.5
362.6
468.0
287.6
412.6
256.3
456.5
340.5
356.3
388.7
356.4
211.7
141.3
377.6
44.9
341.6
188.6
3105
1.9
-4
235.1
55.8
0.84
3.0
-30
441.5
55.1
0.78
3.4
-157
620.9
38.4
0.66
3.5
-275
767.6
14.9
0.60
3.5
-408
770.3
-11.0
0.41
3.2
-351
657.7
-31.5
0.31
3.6
-317
455.5
-42.6
0.25
3.2
-250
189.5
-32.5
0.19
1.9
-98
-112.6
-202.2
-265.9
10-17.8
2825
0.93
2.7
36.2
146.7
6
7
78.5
179.4
AT
B2
-4
265.3
212.6
-26
457.6
354.2
-127
461.0
289.2
-119
422.6
271.6
-133
449.4
339.5
57
356.8
395.7
34
211.7
356.4
67
138.6
373.1
152
44.9
341.6
98
-19.4
2789
204.7
3098
A2
206.3
156.6
tana,
78.5
dPIdx
-189.5
0
We find Fc(p = 3098/2789 = 1.11 which is in good agreement with the value obtained in the first
iteration. The values of AT obtained in the second iteration can thus be used to find the influence of the
internal forces on the factor of safety according to Eq. (450). Inserting these values of 7 , we find:
Slice
1
2
3
4
5
6
7
8
9
10
bm
3.5
3
3.5
3.5
5
"
6
8
o
57.5
51.5
45.0
39.7
33.0
26.3
18.8
12.8
5.0
-4.0
WkN/m
A7kN/m
ueb kN/m
173
390
588
627
825
665
585
543
380
180
-4
-26
-127
-118
-133
57
34
67
152
98
0
25
123
158
210
125
60
30
0
0
'
37
35
"
"
"
"
"
"
AMJR kN/m
106.2
205.5
191.2
195.8
261.4
334.1
314.0
344.5
336.5
194.9
2490
This yields a stabilising moment around of 45-2490 = 111280 kNm/m and a safety factor equal to
Fc<p = 111280/105335 = 1.06.
The result shows that the influence of the internal forces can be neglected and that, consequently, the
simplified solution given by Eq. (451) can be applied. It also indicates that the factor of safety determined on the basis of equilibrium in the horizontal direction gives a slightly higher factor of safety than
moment equilibrium around the centre of the circular-cylindrical failure surface.
Slope
354
5.
stability
GENERAL ASPECTS
across a slope is mainly affected by the situation between the toe and the crest of the slope.
T h e ratio of horizontal to vertical stresses is generally m u c h higer close to the toe than
close to the crest. T h e real stress variation may be quite different from the values assumed
in the analysis, but its effect on the safety factor does not seem decisive.
T h e most important influence on the result of effective stress analysis of slope stability
is of course the variation of pore water pressure with location and depth across the slope.
Therefore, the seasonal variations and their extremes h a v e to b e carefully examined in
order to h a v e a reliable basis for the analysis. T h e m o s t critical time of the year therefore
concurs with periods with high rain falls.
(iii) Retrogressive
Fig. 21), the occurrence of a slope failure often leads to retrogressive slides which, in
consequence, m a y finally e m b r a c e large areas with catastrophic consequences. A typical
e x a m p l e of this p h e n o m e n o n is the earth slide at Rissa, north of Trondheim, Norway, on
April 2 9 , 1978 (a slide that has been d o c u m e n t e d in a film taken by a bystander). Here,
3
a small excavation for an extension of a farm house, comprising 7 0 0 m of soil that was
placed on the waterfront, initiated a slide with a width of around 20 m and a length of
2
the reinforcing effect of root systems of bushes and trees. N a k e d slopes ought to b e
planted with bushes in order to improve the stability of the superficial parts of the slopes.
This is particularly important in silty soil regions subjected to ground frost. Superficial
slides in such regions usually occur during spring when the ground is thawing.
Slope stability can also be improved by soil nailing.
Excavation
355
EXCAVATION
1.
INTRODUCTION
T h e lack of s uitable ground for building purposes and an increasing need of parking space
has entailed a general tendency to provide n e w buildings with an ever increasing n u m b e r
of b a s e m e n t floors. T h e cost of these b a s e m e n t floors will h a v e to b e related to the cost
of excavation in order to reach a cost-effective design. Quite often deep excavations
cause serious p r o b l e m s , giving rise to economical disputes on the contract work. O n c e
the construction of the building has reached ground level, unforeseen p r o b l e m s are considerably lessened. Therefore, an optimal and reliable solution to the excavation problems is an important part in foundation design.
T h e m e a n s and type of machinery anticipated by the contractor in an excavation
contract are very important for the tendering price. Therefore, deviations from reality in
the d o c u m e n t s provided for the tenderers m a y lead to unforeseen claims for compensation
Fig. 253. Example of weathered rock below a seemingly un weathered rock surface.
Excavation
356
DEWATERING
2.1
Hydraulic
ground
failure
There are mainly two causes of hydraulic ground failure: instability d u e to liquefaction
p h e n o m e n a , or instability due to the hydraulic uplift of impervious soil, underlain by
pervious soil in which the pore water pressure exceeds the weight of the overlying
impervious soil.
(i) Liquefaction.
In a physical sense this m e a n s that the soil behaves as a heavy liquid with a unit
weight equal to the unit weight of the soil. Expressed in the terms of porosity of soil or
void ratio e, liquefaction will occur when i - > 1.65(1 - n) = 1.65/(1 + e).
T h e liquefaction condition / = p 7 p w is based on the assumption of n o interlocking
forces between the soil particles and, therefore, in realityparticularly in dense s o i l
the hydraulic gradient m a y exceed p'lpw
357
Excavation
Fig. 254. Hydraulic bottom failure piping due to upward seepage of groundwater. Fine particles
are carried to the bottom of excavation by the seepage pressure and settle in cone-shaped forms around
the seepage channels.
the pore water in sand or silt layers, or in pervious b o t t o m layers, often cause hydraulic
uplift of the b o t t o m of excavation. Alternatively, the reduction of effective stresses due
to excavation entails a reduction in average shear strength b e l o w the b o t t o m of
excavation which m a y lead to failure. It is often difficult to distinguish which of the two
p h e n o m e n a triggers a failure occurrence.
T h e existence of of silt or sand layers in clay deposits always has to b e m a p p e d and
358
Excavation
Fig. 255. Excavation in clay for a waste water pump station immediately after hydraulic uplift had taken
place. Depth of excavation about 11 m; diameter about 23 m. The uplift was caused by too high a pore
water pressure in thin silt layers, wedging into the excavation just below the tip of the encompassing
sheet pile wall.
Excavation
359
Fig. 256. Liquefied silt giving rise to hydraulic uplift of the bottom of excavation is pumped away by
means of an Archimedian screw. Liquefaction caused by stress release during excavation for shopping centre.
excavations w h e r e the 'soil b e a m ' (or 'soil p l a t e ' ) can resist the bending m o m e n t caused
by the uplift pressure. In the case of wide excavations it is not justified.
Example 72: In order to create a dry dock for the construction of oil rig platforms, a circular excavation
with a bottom diameter of 120 m was carried out to a depth of 12.7 m in sand and silt underlain by clay.
The excavation was enclosed by a sheet pile wall driven to a depth of minimum 2m into the clay. Below
the bottom of excavation the clay contained seams of sand and silt. The density of the clay was 1.9 t/
m 3 and its undrained shear strength 40kPa. In order to avoid hydraulic uplift, relief wells (vertical sand
drains) were installed to a depth of 10 m below the bottom of excavation. The area around the site was
mountainous and the decrease in pore pressure in the silt and sand layers below the relief wells, due to
unloading by excavation, could be expected to vanish with time because of inflow of water from the
surroundings. In truth, piping and local hydraulic failure, forming the lake that is visible in the figure
below, occurred when the dock was emptied from water for the construction of the second platform.
Determine the maximum pore water pressure, leading to hydraulic failure of the bottom of
excavation, with regard to the 'plate' stiffness of the 10 m thick clay layer drained by the relief wells
if the short-term tensile (or compressive) strength in the soil is (a) 40 kPa, (b) 80 kPa.
Excavation
360
Solution: Assuming that the clay layer behaves as an elastic material, rigidly fixed along its boundary,
the maximum stresses in the horizontal direction are obtained by the relation:
, 3(1 +
v)(u-\.9gh)R
G h =
4h
at the outer boundary of the excavation.
Introducing v = 0.5 (undrained condition), R = 60 m and h = 10 m, we find oh = 20.25w - 3774 kPa
at the centre and ah = 27.Ou - 5032 kPa at the outer boundary.
In case (a) we have um.dX = 188 kPa and in case (b) w m ax = 1 8 9 kPa. In both cases the critical stress
value is almost the same at the centre and the boundary of the excavation. If the tensile strength is
assumed zero we have w m ax = 186 kPa. The contribution by the plate effect is negligible.
Example 73: A vertical cut in clay, 1.5 m in width, shall be carried out in clay with a density of 1.6 t/
3
m and an undrained shear strength of 20 kPa. The clay is underlain by sand at 5 m depth. The hydraulic
head in the sand layer is 1 m below the ground surface. Determine the allowable depth of excavation
d.dX if the partial safety factor on undrained shear strength is 1.5.
Solution: If the contribution of the shear planes along the sides of the cut are taken into consideration
we have:
( 5 - 4 1 > l ^ + 2 { 5 - J a l) - 2 0 / 1 . 5 = 4^
which yields dal = 4.1 m.
Ignoring the side effects we have d.dl = 5 - 4/1.6 = 2.5 m
If the clay layer, left below the bottom of a cut, carried out to a depth of 4.1 m, is considered as an
elastic beam, rigidly fixed along the sides of the cut, the tensile and compressive horizontal stresses in
the clay along the sides of the cut become equal to:
Ju-\.6gh)-l
Oh -
2h
Inserting the values h = 0 . 9 m, / = 1.5 m and u-Ag kPa, we find oh=32.1
2.2
Damages
caused
by groundwater
lowering
Excavation
361
Fig. 257. Leakage of groundwater into rock tunnels, here visible by the existence of stalactites, may
cause serious problems.
WHm>
groundwater l o w e r i n g > 10 m
5 - 10 m
crush-zones
Fig. 258. Pore pressure decrease in sand beneath a 10 m deep clay deposit due to leakage of groundwater
into a raw water tunnel. N o leak observed in the tail race tunnel. Maximum decrease in pore pressure
observed about 300 m away from the leaking raw water tunnel. A large number of dwelling houses
kilometres away were damaged due to differential settlements caused by the pore pressure decrease.
Excavation
362
6 m
2m
4 m
Au=
10-9.81 kPa
The decrease in hydraulic head after 40 years is 7.6 m at 18 m depth, 3.5 m at 14 m depth and 1.1m
at 10 m depth.
Since in this case - Au = ', the settlement becomes:
s HI.6 + 3.5 + 1.1)9.81/200 = 2.4 m
2.3
Methods
of groundwater
lowering
363
Excavation
in 1838 in
and cobbles, a very high flow rate is required before the critical hydraulic gradient is
reached to lead to liquefaction. Dewatering can then b e carried out from s u m p s placed
Fig. 259. Field of application of the most common methods of groundwater lowering.
Excavation
364
100%
1/
100%
Fig. 260. Flow net and potential lines around excavation with sheet pile walls installed for the purpose
of reducing the hydraulic gradient. Figures indicate the hydraulic head in % of the original hydraulic
head above the bottom of excavation (before groundwater lowering).
Excavation
365
Fig. 2 6 1 . Dewatering of an open excavation by means of two well point systems placed at different
levels.
Excavation
366
level is obtained. If the groundwater level has to b e lowered to greater depth this has to
be carried out in stages b y m e a n s of two or several well point systems placed at different
levels (Fig. 261).
A rough estimate of the groundwater lowering sm achieved by the gravity m e t h o d in the
middle of a circular area with radius R0 surrounded by wells can b e m a d e by the relation:
, = / ^ / / / - % 1
2
(463)
fob
Ro =
(464)
(465)
i0
R= R
9kFt
+~
(466)
w h e r e t - time of p u m p i n g ,
= porosity of the soil.
Example 75: An excavation with an area of 15 m by 10 m shall be earned out to a depth of 3 m below
the groundwater table in silt underlain by bedrock at 20 m depth. In order to avoid problems during
excavation the groundwater table is lowered by installation of well-points immediately outside the
boundaries of the excavation. The aim is to achieve a depression on the groundwater level of 4 m in the
centre of the excavation one week after pumping has started. Determine the required pumping capacity
if the silt has a porosity of 40% and a permeability of 300 m/ year.
1 /2
Solution: We have R0 = (15 1 0 / ) = 6.9 m. The R value after one week of pumping becomes equal
to:
2
1 /2
R = [ 6 . 9 + 930020/(520.4)] = 51 m
The pumping capacity required is obtained from the relation:
367
Excavation
4 = 20-J(202
^ l n
3000
3
6.9
3
(iii) Ejector well points. In ejector well points, also called deep wells, v a c u u m is created
at the tip of the wells by m e a n s of a high-speed flow of water through a j e t nozzle. T h e
water sucked out of the soil is then forced by the water j e t stream u p the return pipe. By
the u s e of ejector well points the lifting height of the water u p the return p i p e can b e u p
to 6 0 - 1 0 0 m. This m a k e s possible a very deep-going lowering of the g r o u n d w a t e r level
and the n u m b e r of wells required can b e reduced accordingly (Fig. 262).
(iv) Electro-osmosis.
Fig. 262. Comparison between stagewise lowering of the groundwater level by ordinary well points and
lowering in one step by means of ejector well points.
Excavation
368
vx = - k e ~ - k j
OX
(467)
(468)
current, we
finally have:
Qw~kepjt
= pjt
(469)
The parameter pe = kepe yields the amount of water transported per electric charge unit.
The consumption of electric charge required to reach a certain desired result depends to
a great extent on the design of the electro-osmotic field (the choice of electrodes and their
location). The total resistance in the electro-osmotic field which governs the interrelation
between amperage and voltage (according to Ohm's and Kirchhoff's laws) is dominated
by the electrode to soil resistance.
In the case of a staff-shaped electrode, the electrode to soil resistance R is given by the
relation (Kupfmller, 1959):
Pe
4L
R =ln()
2nle
De
(470)
369
Excavation
15
\ p = 1.9 t/m
10
. p = 1.8 t/m
6 t/m
= 1.4 t / m ^ " ^ -
10
15
20
Fig. 263. Correlation between electric resistivity of mineral soil and salt content. (Larsson, 1975).
K, = { 1
w h e r e =
l n [ ( c o t ( f l 2 / 2 ) - c o t ( 0 3 / 2 ) - cot(012)}
ll nn((8b4V
/ D^, )) -- l
1
R
)
(471)
n
axctan(nSJle),
40
60
100
Water content,
Fig. 264. Electro-osmotic permeability coefficient of mineral soils as a function of water content.
Excavation
370
Both ke and pe can be determined in the laboratory. Measurements have shown that, for
a given density of mineral soil, there is a close relationship between the electric resistivity
and the salt content of the soil (Fig. 263) while ke is mainly a function of the water content
and the salt content (Fig. 264)
In order to maintain full efficiency of electro-osmotic dewatering, the consumption of
the anodes must be considered (Fig. 265). This can be calculated on the basis of Faraday's
law. Accordingly, the weight AGa (in grammes) set free by the passage of Qe coulombs
is given by the relation:
AGa =
Qe
atomic weight
96494
valence
(472)
For iron anodes, having an atomic weight = 55.9 and a valence = 2, the consumption
AGa (in milligrammes) is 0.29 Qe.
A classical example of the use of electro-osmosis for dewatering purposes is the
successful application designed by Leo Casagrande to achieve a 12 m deep excavation
in very soft, water-saturated silt for a submarine pen in Trondheim during the second
world war (Fig. 266). Previous attempts to reach the required 12 m excavation depth by
the use of a 17 m deep sheet pile cut off led to liquefaction and sheet pile failure already
at 8 m depth of excavation (Casagrande, 1947).
Fig. 265. Iron anode after termination of electro-osmosis. Most of the material is consumed and the
anode is steaming.
371
Excavation
Fig. 266. Failure of sheet pile wall during excavation for submarine pen in Trondheim and solving the
problem by the use of electro-osmosis. The electro-osmotic installation can be seen in the background.
(After Casagrande, 1947).
Example 76: In order to stabilise the slope of an excavation in silty clay, two rows of electrodes are
installed along the slope, of which the anodes are rails with a mass of 25 kg/m and the cathodes, which
are constructed as pumping wells, are made of steel with an outer diameter of 300 m. The distance
between the rows is 4 m, the spacing between the electrodes in the rows is 2 m, and the embedded length
of the electrodes is 10 m. The number of anodes is 10 and the number of cathodes is 9.
Determine the required pumping capacity and how long time it will take before the anodes are
consumed if the electro-osmotic permeability coefficient of the soil is ke = 2-1 ( ) 8 m 2/ s V and the resisitivity
is pe = 6 . Only the electrode to soil resistance needs to be considered. The electric potential anode
to cathode is 30 V.
Solution: The equivalent diameter of the anodes (the specific density of iron is 7.85 t/m 3) is equal to:
d = 2(0.025/7.85) 1 /2 = 0.064 m
The resistance anode/soil becomes:
tfa/s =
210
l n ( ^ ^ ) = 0.61
0.064
372
Excavation
210
ln(^^) = 0.47
0.30
The anode to soil resistance for the whole system is determined by Eq. (471). We have:
= arctan(rc-2/10) which yields:
R& =( 0.61/10){ 1 + ln[cot(0.190)-cot(0.270)-cot(0.337)-cot(0.393)-cot(0.438)-cot(0.475)cot(0.506)cot(0.532)-cot(0.554)]/[ln(8-10/0.064)- 1]} = 0 . 1 4 0
Regarding the cathode to soil resistance for the whole system we have:
Rc = (0.47/9){1 + ln[2662-tan(0.554)]/[ln(8-10/0.3) - 1]} = 0.137 Qm
The total resistance becomes R.d + Rc = 0.277
The electric currency / = 30/0.277 = 108 A and the water carried from the anodes to the cathodes is
equal to:
8
The anode consumption is AG = 0 . 2 9 - 1 0 108-24-3600/10 = 0.27 kg per day and and anode. Each
anode has a mass of 250 kg. Thus the anodes will last maximum 2.5 years.
(v) Pneumatic
method.
373
Excavation
2.4
Reinfiltration
of
water
Excavation
374
3.
U N B R A C E D EXCAVATIONS
3.1
Introduction
After appropriate measures have been taken to avoid stability problems due to unfavourable
groundwater conditions, other causes for instability or excessive ground displacements
in connection with excavation have to be taken into account. T h e excavation has to be
carried out in such a m a n n e r that the shear stresses induced do not approach the shear
strength of the soil. T h e measures required depend on type of soil, depth of excavation
and neighbouring conditions.
3.2
Sloping
sides
In m a n y cases the excavation can be carried out without the need of side supports.
Generally, the only measure to fulfil side stability requirements is to carry out the
excavation with sloping sides.
In cohesionless
soil the slope angle is determined by the angle of repose of the soil
(approximately equal to the internal critical angle of friction). If there is a load acting in
the near vicinity of the slope the stability condition will h a v e to be checked by a detailed
analysis searching for the most dangerous failure surface. In cohesionless soil the shape
of the failure surface can b e assumed to be a logarithmic spiral.
In cohesive
soil the short-term stability of the slopes is determined by the slope angle
and the undrained shear strength of the soil. Since excavation leads to a long-term
reduction of the effective stresses in the soil, long-term stability has to b e determined on
the basis of effective strength parameters. T h e sliding surface can generally be assumed
to h a v e the shape of a circular cylinder. T h e most dangerous sliding surface can be found
by trial and error. In the case of simple geometry and h o m o g e n e o u s soil conditions the
slope stability conditions can be determined by the aid of diagrams.
T h e analysis of the stability of sloping sides of an excavation in simple and more
complicated situations was treated in the section dealing with slope stability, p. 336.
3.3
Cantilevered
walls
Sheet pile walls represent the most c o m m o n type of retaining structure for the support of
the sides of deep excavations. In cases w h e r e the depth of excavation is limited, cantilevered sheet pile walls can be utilised. T h e earth pressure distribution in such a case
depends on the bending rigidity of the wall and the depth of e m b e d m e n t (Rowe, 1951).
Obviously, in the design of a cantilever wall both the conditions of equilibrium and the
limits set to wall m o v e m e n t s will have to be satisfied. F r o m a theoretical viewpoint, the
latter requires good k n o w l e d g e of the soil properties and finite e l e m e n t analysis.
T h e stability against failure can b e treated in a fairly simple way. Accordingly, the wall
is a s s u m e d to b e subjected to classical Mohr-Coulomb type of earth pressure (pp. 278-279).
375
Excavation
P3/(d-zr)
Fig. 267. Earth pressure distribution along a cantilevered wall.
Below the centre of rotation (Fig. 267) the resulting earth pressure is replaced by a force
acting at the centre of rotation. With the symbols given in Fig. 260, the requirements of
equilibrium yield:
Pizl-P2z2
=0
P3 = P2-PX
P3/(D-zr)
=
PpiJF-Par
(473)
w h e r e Ppr = passive earth pressure at the centre of rotation (on the active side),
Par = active earth pressure at the centre of rotation (on the passive side),
F = factor of safety.
Normally, the depth of e m b e d m e n t d is taken as m i n i m u m
1 2 z r.
376
Excavation
1 / 2
4.
B R A C E D EXCAVATIONS
4.1
Introductory
remarks
Excavation
377
Fig. 268. Failure of cantilever sheet pile wall one day after excavation to full depth had been terminated.
378
Excavation
Fig. 269. Schematic picture of the construction of diaphragm wall. In a second step of construction, the
diaphragm wall will be completed and the joints between the panels sealed.
Fig. 270. Diapragm wall constructed as part of the basement walls in a shopping centre. The wall, which
is made water-tight, is carried down to bed-rock at about 10m depth below groundwater and is anchored
by tiebacks.
379
Excavation
Fig. 271. Pile wall, anchored by tiebacks. Open space between the piles covered by shotcrete.
the design. M o s t of the cases w h e r e things go wrong depend on lack of contact between
the designer and the contractor or lack of efficient control.
A wealth of information regarding the analysis and design of anchored structures is
given in the textbook by H a n n a (1982). T h e problems encountered in connection with
deep excavations with side support are related to the earth pressure distribution, as
mentioned above, and to the risk of b o t t o m h e a v e in soft cohesive soils and the risk of
piping in cohesionless soils.
For the sake of simplicity, "anchored w a l l s " will b e used in the following as a term for
both anchored and propped (strutted) walls.
4.2
level
soil. For a cohesionless soil the depth of the wall is determined in order that
the net passive earth pressure is at least 1.3 times the earth pressure required for rotational
Excavation
380
Fig. 272. Earth pressure distribution applied in the design of walls with one row of anchors in
cohesionless soil (left) and cohesive soil. (Sahlstrm and Stille, 1979).
equilibrium. T h e net passive earth pressure applied in the analyst s of rotational equilibrium
is assumed to be distributed as shown in Fig. 272 (Terzaghi, 1943) which yields:
zp = 4 , ( l - / r r ^ )
(474)
- yJ\-F~ )
(475)
Ka)Ydp.
^50
100
200
500
1000
2000
5000
(m/MPa)
Fig. 273. Diagram for determination of the moment reduction factor to be applied in the design of
sheet pile walls with one row of props/anchors.
where Lshp
Ancles = A * m * m
ax
Example 79: To realise a 5 m deep excavation in sand with an internal angle of friction of 30 a sheet
3
3
pile wall has to be installed. The unit weight of the sand is 7= 18 k N / m above, and / = 11 k N / m below,
the groundwater table. Determine the depth to which the sheet pile wall has to be installed if the
groundwater level in the sand is at 5 m depth and the wall is propped at one level, 1.5 m below the ground
2
surface. A load of 20 k N / m is placed on the ground outside the wall. Possible soil/wall friction can be
neglected.
Solution: For a soil with '= 30, we have the earth pressure coefficients Ka = 1/3 and Kp = 3.0. Thus,
the active earth pressure above the bottom of excavation follows the relation:
pa = (20+ 18z)/3 kPa
and the net pressure (active minus passive pressure) below the bottom of excavation:
pnei = 1(20 + 185)/3 + ( - 5X1/3 - 3.0)11 = 36.7 - 29.3(z - 5) kPa
The net pressure is zero for = 6.25 m, i.e. 1.25 m below the bottom of excavation. Choosing a factor
of safety Fs= 1.3 we have:
zp = 0.52dp
a nd
5 2 3
Pmax = - ( - 1 / 3 ) 1 1 ^ = 1 5 . 3 ^ kPa
The earth pressure distribution thus obtained is shown in the figure below.
Moment equilibrium around A requires:
2
2
(20/3>5(5/2 - 1.5) + (185 /6)(52/3 - 1.5) +(1.25 36.7/2)(3.5 + 1.25/3) =
= 0.52d?p.15.3dfp(3.5 + 1.25 + 2-0.52^3) + 1 5 . 3 ^ 0 . 4 8 ^ ( 3 . 5 +1.25 + 0 . 5 2 ^ + 0 . 4 8 ^ / 2 )
Excavation
382
q = 20 k N / m
(ii) Cohesive
soil In the case of cohesive soil, disturbance effects due to pile installation
and/or installation of sheet pile walls m a y cause a considerable reducti on of the undrained
shear strength of the soil (Fig. 274). This entails an increase in the active earth pressure
and a decrease in the passive earth pressure as well as an increased risk of bottom
heave.
Ncbcu
+ 2cwD/(B
+ L)
(478)
+ L)
+q
Normally, the term 2cJ(B
(479)
(480)
The bottom heave factor Ncb for trench excavations can be determined in a way similar
to that of the bearing capacity factor for deep foundations. Assuming D to represent the
383
Excavation
Fig. 274. Example of disturbance effects caused by pile driving and installation of sheet pile wall.
r
_JII I I I I, I I I
Fig. 275. Bottom heave of deep excavations in cohesive soils is analysed in a similar way as bottom
failure of deep foundations.
Excavation
384
BIL = 1
BIL = 0.5
/L = 0
Fig. 276. Diagram for the determination of Ncb {cf. Eq. 480).
depth below the b o t t o m of excavation to the base of the supporting wall, the stability
factor Ncb for an excavation with length L, width and depth H, and for (H + D)IB < 4,
can be expressed by the approximate relation (Fig. 276):
1 H+D
H+D
^ / 7= ( 1 + 0 . 2 - ) { 5 . 1 4 + - [ - ( 8 - ^ )
L
j
707
(481)
For values of (H + D)/B > 4, the value obtained for (H + D)/B = 4 is used, i.e. Ncb - 7.5
for BIL = 0 and Nch = 9.0 for BIL = 1.
T h e shear strength cu in Eqs. ( 4 8 0 - 4 8 1 ) should be taken as the m e a n of the shear
strength values observed to a depth below the b o t t o m of excavation of 0.7. Disturbance
effects due to excavation, installation of the sheet pile wall and/or pile driving should b e
considered (cf Fig. 274).
For w i d e excavations in clay the overall stability should be checked, for e x a m p l e by
circular-cylindrical slide surface analysis.
T h e net passive earth pressure below the b o t t o m of excavation that can be allowed in
the design will b e restricted by the risk of b o t t o m heave. A s s u m i n g a factor of safety of
Fs against wall rotation around the anchor (strut) level, the m a x i m u m net passive earth
pressure is determined by the relation:
Pmax = ^ T T - ( 9 + ? )
(482)
385
Excavation
w h e r e Ncb = the stability factor (In the case of inclined anchors, Ncb is governed b y
the vertical stability of the sheet pile wall, see p . 3 8 9 ) ,
cu = undrained shear strength.
A n allowance should b e m a d e for water pressure against the wall on the assumption that
the g r o u n d w a t e r level coincides with t h e ground surface (as s h o w n in Fig. 2 7 2 ) .
Again the depth D is determined by the rotational equilibrium around point A in Fig. 272.
If the anchors (struts) are prestressed,
(483)
(484)
1.35.
T h e additional earth pressure resultant Qioi - Qe due to prestressing can b e a s s u m e d to
act at the anchor (prop) level.
Example 80: A trench, supported by a sheet pile wall, shall be made to a depth of 3.5 m in clay with
3
a unit weight of 16 k N / m and an undrained shear strength cu = 15 kPa. The width of the excavation is
2
2 m and the length 10 m. The load on the ground surface outside the excavation is 10 k N / m . The wall
is propped at a depth of 1.5 m. Determine the required depth D below the bottom of excavation to which
the sheet pile wall must be installed if the required factor of safety is 1.5. Determine also the strut force.
Solution: The active earth pressurep a = \6z - 2-15 + 10 = \6z - 20 kPa. Water pressure uw = lOz kPa
dominates over the active earth pressure to a depth of 20/6 = 3.3 m. The stability factor Ncb = 1.04-7.5.
With a factor of safety against wall rotation of 1.5, the maximum allowable, stabilising earth pressure
below the bottom of excavation becomes equal to:
Pmax = 7.8-15/1.5 - (16-3.5 + 10) = 12 kPa
1.5 m
^ =^
D
Pmax
Excavation
386
or, in rewritten form:
2
D = 4.5
whence D = 2.1 m and H + D = 5.2 m
The heave stability factor becomes equal to:
7
Ncb = (1 + 0.22/10){5.14 + [(5.6/2)(8 - 5.6/2)]- /3} = 7.6
The safety factor against bottom heave becomes Fs = 7.6 15/(16-3.5 + 10) = 1.73
2
The strut force Qe = 10-3.5 /2 - 12-2.1 = 36 kN/m
4.3
levels
be driven is governed by the depth of excavation below the lowest anchor level and by
the possible risk of piping (cf. p. 357). In the case of cohesive
Fig. 277. Design earth pressure and its redistribution in cohesionless soils. The value of Pa included in
the redistributed earth pressure is chosen as the Rankine earth pressure above point D. (Sahlstrm and
Stille, 1979).
387
Excavation
tot
0.9H+d
Fig. 278. Design earth pressure and its redistribution in cohesive soils. The value of included in the
redistributed earth pressure is chosen as the Rankine earth pressure above point D. (Sahlstrm and Stille,
1979).
tot
(485)
0.9//+ d
w h e r e H and d according to Figs. 277 and 2 7 8 ,
Pa = total earth pressure (including water pressure),
Qtot = 0.SPa +
or:
tot =
0AQpr>Pa,
QPr
QprJPa
> 1.35 ( c / E q s . 4 8 3 ^ 8 4 ) .
A s m e n t i o n e d before, Sahlstrm and Stille (1979) p r o p o s e that an allowance b e m a d e
for the possibility of water pressure in the top part of a clay deposit with a groundwater
level coinciding with the ground surface (as shown in Fig. 272). This m a y s e e m odd since
the total weight of the clay is included in the calculated active earth pressure. However,
the pressure acting against the wall can never b e less than the water pressure.
T h e earth pressure b e l o w the b o t t o m of excavation represents the difference b e t w e e n
active and passive earth pressures. For excavations in clay, w h e r e Ncb < ( + q)/cu,
the
base of the wall should b e carried d o w n to firm soil layers (Fig. 270).
Example 81: A sheet pile wall shall be installed for the purpose of stabilising a 5 m deep excavation
3
in a 9 m thick clay layer underlain of sand. The clay has a unit weight of 17 k N / m and an undrained
3
shear strength of 20 kPa. The sand has a saturated unit weight of 21 k N / m and an internal angle of
friction of 38. The groundwater level in the sand is kept 3.5 m below the ground surface. The width
of the excavation is 4 m and the length 20 m. The wall shall be supported with horizontal struts at depths
388
Excavation
1 m, 3.0 m and 5 m below ground surface. A prestress load of Q =0JPa shall be applied. External load
2
outside the excavation q = 10 k N / m . Determine the required length of the sheet pile wall and the strut
loads. A partial factor of safety of 1.15 should be applied on the angle of friction in the sand.
Solution: The active earth pressure is equal to Pa = llz + 10 - 2-20 = 17z - 3 0 kPa, where denotes the
depth below ground surface. The water pressure uw 10z kPa dominates to a depth of 30/7 = 4.3 m.
The total pressure above the bottom of excavation is equal to:
2
Pa = 104.3 /2 + 17(4.3 + 5.0)/2 - 30 = 141.5 kN/m
whence g t ot = (0.8 + 0.4-0.7)141.5 = 152.8 kN/m
Assuming that the sheet pile wall is installed to a depth of D = 4 m, the safety factor against bottom
heave Fs becomes equal to 1.17 and against hydraulic uplift 17-4/55 = 1.24.
The sheet pile wall has to be driven down into the sand.
The earth pressure in the clay below the bottom of excavation is equal to
pa-pp=
1 7 z - 3 0 - [ 1 7 ( z - 5 ) + 40] = 15 kPa
The design angle of friction in the sand is '= arctan(tan3871.15) = 34
2
The earth pressure coefficients in the sand are Ka = tan (45 - 3 4 7 2 ) = 0.28 and = 1/0.28 = 3.57.
In the sand we thus have:
/ 7 - / 7 = 3 . 5 7 [ 4 1 7 - 5 . 5 1 ( ) + 1 1 ( - 9 ) ] - 0 . 2 8 [ 9 1 7 + 1 0 - 5 . 5 - 1 0 + l l ( z - 9 ) ] = 16.2 + 3.29(z-9)kPa
Thus in our case 2c = 4 m and d = c/2 = 2 m (see Fig. 278)
The redistributed earth pressure pt = 152.8/(0.9-5 + 2) = 23.5 kPa
whence the prop loads: Qx = 23.5(1.0 + 0.5) = 35.3 kN/m
Q2 = 23.5(1.0 + 1.0) = 47.0 kN/m
Q3 = 23.5(2.0 + 1.0) = 70.5 kN/m
The length of the wall is determined by the relation:
2
2-15 = 16.2(D - 4) + 3.29(D - 4 ) / 2
whence D = 5.6 m
The total length required of the sheet pile wall is 10.6 m.
389
Excavation
4.4
Influence
of inclined
tiebacks
W h e n the sheet pile wall is anchored by m e a n s of inclined tiebacks the vertical stability
of the sheet piles m a y b e adventured by the vertical c o m p o n e n t of the anchor loads. This
p r o b l e m has b e e n dealt with by Stille (1976).
Let us a s s u m e that the anchor inclination is (Fig. 279). T h e vertical c o m p o n e n t of the
anchor loads will b e counteracted by wall adhesion. Taking into account the influence on
the earth pressure exerted by wall adhesion, the normal c o m p o n e n t s of the active and
passive earth pressures are given by the a p p r o x i m a t e relations, given by Eqs. 3 6 3 - 3 6 4 :
Pa = + 4 -
NaCu
Pp = JZ+q +
Npcu
w h e r e (Janbu et al,
1956)
JV = 2 v / l + 2 r / 3
Np =
2y/l+2rp/3
r = cjcu
T h e wall adhesion parameter r can b e positive or negative d e p e n d i n g on the direction
of the relative displacement between soil and sheet pile wall. T h u s , w h e n the soil is
m o v i n g d o w n w a r d s in relation to the wall, r b e c o m e s positive on the active side of the
wall and negative on the passive side, and vice versa.
A s s u m i n g that the forces are acting on the sheet pile wall as s h o w n in Fig. 2 7 9 , the
following equilibrium conditions are obtained:
(Pp-Pa)une-(Cp-Ca)cose
=0
(486)
Fig. 279. Assumptions behind the analysis of the stability of sheet pile walls with inclined anchors.
Cohesive soil.
Excavation
390
+ D-
//,),
0.5yD+NpCuD,
Ca = racu(H
C
(487)
Ca) = 0
Rsme-(Cp-
rc
D-H^,
pu >
Hi = NacJy.
Introducing the stability n u m b e r Ncb = yH/cu and the values of Pp, Pa, Cp,Ca and / / , into
Eq.(486), Ncb is obtained by solving the following equation of the second degree:
(488)
where
Na{\+HID)+Np
[rp-ra{\+HID)]cote
1+0.5H/D
B =
Na(2racote-Na)
1+2D/H
R =
Hc
rnD
ra(l+--
Na
(489)
)]
V2/3
//
-i
0.5
//
/1/3
/I72/3
I Vo.5
^Vl/3
N* 4
/ -
<P
= 0
I S )
J .
H~3
~ =1
0
Is*
- = 0.5
1
RcO
Hcu
Fig. 280. Values of stability number Ncb as a function of anchor load and wall adhesion for anchor
inclination = 45.
391
Excavation
B y a s s u m i n g t h e v a l u e of rp, Ncb
and Np = 2(1 + 2 r / / 3 )
l/2
and Np =
1 / 2
].
Example 82: For the side support of a 4 m deep excavation in a clay layer with a unit volume weight
3
of 16 k N / m and an undrained shear strengh of 20 kPa a sheet pile wall is installed to a depth of 7.3 m.
The wall is anchored at a depth of 2 m by tiebacks with an inclination of 45. The anchor force is 110
2
kN/m. External load outside the excavation q = 10 kN/m . Determine the factor of safety on the assumption
that we can allow a value of rp = 0.5. Determine also the earth pressure distribution and the factor of
safety against rotation around the anchor level.
Solution: W e have Rcos6l(cuH) = 110-cos457(20-4) = 0.97 and HID = 4/3.3 = 1.2.
For a value of HID = 1, rp = 0.5 and Rcos6l(cuH) = 1, we have Ncb ~ 4.3.
Introducing this value into Eq. (489) we find:
1/2
110 = (4-20/sin45)[0.5-3.3/4 - ra(\ + 3.3/4 - 2 ( l + 2 r t / 3 ) / 4 . 3 ]
whence ra = - 0.39
Inserting this value of ra into Eq. (489) we find Ncb = 4.28.
Another iteration procedure yields the same values of Ncb and ra.
The safety factor Fs = 4.28-20/(16-4 + 10) = 1.16
1 72
The active earth pressurep a = I6z + 10- 2-20(1 - 2-0.39/3) = I6z - 24.4 kPa. Water pressure uw
= 9.8z kPa dominates over active earth pressure to a depth of 24.4/6.2 = 3.9 m.
1 /2
The passive earth pressure pp = 16(z - 4) + 2-20(1 + 2 - 0 . 5 / 3 ) = 16(z - 4) + 46.2 kPa.
Net earth pressure at the bottom of excavation:
p p - P a = 46.2 - 16-4 + 24.4 = 4.9 kPa.
Rotational stability yields:
2
4.9-3.3(3.3/2 + 2) = 9.8(4 /2)(2-4/3 - 2)FS
whence Fs= 1.13.
10 k N / m
Excavation
392
4.5
Design
anchors
(i) Design of wale beams. In the design of the wale b e a m s , consideration m u s t b e paid to
the possibility of anchor failure. Such anchor failure will cause a redistribution of the
earth pressure against the sheet pile wall and induce a load increase in the anchors nearby.
T h e consequence of anchor failures has been investigated by Stille (1976). In 13 cases
of simulated anchor failures, the load increase in adjacent anchors was less than 2 5 % . In
these cases the earth pressure, due to prestressing of the anchors, exceeded the Rankine
earth pressure. Generally, the load increase was very small, particularly w h e n the anchor
failure took place in the lower anchor rows.
Obviously, considering the insignificant effect of anchor failure and normal safety
requirements, the possibility of a d o m i n o effect, leading to a total collapse of the sheet
pile wall, seems negligible. However, the possibility of anchor failure cannot be
neglected and ought to be considered in the design of the w a l e b e a m . Sahlstrm and Stille
(1979) r e c o m m e n d that the design of the wale b e a m b e based on a modified version of
the limiting creep stress method. Assuming that the load acting on the w a l e b e a m before
anchor failure is q, the m a x i m u m allowable bending m o m e n t , M m
ax
can be determined
by the relation:
Mm
ax
= ^-<1.5aa
lo
ll
Wx
(490)
393
Excavation
/// a . //c ^
'V = s/r ^
//| ^
/yan0'|jjj|
7 ^ m j
/ > l f t al n 0 '
\W\
2&
Fig. 281. Anchoring of retaining wall by means of a horizontal tie rod fixed to a vertical plate.
T0exp(-Ax/d)
(491)
(492)
P=
nd T(\
- [ 1 - e x p ( - A L Id)}
(493)
We thus realise that the higher the displacement m o d u l u s the shorter the length taking
part in carrying the load and the higher the bond stresses. Obviously, yielding will take
place in the b o n d in the upper part of the anchor already at an early stage of the loading
and progress deeper d o w n with increasing load.
Soil anchors h a v e to b e formed with due regard to the soil characteristics with the aim
to producing a reliable and cost-effective solution.
T h e m o s t simple type of soil anchors consist of vertical plates buried in the soil and
fixed to the sheet pile wall by horizontal tie rods. T h e anchoring force is determined as
the difference b e t w e e n passive and active earth pressures acting against the plate. For
superficial anchor plates the vertical stability has to checked b y the relation (Fig. 281):
Excavation
394
W + 2 tm5a
+ Plp tan0'
(494)
= normal component of passive earth pressure against the overlying soil section,
and are the angles of friction at the active and passive plate/soil interfaces.
The upward force on the passive side can be reduced by providing the passive plate
surface with a friction-reducing coating.
The distance between the sheet pile wall and the anchor plate is determined by the
condition that the passive failure zone in front of the anchor plate and the active failure
zone behind the sheet pile wall should not intersect. The overall stability also has to be
checked.
Nowadays, these types of anchor are more or less completely replaced by inclined
tension anchors. In cohesionless soil, the anchoring zone is generally enlarged by
injection of grout that permeates into the granular material, but the form and homogeneity
of the grouted zone thus achieved is difficult to control. In order to produce a well and
easily controlled anchor body in any type of soil, so-called expander anchors (Fig. 282)
have been developed and used with great success under various soil conditions. T h e
expander body consists of a steel container which is folded into the shape of a cylinder
with square cross section. The expander body is attached to a steel tube (of the same kind
as those used as micro-piles) and is then inserted into the soil to the required depth, either
by hammering it into the soil or by preboring. After installation, the body is expanded by
grout injection through the steel tube. By recording the volume increase vs. grouting
pressure, information is obtained about the soil conditions and the bearing capacity of the
anchor can be estimated according to the principles given for the pressuremeter test.
The expander bodies now on the market, in their folded shape before installation, have
dimensions varying from 1 m in length and 60 m m in width (EB 300) to 3 m in length
and 110 m m in width (EB 800). The expanded diameter of E B 300 is around 300 m m and
of E B 800 around 800 m m .
The first type of inclined anchors in clay, used in practice, consisted of H E A steel beams
driven with 45 inclination deep into the clay and fixed to the sheet pile wall by tendons
(Fig. 283). Nowadays, expander anchors form a cost-effective solution also in the case
of cohesive soils.
The grout injection of the expander anchor brings to mind the pressuremeter test.
Actually, results of pressuremeter tests form an excellent basis for determination of the
bearing capacity and the deformation behaviour in loading. As shown by Sellgren (1991),
the displacement s for the tension load can be calculated by the relation:
395
Excavation
Fig. 282. Expander bodies EB 800 (folded and expanded) and EB 300 (expanded).
s=
i-Q/Qf
(495)
Fig. 283. HEA steel beam, installed with an inclination of 45 to the horizontal, to serve as an anchor
for a sheet pile wall in soft, high-plasticity clay.
Excavation
396
= ^
EtAt
+ ^
9DaEs
(496)
Fig. 284. Ultimate load for various types of expander bodies determined by the results obtained in
expansion of the bodies (Sellgren, 1992).
So/7
improvement
397
SOIL I M P R O V E M E N T
1. I N T R O D U C T I O N
T h e a i m of soil i m p r o v e m e n t is to bring about a condition w h e r e the g e o m e c h a n i c a l
properties of the subsoil b e c o m e good e n o u g h to instigate cost-effective solutions to the
foundation p r o b l e m s and m i n i m i s e m a i n t e n a n c e costs. Soil i m p r o v e m e n t can be a m e a n s
to avoid expensive piling and to reduce the risk of d a m a g e to adjacent buildings during
foundation w o r k s .
In the case of cohesive soils, soil i m p r o v e m e n t is most c o m m o n l y used to reduce, or
to eliminate, l o n g - t e r m settlement which otherwise can create serious p r o b l e m s . For
e x a m p l e , in the architectural planning of buildings, the ground level is often raised and
the groundwater level often lowered by drainage through s e w e r a g e and water pipes. In
cases w h e r e the subsoil consists of compressible soil this m a y cause serious p r o b l e m s in
the connection b e t w e e n building and environs (Fig. 285) unless the deformation properties of the soil h a v e been i m p r o v e d in advance.
In urbanised areas, land with g o o d foundation properties has been m o r e or less used
up for building purposes. Therefore, soil that w a s originally considered too bad has
started being utilised. T h e r e is a tendency to place, first of all, industrial activities to such
Fig. 285. Large settlement of the ground encompassing buildings on piles brings about access problems
and may cause breakage of connecting sewerage and water pipes.
So/7
398
improvement
bad areas whereas dwellings are mostly concentrated to areas with better foundation
properties. T h e problems encountered will certainly increase with increasing need of
ground for building purposes. Obviously, soil i m p r o v e m e n t techniques will b e c o m e
increasingly important in the future.
(i) Improvement
by compaction.
in connection with the construction of earth dams which due to their i m m e n s e heights (up
to 300 m ) and the catastrophic consequences of failure require extremely careful
w o r k m a n s h i p and control. T h e compaction technique is well established and detailed
instructions for the compaction process exist in most countries.
N o w a d a y s , compaction of fill and natural soil has also b e c o m e quite c o m m o n in
connection with industrial building and house-building activities in order to find costeffective foundation solutions. Recently, the need of land reclamation for industrial and
harbour development has urged the development of deep-compaction technique onward
and compaction can now be carried out to depths of 3 0 - 4 0 m below the compaction
surface.
Compaction has two aims: Firstly it should bring about a densification of the soil and
secondly it should crush sharp edges and sharp contact points between the soil grains.
Crushing leads to an increase in the size of the contact areas and, consequently, to a
decrease in contact stresses between the grains which in turn yields a lower compressibility.
T h e crushed material also fills the voids in the soil and results, as does the reorientation
of the grains, in a v o l u m e decrease.
Densification by reorientation of grains is best achieved by vibrations while crushing
is best achieved by impact. Reorientation of grains i s the foremost reason for densification
in fine-grained soils. D u e to small grain size, the n u m b e r of contact points b e t w e e n the
grains per unit v o l u m e is very large (for example, in a case w h e r e the grain size is 2 m m ,
2
the n u m b e r of contact points per m cross-sectional area is about 1 0 ) , and the contact
stresses thus b e c o m e insignificant. In a soil with large grain size, such as boulder and
cobble soil, crushing of grains is the foremost reason for densification. This is particularly
the case for blasted rock fill. On the one hand, the n u m b e r of contact points per unit
v o l u m e is quite small (in a case w h e r e the grain size is 6 0 0 m m , the n u m b e r of contact
2
points per m cross-sectional area is about 10) and on the other hand the grains have not
been subjected to the wear of nature and time.
In water saturated soil the v o l u m e decrease achieved by compaction is always
accompanied by escape of an equally large volume of water. In soils with low permeability,
the water escape m a y take place very slowly and compaction of such soils therefore
involves a risk of liquefaction p h e n o m e n a occurring. Even if the soil is not originally
water saturated there m a y b e a risk of liquefaction provided that the v o l u m e decrease
during compaction is large e n o u g h for the soil to pass into a state of full water saturation.
T h e choice of compaction m e t h o d and e q u i p m e n t obviously depends on the grain size
So/7
399
improvement
distribution and the content of fines and cobbles and boulders. W h e n the content of fines
is large then the degree of water saturation is of significant importance.
(ii) Improvement
by preloading.
by chemical
and
electrochemical
treatment.
Chemical
and
SHALLOW COMPACTION
water
content
Soil
400
10
15
Water content w
20
improvement
25
(%)
Fig. 286. Dry density as a function of water content at laboratory compaction. Compaction is carried
out at varying water contents in order to enable a full presentation of pd vs. w. Lines represent water
saturation for different values of particle density.
pd instead of bulk
density p i s that it gives us an indirect m e a s u r e of the porosity (void ratio) of the soil (see
p. 14). According t o E q . (11), p. 14, the bulk density at the water content w in question
is obtained by:
(497)
p = Pd(\+w)
Increasing flocculation
- l/p8)
(498)
Increasing dispersion
Water content w
Fig. 287. Influence of compaction energy on optimum water content and dry density. Compaction wet
of optimum gives a more parallel particle orientation than compaction dry of optimum. Large
compaction energy also leads to a more parallel particle orientation.
So/7
401
improvement
Fig. 288. Example of the influence on the optimum water content of compaction energy. Notice that
loose infilling of material at optimum water content according to laboratory compaction may result in
minimum dry density.
However, from a statistical viewpoint this is not always the case due to deviations from
the a s s u m e d value of pg (cf. Fig. 288).
Example 83: As a result of laboratory compaction, the optimum water content of the soil was found
3
equal to w o pt = 7% and the dry density equal to pd = 2.18 t/m . Determine the porosity and the degree
of saturation of the compacted soil.
3
Solution: Assuming that the particle density of the soil is pg = 2.7 t/m , we find by Eq. (11):
2.18 = 2.7(1 - n), whence =0.19 (19%)
The water content at full saturation, according to Eq. (498), is:
ws = 1/2.18 - 1/2.7 = 0.088, whence Sr = 0.07/0.088 = 0.79 (79%)
So/7
402
and wet of
improvement
optimum
for soil compacted wet of o p t i m u m except at very high pressure. T h e compression curve
dry of o p t i m u m resembles that of undisturbed clay and wet of o p t i m u m that of disturbed
clay. T h e consolidation process is quicker dry of o p t i m u m .
Shear strength.
So/Y
403
improvement
high bearing capacity and small settlement while it ought to b e performed wet of o p t i m u m
to achieve low hydraulic conductivity.
2.3 Compaction
equipments
equipments.
A m o n g the static c o m p a c t i o n e q u i p m e n t s w e h a v e
heavy-weight caterpillars, bulldozers, scrapers and trucks etc. T h e s e are not primarily
meant for c o m p a c t i o n and, therefore, no instructions are given for their u s e as compaction
tools. All the same, the u s e of bulldozers especially can give quite satisfactory compaction
results depending upon the fact that they induce high horizontal in situ stresses in the fill.
W h e n speaking about static compaction e q u i p m e n t s w e normally allude to rollers of
various t y p e s s m o o t h rollers, sheepsfoot rollers, padfoot rollers, grid rollers and
rubber-tired or pneumatic-tired rollers (Figs. 2 9 0 - 2 9 2 ) . T h e rollers can either b e selfpropelled or tractor-drawn or coupled together to form so-called t a n d e m rollers as well
as coupled together sideways.
In order to p r o d u c e as deep a compaction as possible the weight of the rollers has gradually been increased and n o w a d a y s weights of u p to 15 t for smooth steel rollers, 3 0 1 for
sheepsfoot and padfoot rollers and 50 t for rubber-tired rollers are c o m m o n . In special
Fig. 290. Padfoot rollers are recognised by feet protruding from the cylindrical steel shell of the roller.
The feet of the padfoot roller are shorter and larger than those of sheepsfoot rollers.
Soil
404
improvement
Fig. 291. Rubber-tired rollers are utilised for the compaction of fine-grained soils especially clay with
low shear strength.
cases even heavier rollers are utilised and their weight can be increased by the use of
ballast.
The compaction effect of smooth rollers is achieved by high static contact pressure
induced by the line load of the roller. Sheepsfoot and padfoot rollers, and to a certain
extent also grid rollers, exert high contact pressure and kneading together. The contact
pressure under the feet is, of course, considerably higher than under the steel cylinder
itself.
Rubber-tired rollers also bring about compaction by the combined effect of static
pressure and kneading. High wheel pressure should be used to achieve deepest possible
compaction.
The depth of compaction accomplished by the use of static rollers is fairly small
(generally limited to 0.2-0.3 m) and their compaction capacity is therefore low.
(ii) Vibratory
rollers. All the static rollers mentioned above also exist as vibratory
compaction equipments.
Vibratory rollers generally work in a frequency range of 2 0 - 8 0 Hz with nominal
amplitudes varying from 0.3 to 2.5 mm. The vibrations are accomplished by means of one
or several rotating eccentrics. (The nominal amplitude is obtained theoretically as the
ratio of moment of eccentric weight to mass of roller. The real amplitude at certain
frequencies can be as much as 5 0 - 1 0 0 % higher than the nominal one, depending on
resonance phenomena).
The vibratory action of the roller induces a long sequence of compression waves into
the subsoil. The result of compaction by means of vibratory rollers is dependent on
(Forssblad, 1981):
So/7
405
improvement
20
40
"5 60
80
100
16
Number of passes
Fig. 293. Average settlement of the surface of 1.5 m thick rock fill vs. number of passes of a 5 1 smooth
vibratory roller (Lindblom,1973).
So/7
406
improvement
compression and tension w a v e s in the soil. T h e principle behind the oscillatory roller
(Fig. 294) is to induce transversal shear w a v e s into the subsoil w h i c h is very favourable
from the point of view of compaction. T h e soil particles are sheared into a denser state
and loosening of the soil surface due to reflected tension w a v e s does not occur.
C o m p a c t i o n is faster and the wear of the e q u i p m e n t is reduced (Thurner, 1992). According to Thurner, the layer thickness should not exceed 0.4 m.
(i v) Vibratory plate compactors.
of fill in trenches or in places w h e r e the space is too limited to allow the use of rollers and
for complementary compaction of loosened surface layers after vibratory roller compaction
has been terminated.
M o d e r n vibratory plate compactors are provided with two counteracting, adjustable
eccentrics. In this w a y the direction of the centrifugal force can b e varied and the
compactor can b e m a d e to m o v e forward or b a c k w a r d with a speed of u p to 24 m/min.
Fig. 294. The oscillatory roller is provided with eccentrics arranged in a way to induce transversal shear
waves into the soil.
Soil
407
improvement
The adjustment of the eccentrics is carried out by hydraulic action by the use of a finger
tip controlled back-and-forth lever on the handle. The amplitude is high enough to give
good compaction results to fairly great depth.
2.4 Practical
application
In the following a brief summary will be given regarding the applicability of the various
equipments for compaction of different types of fill (see also Hausmann, 1990). The
limits of the layer thickness that can be accepted to achieve a good compaction result is
governed by the type of equipment and geotechnical characteristics of the fill. The type
of equipment and the limiting height of the lifts for different fill types can be chosen
according to the following general outlines.
Blasted (crushed)
2 - 3 m. The m a x i m u m width of the rock fragments should not exceed 2/3 times the layer
thickness. The vibratory roller should be allowed to act directly on the rock fill surface
without an intermediate layer of finer material. Spreading of rock fill material by the aid
of heavy bulldozers generally generates good compaction.
Sand and gravel. Vibratory plates using a layer thickness of 0.5-1.5 m. If static rollers
are used the layer thickness must be reduced to a maximum of about 0.25 m. The best
result is obtained if the fill is water saturated or nearly dry. Compaction of fine sand,
however, should be performed at a water content close to optimum.
Silt and well-graded
padfoot rollers. The water content should be close to optimum. The layer thickness should
not exceed 0.5 m. If static rollers are used the layer thickness should be limited to a
m a x i m u m of 0.2 m. Rainy weather obstructs or makes impossible compaction work.
Clay. Rubber-tired rollers, padfoot rollers or sheepsfoot rollers. If rubber-tired rollers
are applied the wheel pressure should be adjusted with regard to water content and shear
strength of the fill. Sheepsfoot rollers ought to be used if the undrained shear strength of
the clay fill exceeds 200 kPa.
2.5 Compaction
in winter
time
Compaction in winter entails special problems that have to be taken into account. Thus,
pore water in the soil will freeze to ice and form an obstacle to void ratio reduction. Even
apparently dry soil contains enough condensed humidity to give a similar effect. The soil
particles will be more or less locked in their positions or will be covered by a frozen ice
film that prevents direct contact between the particles. The problems increase with
increasing content of fines (Fig. 295).
In order to accomplish a good compaction result it is very important to supply the soil
with enough heat during the compaction process to prevent freezing of the soil. To reduce
So/7
408
improvement
+20C
-0.5C
- -2C
-5C
-10C
10
15
Fig. 295. Compaction of frozen soil becomes increasingly difficult with decreasing temperature. The
best result is obtained if the soil is completely dry. (After Forssblad, 1981)
methods
Generally, the purpose of compaction is to achieve as high a density (as small a void ratio)
of the compacted soil as possible. High density is considered to be coupled with good
geotechnical properties so the most common control method is based on the results of
density observations. Samples which are taken at the surface of the compacted fill, and
whose volumes are determined by some standardised procedure (for example by means
409
So/7 improvement
of water or sand volumeters), are dried to d e t e r m i n e their dry densities and the values
obtained are c o m p a r e d with the m a x i m u m dry density d e t e r m i n e d beforehand by the
Proctor m e t h o d , for e x a m p l e .
As an alternative to the use of volumeters the use of nuclear density m e a s u r e m e n t s
h a v e b e c o m e quite popular. This m e t h o d gives a quick a n s w e r (and takes only o n e tenth
of the time required by the use of a sand volumeter) and requires less m a n u a l input.
Moreover, the dry density is obtained directly w h i c h is very a d v a n t a g e o u s in a field
control w h e r e an i m m e d i a t e answer is required.
T h e requirements to b e placed on the c o m p a c t i o n result naturally h a v e to b e adjusted
with regard to the p u r p o s e of compaction. Forssblad (1967) r e c o m m e n d s the following
guiding values of lowest allowable degree of c o m p a c t i o n (Rd = p ^ / p ^ m a x )
subgrade for streets, roads and airports
95%
90%
earth d a m s
90-95%
fill in trenches
90%
90%
90-95%
T h e s e values are based on long experience and are considered as a guarantee that the
result is satisfactory.
However, in m a n y cases better control criteria can be established to c h e c k that the
result satisfies the conditions set up. In situ m e t h o d s for determination of settlement and
strength characteristics of the fill after compaction has been terminated are m a n y times
preferable ( H a n s b o & P r a m b o r g , 1980).
N o w a d a y s , d y n a m i c compaction m a c h i n e s , such as vibratory or oscillatory rollers,
can b e e q u i p p e d with a c o m p a c t i o n d o c u m e n t a t i o n system (Thurner & S a n d s t r m , 1991 ;
Thurner, 1992). Such a s y s t e m consists mainly of a so-called c o m p a c t i o n meter, a
speedometer and a c o m p u t e r unit with an L C D display. T h e c o m p a c t i o n meter provides
continuous information about vibration frequency and roller speed and presents a
compaction meter value, representing the ratio of the amplitude of the first h a r m o n i c to
the amplitude of the fundamental tone of vertical vibrations. T h e c o m p a c t i o n m e t e r value
gives a m e a s u r e of the resistance to compaction and consequently of the result obtained.
3.
DEEP COMPACTION
Soil
410
improvement
Heavy
tamping
for a long time on a small scale. Thus, the Proctor m e t h o d used on a laboratory scale is
based on the s a m e principle as well as manual tampers and soil r a m m e r s used in the field.
However, the depth effect of the latter e q u i p m e n t s is quite small because of insignificant
mass (generally less than 100 kg). Heavy tamping, as a full-scale m o d e r n compaction
technique, was developed and introduced by the French engineer Louis M n a r d and his
c o m p a n y T L M (Technique Louis M n a r d ) . F r o m a modest start with a m a x i m u m mass
Fig. 296. Heavy tamping applied at Nice. The mass of the falling weight is 190 t and the drop height
(free fall) 25 m. As can be seen, the weight is made by steel plates bolted together. By courtesy of TLM.
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improvement
aspects.
dcp = njmh
(499)
w h e r e m = m a s s of the p o u n d e r in t,
h = drop height in m,
= empirical constant varying b e t w e e n 0.3 and 1.0.
T h e choice of value depends upon the grain size distribution of the soil and the degree
of saturation.
Fig. 297. Heavy tamping can be performed as shown here, dropping the weight in a direct sequence
successively at each intended point, or, alternatively, first in every second point of design and then in
the points in between not yet compacted. By courtesy of TLM.
412
Soil
improvement
For cohesionless soil has been found to vary b e t w e e n 0.5 and 1. T h e lower b o u n d a r y
value concerns cases w h e r e the loss of energy nearest to the soil surface is considerable.
This is for instance true for very coarse-grained soil (boulder size) w h e r e , at the beginning
of compaction, a large part of the compaction energy is c o n s u m e d in fragmentation. As
previously mentioned, the n u m b e r of contact points per unit v o l u m e of coarse-grained
soil is smaller the larger the particle size (p. 398). T h e stress shock i n d u c e d b y the impact
cannot therefore b e resisted by the material and is the cause of fragmentation. It is
advisable, also in the case of fine-grained material to a s s u m e the lower b o u n d a r y value.
In the case of silt and clayey soils, values in the range of 0.3 to 0.5 are r e c o m m e n d e d .
T h e view is s o m e t i m e s expressed that heavy t a m p i n g can b e u s e d also for consolidation
(compaction) of clay. However, it is very doubtful if h e a v y t a m p i n g on clay has any
favourable effect at all. T h u s , n o remaining v o l u m e decrease of the soil can b e obtained
unless water and/or gas are expelled from the voids in the soil. Since the permeability of
clay soils is extremely low and, consequently, the time required for the water and/or gas
to escape is very long, ' c o m p a c t i o n ' only results in shear deformations and not in a
v o l u m e decrease. (In unsaturated clay, gas bubbles will certainly b e c o m p r e s s e d under
the impact of the b l o w but will quickly retain their original size). Moreover, the
disturbance effects of h e a v y tamping will induce high excess p o r e pressure in the clay
which starts u p a consolidation process a c c o m p a n i e d by a long-term v o l u m e decrease
Fig. 298. The depth and the outlook of the craters formed by the impact of the pounder give a good
indication of the effect of compaction. A good sign of effectiveness is that there is no heave of soil around
the craters.
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(Hansbo, 1978). Overconsolidated clay m a y thus turn into underconsolidated clay that
after consolidation b e c o m e s normally consolidated with a lower void ratio than in its
original state. Consequently, h e a v y t a m p i n g should not b e used on clay with a high degree
of saturation and the water content of the clay should not b e a b o v e the plastic limit.
(iv) Application.
in which the p o u n d e r is dropped successively in the points settled in the design, usually
in a square pattern. T h e spacing between the points of compaction is chosen in order to
get as even a compaction result as possible. B y increasing the compaction energy at points
which will b e subjected to heavy loads (beneath columns and walls), the risk of
detrimental differential settlements can b e eliminated. C o m p a c t i o n is generally carried
out in several passes. After each pass the craters formed in the ground at each point of
compaction are filled with soil before the next pass of compaction is started. S o m e t i m e s
it m a y b e suitable to start compaction in a m o r e open pattern than according to the design
and then finalise compaction in agreement with the design. To begin with, when
compaction is started, the imprint of the p o u n d e r is generally considerable but will
successively b e c o m e reduced with the increasing n u m b e r of b l o w s . C o m p a c t i o n is
continued until the imprint of the impact is reduced to an acceptable upper limit or until
the compaction results m e e t the requirements for the subsoil based on in situ control
m e t h o d s . A h e a v e of soil around the imprints of the p o u n d e r is an indication that the
kinetic energy of the blow is partly or, if the worst c o m e s to the worst, completely
consumed by shear deformations and, therefore, thatthecompactionresultisunsatisfactory.
T h e appearence of the imprints is a good indication of the possibilities of achieving a good
compaction result (Fig. 298).
W h e n the compaction requirements at depth are fulfilled, heavy tamping is usually
terminated by a so-called ironing pass in which the pounder is dropped from about 1-2
m height in a very dense pattern to compact the superficial part of the soil (Fig. 299).
T h e densification of the subsoil in the course of heavy tamping is often accompanied
by a sudden increase in pore pressure which in the case of fine-grained soil, such as silt
and silty sand, m a y cause liquefaction and create vertical flow channels. T h e flow
channels in combination with vertical fissures, caused by horizontal tension stresses,
accelerate the dissipation of excess pore pressure. In addition, the excess pressure will
h a v e a p e a k b e l o w the point of compaction and therefore three-dimensional outflow takes
place. This explains w h y heavy tamping can be used as an effective m e a n s of compaction
also in the case of saturated silt.
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Fig. 299. Ironing of the surface of compacted silt and sand fill at the Vnern terminal (Hansbo, 1974
and 1978)
3.2
Vibrocompaction
of the soil particles into a denser state by m e a n s of induced vibrations in the horizontal
or the verticafdirection.
(ii) Equipment.
and the vibro-wing or vibro-rod method. Vibroflotation is the oldest m e t h o d and has been
used since the 1930s while the vibro-rod has been developed fairly recently in Japan and
the vibro-wing even m o r e recently in Sweden.
T h e vibrating unit used for vibroflotation
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and
vibro-
wing m e t h o d s . In this case the vibrations take place in the vertical direction and not in the
horizontal direction as w a s the case with vibroflotation. T h e vibro-wing is a further
d e v e l o p m e n t of the vibro-rod w h e r e the proturberances attached to the vibro-rod are
e x c h a n g e d for steel fins with a length of 0.8 m at 0.5 m spacing (Fig.301). T h e vibro-rod
and the vibro-wing are operated by conventional piling rigs with a h e a v y vibratory
h a m m e r w h o s e frequency is in the range of 5 - 2 0 H z for the Franki tristar p r o b e and
typically about 2 0 H z for the vibro-wing.
T h e Franki tristar p r o b e (Fig. 302) is another vibro-compaction tool. It has three long
steel plates, 5 0 0 m m in width and 2 0 m m in thickness, attached to a 1 5 - 2 0 m long steel
rod at 120 angles to eachother. Additional 300 m m long steel ribs, 5 0 m m in width and
10 m m in thickness, are fixed on top of the steel plates at 2 m intervals in order to i m p r o v e
the compaction efficiency.
A new technique has recently been developed according to w h i c h c o m p a c t i o n is
carried out at the resonant frequency of the subsoil (Massarsch, 1991 ). A d y n a m i c probe
416
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improvement
Fig. 301. Compaction of sand fill by means of the Vibro-wing. By courtesy of NCC.
Fig. 302. Compaction of sand by means of the Franki tristar probe. By courtesy of Mller Geosystem.
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Fig. 303. Upper and lower boundaries of grain size distribution curves for soils suitable for compaction
by means of vibration and blasting.
() Application.
distribution curve falls within the limits given in Fig. 3 0 3 . In the case of vibro-wing or
vibro-rod, the soil should preferably b e water saturated. This is not required in the case
of vibroflotation w h e r e water jetting can b e used. Vibrocompaction is generally carried
out in an equilateral triangular pattern with a spacing b e t w e e n the points of compaction
varying from about 5 m in fine sand to about 1.5 m in coarse sand.
T h e u p w a r d flow of flush water in vibroflotation carries fine material in the soil u p to
the ground surface w h e r e it settles. This material is generally taken a w a y and replaced
by coarse material w h i c h sinks d o w n into the soil around the vibrator and forms well
c o m p a c t e d c o l u m n s with high bearing capacity. A similar principle is used in so-called
vibro-replacement w h e r e cohesive soil around the vibrating unit is r e m o v e d by m e a n s of
jetting and replaced b y a c o l u m n of coarse material c o m p a c t e d b y vibration.
3.3
Blasting
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Fig. 304. Liquefaction phenomenon taking place after blasting in loose sand. By courtesy of L. Kok.
G=K(Owy
(500)
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improvement
correspondingly from 1.05 to 2.5. At a gas content of 1%, ~ 250). The influence
of gas content on and shows that even a small amount of gas will strongly
reduce the shock wave stress level, and consequently reduce the effectiveness of
blasting.
The size C of the explosives (in kg T N T ) can be taken as (Ivanov, 1978):
3
C = 0.055 dch
(501)
1/3
(502)
1/3
1/3
t o l . 5 3 + 0.771n(C /#)
Liquefaction occurs when Aula ' 0 tends to unity. This gives a mean value for the radius
of liquefaction:
Rliq
with a m a x i m u m of 4.7 C
1 /3
= 2.8C
1 /3
(503)
1/3
and a minimum of 2 . 0 C .
ReJf=kC
(504)
1.5</cft = 3.9Ci'3
(505)
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soil itself. In this case, the charge is placed at a height h (in m ) a b o v e b o t t o m according
to the relation:
h =
0.35C
(506)
and the size of the charge (in kg) according to the relation:
C =
0.1//
46
(507)
305-306.
2/3
1/4; 1 Z 2 - 3 / 4
common)
I 2
I 6
C kg T N T
Fig. 305. Design diagram for blasting in water saturated sand. Legend: Reff=
of compaction; dch = depth of charge.
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10
20
30
40
50
C, kg TNT
Fig.306. Design diagram for blasting in water. Legend: Re^= effective radius; dcp - depth of compaction;
H = required depth of water; h = height of charge above sand bottom.
Fig.307. In modern blasting technique, special rigs for installation of explosives are utilised (left). The
explosives are enclosed in thermo-shrinked plastics for direct placement in the detonation holes. By
courtesy of L. Kok.
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piling
buildings often give rise to detrimental building settlements. T h e settlements are caused
by both soil vibrations induced by pile driving and soil displacements corresponding to
the pile v o l u m e inserted into the soil. This p h e n o m e n o n is taken advantage of in
compaction piling, w h e r e the main object of pile driving is to achieve compaction. T h e
piles also serve as reinforcement of the soil.
(ii) Application.
with concrete piles. W h e n timber piles are used the pile cut-off should b e satisfactorily
below the lowest groundwater table in order to avoid rotting. T h e pile spacing is generally
chosen at 1.5-2.5 m. T h e compaction effect, according to experience, extends to a
distance from the pile periphery of about 3 times the pile diameter ( s o m e w h a t m o r e
nearest to and b e l o w the pile tip). T h e increase of average density due to compaction
piling can b e estimated by m e a s u r e m e n t of the settlement of the soil surface and the pile
v o l u m e driven into the ground.
Soil
423
improvement
control
In the case of deep compaction, the control methods used to check the results of shallow
compaction are no longer applicable. A rough indication of the result is obtained by
studying the settlement of the soil surface during compaction, and this should always be
done since it gives us information of the relative compression of the subsoil due to
compaction. However, other characteristics of the soil after compaction will have to be
determined by more sophisticated methods. In this case, it is necessary to use in situ
methods, such as sounding (for example C P T or SPT), preferably in combination with
pressuremeter or dilatometer tests (Fig. 308). Sounding results give a good picture of the
homogeneity of the soil and may be used as an indirect measure of the deformation and
bearing characteristics of the compacted soil (p. 103). A safer measurement is obtained
by the use of the pressuremeter (Fig. 53) or, where this is possible, the dilatometer (Fig.
56).
In case neither sounding nor pressuremeter or dilatometer tests can be performed, for
instance in rock fill, the results obtained can be checked by measuring the surface wave
velocity. The surface wave velocity serves as a basis for determination of the average
dynamic shear modulus of a surface layer, with a thickness equal to the wave length of
the surface wave (cf. p. 87). By varying the wave length a good picture can be obtained
regarding the variation with depth of the shear modulus.
qc, MPa
Epr MPa
pl, MPa
Fig. 308. Influence of heavy tamping (12 t pounder dropped from 12 m height) on the characteristics
of silty sand at Vnerterminalen, Karlstad (Hansbo et al, 1974). Cone penetration test carried out before
compaction, after 2 passes and after 4 passes of heavy tamping. Pressuremeter values obtained before
and after termination of compaction.
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424
Ground pressure, MPa
0
improvement
0.20'
0.25
Fig. 309. Ground pressure vs. settlement curves obtained by measurements of the deceleration of a 40
2
t pounder (bottom area 4 m ) as compared to the results of corresponding plate loading test (bottom area
2
of plate 4.9 m ) . 2nd test (right) carried out after complementary compaction on the test spot (four blows
from 20 m height and one the last from 30 m height).
PRELOADING
4.1
The preloading
(i) Application.
technique
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Fig. 310. By taking advantage of occasional overloading the consolidation settlement for the design load
can be attained in relatively short time. The broken curve indicates the consolidation process under the
design load in case overloading is not utilised.
stresses induced by preloading on the one hand and final loading on the other is as equal
as possible. In order to avoid misinterpretation of the results of preloading and to m a k e
possible an effective check-up of the consolidation process (for instance the required time
of preloading) it is necessary to k n o w in detail the geological and geotechnical conditions
at the site. T h e planning and execution of preloading m u s t b e carried out by specialists
and a careful follow-up of the consolidation process is a must.
Preloading is mostly used to eliminate settlement in highly compressible cohesive
soils but is s o m e t i m e s used also in the case of coarse-grained soils.
In the former case, the time of preloading should be long e n o u g h to eliminate primary
consolidation settlement and, at least partially, secondary settlement for the final load
(Fig. 310). T h e real process of consolidation ought to be carefully monitored with both
settlement gauges and pore pressure meters and the results obtained should be checked
against theoretical prediction. If there is good agreement preloading can be terminated
w h e n the consolidation settlement predicted for the design load has been reached (or
preferably e x c e e d e d ) .
W h e n preloading is applied on coarse-grained soils, it is suggested that the loading
time b e chosen so as to reduce creep to a m a x i m u m of 0.02 m m / m i n .
4.2 Design for preloading
(i) Homogeneous
of low-permeable
soil conditions.
soils
soils is determined on the basis of the consolidation theory (p. 324). Generally, earth fill
is applied as a m e a n s of preloading but alternative m e t h o d s , such as occasional
groundwater lowering, h a v e also been utilised. T h e width of the preloaded area is usually
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improvement
large in comparison with the thickness of the compressible soil layer. Therefore, onedimensional consolidation can be assumed to take place (cf. E x a m p l e s 6 2 - 6 3 , p. 328).
It is important to check that the required degree of consolidation is reached throughout
the soil layer. T h u s , the consolidation time cannot be based on the average degree of
consolidation determined according to Terzaghi 's consolidation theory. T h e degree of
consolidation Uv reached at a certain depth can easily b e obtained by the numerical
m e t h o d presented on pp. 3 2 5 - 3 2 7 , or, in the case of equally large excess p o r e pressure
throughout a h o m o g e n e o u s soil deposit, by the relation:
1 - Uv = - s i n ( W - ) e x p ( - / V 7 v )
h
w h e r e - / 2 , 3/2, , (2m + 1)/2, ,
(508)
(ra is an integer),
Speeding
(i) General
up consolidation
aspects.
by vertical
drains
100
0.06
sand
gravel
60
Grain size d, mm
Fig. 311. The grain size distribution boundaries of ideal sand suitable for sand drains.
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427
Fig. 312. The sand wick represents a type of prefabricated small-diameter sand drain.
428
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T h e cardboard wick, invented by Kjellman in the late 30s, is the prototype of all the
m o d e r n band drains existing on the market. It is an interesting fact that sand drains were
first installed for stabilization purposes only 5 years earlier than the cardboard wicks
(Johnson, 1970). B a n d drains and sand drains are thus of very nearly the same age.
T h e overall features of the different proprietary drains are very nearly the same. A
successive development is taking place of both core and filter design, mainly for the
purpose of cost reduction. N o w a d a y s , all types of filter are m a d e of synthetic material.
S o m e filters are in the shape of a loose sleeve enclosing the core while others are fixed
to the core. Besides various types of filter, one and the s a m e trade m a r k m a y represent
a n u m b e r of different drain cores.
N o w a d a y s the various drain m a k e s differ only in respect of the channel system in the
Fig. 313. A considerable number of band drains are marketed. Shown here from top to bottom and from
left to right are: Cardboard wick (the prototype of all band drains), Geodrain, Castle Board, Alidrain,
PVC, Desol, Mebradrain and Colbond.
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improvement
drain core and the filter characteristics, whereas their geometrical properties are m o r e or
less the s a m e (see H a n s b o , 1992).
Sites in need of vertical drainage quite often h a v e a low bearing capacity. Consequently
there m a y b e a need of reinforcing the ground surface, for e x a m p l e by m e a n s of
geotextile, in order to enable the machinery for drain installation to enter the site without
danger.
T h e drainage layer should b e at least 0 . 3 - 0 . 5 m in thickness. T h e material in the drainage blanket shall h a v e good drainage properties; otherwise it is necessary to i m p r o v e
horizontal drainage by connecting the vertical drains with a horizontal drainage system
in order to avoid the build-up of back-pressure in the drains which w o u l d delay the
consolidation process.
(ii) Drain installation
techniques.
in accordance with the directions given in the design. M o s t l y the drains are installed in
an equilateral triangular pattern or in rows with a drain spacing equal to the spacing
between the drain rows. Irrespective of which of the two drain patterns is chosen, the
n u m b e r of drains required to achieve a certain rat of consolidation will b e the same. It is
important that the drains after installation do not deviate appreciably from the vertical
direction.
For sand drains, several installation procedures are used. T h e m o s t c o m m o n types of
sand drains with reference to the installation m e t h o d are driven closed-end mandrel sand
drains, continuous flight hollow stem augered sand drains, internally jetted sand drains,
rotary jet sand drains and dutch jet-bailer sand drains. A m o n g these, the driven closedend mandrel causes causes m o r e disturbance of the soil than the others. A c c o r d i n g to
L a d d (1976), the continuous flight hollow stem auger and the j e t bailer auger m e t h o d s
h a v e advantages over the internally jetted and the rotary j e t m e t h o d s of drain installation
in that they cause a lesser disturbance and do not require as close a supervision as the other
installation m e t h o d s .
Contractors w o r k i n g in the field of b a n d drain installations h a v e generally developed
their o w n type of e q u i p m e n t for drain installation. A c o m m o n feature is that the drains
are held inside a steel mandrel which protects the drain from being d a m a g e d during
installation. D u e to the small cross-sectional area of the b a n d d r a i n s 3 - 7 m m in
thickness and 100 m m in w i d t h t h e m a n d r e l for their installation can b e m a d e very
slender and, therefore, the disturbance effects during installation can b e greatly reduced.
Two principles of installation can b e distinguishedthe so-called static and d y n a m i c
installation m e t h o d s . In the first case, the m a n d r e l with the drain inside is p u s h e d into the
soil by static pressure, while in the second case it is driven into the soil b y m e a n s of a
gravity h a m m e r or a vibratory driver. Both static and d y n a m i c installation rigs of various
types exist, from conventional piling rigs to quite advanced rigs such as those shown in
Fig. 314. Floating rigs exist by w h i c h drains can b e installed from o p e n water, Fig. 3 1 5 .
430
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improvement
Fig. 314. Rigs for installation of band drains: static, to a depth of more than 30 m (left); dynamic to a
depth of more than 40 m.
Fig. 315. Rig for installation of band drains from open water.
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431
Fig. 316. Cutting the drain after installation (top). An anchor is the fixed to the drain. The anchor
prevents soil intruding into the mandrel during drain installation and keeps the drain in place when the
mandrel is withdrawn.
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improvement
can be cut just above the drainage blanket placed on the bottom. In deep water this
technique can save a considerable length of drains.
T h e present types of mandrel are often slender. Therefore, w h e n they are inserted into
the soil they m a y deviate considerably from the vertical, particularly in the case of deep
installations. Since it is essential for a well-functioning drain system that the prescribed
drain spacing is maintained throughout the drained soil layer such a deviation from the
vertical can h a v e a negative and unpredictable effect on the consolidation process at great
depth. Therefore, either the mandrel should be equipped with an inclinometer that gives
information about the horizontal position of the drain at various depths or the mandrel
should be stiff enough to ensure verticality of the drains.
4.4 Design
(i) Theoretical
of drain
installations
assumptions.
D = 2</7.
(509)
Fig. 317. Terms used in the analysis of vertical drains. D = diameter of soil cylinder dewatered by a drain,
d= drain diameter, ds = diameter of zone of smear, / = length of drain when closed at bottom (21 = length
of drain when open at bottom), qw = discharge capacity of drain.
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improvement
hz=
hAo
e x
P(-
8 7
)/)
Th - cht ID
- time factor,
(510)
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improvement
t = time of consolidation,
D = diameter of drained circular cylinder,
= 1() + (kf/ks)lns
- 0.75 + (2 -
z)kf/qW9
n=D/d,
s =ds
Id,
where kw = permeability in
the vertical direction of the drain and Aw = cross-sectional area of the drain).
Since band drains have a rectangular cross-section it is necessary to transform them
into circular-cylindrical drains with equivalent draining capacity. Kjellman (1948) stated
that "the draining effect of a drain depends to a great extent upon the circumference of
its cross-section, but very little upon its cross-sectional a r e a " and that "certain
considerations show that the cardboard wick is as effective as a circular drain with a 1-in.
radius". This statement was later confirmed by the aid of finite element analysis (Hansbo,
1979, 1981). Thus, band drains and circular drains can be assumed to be equally efficient
provided that their circumference is equal, in other words:
deq = 2(b+t)ln
(511)
presented above is based on the assumption that Darcy's law is valid. This may not always
be the case. We know from soil physics that the pore water in claywhich in reality
consists of water molecules and ionsis more or less strongly bound to the mineral
surface of the clay particles. Moreover, in the pore water there are most probably a great
number of microscopic mobile mineral particles and/or organic matter which are bound
only by sorption and hydrodynamic forces and which do not belong to the load-carrying
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improvement
Fig. 318. Comparison between approximate and rigorous solutions of consolidation of clay provided
with vertical drains. Drain characteristics: = 5; s = 1; HD = 16; kjkh = 1730. (Mesri, 1990).
clay skeleton. Since the binding forces increase in the direction of the mineral surfaces
it is but logical to a s s u m e that h y d r o d y n a m i c forces will h a v e an influence upon the
apparent porosity of the clay. T h e higher the h y d r o d y n a m i c forces the larger the flow
channels, but only u p to a certain limit. This indicates non-validity of D a r c y ' s law at small
hydraulic gradients.
An alternative solution to consolidation by vertical drains, b a s e d on non-validity of
Darcy's law w a s derived b y H a n s b o (1960). In this solution the flow of p o r e water is
assumed to follow the relation given by Eq. (17), p . 22, a s s u m i n g = 1.5:
=
(512)
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improvement
1 "o
]/l-Uh
1)
(513)
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improvement
(i ) Partially penetrating
rate of consolidation of the clay beneath the drain tips is slower than in the clay penetrated
by the drains. This will result in a hydraulic gradient between the undrained and the
drained parts of the clay w h i c h will delay the consolidation process to a certain height
above the drain tips but, on the other hand, speed u p the consolidation process to a certain
depth below the drain tips. As shown by Runesson et al. (1985) the delay in the consolidation
process within the z o n e penetrated by the drains is noticeable only to a very limited height
above the drain tips. F r o m a practical viewpoint, it can be neglected w h e n the height
above the drain tips e x c e e d around 2 0 % of the drain length.
(v) Multilayer
L1
L1
f \
L2
L2 \
1
Single-layered LI
Two-layered
Single-layered L2
Fig. 320. Consolidation of multi-layered anisotropic soil by vertical drains with well resistance.
Approximate solution according to Eq. (510).
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improvement
(vi) Zone of smear. T h e method applied for the installation of the drains is of p a r a m o u n t
importance for the extent and characteristics of the zone of smear. Concerning sand
drains, installation by careful jetting or by the Dutch jet bailer auger m e t h o d seems to
cause a m i n i m u m of disturbance, while installation by m e a n s of a closed-end mandrel,
particularly if driven into the soil by drop h a m m e r s or vibratory h a m m e r s , seems to cause
a m a x i m u m of disturbance. In the former case, smear can be ignored and h e n c e ds - d.
In the latter case, the effect of installation is very similar to that of conventional pile
driving. Investigations including both drains installed by m e a n s of a closed-end mandrel
and driven piles (Holtz & H o l m , 1973; Akagi, 1976) indicate that the zone of smear can
be assumed to have a diameter of ds ~ 2d. There are undoubtedly cases where the remoulding
effect m a y reach further out in the surrounding of the drains which has been indicated by
an overall reduction in shear strength. However, as a rule of t h u m b , for driven closed-end
mandrel sand drains w e can put s = 2.
T h e extent of the zone of disturbance caused by installation of band drains will be
dependent of the type of mandrel used for installation and the size of the anchor fixed to
the tip of the drain during installation. As a rule of t h u m b , ds can be chosen on the same
premises as in the case of driven sand drains, i.e. the cross-sectional area of the mandrel
(also taking into account the area of the folded anchor) is replaced by an equally large
circular area w h o s e diameter is then doubled. T h e s value is the ratio of this ds value to
the equivalent diameter of the drain.
F r o m a practical viewpoint, the question regarding which diameter to choose in the
design is also coupled with the question regarding which permeability to choose for the
zone of smear. A s s u m i n g a value of s equal to 2, the permeability ratio k}/ks can be assumed
equal to k}/kv.
On this assumption, a design diagram of the type shown in Fig. 321 can
be utilised.
(vii) Well resistance.
depends mainly upon its long-term discharge capacity. Too low a discharge capacity can
seriously delay the consolidation process, especially in a case where the drains are
installed to great depth. It is specially important that the discharge capacity is not too low
in the beginning of the consolidation process while it is of lesser importance at a later
stage of the consolidation process.
To ascertain a high enough discharge capacity of sand drains, the grain size distribution
curve of the sand should fall within the limits given in Fig. 3 1 1 .
T h e discharge capacity of a certain band drain is generally given by the proprietor of
the drain. However, before a certain drain m a k e is selected for a j o b , evidence of the drain
fulfilling the discharge capacity requirements placed upon the drain in the design
assumptions ought to be given. Generally, appropriate laboratory testing can give sufficient evidence of the discharge capacity to be expected under field conditions (Hansbo,
1983), but in case of a drain never before m a d e use of in practice, full-scale field tests are
So/7
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improvement
f=-t
( y e a r s ) / l n ( l - Uh)
fch
D^S
Fig. 321. Graphs for design of band drain installations. Negligible well resistance. Zone of smear ds - Id.
Permeability ratio kh lks = kh lkv. From the design value of average degree of consolidation Uh we find
the/value. Then t h e / c A value gives the drain spacing ( the D value).
E f f e c t i v e lateral p r e s s u r e , kPa
Fig. 322. Results of discharge capacity tests for different band drains carried out on a laboratory scale.
Drains enclosed in soil. In the ENEL tests (Jamiolkowski et al., 1983) the drains were tested in full scale,
in the CTH tests (Hansbo, 1983b) and in the KU tests (Kamon, 1984) with reduced width (40 and 30
mm, respectively). Legend: A = Alidrain, BC = Bando Chemical, CB = Castle Board, C = Colbond, G
= Geodrain and M = Mebradrain. (p) indicates filter sleeve of paper.
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improvement
r e c o m m e n d a b l e . Full-scale field tests also serve the purpose of showing that the drains
are strong e n o u g h to resist the strains subjected to t h e m during installation. Moreover,
compression of the soil in the course of consolidation settlement entails folding of the
drain which m a y bring about clogging of the channel system. T h e latter effect is difficult
to discern in a laboratory test.
As a general rule, investigations on a laboratory scale of the discharge capacity of a
drain ought to be carried out in a way so as to simulate field conditions as closely as
possible. Such tests, performed in several laboratories (see, for e x a m p l e , H a n s b o , 1983),
h a v e revealed a great influence on the discharge capacity of the lateral consolidation
pressure (Fig. 322). T h e reason is that the filter sleeve is squeezed into the channel system
which entails a reduction of the channel area, or that the channels themselves are
squeezed together.
T h e influence of well resistance for drains with a discharge capacity above 100 mVyear
can b e neglected unless the drains are very long (/ > 20 m ) .
(viii) Filter sleeve. T h e filter permeability required can be estimated by considering the
filter as a zone of smear with a diameter ds equal lod+2t
of the filter. By studying the influence of the filter thickness on the course of consolidation
w e find that the permeability need not b e higher than the permeability of the soil in which
the drains are installed. It is the discharge capacity of the channel system that is decisive,
not too low a permeability of the filter enclosing the channel system. On the contrary, too
Fig. 323. Influence on discharge capacity of filter deterioration (Koda et al., 1986). Tests on Geodrains
with filter sleeves of synthetic material (broken lines) and paper (full lines) which were pulled out of
the soil after different lengths of time after installation (number of days given in figure).
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improvement
high a permeability m a y entail a risk of clogging of the drains. Fine soil particles m a y
penetrate through the filter m e s h and finally lead to clogging (siltation).
With time a certain deterioration of the filter will h a v e to b e expected due to
bacteriological activity or fungi attacks. A n interesting investigation of the influence of
filter deterioration on the discharge capacity of Geodrains w a s publicized by K o d a et al.
(1986). As can b e seen from Fig. 3 2 3 , both paper and synthetic material are subjected to
deterioration. A previous investigation on paper filter after two years in organic soil at
Porto Tolle in Italy, carried out by the R o d i o Company, s h o w e d serious deterioration
s y m p t o m s (Hansbo, 1986). Fortunately, w h e n deterioration is beginning to affect the
discharge capacity of the drains the consolidation process is generally near its termination
and then a decrease in discharge capacity has a very small influence on the consolidation
process.
4.5
Practical
(i) Preloading.
aspects
of vertical
drainage
Soil
442
improvement
Fig. 324. Correction of the consolidation curvewith regard to gradual application of the load. Broken
curve indicates the consolidation process caused by instantaneous loading. Full line curve represents
consolidation process by gradual application of the load.
and limitations.
design are related to the drain characteristics and the m e t h o d of installation (disturbance
effects) on the one hand and the consolidation characteristics of the soil on the other. T h e
most important drain characteristic is the discharge capacity u n d e r long-term conditions.
T h e discharge capacity of the drain should b e high e n o u g h and the drain m u s t not b e
clogged d u e to the channel system being silted u p in the course of consolidation which
places certain requirements on the filtration of water squeezed into the channels of the
core. Finally, the strength of the drain must b e sufficient so that the drains do not b r e a k
during installation.
M a y b e even m o r e important for the prediction of the consolidation process are the soil
parameters governing Eq. (510), in particular the coefficient of consolidation in horizontal
pore water flow b u t also the hydraulic conductivity in the z o n e of smear. A correct choice
of the design parameters n o doubt represents the m o s t important part of the design of a
vertical drain installation, considerably m o r e important than the question regarding
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of consolidation.
water flow is usually larger than the coefficient cv in vertical p o r e water flow determined
by conventional o e d o m e t e r tests. As the M value is equal in both cases, then obviously
the ratio Cj/cv = k/kv
horizontal layers in the soil. K n o w i n g the M value from the results of o e d o m e t e r tests,
the ch value can be determined by measuring the permeability in the horizontal direction
of the soil sample. A n even better w a y of determining the ch value is to use the piezocone.
As is w e l l - k n o w n , during penetration of clay by the p i e z o c o n e a high excess p o r e water
pressure is induced. A fairly reliable value of the coefficient of consolidation can b e
obtained by interrupting the penetration and studying the rate of dissipation of excess
pore water pressure.
In practice, normally only the value of cv is given and therefore ch has to be estimated
by simple j u d g e m e n t based on studies of the soil structure, in particular the frequency and
thickness of pervious layers. If the spacing between such layers is small the clay between
will consolidate mainly by outflow of water in the vertical direction to the pervious
layers. Consequently, in such a case the ch value is governed by the properties of the pervious layers and not by those of the m o r e or less impervious clay.
For seemingly h o m o g e n e o u s soil it can generally be a s s u m e d that c}/cv
(iv) Checking
= 2.
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improvement
Solution: First calculate the consolidation process on the assumption that ch = 0.5 m /year throughout
2
the clay layer, and then on the assumption that ch - 0.2 m /year throughout the clay layer.
The equivalent drain diameter is:
d = 2 ( 1 0 0 + 4)/ = 0.066 m
The diameter of the zone of disturbance ds 0.15 m
The diameter of the cylinder dewatered by a drain is:
D=2/hVdK=
1.47 m
We have in the first case:
-3
Pi = ln( 1.47/0.15) + 21(0.15/0.066) - 0 . 7 5 +(20 - )0.03/20 = 3.174 + 4 . 7 U ( 2 0 - ) 1 0
and in the second case:
2 = 3.174 +(20 - )0.05/20 = 3.174 + 7.85z(20 - )10-
Eq. (512) and Fig. 318 yield the following values of Uh with depth:
z, m:
U%:
12.0 12.0
z, m:
15
Uh%:
37.5 37.4 37.3 37.4 37.5 37.8 38.2 38.8 39.5 40.4 41.4 42.9 44.1
1
9
2
10
3
11
4
12
5
13
6
14
8
16
17
18
19
At 8 m depth Uh can be taken as the average ofg 12.0 and 37.5, i.e. Uh = 24.8%
20
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improvement
Example 85: A 30 m thick clay layer, underlain by sand, shall be consolidated for a design load of 100
2
k N / m in one year. For the purpose band drains of good quality, 100 mm in width and 4 mm in thickness,
shall be installed with the drain tips penetrating into the sand layer blow the clay. For preloading, gravel
3
fill is available with a density of 1.8 t/m . The cost of drain installation, including the drain material,
has been offered at a price of l/m drain and for placement and removal of temporary fill at a price of
3
8 /m of fill material. The drainage layer, 0.5 m in thickness, to be placed on top of the clay surface
2
is offered at 5/m . Determine the optimal drain spacing for the project, if the clay has the following
2
characteristics: coefficient of consolidation ch = 1.5 m /year; permeability kh = 0.05 m/year. The discharge
3
capacity of the drains can be estimated at qw =500 m /year. The installation is assumed to cause a zone
of smear with a diameter of ds - 22d and a permeability ks = k/3.
Solution: The equivalent drain diameter is 0.208/ = 0.066m. Assuming that the drains are installed in
an equilateral triangular pattern, x m in spacing, the diameter of the dewatered cylinder becomes equal
to D = 1.05JC. Assuming further that the settlement during consolidation is directly proportional to the
size of the load and the average degree of consolidation U, we have:
J_ <i + i
A<
u
2
where spq = primary consolidation settlement under the load q = 100 kN/m
Aq = temporary overload
Obviously, for the condition U = 100% to be fulfilled it is required that Umin = q/(q + Aq). Since the
lowest consolidation degree is obtained in the middle of the clay layer we have to choose in our analysis
of the effect of well resistance = / = 15 m as given by Eq . (510).
The analysis gives the following result:
1.2
Drain spacing, m
1.1
uh%
2
Temporary surcharge Aq, kNm
Temporary fill, m
2
Drain costs, / m
2
Cost of fill, / m
2
Cost of drainage layer, / m
2
Total cost, / m
The most cost-effective drain spacing
86
91
10
16
0.6 0.9
29.6 24.8
4.8 7.2
5.0 5.0
39.4 37.0
is found to
1.3
81
24
1.3
21.2
10.4
5.0
36.6
be 1.3
1.4
1.5
75
33
1.8
18.3
14.4
5.0
37.7
m.
69
45
2.5
15.9
20.0
5.0
40.9
Example 86: Determine the average degree of consolidation, 9 months after load application, by Eq.
(510), on the one hand, and by Eq. (512), on the other, for a thick clay deposit provided with vertical
band drains, 100 mm by 4 mm, with a spacing of 1.0 m in equilateral triangular pattern. The drains have
a high discharge capacity and, therefore, well resistance can be ignored. The coefficient of consolidation,
2
as determined by oedometer tests, is cv = 0.5 m /year. A fill with large extension has been placed on the
2
ground representing a load q = 30 kN/m .
Solution: In the analysis, based on Eq. (510), the value of ch can be assumed equal to 2 c v. The diameter
of the zone of disturbance ds can be assumed equal to 22d and the ratio khlks = 2. In the analysis, based
on Eq. (513), can beassumed equal to 0.75c v.
The diameter D=2^JO.5/3/K
= 1.05
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improvement
(2) The value of a, for Did = 1.05/0.066 =16, according to Fig. 317, is 0.24. Eq. (513) yields:
, = - + A
JILy*
= _( 1 +
Case
'
7 5
'
0 5
= ,.85 (85%)
f ^ ^ y
1 2 - 0 . 2 4 - 1.05 V 1.05-9.81
aD\DyJ
4.6
histories
A large n u m b e r of case histories h a v e been publicised [see, for e x a m p l e , the special issue
of Gotechnique of M a r c h 1981 on vertical drains and the publications by Lo ( 1991 ) and
H a n s b o (1992)]. Two case histories of special interest will be presented here: the test field
at Sk-Edeby, Sweden, and the test field at Tianjin harbour, China.
(i) Test field at Sk-Edeby,
of Stockholm, was established in 1957 and is one of the oldest monitored test fields for
the study of vertical sand drain and band drain behaviour. It contains five circular test
areas, three of which provided with sand drains , 0.18 m in diameter, with drain spacings
varying from 0.9 to 2.2 m, and one with band drains, type Geodrain, with drain spacing
equal to 0.9 m. In order to distinguish the effect of vertical drains from the effect of onedimensional consolidation, one of the areas was left undrained. T h e test areas were
2
loaded with 1.5 m gravel, corresponding to a load of 27 k N / m except for test area III
2
which was loaded up to 2.2 m gravel (36 k N / m ) T h e geology at the site and the geotechnical
properties of the different test areas w e r e described in detail by H a n s b o (1960). T h e
results obtained h a v e been presented in several publications (Holtz & B r o m s , 1972;
Larsson, 1986; H a n s b o , 1987).
In short, for the clay b e t w e e n 2.5 and 7.5 m depth, selected for study in order to avoid
the influence of the dry crust and of irregularities in soil characteristics near firm bottom
(at 8 - 1 5 m depth), the coefficient of consolidation c v determined by oedometer tests
C o n s o l i d a t i o n time
months
years
Fig. 325. Consolidation process for the 5 m thick clay layer between 2.5 and 7.5 m depth.
So/7
447
improvement
2
0.43 to 0.56. T h e coefficient of consolidation in horizontal pore water flow ch, w h i c h was
determined in an o e d o m e t e r test w h e r e drainage was allowed only through a central drain,
2
diameter, can be estimated on the basis of sand permeability at about 100 m /year. Assuming
a zone of smear ds - Id = 0.36 m and a ratio kh lks = 4 (equal to ch lcv
) an acceptable
a v
Drain spacing
Time, years
flot. %
uv,%
uh.%
ch m /year
2
A, m /year
1/6
34
1
33
0.85
0.07
1
80
10
78
0.53
0.06
2
94
19
92
0.45
0.07
1/2
1.5 m
1 4
2.2 m
1
36
60
95
5
10
34
33
56
92
0.89 0.91 0.70
0.14 0.16 0.20
33
10
25
0.75
0.16
9
69
34
53
0.50
0.12
93
51
85
0.55
0.19
According to Eq. (513) the consolidation process depends on the m a g n i t u d e of the load
placed on the ground. This is not the case with Eq. (510). In test area the load was 36
2
k N / m . F r o m the test results in area III w e find the best agreement b y choosing the values
given in Table 39.
TABLE 39.
2
Best fit of ch and values in Area III (load 36 kN/m ).
Drain spacing
Time, years
ftot.*
Uy.%
uh,%
ch m /year
2
A, m /year
1.5 m
1/6
20
1/2
42
5
39
1.1
1
65
10
61
1.0
2
87
19
84
1.0
1
19
1.4
0.18
0.16
0.17
0.21
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improvement
J
-1 ti-
I
li
i
Loading time, years
\
\
H ~~
0 6 tn
Fig. 326. Observed compression of clay layer situated between depths of 2.5 and 7.5 m in Test area V
at Sk-Edeby. Band drains, type Geodrain, with 0.9 m spacing. Broken line represents theoretical
2
correlation based on Eq. (515) with = 0.15 m / year.
T h e values are m o r e consistent than the ch values throughout the consolidation process.
T h e values w h i c h include the influence of disturbance are very close to the cv values
2
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449
improvement
Fig. 327. Observed settlement vs. time relationship in the Tianjin pilot test.
m /year.
T h e settlement vs. time of consolidation obtained in the test areas with 1.3 m drain
spacing, using a surcharge on the one hand and v a c u u m on the other, is shown in Fig. 327.
As in the previous case record a good agreement is obtained between measured
2
So/7
450
5.
GRANULAR COLUMNS
5.1
Aim and
improvement
application
Large-diameter sand drains are often considered to act as reinforcing and stiffening
elements in the soil which contribute to reducing settlement. However, in the case of
ordinary sand drains, which are usually in a loose state, this effect can b e disregarded. T h e
purpose of granular columns is both to increase stability and to r e d u c e settlement. They
also serve as drains and, consequently, contribute to speeding u p excess p o r e pressure
dissipation. D u e to dilation characteristics and drainage capacity, granular columns are
also used as a m e a n s to reduce liquefaction potential.
T h e most c o m m o n types of granular columns are so-called stone c o l u m n s and sand
compaction piles. A m o n g the installation methods utilised, the cased borehole method,
the J a p a n e s e vibro-composer method and vibro-replacement m e t h o d are the most
c o m m o n ones.
According to the cased borehole method, gravel and/or sand is filled into the casings
and compacted during the withdrawal of the casing by m e a n s of a 1 . 5 - 2 1 gravity h a m m e r
dropped from a height of 1 to 1.5 m. T h e granular material can also b e c o m p a c t e d by redriving the casing several times during withdrawal.
By using the vibro-composer method, large-diameter sand c o m p a c t i o n piles, up to 2
m in diameter, can b e produced. T h e sand c o m p a c t i o n piles are constructed by driving
the casing to the required depth by m e a n s of a heavy vibratory h a m m e r , located at the top
of the casing. During the driving operation, the casing is closed at the b o t t o m by means
of a sand plug. T h e casing is filled with a specified v o l u m e of sand and the casing is then
repeatedly extracted and partially redriven by m e a n s of the vibratory hammer, starting
from the bottom, in order to c o m p a c t the sand. T h e operation is repeated until a well
c o m p a c t e d sand column is obtained.
T h e vibro-replacement method is mainly applied in silt soils. T h e granular columns are
created in a w a y similar to vibroflotation (see p . 415). T h e process can b e either wet or
dry. In the wet process, a hole is formed in the ground by jetting the vibroflot cylinder
down the the required depth with water. W h e n the cylinder is w i t h d r a w n a hole is formed
in the ground with a larger diameter than that of the cylinder. T h e uncased hole is filled
with gravel and densified by vibroflotation.
In the dry process no water jetting is used when the hole is formed by the cylinder. T h e
dry process can only b e used where the uncased borehole can stand open without the
support of water.
5.2
Design
aspects
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improvement
columns
and surrounding
the piles are generally installed in a triangular or square pattern. T h e contributory area of
the soil surrounding the granular pile can b e approximated to a circular cylinder with a
cross-sectional area equal to the area A enclosed b y four neighbouring granular piles (see
p. 4 3 2 ) , i.e. the diameter of the cylinder D - lyjA
In .
(514)
^ t f c o l ^ + ^ s o i r O ~ s)
Introducing m = McoXIM^OX
w e find:
a ml =
tfS0il
(ii) Bearing
capacity.
m
q
1 + (m -
\)as
(516)
1 + (m -
\)as
internal angle of friction of the column material and the shear strength of the surrounding
clay. In principle, the failure condition can b e c o m p a r e d with that occurring in the triaxial
test w h e r e the radial, confining pressure ar is governed b y the horizontal overburden
pressure
and the undrained shear strength cu of the clay. A s the top part of the ground
is generally weathered (dry crust formations) with high shear strength, failure will
normally take p l a c e at depth in the form of bulging within the z o n e w h e r e the shear
strength of the soil is at its m i n i m u m .
In the case of cylindrical expansion the radial stress at failure orj, according to the
theory of plasticity, can b e expressed by the relation:
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452
+ ln
improvement
(518)
2cu(\+vc)
between
+ 5cu and
6cu. In m o s t cases in practice it can be assumed that the governing, effective confining
pressure at failure is equal to "' r /= c r ^ ) + 5cu. Introducing this value into the MohrC o u l o m b failure criterion (pp. 278-279), we find the ultimate, effective column stress <fjQ0\
from the relation:
1 + sin0'
1 -sin0
(519)
Soil l a y e r s
Consistency limits, %
U n i t weight]
3
Natural water content, % [kN/m
20
B r o w n to r e d d i s h
brown weathered
clay
D a r k g r e y soft c l a y
often w i t h d e c o m posed w o o d and
sandy seams
40
60
80
100
15
16
Undrained shear
s t r e n g t h cu,
kPa
(field v a n e test)
10
20
30
40
Soil
453
improvement
38.2
2
36.9
3
35.6
4
37.7
5
43.3
32.7
30.2
21.3
30.7
35.6
l
29.2
2
27.6
3
26.1
4
28.6
5
37.0
The measured mean value of all the tested columns is 30.1 kN while the calculated mean value is 29.7 kN.
improvement.
increasing slope stability conditions. In this case failure is assumed to take place along
a curved (normally circular-cylindrical) failure surface in which the shear strength of the
columns and the surrounding soil is fully mobilised. With regard to the fact that the stressstrain behaviour of the columns and the natural soil up to failure may be completely
different from eachother this assumption is very questionable.
According to a method of analysis of a granular column improved slope, suggested by
Chambosse & Dobson (1982), named the lumped parameter method, the driving and
stabilising forces are first determined for the natural, unimproved soil (see Section on
slope stability). The additional effects of the granular columns on driving and stabilising
forces are then calculated and added to the respective values of unimproved soil. T h e
safety factor is the sum of the stabilising forces divided by the driving ones.
(iii) Settlement.
A f i e o i [l l + ( m - l ) f l j]
(520)
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454
improvement
where = the overall vertical stress increase at depth due to the load q,
h = depth of the clay layer (length of the columns),
M s o i lz = the compression modulus of the soil at depth z.
Alternatively, the settlement is calculated as:
sp = l
^ d z
J
"
0 M z c ol
(521)
=
J
0 Mzwl[l
{m-l)as]
If the extension of the loaded area is large in comparison with the thickness of the clay
layer (spread load), can be assumed equal to q.
In the case of partially penetrating columns, the compression of the unstabilised soil
layer underneath the tips of the columns will, of course, have to be added to the settlement
given by the above relations.
The course of settlement can be determined in a similar way as was done for vertical
drains. However, in this case the diameter of the columns is generally too large for using
the simplified solution given for small-diameter vertical drains. Assuming that the
installation of the granular columns causes a zone of disturbance with reduced permeability
ks and a diameter ds (as in the case of vertical drain installation) a more correct solution
of the consolidation process will take the form (cf. Eq. 510):
Uh = l - e x p ( - )
where Th = ch
(522)
t/D ,
kh
= - T - r O n - + f \ns-0.75
-l
s
ks
1 1
+ - - )
An
1
- - J ,
kh
+ (21 - ) (I
qw
= Did,
s=
ds/d.
2
Example 87: Determine the primary consolidation settlement due to a spread fill of 25 kN/m on a 10
m thick, granular column-improved clay layer if the compression modulus of the clay M c l ay = 300 kPa.
The granular columns, which are 0.5 m in diameter and have a compression modulus M c ol = 2500 kPa,
are placed in an equilateral triangular pattern with a spacing of 1.3 m. Determine also the time required
to reach half the final settlement if the coefficient of consolidation of the clay in horizontal pore water
2
flow is 0.5 m /year. The well resistance of the granular columns and the zone of smear can be neglected.
2
1 / 2
/ JK
- 1.365
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455
improvement
The time required to reach half this settlement (U = 50%) can be estimated by Eq. (522). For a value
of = Did = 1.365/0.5 =2.73 we have:
= [2.73 2/(2.73 2 - l)][ln(2.73) - 0.75 + 2.73~ 2 - (42.73 4)- 1]= 0.443
whence:
Th = - ln(0.5)0.443/8 = 0.0384
which yields:
t = 0.0384 1.365 2/0.5 = 0.143 years - 50 days
6.
STABILISATION BY A D M I X T U R E S
6.1
Superficial
stabilisation
Admixtures are often used for the purpose of increasing the bearing capacity of the
ground surface or of the bottom of excavations in soft soil conditions. The most common
additives utilised in clay soils are lime and cement which bring about a strength increase
by chemical reactions. Other common additives are bitumen and, for preventing leakage,
bentonite.
The stabilisation caused by cement and lime is achieved by ion exchange. The effect
of ion exchange can be easily observed, for example by mixing common salt (sodium
chloride) with a remoulded quick clay. The consistency of the clay will convert from a
liquid state into a plastic state. Thus the salt has drastically changed the shear strength as
well as the sensitivity of the clay. On the other hand, mixing sodium pyrophosphate with
clay of normal sensitivity will convert the clay from a plastic or viscous state into a liquid
state (the clay.has turned into a quick clay).
Fig. 328. Water saturated soil with a high silt content easily turns into a liquid state. The shear strength
of the soil becomes strongly reduced.
Soil
456
improvement
T h e influence of ion e x c h a n g e on the undrained shear strength of clay varies with the
type of ions involved. T h e best effect is reached by a l u m i n i u m ions, while the effect of,
for e x a m p l e , calcium ions is quite limited. T h e i m p r o v e m e n t obtained by the use of
admixtures is very m u c h dependent on the efficiency of the mixing process.
Briefly, lime is utilised for stabilisation of clay and silt (Figs. 3 2 8 - 3 2 9 ) . T h e favourable
effect of adding lime is caused by ion e x c h a n g e in the first place and, in the second place,
by cementation. T h e shear strength is improved and the plasticity index of the soil and
the o p t i m u m water content are increased. T h e best effect is obtained by m e a n s of
quicklime.
Cement is used as a m e a n s of stabilising practically any type of inorganic soils. T h e best
result is obtained in well graded soils with less than 5 0 % fines and a plasticity index IP
< 2 0 % . Sand and cement should be mixed dry of o p t i m u m , clay and cement s o m e w h a t
wet of optimum. T h e soil-cement mixture should be kept moist during curing. T h e
increase in shear strength is speeded up by high temperature. Certain additives, such as
lime, calcium chloride and, even better, sodium sulphate i m p r o v e the result and are costeffective. T h e a m o u n t of cement needed is about 5 - 1 0 % (by weight) in gravel, 7 - 1 2 %
in sand, 1 2 - 1 5 % in silt and 1 2 - 2 0 % in clay.
T h e resulting shear strength varies from about 350 kPa in silty clay to about 10 M P a
in well grade gravel.
Bitumen
(for e x a m p l e asphalt and tar) is used mainly for stabilisation of sand and
gravel. Asphalt, which is most c o m m o n l y used, binds the particles together and prevents
water absorption (important in the case of swelling material). T h e u s e of proper additives,
such as phosphorus pentoxide, will increase the shear strength (an admixture of 2 %
phosphorus pentoxide + 5 % asphalt was found to p r o d u c e a threefold increase of the shear
strength of silt).
Fig. 329. The superficial soil shown in Fig. 328 is treated with a lime admixture. Due to ion exchange,
and to the capacity of binding surplus water exerted by lime, the ground surface becomes firm and
obtaines a high bearing capacity.
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6.2
457
improvement
Lime and/or cement
columns
A stabilising effect on soft clay and silt to great depth cah b e attained by forming soil
columns m i x e d with lime or c e m e n t or armixture of lime and cement. G y p s u m is also used
in certain conditions. In the following, the c o l u m n s w i l l b e referred to as lime c o l u m n s ,
irrespective of the admixture utilised.
L i m e c o l u m n s are utilised for stabilisation purposes (for instance of trench walls) and
for the p u r p o s e of reducing and speeding u p consolidation settlement. T h e pattern of
installation will h a v e to b e chosen with regard to the a i m of stabilisation.
(i) Installation
techniques.
Fig. 330. Equipment for installation of lime columns provided with two containers for storage of lime
and/or cement.
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improvement
the container, the feed pressure, impeller delivery, type of mixing tool, a m o u n t of
admixture to be used, etc. are settled. For each column that has been installed a check tape
is obtained from a data printer. This m a k e s it possible to j u d g e whether or not a
readjustment of the installation technique is required and if extra c o l u m n s will have to
be installed.
In Japan a so-called deep-mixing method has been developed, comprising up to eight
mixing units, by which lime or cement columns can b e created to a depth of m a x i m u m
4 0 m. T h e mixing units overlap and in this w a y a cross-sectional area of the columns can
be obtained of u p to about 9.5 m 2 , depending upon the n u m b e r and diameter of mixing
units. Under-water installations can also be m a d e .
(ii) Choice of admixture.
settled for the project. T h e choice is generally based on laboratory investigations in which
various amounts of lime, cement or g y p s u m are m i x e d with the soil to be stabilised.
However, it has to be recognised that uncritical j u d g m e n t of laboratory results generally
entail an overestimation of the results to be expected from field installations. A main
difficulty met with in field conditions is to check the in situ characteristics of the columns
installed, such as shear strength, homogeneity, deformation m o d u l u s and permeability.
At present the control of the results achieved is generally performed with the aid of a
special sounding tool w h o s e penetration resistance is m e a s u r e d w h e n pressed down in
the centre of the column. However, by this m e t h o d only a general view of the shear
strength and of the h o m o g e n e i t y of the columns can be obtanied with reasonable degree
of accuracy. Deformation moduli and permeability characteristics are generally appreciated
only on the basis of laboratory tests.
T h e admixture to be used depends on the soil type. Q u i c k l i m e has a better effect than
slaked lime but is problematic to handle. A mixture of cement and lime results in a higher
Fig. 331. Mixing tool for the production of lime or cement columns
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459
improvement
strength and lower compressibility than is achieved with lime alone. G y p s u m improves
the growth of strength in organic soils. In silt with a clay content below 2 0 % , it is
r e c o m m e n d e d to add fly-ash. T h e o p t i m u m lime or lime/cement content increases with
increasing plasticity of the soil. Normally, a content corresponding to 6-8 wt.% of the soil
is sufficient. T h e strength increase of the stabilised soil takes p l a c e very rapidly in the
beginning but slows d o w n with time. T h e increase m a y continue for years.
T h e texture of lime stabilised soil b e c o m e s grainy and reminds of dry crust clay. T h e
columns are often quite i n h o m o g e n e o u s depending upon insufficient mixing. At the tip
of the c o l u m n s a r e m o u l d e d zone of soil will b e formed by the mixing tool, m a y b e without
i m p r o v e m e n t effects being achieved.
(iii) Design
depends on the shear strength in planes of w e a k n e s s that are m o s t likely to exist in the
columns. T h e ultimate load is generally calculated on the assumption that failure takes
place in drained condition. T h e angle of internal friction in lime c o l u m n s , determined on
a laboratory scale, has been found to vary between 30 and 40. A s s u m i n g that the column
has the strength characteristics </>'= 30 and c - 0 and that the creep strength (p. 34) of
the surrounding clay is c c r c l a ,y the bearing capacity of a lime c o l u m n , according to Eq.
(519), b e c o m e s equal to:
tf/col
( / +
c r clay)
5 c
)3
r0a
+ $ccr
clay + ^ccr
col
(524)
Fig. 332. Lime columns installed for stabilisation of trench walls. Lime/cement columns used for
stabilisation of the sides of deep excavations should be installed in several parallel rows.
So/7
460
improvement
stabilise slopes or trench walls (Fig. 332). In the latter case, they are then generally
installed in a number of rows on both sides of the trench. The stability of the trench is
analysed on the basis of the strength properties of the untreated soil and those of the
columns. Failure is assumed to take place along a plane, or curved, failure surface in
which the shear strength of the columns and the surrounding soil is fully mobilised. With
regard to the fact that the stress-strain behaviour of the columns and the natural soil up
to failure may be completely different from eachother this assumption is very questionable.
(v) Design with regard to settlement.
stabilised soil is analysed in the same way as previously described for granular column
stabilised soil, Eqs. (520-522). In the case of lime columns, however, it is necessary to
check that the stresses in the column is below its failure load determined by its creep
strength. The part of the load that induces stresses in the columns exceeding its creep
strength will have to be carried by the untreated soil.
The most difficult problems of settlement analysis are mainly related to difficulties of
determining the true value of the creep strength the compression modulus and the
permeability of the columns. By experience, the compression modulus of lime columns
is often assumed equal to (75 25) times the shear strength ^ the column. The permeability is generally assumed equal to 100-1000 times the permeability of clay.
An illustrative example of the scatter of values to be expected regarding the properties
of lime/cement columns was reported by Pramborg & Albertsson (1992). In this case the
TABLE 40.
Characteristics of stabilised soil determined by different methods.
Measured value
Permeability,
minimum
maximum
1.6
ll
average
m/year:
(Pressure-permeameter)
4.4
218
104
305
204
Pressuremeter (M 3 Epr)
21.9
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461
improvement
2 00
In this case, the specific discharge capacity qw of a column, 0.5 m in diameter, is found
3
to b e only about 0.9 m / y e a r , and of a column, 1 m in diameter, about 3.6 mVyear. Thus,
in the analysis of the course of settlement of a soil, stabilised with lime/cement columns,
the influence on the consolidation process of well resistance qw cannot be neglected. On
the other hand, the installation of the columns is m a d e in a way which does not normally
give rise to disturbance (ds - d).
2
Example 88: In the test field just described, a fill of 1.5 m sand and gravel, corresponding to 27 kN/m was
placed on the ground surface after the installation of the lime/cement columns. Determine the settlement
in the test field in the two cases, ( 1 ): 0.5 m columns with 1.3 m spacing and (2): 1.0 m columns with 1.8
m spacing. The groundwater level is at 1 m depth. The lime/cement columns are assumed to have the
characteristics given by the pressuremeter tests. Average values can be applied except in the determination
of column stress leading to creep failure of the column. This is governed by the minimum shear strength
of the clay, 5 kPa, at 2 m depth.
Solution: First calculate the final primary
settlement.
2
462
Soil
improvement
ELECTROCHEMICAL
STABILISATION
Electro-osmosis
Electro-osmotic dewatering can also be used for the purpose of consolidating silt and
clay. In the case of one-dimensional consolidation in the direction in an electric field
with constant voltage V over two rows of electrodes with large extension, the consolidation
equation can be written (Esrig, 1968):
dV
kh d u
dx
Yw dx
1 du
M dt
coefficient,
463
So/7 improvement
Yw - unit w e i g h t of water.
Introducing the auxiliary varaiable = (ke/k)ywV+
the form:
*
w h e r e ch = k^Aly^
( 5 2 6 )
direction).
A s s u m i n g that the anodes are arranged circularly around the cathode w e have:
" T
p dp
T 2
dp
i :
at
dx
kh
dx
Fig. 333. Variation of electric potential and pore pressure gradient between anodes and cathodes at
equilibrium.
So/7
464
improvement
A s s u m i n g that the potential gradient between the row of anodes and the row of
cathodes is linear the average effective stress increase can b e expressed as:
A^
/
av
= ( f t e y w / * A ) W 2 - A i i c a t h.
(529)
versa).
2
Ion
diffusion
D u e to the effect of ion diffusion the characteristics of the soil can b e thoroughly changed.
Thus, depending on the type of ions diffused the strength and character of the soil may
change drastically for the w o r s e or the better. T h e rate of ion diffusion is normally very
slow and can, for example, be calculated in the case of potassium chloride. T h u s , if KCl
is poured into a vertical hole in the soil, the rate of diffusion can be calculated by the
relation:
de
\ d
dc
- - - ( D p )
dt
dp
dp
(530)
So/7
improvement
465
anodes in wells or drains filled with electrolytes containing suitable ions and utilising the
electro-osmotic potential of enforcing and accelerating the ion escape into the surroundings.
A l u m i n i u m and p o t a s s i u m ions are well fitted for the purpose.
8.
SOIL INJECTION
Fig. 334. T u b e manchette' for guided injection of soil. A steel pipe, provided with peripheral
holes,.generally spaced 330 mm vertically apart, is inserted into a prebored hole in the soil. The annular
space between the pipe and the borehole is filled with semi-plastic material. The injection agent is
introduced through an injection tube, the ends of which has packers on both sides of the orifice in order
to make possible injection through any selected sleeve in the tube manchette.
So/7
466
improvement
size distribution of the soil. Roughly speaking, injection material utilised in coarse sand
should h a v e an upper grain size limit of 0.04 m m , while injection of m e d i u m or fine sand
should b e carried out with colloidal material and in coarse silt with material having a
viscosity close to that of water.
T h e injection is generally carried out by the aid of compressed air or injection p u m p s
through tubes provided with peripheral holes. According to a m e t h o d developed by the
French contractor Soltanche, a sleeve device, k n o w n as 'tube m a n c h e t t e ' (Fig. 334)
is utilised. A casing, 75 m m in diameter, is first installed in the soil. Then a tube
manchette, 38 m m in diameter and provided with peripheral holes, spaced 330 m m
vertically apart and closed by means of rubber sleeves, is inserted into the casing. T h e
annular space between the tube manchette and the casing is filled with semi-plastic
material after which the casing is withdrawn from the soil. T h e injection tube has a
diameter of 25 m m and is provided with packers on both sides of the orifice. Before
injection the orifice of the tube is placed just inside the sleeve w h e r e injection is to be
m a d e . B y increasing the injection pressure the sleeve opens as does also the enclosing
plastic material and the injection material penetrates into the soil at the selected level.
Injection generally starts from bottom. T h e m e t h o d of injection j u s t described has the
advantage of m a k i n g possible injection at any selected level as well as re-injection
without the need of new boreholes.
8.1
grouting
Ht+r )
3vl/3
(531)
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467
improvement
Fig. 335. Limiting grain size distribution curve of grout for soil injection.
For a cylindrical injection source, such as t h e tube m a n c h e t t e ' , of length / and diameter
d, w e can a s s u m e that the surface area of the sphere approximates the e n v e l o p e surface
area of the cylinder, Le. 4nr
1 / 2
In order to i m p r o v e its efficiency, the grout is often activated by adding bentonite, flyash or water glass to the c e m e n t grout. T h e grout can also b e activated by heating it to 2 5 35 C or b y m e c h a n i c a l treatment in special mixers. In this w a y it is possible to apply a
lower injection pressure or to enlarge the radius of influence.
8.2
Chemical
grouting
with
hardener
Injection of soil w h o s e grain size varies from m e d i u m sand to coarse silt requires the use
of diluted chemical grout, such as silicates (water glass) and p o l y m e r s ( c h r o m e lignins,
synthetic resins, plastics), which sets by m e a n s of catalysts (hardeners). T h e lower limit
of soil permeability for chemical grouting to b e successful according to Littlejohn (1992)
is 1 /s (30 m/year).
T h e radius of influence depends on the viscosity of the c h e m i c a l grout, the a m o u n t of
chemical grout injected per time unit and the jelling time of the grout. T h e jelling time
ought to b e short in comparison with the total injection time at the injection level in
question, particularly if the soil contains coarse-grained layers or if there is a groundwater
current. A rough approximation of the radius of influence is given by the relation (Karol,
1960):
So/7
468
R - 0.6(^^)
improvement
1 /3
(532)
Permeability, mm/s
Radius of influence, m
0.2-1.2
1.2-2.3
2.3-5.8
5.8-9.0
0.3-0.4
0.4-0.6
0.6-0.8
0.8-1.0
0.03-0.06
0.06-0.12
0.12-0.23
0.23-O.58
0.3-0.4
0.4-0.6
0.6-0.8
0.8-1.0
So/7
improvement
469
than cement or water glass. D u e to very low viscosity (tending to the viscosity of water)
they can b e injected into very low-pervious, granular soils, such as coarse silt. A m o n g the
polymers used can b e m e n t i o n e d chrome-lignin and acrylamide.
T h e jelling time of chrome-lignin, which is a byproduct in the cellulose industry, can
be controlled b y m e a n s of various catalysts and the final product is lignin-sulphonate. T h e
strength of the grouted soil is low, about 0.5 M P a .
A c r y l a m i d e is a m o n o m e r of calcium acrylate with either s o d i u m sulphate or
a m m o n i u m persulphate catalyst. Its polymerisation depends on temperature and the
concentration of solution and catalyst. T h e strength of the grouted soil is of the order of
magnitude of 100 kPa.
T h e lower limits of various injection grouts with reference to the soil permeability is
shown in Fig. 336.
1000
Fig. 336. Injectability limits for various grouts with reference to soil permeability (Cambefort, 1967)
470
8.3
So/7
Bitumen
(i) Bitumen.
and bentonite
improvement
injections
particle size ( 1 - 2 ) and low viscosity and can, therefore, b e injected easily in finegrained soils, such as coarse silt. T h e grout drives a w a y the p o r e water without mixing
with it. T h e bitumen solution sets by the effect of the hardener and sticks to the grains in
the soil.
(ii) Bentonite.
swelling in consequence. T h u s , for e x a m p l e , 'Volclay ' bentonite swells 12 times its original v o l u m e from dry to water saturated condition. Bentonite is injected in sand and
gravel for the purpose of creating barriers against water leakage.
At injection of coarse gravel, coarsely-ground bentonite is m i x e d with bentonite slurry.
This prevents the coarse bentonite grains from forming agglomerates which w o u l d m a k e
p u m p i n g of the grout impossible. T h e coarser grains in the bentonite slurry will gradually
swell and cause clogging.
Injection of fine to m e d i u m gravel can also b e carried out with a s o m e w h a t lumpy
bentonite slurry, though with smaller grain size than in the previous case. L u m p s cannot
be allowed at injection of sand. T h e m o r e fine-grained the soil, the m o r e dilute the
bentonite slurry.
Bentonite slurry is sometimes mixed with gravel, sand, sawdust, metal filings, or the
like in order to i m p r o v e the possibilities of creating water-tight barriers in current water.
A bentonite slurry with a bentonite content of only 8 % quite simply keeps sand in
suspension.
9.
THERMAL IMPROVEMENT
9.1
Burning
So/7
improvement
471
9.2
Freezing
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472
"
-20
- 10
- 30
-40
improvement
- 50
Temperature of sample, C
Fig. 338. Influence on compressive strength (unconfined compression tests) of ice and soil samples due
to change in temperature.
Freezing can b e carried out by the aid of a m m o n i a gas, carbon dioxide or liquid
nitrogen.
Freezing by the add of carbon dioxide is usually carried out in the following way. T h e
carbon dioxide is first compressed to a gas pressure of about 6 - 7 M P a . This causes a
temperature increase to about + 4 5 C . T h e gas is then conveyed to a condensator where
it turns into a liquid at a temperature of about + 2 0 C . N o w the liquid carbon dioxide is
allowed to pass through a narrow orifice into the freezing pipes. This causes a fall in
pressure to about 0.5 M P a and this, in turn, entails gasification a c c o m p a n i e d by strong
absorption of heat from the surroundings w h i c h freeze. T h e gas is then returned to the
compressor for a new cycle to begin.
10.
REINFORCED SOIL
10.1 General
aspects
Reinforcement of soil to m a k e possible the u s e of very steep slopes along road cuts or to
reduce earth pressure against retaining structures is an old technique(Figs. 3 3 9 - 3 4 0 ) but
has gained n e w and interesting applications in soil m e c h a n i c s . T h e reinforcement is
generally carried out b y m e a n s of various geosynthetic materials or, m o r e seldom, b y the
u s e of steel. T h e behaviour of reinforced soil depends on the strength and stress/strain
characteristics and the long-term resistance of the reinforcement materials, their design
(strips, grids, m a t s , m e s h e s , strands, etc.) and positioning and the friction along the
contact area b e t w e e n reinforcement and soil.
So/7
improvement
473
Fig. 340. Fascines represent another old type of soil reinforcement, here in a modern version.
474
So/7
improvement
yll
Fig. 341. Mhr's strain circle utilised for positioning of soil reinforcement. The reinforcement should
preferably be placed in the direction of pure tension 3.
the relative density (i.e. the larger the dilatency - ) , the larger the pure tension 3 (the
centre of the M h r strain circle shown in Fig. 341 will b e displaced to the left). This
tendency towards increasing tension due to dilation is counteracted by the reinforcement.
Dilation also increases the earth pressure against the reinforcement bodies which
contributes to improved co-operation between soil and reinforcement. T h e latter obviously
b e c o m e s m o r e effective in dense than in loose or m e d i u m dense soils.
T h e reinforcement can be installed in two different w a y s either in connection with
filling operations or directly in natural soil, for e x a m p l e in cut slopes. T h e latter technique
is referred to as soil nailing.
10.2
Design
So/7
improvement
475
476
Design
aspects
GEOMECHANICAL VIEWS
1.1 Accuracy
in methods
ofcalcufation
one of the corner-stones in foundation engineering. This is understandable from the point
of view that ground failure would entail very serious c o n s e q u e n c e s for the building and
the people working or living in the building. However, in the case of building foundations
the main p r o b l e m in the design is not to ensure a certain factor of safety against ground
failure (except, of course, for foundations adjacent to slopes) but to m a k e sure that the
total and differential settlements will not cause d a m a g e to the building. By tradition, a
high e n o u g h factor of safety against failure was considered as the only criterion necessary
to avoid d a m a g e due to settlement. Only in the case of soft, highly compressible soils,
settlement analysis was normal part of the design.
T h e determination of the bearing capacity, however, is important from quite another
aspect. T h u s , a high-enough safety against ground failure is needed for the soil to b e h a v e
as a pseudo-elastic material applicable to the computational m e t h o d s used in settlement
analysis.
Design
All
aspects
(ii) Stability.
the working load can be determined with acceptable accuracy on the basis of the theory
of elasticity. In granular soils, the stress distribution is influenced b y the fact that the soil
cannot take tensile stresses. However, in practice the difference in result obtained for a
particulate m e d i u m and for an elastic m e d i u m is negligible.
(iv) Settlement.
the loading condition does not result in essential plastic deformations of the soil. F r o m
a practical viewpoint, settlement analysis for foundations on n o n - c o h e s i v e soils cause
problems mainly for the reason of difficulties in determining relevant elastic properties.
N o n - c o h e s i v e soil is a particulate material and, hence, far from b e h a v i n g as an elastic
material; undisturbed sampling is practically impossible and the deformation parameters
determined on a laboratory scale therefore quite misleading. However, m o d e r n in-situ
methods h a v e greatly i m p r o v e d the accuracy of prediction.
Settlement analysis for cohesive soils is facilitated by the fact that undisturbed sampling is possible. This in turn facilitates the evaluation of requisite parameters for settlement analysis. Calculation m e t h o d s may often seem too simplified to b e trusted, but the
use of more sophisticated analytical solutions may not be worthwhile with regard to the
complexity of nature. Moreover, the deviations between the results obtained with simplified
and more sophisticated approaches, using the same input parameters, are generally negligible.
An illustrative example of this is the simplified Terzaghi consolidation equation (Eq. 29):
du
du
Design
478
aspects
du
M d
du
du
M dk du ?
~dt
Ywdz
dz
dz
Ywdu
dz
Let u s further a s s u m e that w e have to deal with a 1 0 m thick clay layer drained at top
and b o t t o m with an induced excess pore water pressure u0 = 1 0 k P a (constant with depth).
T h e compression m o d u l u s of the clay M=200
k0exp(-k)
M
k =
whence
(533)
dk
M
du
T h e coefficient of c o n s o l i d a t i o n cv = ^ Q M / ) ^ ~ 0 . 0 3 - 2 0 0 / 1 0 = 0 . 6 m / y e a r .
T h e excess pore water pressure variation throughout the clay layer according to the
simplified and m o r e accurate relation, solved by c o m p u t e r analysis, is s h o w n in Fig. 3 4 2 .
A s can b e seen, t h e difference in results is negligible from a practical viewpoint.
E x c e s s pore water pressure, kPa
0.5
1.0
1.5
2.0
\
\
\
\
\
1
1
/
/
10
Fig. 342. Remaining excess pore water pressure after 50 years of consolidation according to Terzaghi's
simplified solution (broken line; average degree of consolidation 95%) and the more accurate solution
taking into consideration a decrease in permeability due to a decrease in void ratio during the
consolidation process (average degree of consolidation 90%). Initial excess pore water pressure u0 = 10
kPa. Clay layer drained at top and bottom. Terzaghi's simplified solution would yield the same result
2
2
as the more accurate one provided that c v is 0.435 m /year (J: r ed = 0.0217 m/year) instead of 0.6 m /year.
Design
1.2
479
aspects
Probabilistic
approach
fx M
1
l -
= 7 = exp[--(
-)
Gyj2%
(534)
w h e r e = E(X) - m e a n value of X
a - D(X) = standard deviation of X
A useful m e a s u r e of the scatter of a r a n d o m variable is its coefficient of variation V(X)
= al.
T h e standardised form of normal distribution,
1
()
= j = e x p ( - ),
OyJlK
(),
which is the area b e l o w the normal distribution curve from t = -<> to t = JC, is obtained by
(Fig. 344):
Design
480
aspects
&x(x)
1
. t
= = exp(--)dt
/ 2 "~
(535)
()
1
t
= 0.5 + - = P e x p ( - - ) A
0
2
/2
(536)
= - 7 = | * ( - ^ - )
V2k
(537)
which generally forms the basis for tabulated values of the cumulative-distribution curve.
(i) Safety margin. T h e classical measure of fitness of a structural element to perform an
i m p o s e d d e m a n d is the factor of safety F, defined as the quotient of capacity and demand
( F = CID). Failure is indicated where F < 1. Lately, a new additional approach has c o m e
into use based u p o n probability theory. Accordingly, the fitness is measured by the safety
margin SM, defined as capacity minus demand (SM-C-D).
Design
481
aspects
are a b o v e zero
(Fig. 346). Obviously, increases in imply increases in the reliability and the safety
margin. T h e reliability index tells us that the probability of failure is:
aSM
1
where {) = =
2
Gsm
(539)
\ tx^(--)dt.
Safety margin,
SM
Design
482
aspects
for
where y =/(1
++
M+
=p (y+
1,
+ p+ ~(y
-)
(540)
2),
= (i + p)/4,
p+ + = pr-
p+- = / r + = ( l - p ) / 4 .
In these equations, is the correlation coefficient between xx and x2. T h e weighting
factors/? can vary from - 1 t o + 1 . Should there be no correlation, then = 0 and consequently
1/4.
p=
R o s e n b l u e t h ' s generalised version of the point estimate method (see Harr, 1987) is
valid for any n u m b e r of r a n d o m variables. For example, for a function of three variables
y =fijcl,x2y)c3)
j
p+
w e have:
=f(pxl
=
=r
p
+ +
= (1 + p , 2 - p 2 3 -
p+ + - = pr-+
+
],2
2, 3
3\
(1 + p 1 2 + p 2 3 + p 3 , ) / 2 3 ,
p - = / r + - = (l - p
~ = /r ++= ( l - p
2- p
2 3
+ p
1 2
p 3 1) / 2 3 ,
+ p 3 1) / 2 3 ,
2
- p 3 1) / 2 3 .
E(y )
=p
++
(y
+ + M
+p
-(y
~) + + / ? - - ( y - - )
+ p ~(y~ ~ ~)
(541 )
T h e sign of the correlation coefficients is determined by the sign of the product of the
corresponding + and - of the weighting factors, for example (+)(-) = (-) and (-)(-) = (+).
T h e expected value E(y) = py of the function is obtained for M - 1 while the variance
2
GY is obtained for M - 2.
Example 90: A strip footing with width b = 1 m is subjected to a vertical load Q = 500 kN/m. The footing
is founded at a depth dj- 1 m in sand with an internal angle of friction ' = 30 and a cohesion intercept
/
3
c = 10 kPa. The groundwater level is at 1 m depth. Above the water table we have = 18 k N / m and
3
below Y =11 kN/m . Determine the reliability index and the probability of failure according to GPEM
if the coefficients of variation are V(Q) = V(f)=0, ( ) = 20% and V(c 0=40%. The correlation coefficient
is p i c ^ O = - 0.5, i.e. if (//increases, ^decreases, and vice versa.
Solution: The safety margin is defined by the relation (p. 126):
SM = 0.5 YbNY+ yjjNq + c'Nc - Qlb
The probability is determined by the safety margin SM < 0. We have to consider two random variables
0'and (/with standard deviations
= 6 and oc = 4 kPa. The bearing capacity factors become N + = 23.8,
483
Design aspects
+
/+
Nq = 37.7, Nc = 50.5, c = 14 kPa and yVy- = 7.65, N~ = 9.60, Wcr = 17.1, c'- = 6 kPa. Furthermore,
+
we have p + =p-- = ( 1 + p c 0) / 4 = 1/8 and p+ - = / ? - + = ( 1 - p c 0) / 4 = 3/8. Hence,
++
++
+
2
p SM
= (0.5-11-23.8 + 18-37.7 + 14-50.5 - 500)/8 = 127.1
p+ +(SM +) = 129159.0
+
+
+ 2
p SM+= 3(0.5-11-23.8 + 18-37.7 + 6-50.5 - 500)/8 = 229.7
p ~(SM -)
= 140683.6
+
+
+
+ 2
= 3(0.5-11-7.65 + 18-9.6 + 14-50.5 - 500)/8 = 158.2
p- (SM~ )
= 66741.9
p- SM~
2
p--SM = (0.5-11-7.65 + 18-9.6 + 6-50.5 - 500)/8 = 2.2
p(SM)
= 39.9
2
The mean value pSM = lp(SM) = 517.2 and () = 336624.5
2
2
The variance (oSM)
= 336624.5 - (517.2) = 69128.7 which yields oSM=262.9
kPa and the reliability
index = pSMloSM
= 517.2/262.9 = 1.97.
Hence, the probability of failure (SM < 0) = 2.4%
Example 9 1 : A square footing with width b = 2 m is founded on the ground surface of a sand deposit
of great depth. The deformation characteristics of the sand are characterised by Poisson's ratio v = 0.3
and the pseudo-elastic modulus = 15 MPa. Determine the probability of of the settlement exceeding
2
5 = 1 5 mm for an average foundation pressure q = 0.1 M N / m if the coefficients of variation are V(v)
= 30%, V(E) = 20% and V(q) = 10%.
Solution: The safety margin can be expressed by the relation (pp. 141-142):
SM = 0.015 -reqblE
where 0.015 is the settlement limit (in m),
Te is a function of v, depth of soil deposit and ratio of foundation length to foundation width.
The probability is determined by SM < 0. The standard deviations become - 0.09, = 3 MPa and
2
+
+
q = 0.01 M N / m . In our case we find / = 0.728 and ~ = 0.820. Moreover, q = 0.11, q~ = 0.09, E
= 18 MPa and E~ = 12 MPa. We have:
3
+ +
+2
+ + +
(SM
) = 0.0372-10= 0.015 -0.728-0.11-2/18 = 0.00610
SM
3
+
+
+
+ 2
(SM - ) = 0.0596 1 0 = 0.015 -0.728-0.09-2/18 = 0.00772
SM 3
(SM+- " ) 2 = 0.0166-10SM+-- = 0.015 -0.728-0.09-2/12 = 0.00408
2
(SM-- " ) = 0.0073-10-3
SM = 0.015 -0.820-0.09-2/12 = 0.00270
+
+
(SM~ " ) 2 = 0.0000-10-3
SM- - = 0.015 -0.820-0.11-2/12 = - 0.00003
+
(SM- + ) 2 = 0.0248-"
= 0.015 -0.820-0.11-2/18 = 0.00498
SM+
(SM + ) 2 = 0.0462-10-3
SM-~ = 0.015 -0.820-0.09-2/18 = 0.00680
+
+
2
(SM
SM+ + - = 0.015 -0.728-0.11-2/12 = 0.00165
- ) = 0.0027-10-3
2
3
(5 )/8 = 0.00425 = pSM and ( ) /8 = 0.02432-10"
The variance (oSM)
= 0.00425/0.00250 = 1.70.
2.
FOUNDATION
2,1
Design
REQUIREMENTS
criteria for
buildings
settlement.
settlements
m u s t b e restricted
Design
484
aspects
Fig. 347. Settlement of a block of old buildings founded on deep clay deposits. Original situation of the
buildings indicated with black lines. (By courtesy of BAAB).
damage
limits.
Design
485
aspects
Fig. 348. Differential seulement of a small family house due to influence of road embankment causing
consolidation settlement.
According to Bjerrum (1966), the empirical limiting values of the angle of inclination
can b e estimated from Fig. 3 5 1 . Bjerrum also presented an empirical correlation
between the m a x i m u m differential and the m a x i m u m total settlements of buildings
founded on sand and on clay (Fig. 352).
(ii) Design
settlement
limits.
Design
486
aspects
11
<
<
>
4>
'
ML j
_=H
2
4
Number of storeys
Fig. 350. Inventory of damages due to differential settlements (Rethaty, 1964). Filled circles represent
damage to frameworks; open circles damage to partition walls and Wallings. Line ( 1 ) denotes lower limit
for damage to frameworks; line (2) lower limit for damage to partition walls and Wallings.
.5
*'3I
0.001-0.005
0.003
0.001
0.0025-0.004
0.003
0.002
0.005
.S
13
6
Rotation, %o
10
Fig. 351. Empirical limiting values of of angle of rotation according to Bjerrum (1966).
Design
487
aspects
10
20
30
40
10
15
Maximum settlement, cm
Fig. 352. Maximum differential settlements Asmax and maximumangle of rotation or building founded
on clay (left) and sand according to Bjerrum (1966).
criteria for
tanks
Tanks are large cylindrical steel containers built for storage and distribution of liquids
(petroleum products, molasses, etc.). T h e y are mostly grouped together in tank farms in
close proximity to w a t e r w a y s used for the transportation and delivery of the products to
b e stored in the tanks. In c o n s e q u e n c e , they are located in places w h e r e subsoil conditions
are often very poor.
Design
488
aspects
T h e cost of foundation of tanks represents the major part of the building cost. In order
to m i n i m i s e these costs, foundation methods may be accepted which, according to
prediction, will entail settlement that verge on adopted criteria for permissible settlement.
If in reality settlements should tend to exceed the limits given in the design criteria, there
is always a possibility to re-level the shell or the b o t t o m plate of the tank.
(i) Damage
problems.
Radial deformation of shell (or shell out-of-roundness). In the case of floating roof,
shell out-of-roundness m a y lead to binding of the floating roof during operation, or to a
gap between the tank shell and the floating roof. In the case of fixed cone roof tanks, shell
out-of-roundness m a y result in upper shell course buckling.
Rupture of shell, or bottom plate, or bottom plate-shell connection
(ii) Design settlement
982) and
by Malik et al. (1977), represent the most recent and well-founded design criteria. T h e
criteria are divided into tilt, differential settlement of shell and differential settlement of
bottom plate.
Tilt. To avoid spilling oil from the floating roof or stressing the roof of cone roof tanks,
the following condition should b e satisfied (Fig. 352):
(542)
5m!a<2Ahd
w h e r e Ahd is the design freeboard.
Moreover, the change in diameter caused by rotation of the shell relative to the roof
must not exceed the tolerance of the seal A/? t o l, w h e n c e :
5 n i < 2 / D A / ? 1 0,
Differential
settlement
(543)
as being the best in respect of ovalness resulting from out-of-plane distortion of floating
roof tanks. Accordingly (Fig. 353):
Sl
"
~^~
w h e r e N= total n u m b e r of survey points spaced at equal angles along the shell perimeter,
Si = out-of-plane settlement in the /:th survey point.
However, it is suggested that large tanks can stand m o r e differential settlement than
is indicated by this relation.
( 5 4 4
Design
489
aspects
Aspects of performance
Mode of failure
Planar
tilt
Overtopping of shell
L o s s of roof seal
Non-planar
tilt
Overstress of shell
Rupture from d i s h - s h a p e d settlement
Non-planar settlement
deformed tank wall
Rupture of c o n n e c t i o n as shell
bridges over soft spots
gap.
d e f o r m e d annular ring
Fig. 353. Aspects of the performance and modes of failure for oil tanks.
T h e out-of-plane settlements, i.e. the settlements relative to planar tilt, are obviously
an important part of the design criteria for tanks.
T h e planar tilt is given by the first h a r m o n i c of the Fourier series:
Z ^ A o + A ^ o s C ^ + jS)
w h e r e { = 2ni/N (i = 1, 2, 3, - , N),
A0 = (1/0;, = m e a s u r e d average settlement of shell,
(545)
Design
490
! = - y/;
(Pi
cos
q>i
) + , (p, sin
aspects
) ,
2AXID.
i = Pi -
T h e criterion for verstress of the shell, based on experience of 90 case records, is given
by the relation:
2
1 1 )
A
' - ^ r
<
546)
settlement
of bottom plate.
r e c o m m e n d e d that:
wl +
vv <
031afD
(547)
}
E(FS)
( 5 4 8
/ 2.25 afDld
S
"
EH(FS)
3 4/
T h e value of the safety factor (FS) is chosen in the s a m e way as in the case of dishshaped settlement.
( 5 4 9
Design
491
aspects
Important
(i) Soil-structure
aspects
of foundation
interaction.
design
the ground in connection with the building activity. T h e analysis of the settlement due to
the weight of the building is generally carried out by the geotechnician on the basis of the
building loads presented to h i m by the structural engineer. Now, the general procedure
adopted by the structural engineer in his analysis is to a s s u m e that the building is resting
on unyielding ground. However, differential settlements will give rise to changes in the
loading condition and consequently the settlements calculated by the geotechnician on
the premises given by the structural engineer will not b e correct. L o a d s on c o l u m n s that
settle m o r e will partly be carried over to columns that settle less. H e n c e , the resulting error
is generally on the safe side as regards the a m o u n t of calculated differential settlements.
T h e errors in calculated internal forces, however, m a y b e serious. Calculated bending
m o m e n t s m a y h a v e an opposite sign to that occurring in reality which m e a n s that
reinforcement bars m a y b e placed on the wrong side of the beam. A d v a n c e d computer
p r o g r a m s will probably take care of these problems in the future by m e a n s of a successive
iteration process. Of course, several difficulties will be encountered with regard to the
gradual build-up of the structural stiffness during the construction period. Fortunately,
the structure has an admirable capacity to adjust itself to its design. For e x a m p l e , in a
concrete structure fine cracks will appear at w e a k sections, plastic hinges will be formed,
and so forth. M o s t important of all, a considerable a m o u n t of a c c o m m o d a t i o n will have
taken place during the construction period.
(ii) Other causes of settlements.
often take place d u e to goundwater lowering or to changes in ground level. This fact is
often ignored and attention is mainly paid to the foundation of the buildings, in particular
whether piling is required or not. T h e differential settlements often observed between
piled buildings and the surrounding area (Fig. 224) m a y cause serious trouble, for
instance, b r e a k a g e of connecting sewers.
T h e influence of building activities on the groundwater condition can b e quite hard to
predict. For e x a m p l e , groundwater leakage into tunnels and caverns in b e d r o c k often
entail serious g r o u n d w a t e r lowering (Fig. 257). Therefore, if tunnels or caverns are
situated in areas w h e r e the overlying soil consists of clay or other compressible soil
deposits, leakage m u s t be prevented in o n e w a y or the other or water infiltration be
arranged in order to maintain the existing state of pore pressure distribution in the area.
Observations h a v e shown that it is extremely difficult to predict w h e r e the groundwater
492
Design
aspects
will b e mostly affected by tunnels in bedrock. In the case shown in Fig. 258 (p. 361) with
a raw water tunnel in b e d r o c k passing a crush-zone, water l e a k a g e entailed a pore
pressure decrease of over 100 kPa in a sand aquifer underneath clay within a fairly limited
area, with its centre about 300 m away from the leaking tunnel line. Obviously, theoretical
m o d e l s for calculation of the depression in groudwater level around a tunnel or a rock
cavern, based on condition of symmetry, are useless.
L e a k a g e can b e prevented by grout hole drilling and cement injection (p. 466).
Infiltration is often connected with difficulties of various kinds (see p. 373).
In semi-arid regions, desiccation during lengthy dry periods can lower the groundwater
table beneath u n c o v e r e d areas to considerable depth. D u e to desiccation, cracks are
broken u p in the soil and are filled with water during the rainy season. If the soil contains
swelling minerals (smectite), a consequential swelling and h e a v e will take place. Beneath
buildings and other covered areas the loss of water is very m u c h smaller than in adjacent,
uncovered areas and, hence, a long-term increase in the water content in the soil below
central parts of the covered areas will take place. T h e consequential h e a v e is m o r e or less
independent of the external loading condition. T h e differential settlements caused by
h e a v e inside buildings and by settlement outside can entail serious d a m a g e to buildings
(Fig. 355).
Retaining structures supporting frost-susceptible soil are subjected to a considerable
increase in earth pressure in wintertime (Fig. 356). H e n c e , it is very important that frostsusceptible soil adjacent to the retaining structure should be replaced with frostinsusceptible soil. Frost h e a v e is a c o m m o n cause of differential settlement in buildings
which are founded at smaller depths than the depth of frost penetration or subjected to
frost penetration which is insufficiently protected against.
Design
aspects
493
on the stability of of the building area is another important matter to consider. T h e bearing
capacity of the of the foundation itself is generally guaranteed by the application of safety
factors in the design and is therefore seldom in jeopardy. M a n y times, however, the
overall stability of the building site or its neighbourhood can be j e o p a r d i s e d due to
unforeseen influence on the overall long-term stability condition. It is thus extremely
Fig. 357. Slide in built-up area in Norway. The slide scar indicates a thick and strong dry crust formation
which may have been misleading in the judgment of the stability condition.
494
Design
aspects
important that the stability conditions are carefully investigated wherever the building
activities take place on sloping terrain or in immediate vicinity of slopes with clay and
silt deposits (Fig. 357).
In seismic regions with loose sand or silt, complete liquefaction of large soil masses
have taken place. In 1920, in the Chinese province Shansun, an earthquake was reported
which induced an earth slide in loess soil comprising an area, 480 k m in length and 160
k m in width, resulting in about 100,000 casualties. Piping phenomena in connection with
strong earthquakes are not as disastrous but quite spectacular (Fig. 358).
The vibrations induced by blasting can also entail liquefaction, although in more rare
cases. However, blasting in the vicinity of slopes where the stability is at stake should
certainly be avoided (Fig. 359).
Fig. 359. Slide triggered by rock blasting on the west coast of Sweden.
Design
495
aspects
In certain areas, formation of sinkholes in soluble rocks and sensitive soils are
experienced. D r a m a t i c e x a m p l e s of the catastrophic consequences that m a y follow upon
sinkhole formations h a v e been presented by Wagener (1989).
Besides long-term settlement caused by mining operations there m a y also b e a risk of
ground failure. A n e x a m p l e of such a failure due to the collapse of the roof of an old iron
m i n e w a s given in Fig. 8 (p. 17).
3.
INTERNATIONAL CODES
Scope of the
Eurocode
Geotechnical
E u r o c o d e 7 aims at creating standards for laboratory testing and for field testing and
sampling and at establishing c o m m o n rules for geotechnical design, including also
specific geotechnical structures, such as ground anchors, soil i m p r o v e m e n t , dewatering,
bridges, off-shore structures, etc. E u r o c o d e 7 is expected to b e finalised and approved of
in 1997.
3.2 Design philosophy
of
Eurocode
(i) Limit states. T h e design of structures should b e carried out with the view to satisfying
the requirements connected with states, b e y o n d w h i c h the structure no longer satisfies the
design performance criteria. T h e r e are two limit states that h a v e to b e considered in the
design, n a m e l y the ultimate
Design
496
aspects
parameters taking part in reaction. In geotechnical design, not only the external forces are
considered as action but also the weight of the soil and rock, the internal stress situation
(effective stresses, pore pressure, etc.), changes in stress situation (for instance due to
excavation), influence of vegetation or climatic variations, m o v e m e n t s (for instance due
to consolidation, decomposition, solution, creep), temperature effects, etc.
For the ultimate limit state, the design values of actions should either b e assessed
directly or b e based on characteristic values modified by a certain partial safety factor.
T h e characteristic values depend u p o n the type of supported structure. For soil and rock
density, the characteristic value is defined as for soil properties in general. T h e values of
the partial safety factor are chosen with due regard to the uncertainty in the values of
action.
For the serviceability limit states, the design values can generally b e taken equal to the
characteristic values.
(iii) Soil parameters.
[l-l.64V(X)]E(X)
(550)
Design
497
aspects
Fig. 360. Failure of earth fill cofferdam caused by insufficient bearing capacity. Subsoil consisting of
clay with silt layers. An earth fill cofferdam is a typical example of a structure belonging to category 3.
(iv) Geotechnical
categories.
basis for geotechnical design depends upon the type of structure and risks involved. In
the E u r o c o d e , three geotechnical categories are defined:
category 1 , including small and relatively simple structures 'for which it is possible to
ensure that the fundamental requirements will b e satisfied on the basis of experience and
qualitative geotechnical investigations' and w h e r e there is negligible risk for property
and life,
category 2, including 'conventional types of structures and foundations with no
abnormal risks or unusual or exceptionally difficult ground or loading c o n d i t i o n s ' ,
Design
498
aspects
ISO
standards
References
499
REFERENCES
Akagi, T., 1976. Effect of displacement type sand drains o n strength and compressibility
of soft clays. Ph. D . Thesis, University of Tokyo.
Andrasson, . & H a n s b o , S., 1977. C o m p a c t i o n control by d y n a m i c m e t h o d s . V g - och
Vattenbyggaren, N o . 8 - 9 .
Andrasson, , 1979. Deformation characteristics of soft, high-plastic clays under
d y n a m i c loading conditions. Ph. D. Thesis, C h a l m e r s University of Technology,
Gothenburg, D e p . of Geotechnical Engineering.
Andrasson, L., 1974. Frslag till ndrade reduktionsfaktorer vid reduktion av vingborrbestmd skjuvhllfasthet m e d ledning av flytgrnsvrdet. (Proposal for altered correction
factors w h e n reducing field vane shear strength on the basis of the liquid limit value).
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Baguelin, F., Jzequel, J. F. & Shields, D. H., 1978. T h e pressuremeter in foundation
engineering, Trans. Tech. Publications, Paris.
Barden, L. & Sides, G., 1971. Sample disturbance in the investigation of clay structure.
G o t e c h n i q u e 2 1 , N o . 3, 2 2 1 - 2 2 2 .
Barkan, D . D., 1962. D y n a m i c s of basis and and foundations. M c G r a w - H i l l B o o k
C o m p a n y , N e w York.
Barker, J. ., 1 9 8 1 . Dictionary of Soil m e c h a n i c s and F o u n d a t i o n E n g i n e e r i n g .
Construction Press, L o n d o n and N e w York.
Barron, R. ., 1944. T h e influence of drain wells on the consolidation of finegrained
soils. (R. I.) Ph. D. Thesis, Providence, U. S. Engineering Office.
Barron, R. ., 1948.Consolidation of fine-grained soils by drain wells. Proc. A S C E ,
Paper N o . 2 3 4 6 (with discussions).
Barton, J O, 1982. Laterally loaded m o d e l piles in sand: Centrifuge tests and finite
element analysis. Dissertation, Univ. of C a m b r i d g e .
Beigler, S.-., 1976. Soil-structure interaction under static loading. Ph. D . Thesis,
C h a l m e r s University of Technology, Gothenburg.
Berezantsev, V. G., Khristoforov, V. S. & Golubkov, V. N., 1961. L o a d bearing capacity
and deformation of piled foundations. Proc. 5th Int. Conf. Soil M e c h . F o u n d . Eng.,
Paris, Vol. 2, 1 1 - 1 5 .
Bergau, W., 1959. M e a s u r e m e n t s in grain silos. Swedish Geotech. Inst., Proc. N o . 17.
Bergdahl, U. & Eriksson, U., 1983. B e s t m n i n g av j o r d e g e n s k a p e r m e d s o n d e r i n g e n
litteraturstudie. (Determination of soil characteristics by s o u n d i n g a literature survay).
S w e d i s h Geotech. Inst., Report N o . 22.
500
References
References
501
502
References
Dubin, . & Molin, G., 1986. Influence of a critical gradient on the consolidation of clays.
In R. . Yong & F. C. Townsend (Editors), Consolidation of soils: testing and
evaluation. A S T M Special Tech. Publ. 892, 3 5 4 - 3 7 7 .
D u n c a n , J. & C h a n g , C . - Y , 1970. Nonlinear analysis of stress and strain in soil. A S C E ,
J. Soil M e c h . Found. Eng., Vol. 96, N o . S M 5.
Du Thinh, ., 1984. T h e built-in stiffness of footings in sand. Ph. D. Thesis, C h a l m e r s
University of Technology, Gothenburg.
Ekstrm, J., 1989. A field study of model pile group behaviour in non-cohesive soils.
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References
509
510
References
References
511
Subject
512
index
SUBJECT INDEX
Actions, 496
Active earth pressure, 270, 279
Admixtures for soil stabilisation purpose:
bitumen, 456
cement, 456
cement columns, 457
lime, 456
lime columns, 457
Anisotropy
strength, 34
structural, 4
Angle of intersection:
between failure surface and ground surface, 274
between failure surface and wall, 276
Angular frequency, 243
Atterberg limits, see Consistency limits
Subject
513
index
plastic limit, 48
shrinkage limit, 48
Contact pressure:
approximation for rafts, 120
influence of superstructure, 124
rigid footing on elastic medium, 107
raft on elastic medium, 119
subgrade coefficient theory, 118
Continuous sampling, 71
Coring, 169
CPT, see cone penetration test
CPTU, see piezocone test
Creep, 9, 32
Creep limit, 32
Creep strength, 34, 101
Critical damping, 245
CRS oedometer tests, 78
Damping:
critical damping, 245
geometric damping, 263
hysteretic damping, 183
internal damping, 261, 263
overdamping, 245
radiation damping, 183
underdamping, 246
viscous damping, 184
Darcy's law, 21
Deformation properties of soil:
correlation with sounding resistance, 90
dilatometer test, 82
direct shear test, 74
empirical strength correlations, 89
empirical stress correlations, 87
oedometer test, 76
pressuremeter test, 80
strain dependence, 26
stress dependence, 28
triaxial test, 75
time dependence, 28
Degree of water saturation, 12
Design crieria:
buildings, 483
calendering machines, 487
tanks, 487
Design of vertical drain installations, 4 3 2 , 4 3 6 , 4 3 9
Design settlement limits, 485, 488
Deviation from Darcy's law, 22
Diameter of influence of vertical drains, 4 3 2
Diaphragm walls, 233, 378
514
Diaphragm walls:
bearing capacity, 234
settlement, 240
Dilatometer, 82
Dilatometer modulus, 84
Direct shear test, 74
Discharge capacity of drains, 432, 434
Discontinuities, 54
Displacement piles, 160
Dissolved matter, 11
Drain installation techniques, 429
Drained strength parameters, 32
Driven piles, 160, 175, 199
Dry density, 14, 400
Dry of optimum, 402
Dynamic compression modulus, 86
Dynamic consolidation, see Heavy tamping
Dynamic factor, 248
Dynamic preloading of piles, 196
Dynamic shear modulus, 86
Earth pressure:
active, 270, 279
at rest, 271
distribution, 279, 295
enforced wall rotation, 294
graphical determination, 285
horizontal ground: smooth vertical wall, 278
horizontal ground: rough vertical wall, 280
leaning wall: inclined ground surface, 283
passive, 270, 279
Earth pressure at rest:
line load, 272
point load, 273
Earthquake, liquefaction due to, 34, 494
Echo sounding, 67
Effective strength parameters, 32
Effective stress, 20
Ejector wells, 367
Elastic modulus:
definition, 23
determination, 74
Electric resistivity method, 67
Electro-chemical stabilisation:
electro-osmosis, 4 6 2
ion diffusion, 464
Electro-osmosis:
consolidation, 462
dewatering, 367
electric potential gradient, 368
Subject
index
Failure surface:
intersection angle with ground surface, 274
intersection angle with wall, 276
Ktter's equation, 125
Fall-cone test, 96
False refusal, 167
Fascines, 473
Field vane test, 99
Filter requirements for band drains, 4 4 0
Flow net, 364
Subject
index
Footings:
bearing capacity, see Bearing capacity of footings
contact pressure, see Contact pressure
settlement, see Settlement of footings
Forced, damped vibrations, 246
Foundation requirements, buildings:
design settlement limits, 485
empirical damage limits, 484
Foundation requirements, tanks:
design settlement limits, 488
Freezing, 471
Frequency:
angular, 243
circular, 243
natural, 244
Frost activity, 52
Frost susceptibility, 53
Geometrical damping, 263
Geotechnical categories, 496
Grain density, 13
Grain size:
distribution, 39
fractional groups, 39
fractional limits, 39
Granular columns:
bearing capacity, 451
design, 451
load share columns/surrounding soil, 451
settlement, 453
slope stability improvement, 453
Gravity walls, 298
Gravity wells, 365
Ground anchors, 392
Heavy tamping, 410
Helenelund's graphical method, 324
Hydraulic ground failure, 356
Hydraulic radius, 303
Hydraulic time lag, 73
Hydraulic uplift, 357
Hydrodynamic condition, 21
Hydrodynamic time lag, see Primary consolidation
Hydrostatic condition, 21
Ignition loss method, 44
Illite, 7
Impact, 249
Inclined anchors, 378, 389, 394
Internal damping, 263
International codes, 495
515
Ion diffusion, 464
ISO standards, 38, 498
Karst formations, 16
Kotter's equations, 125
Laminar flow, 22
Laterally loaded piles:
ultimate resistance in non-cohesive soil, 202
ultimate resistance in cohesive soil, 206
deflection, 209
Lime/cement columns:
bearing capacity, 459
choice of admixture, 456
installation techniques, 457
settlement, 460
stabilisation of trenches/slopes, 460
Lime stabilisation, 456
Limit state design, 495
Linear vibration theory, 242
Liquefaction, 22, 356
Liquidity index, 49
Macro-structure of soil:
anisotropy, 9
non-homogeneities, 10
Macro-structure of rock:
appertures, 16
argillaceous zones, 16
discontinuities, 16
fabrics, 15
joints, 16
karst formations, 16
shear zones, 16
Mass points:
definition, 255
displasement, 256
Massive rock, 16
Mnard pressuremeter, 80
Metal foil sampler, 72
Micro-structure of soil:
aggregates, 7
bridges, 7
clay minerals, 4
crystalline, rock-forming materials, 8
domains, 7
organic material, 8
Microstructure of rock, 15
Mohr-Coulomb failure criterion, 27, 32, 278
Moment reduction, sheet pile walls, 380
Subject
516
Montmorillonite, 6
Non-displacement piles, 169
Non-homogeneity, 10
Non-linear vibrations, 252
Oedometer modulus:
definition, 26
determination, 76
Oedometer test, 76
Optimum water content, 399
Organic material, 8
Oscillatory rollers, 406
Overconsolidation ratio, 88
Overdamping, 245
Padfoot rollers, 403
Particle density, see Grain density
Particle size, see Grain size
Peat:
amorphous, 44
fibrous, 44
pseudo-fibrous, 44
Phase shift, 247, 249
Piers:
bearing capacity, 234
settlement, 240
Piezocone tests, 65
Piezometers, 68
Piles:
bearing capacity, see Bearing capacity of piles
buckling, see Buckling of piles
laterally loaded, see Laterally loaded piles
settlement, see Settlement of piles
Pile group design, 227
Pile loading test, 174
Pile loading test, interpretation:
Brinch Hansen's 80% criterion, 175
creep load, 175
Polish method, 175
Piled raft design, 229
Piston sampler:
inside clearance ratio, 71
edge taper angle, 71
Plasticity index, 49
Plate loading test, 85
Poisson's ratio:
definition, 25
determination, 74
Pore gas pressure, 20
Pore pressure measurements, 67
index
Subject
index
517
Silo pressure:
eccentric discharge gate, 303
Janssen-Knen equation, 302
Skempton's pore pressure equation, 23
Slope stability:
Bishop's rigorous method of analysis, 343
Bishop's simplified method of analysis, 343
circular-cylindrical failure surface, 336
composite failure surface, 345
effective stress analysis, 340
Janbu's correction factor, 348
Janbu's general procedure of slices, 347
logarithmic failure surface, 344
undrained analysis, 336
Soil anchor plates, 393
Soil classification:
grain size distribution, 42
lime content, 43
organic content, 43
Soil injection:
bentonite, 470
bitumen, 470
cement and cement-clay grouting, 466
chemical grouting with hardener, 467
injectability limits, 469
polymers, 469
water glass, 468
Soil investigations:
cone penetration test (CPT), 64
dilatometer test, 82
direct shear test, 74, 93
dynamic probing, 61
echo sounding, 67
electric resistivity method, 67
fall-cone test, 96
field vane test, 99
geophysical methods, 65
oedometer test, 76
piezocone test (CPTU), 65
plate loading test, 85
pore pressure measurements, 67
pressuremeter test, 80
sampling, 69
seismic method, 65
standard penetration test (SPT), 61
triaxial test, 75, 95
unconfined compression test, 95
weight sounding, 63
Soil/structure interaction, 124, 219, 491
Specific density, 13
518
Spread foundation:
bearing capacity, 125
contact stresses, 107
settlement, 139
Spring constants:
horizontal vibrations, 261
rocking-sliding mode of vibration, 260
torsional vibrations, 261
vertical vibrations, 258
SPT, see Standard penetration test
Standard penetration test, 61
Static rollers, 403
Strain hardening, 73
Strain softening, 73
Strength correlations:
penetration resistance, 102
preconsolidation pressure, 101
Strength parameters, definitions:
effective, 32
drained, 32
total, 32
undrained, 33
Strength properties of soil, determination:
direct shear test, 93
fall-cone test, 96
field vane test, 98
pressuremeter test, 80
triaxial test, 75
unconfined compression test, 96
Strength properties:
non-cohesive soils, 46
cohesive soils, 46
intermediary soils, 47
Stress distribution:
circular flexible load (e.g. oil tanks), 112
embankments, 110-112, 323
raft foundations, 117-124
rocking mode of vibration, 252
spread foundations, 107-117
Stress wave analysis, 162, 179
Structural discontinuities, 16
Subgrade coefficient theory, 118
Superstructure, influence of, 124
Suspended matter, 11
Svidyn analysis, 182
Swedish Standard Piston Sampler, 70
Tanks:
damage problems, 488
design criteria, 487
Subject
index
Subject
index
519