Design of Foundations for Buildings and Bridges using Nonlinear Soil
Model
A. A. Alekhin
1
, G. S. Lobanova
2
and V. V. Orlov
2
1
Ural Federal University, Yekaterinburg, Russia
2
Ural State University of Railway Transport, Ekaterinburg, Russia
Keywords: Soil base of structures, soil, deformation, physical nonlinearity, mathematical model, model parameters,
determination of parameters.
Abstract: The article substantiates the need to design foundations of bridge supports on non-rocky soil base taking into
account physically and geometrically nonlinear features of soil deformation. Dusty-clayey soils (sandy loams,
loams, clays, and their older and stronger formations - saprolites) of semi-hard, hard, and often tough-plastic
consistency, serve as a reliable and economical soil basis for various structures, including bridges. According
to Article 16 of Technical Regulations on the Safety of Buildings and Structures (Federal Law № 384-FZ), as
well as paragraphs 5.1.11 of SP 22.13330.2016 ("Soil bases of Buildings and Structures") in this case design
must be carried out taking into account the nonlinear deformation of soils. The use in design of a theory
(design model) that corresponds to physical essence of soils instead of nominal for them, physically linear
(Hooke) model, makes it possible to increase radically reliability of structures and obtain at least a twofold
saving in the cost of foundations.
1 INTRODUCTION
Soils: hard rocks and non-hard soils (sands, sandy
loams, loams, clays) are the only solid natural
environment, one of four (except liquid, gaseous and
electromagnetic) natural environments at Earth and,
obviously, at other planets. From an engineering point
of view, soil base is bearing element of all structures,
as a rule, in natural, and sometimes in an artificially
improved state. If soil base consists of non-rocky
soils, then it turns out to be the most deformable load-
bearing element of the structure. But often due to
technical, economic, hydrological reasons, it is
impractical (for example, when dense soils or rocks
are very deeply buried) to pass through non-rock soil
by drilling or driven piles. In the case of using pile
foundations on friction piles (not resting on rocky
base), as well as non-piled foundations, it is necessary
to predict (calculate) their settlements and other
deformations of foundations (for example, their
incline and difference in settlement with neighboring
foundations, which is especially sensitive when
providing reliability and working capacity of bridge
spans. This situation imposes increased requirements
on resolving capabilities of soil base design model,
but in fact, on design soil model, in other words,
mechanical and mathematical deformation model of
non-rock soil (for rock soil at main positions it is
identical to deformation model of concrete, metal and
rubber, which means to Hooke-Young's linear
deformation theory) must corresponds as much as
possible to real mechanics of non-rocky soil,
determined in turn by its physical nature, which is so
complex and multifactorial that even generally
accepted in mechanics principle of replacement in
deformation theory of real soil structure by an ideal
continuous medium, which means introduction
instead of real forces between particles of stresses, as
forces acting on an infinitely small (idealization) area
and relative deformations in an infinitely small
(idealization) volume, creates an error in predicting
soil deformations by about 25...30% (for metals, for
example, as much more homogeneous formations, the
error of such replacement is up to 5%).
More serious errors arise in prediction of internal
forces (stresses) and, because of this, in external force
interactions of neighboring objects (for example, in
pressure of foundation on its soil base). But when
designing, not only quantitative values are important,
but even more important is character of their
distribution, for example, unevenness, and this is
determined the more reliably, the more adequate is
Alekhin, A., Lobanova, G. and Orlov, V.
Design of Foundations for Buildings and Bridges using Nonlinear Soil Model.
DOI: 10.5220/0011582300003527
In Proceedings of the 1st International Scientific and Practical Conference on Transport: Logistics, Construction, Maintenance, Management (TLC2M 2022), pages 239-244
ISBN: 978-989-758-606-4
Copyright
c
2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)
239
mechanical and mathematical model to mechanics of
soil, by the way, in this case, quantitative values are
also more reliably determined due to decrease in
absolute terms of error from replacing real
environment in design with continuous one.
Approximately same assessment of the complexity of
geotechnical design belongs to famous Anglo-Polish
engineer and geotechnical scientist Zenkevich, a
specialist in computer-aided design of soil bases of
offshore oil platforms, who actually excluded an
accurate prediction of soil deformation during
loading even in the case of using the most advanced
soil model, but at the same time who pointed out the
need for ability of design specialists to model well the
mechanical properties of soil which are necessary to
solve the problem (Zienciewich, 1978).
Unfortunately, in the 1920s, when important problem
of reliable design of soil bases for large and complex
industrial and civil objects arose, no other
deformation theories except Hooke's theory (the
theory of linear, more precisely, linear elastic
deformation), especially for soil, did not exist
(Terzaghi, 1961). However, back in 1798, the Swiss-
Russian mathematician Fuss proposed an engineering
method for predicting depth of wheel rut of carriages,
carts and cannon carriages, which was important for
Petersburg soils (Fuss, 1798), implemented later, in
1872 (Fuss, 1798) by Saxon scientist Winkler in a
linear formula for predicting settlements of railway
sleepers and, accordingly, deflections of railway
tracks (Winkler, 1872) at the place of pressure
application P: S = P / C
z
(here C
z
= const is a
coefficient of proportionality, which Winkler called
“bed coefficient” on not quite correct analogy with
stiffness of sofa springs). But in the 1920s, after many
checks (Kurdyumov, 1894; Minyaev, 1916; Gerner's
experiments with a round pressure area, 1932;
Bernatsky, 1935), this formula was rejected as a
possible mechanical and mathematical deformation
soil model for design of structures (Terzaghi, 1961):
firstly, due to the absence of a relationship between
relative deformations εij and stresses σij for an
elementary but representative volume of soil medium
(sample), which reduces Fuss-Winkler formula to
some isolated boundary condition which is not in
agreement with mechanism of internal deformation of
soil massif, that, in turn, does not allow to carry out a
full-fledged analysis of its deformation, for example,
analysis of the effect of load influence on adjacent
sections of soil massif and on neighboring structures;
secondly, the hypothesis of constancy of the stiffness
coefficient C
z
was not confirmed, which is a
consequence of previous defect of Fuss-Winkler’s
formula. In this regard, the use of Fuss-Winkler’s
formula for calculating deformations of soil base in
some widely advertised and currently used programs
(for example, in programs "LIRA", "SCAD" and
others) for any method of determining the value of
stiffness coefficient C
z
contradicts to basic principles
of mechanics and is explained by failure of
developers of these programs in their attempts to
apply correct soil base model. The presence of this
formula in SP 22.13330.2016 ("Soil bases of
buildings and structures") (Gosstroyizdat of Russia,
2017) is some kind of temporary compromise, which,
however, for example, in Yekaterinburg has already
led to emergencies associated with incorrect
calculation of soil bases at several objects. In fact,
Fuss-Winkler formula, due to its purpose and method
of derivation, can be correctly used only for an
approximate, and therefore actually estimated
forecast of lateral displacement of a driven pile
(actually more is not required) and for approximately
same type of analysis of elastic work of soil under
action of a not very intensive dynamic load from
industrial equipment (but not from a much more
intensive train load). In general, and it should be well
known to engineers, that veracity and reliability of
geotechnical calculations as for any other mechanical
calculations, are ensured by using four groups of
resolving relations: 1) equilibrium-motion equations,
Newton, 1650; 2) geometric relations of compatibility
of deformations and displacements in the framework
of the theory of continuous medium, Cauchy, 1820s;
3) generalized physical relationships between relative
deformations and stresses, Genki, 1920s; and, most
importantly (Bell, 1984), 4) obtained from results of
special experiments (tests), relations between relative
deformations and stresses in a conventionally
elementary (small), but representative volume
(sample) of a solid formation, including for soil
medium. These relations in turn determine type and
value of rigidity of this solid formation, in this case of
soil. But Fuss-Winkler/s formula S = P / C
z
does not
belong to any of these four groups of resolving
relations, even as defining stiffness relation, despite
its outward likeness to Hooke-Young stiffness
relation ε = σ / E, including the constancy in both
formulas of coefficients of proportionality C
z
and E.
As indicated above, there is no in Fuss-Winkler
formula to contrast to Hooke-Young formula direct
connection between values included in Fuss-Winkler
formula and values included in other resolving
relations. It is for this reason that Austrian-American
geotechnician Terzagi, founder of International
Geotechnical Society, in absence of other
deformation theories, as well as in full absence at that
time devices for obtaining defining relationships
TLC2M 2022 - INTERNATIONAL SCIENTIFIC AND PRACTICAL CONFERENCE TLC2M TRANSPORT: LOGISTICS,
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between relative deformations and stresses for soils
and effective computing means, has proposed in
1920s (Terzaghi, 1961) to apply for predicting
deformations of soils had being applied at that time
for all materials, Hooke's theory (now known as
theory of linear deformation), which mathematically
closed above mentioned system of resolving
relations. During that period of large-scale
development of industrialization and housing
construction all over the world, adoption of
mathematically correct design model for soil and soil
base has much intensified building design in all
directions. But contradiction that arose due to
insurmountable circumstances at the beginning of
XX-th century (primarily due to the lack of a
mechanical and mathematical design model adequate
to mechanical properties of soils) and turned into a
serious problem at the beginning of XXI-th century in
condition of construction of high-rise and other
uniquely complex objects, and especially bridges, is
that for calculating deformations of soils was adopted
mechanical and mathematical deformation model of
incompatible with them on physics and mechanics
metals, moreover, in many aspects this unreliable and
dangerous approach is still preserved, including, and
that is especially seriously threatens to technical
safety, in regulatory documents, including SP
35.13330.2011 "Bridges and Pipes", although since
2009 this unacceptable situation was actually blocked
by article 16 of Federal Law No. 384-FZ (Technical
Regulations on Safety of Buildings and Structures),
which requires the use in geotechnical design of
adequate for non-rocky soils physically nonlinear
deformation model (Federal Law № 384-FZ, 2010).
2 MAIN FEATURES OF
PHYSICALLY NONLINEAR
SOIL DEFORMATION
Obviously contradictory proposal of Terzaghi about
adoption for calculation of deformations of soils with
loose internal connections deformation theory of,
absolutely opposite to them in physical and
mechanical properties, dense materials (metals and
even rubber) could not but raise numerous questions
of geotechnical specialists, that even demanded later,
in 1948 year, when Terzaghi's proposal, again due to
insurmountable circumstances (absence, on one hand,
a proven, soil deformation model and accessible, with
the necessary power of computing means, and on
other hand, urgent need on prompt restoration of
destroyed by war and construction of new objects),
nevertheless, was adopted in USSR (Gersevanov,
1948), to apply this illogism, namely, having
identifying value of Young's modulus E, which is
consistent for materials, with inconsistent, as it was
later found out for soils (Lushnikov, 1969; Ruppeneit,
1973), value of modulus of deformation with same
designation E. although the physico-mechanical
essence of these two values is absolutely different
what is more, they are not even close analogs, as it
was actually assumed in 1948 (Gersevanov, 1948).
Young's modulus E by definition (according to the
technology of converting results of experiments of
Hooke and his folowers with metals and other similar
materials into mechanical and mathematical formula
of Young) reflects exclusively only direct
proportionality between applied unidirectional force
and resulting unidirectional deformation (and even
adopted later, more general in relation to Young's
formula, Genki's ratios with coefficient Poisson ν did
not change physical and mechanical essence of
Young's modulus E); modulus of soil deformation E
as a constant for any method of its determination, due
to the nonlinear features of deformation of soils,
inevitably includes an element of disproportionality
between applied forces and resulting deformation,
moreover, multidirectional, which just creates effects
of physical and geometric nonlinearity, and also
effects of contraction and dilatancy, one of a
manifestations of which is, for example, a multiple
difference in values of modulus of deformation
obtained from different methods of soil compression
testing (settlement plate, pressuremeter, odometr)
(Gersevanov, 1948; Lushnikov, 1969; Ruppeneit,
1973), which, in principle, cannot be when
determining Young's modulus, for example, for
metal. True, unreliable are formulas themselves
(Lame and Schleicher) for calculating the modulus of
soil deformation according to GOST 20276-2012
("Soils. Methods for in-situ determination of strength
and deformability characteristics") (GOST 20276–
2012, 2013), since they are derived from the
deformation theory not of soil, but of metal, that is,
from Hooke's theory of linear deformation, There are
also many other serious inconsistencies from
application of physically linear theory of deformation
to physically nonlinear soils, including, recently
noted even in textbooks, significant differences for all
types of buildings and for all types of soils between
calculated and actual values of deformations of soil
bases (Ukhov, 2002). At the same time, back in 1939
... 1940 Leningrad scientist Botkin, obviously
disagreeing with Terzagi's proposal to accept the
theory of deformation of metals and rubber for
calculating deformation of soils, performed tests on
Design of Foundations for Buildings and Bridges using Nonlinear Soil Model
241
samples of sand and clay soils in a special triaxial
compression device recently created in Germany -
stabilometer, received deformation graphs of sandy
and clay soils (Botkin, 1939; Botkin, 1940), depicted
in the form of diagrams in Figure 1b (Botkin, 1939;
Botkin, 1940). Thus, firstly physically nonlinear type
of deformation of soils was investigated, which was
different from the physically linear type of
deformation proposed by Terzaghi, also shown
diagrammatically in Figure 1a.
a) physically linear (for structural materials
and rocks)
b) physically nonlinear (for non-rocky soils:
sands and clays)
Figure 1: Types of deformation.
Names of types of deformation (that time
physically nonlinear type of deformation was known
and was studied only for soils; for rocks, by analogy
with already studied concrete, metals and rubber, the
physically linear type of deformation was adopted)
were given in 1950s according to the form of given
diagrams, but physically, for example, physically
nonlinear deformation, consists in dependence of
rigidity (resistance to deformation) of a solid medium
or solid material on their internal stress-strain state
(herein after SSS) and decrease of this rigidity as
stress-strain state state approaches to the limit state
for a given solid formation, namely to it strength
(resistance to destruction). Physical essence of
physically linear (Hooke's) deformation, on the
contrary to nonlinear deformation, consists in
independence from internal stress-strain state, i.e., in
the constancy of the rigidity of a solid medium or
solid materials throughout the entire deformation
process due to density and strength of their internal
bonds. In fact, in deformations of all solid formations
(natural and artificially improved soils, as well as
solid materials, there is a factor of physical
nonlinearity to one degree or another, sometimes in
certain areas of deformation (for example, area of
yielding for steel). But only in soils, both in natural
and in artificially transformed states, due to the
fragility and looseness of internal bonds physically
nonlinear type of deformation, in its essential
understanding, is present at all stages of deformation
without exception. Thus, it is obviously that design
model for a soil should be at a minimum reflects
physical and geometric nonlinearities, which is quite
definitely declared by paragraphs 5.1.11, 5.1.12, 5.3.3
and paragraphs of Appendix C of SP 22.13330.2016
“Soil bases of buildings and structures
"(Gosstroyizdat of Russia, 2017), and especially by
Article 16 of Technical Regulations on Safety of
Buildings and Structures (Federal Law № 384-FZ,
2010), which excludes the use of formulas of the
theory of a linearly deformable soil with Young's
modulus (modulus of deformation) E for designing
soil bases, which from the standpoint of real soil
deformation as physically nonlinear medium and
hence requirements of above paragraphs of SP
22.13330.2016 and Federal Law № 384-FZ, cannot
be used at designing of soil bases, especially since
this value at different points of soil base, as in plan
and in depth due to the factor of physical nonlinearity
(dependence of stiffness on stress-strain state) is
significantly different: naturally, the question arises
about the localization of soil base section, for which
value of modulus of deformation E is given in
reports of geological engineers. There is no answer
within the framework of a nominal, and therefore
unreliable for soil, physically linear model (the theory
of a linearly deformable medium). But this answer
can be easily found within the framework of a
physically nonlinear model adequate to mechanical
properties of soil. Phenomenological formulas for
stiffness characteristics of this theory were first
derived by Botkin: for modulus of volume change
TLC2M 2022 - INTERNATIONAL SCIENTIFIC AND PRACTICAL CONFERENCE TLC2M TRANSPORT: LOGISTICS,
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(bulk modulus) K = σ / ε = σ
1-α
/A
0
and for modulus
of change of shape (shear modulus) G = σi /ε
i
= (σ
u
-
σ
i
) / B = (Aσ+C) / (B+ (Here u = Aσ + C – strength
condition for soil according to Mises-Botkin version).
Thus, for soils, it is incorrect to determine values of
the parameters of the physically linear model E
(deformation modulus), which are different at
different points of the soil base and ν (Poisson's ratio,
which, by the way, for soils is almost impossible to
determine by direct measurements, but it is necessary
to determine values of parameters-constants of a
physically nonlinear model (for example, Botkin's
model: A0, α, A, B, C. Since values of these
parameters, as well as the parameters of any soil
model (and now there are quite a lot of them,
depending on the problems to be solved (Fedorovsky,
1985)), depend on the natural state of soil, including
its natural stress state, then their true significations
must be determined from the results of in-situ static
tests with the simplest scheme and the least disturbing
the natural state of soil. Brief mathematical
description of the algorithm of the method
(Fedorovsky, 1985) for Botkin soil model, as the most
verified in accordance with paragraph 5.1.12 of SP
22.13330.2016 ("Soil bases of buildings and
structures") (Gosstroyizdat of Russia, 2017) Using
true values of parameter stresses and strains of soil
masses and corresponding displacements of
foundations are determined. For rigid foundations of
columns and bridges supports, as well as for rigid slab
foundations, this problem has been completely
solved; it was also solved to determine the
deformations and stresses under the embankment,
which is important for analysis of karst problem. At
present, post-graduate students of Bridges and
Transport Tunnels Department are solving a similar,
but at the same time, due to computational features,
most difficult problem for slab foundations of any
rigidity. Comparison of results of geotechnical
calculations for bridge foundation using physically
nonlinear and physically linear (Hooke's)
deformation theories showed that at a pressure on soil
base of 400 kPa (40 t/m2), for first case average
settlement of foundation is 6.3 cm, and for second
case – 7. 7 cm. Thus, settlement according to
physically linear theory exceeded settlement
according to physically nonlinear theory by 20%,
which is explained by fact that at physically linear
case increase of soil stiffness with increasing depth is
not taken into account. Taking into account the still
unexplored uncertainty for soil on replacement of
granular soil medium to ideally continuous
(according to some data, it can be up to + 25%, and in
the case of using a linear model that is inadequate to
the soil, naturally more), the total difference in the
calculated settlements can reach 50% The nature of
the distribution of contact pressures under foundation,
as well as in the soil mass differ significantly in
physically nonlinear and physically linear
calculations, and m in both cases, they are uneven and
not equal to average pressure under bridge support
(see Fig. 2). unequal to average pressure from the
bridge (see Fig.2). The unevenness of diagram at
nonlinear case is explained by the dependence of soil
stiffness on stress state, which is adequately reflected
by physically nonlinear model. Nonlinearity of the
Figure 2: Diagrams of contact pressures Pk under the base of the bridge foundation according to nonlinear (solid line) and
linear (dashed line) calculations at ground pressure P = 400 kPa.
Design of Foundations for Buildings and Bridges using Nonlinear Soil Model
243
diagram at linear case is at more significant values of
pressure along the edges than in the nonlinear case is
explained by increased distribution ability of linear
model, inadequate to soil, in entire space of acting
forces compared to nonlinear one (for analogy, one
can compare distribution ability of rubber and
plasticine). The unevenness of contact pressures
seriously affects at the roll of bridge supports, which
is sensitive for bridge spans, therefore, the most
accurate determination of contact pressures is
extremely important for design of bridges, especially
railway ones. At the same time, it must be mind that
it is the physically nonlinear deformation that is main
mechanical feature of non-rocky soils.
3 CONCLUSIONS
To improve traffic safety deformations of the soil
base of bridge supports must be calculated using
physically and geometrically nonlinear soil model, as
required by Federal Law № 384-FZ (Technical
Regulations on the Safety of Buildings and
Structures).
The use of a physically and geometrically
nonlinear soil model makes it possible to obtain the
most reliable prediction of deformation of ground
embankments and its soil bases in occurrence of karst
cavities and other defects in them.
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