ABSTRACT
A mathematical technique is hereby advanced for investigating the bearing
capacity and associated normal stress distribution at failure of soil
foundations. The stability equations are obtained using the limit equilibrium
(LE) conditions. The additions of vertical, horizontal and rotational equilibria
are transformed mathematically with respect to the soil shearing strength,
leading to the derivation of the equation of the functional Q, and two integral
constraints. Generally, no constitutive law beyond the conlomb’s yield criterion
is incorporated in the formulation. Consequently, no constraints are placed on
the character of the criticals except the overall equilibrium of the failing soil
section. The critical normal stress distribution, dmin, and consequently the
load, Qmin, determined as a result of the minimization of the functional are the
smallest stress and load parameters that can cause failure. In other words, for
a soil with strength parameters c, ø, ૪, and footing with geometry B, H, when
stress d < dmin (c, ø, ૪, B, H) and load Q <Qmin (C, ø, ૪, B, H) foundation is
stable. Otherwise the stability would depend on the constitutive character of the
foundation soil. In the mathematical method employed, the stability analysis is
transcribed as a minimization problem in the calculus of variations. The result
of the analysis shows, among others, that the Meycrhoff and Hansen’s
Superposition approaches can be derived using the technique of variational
calculus, and consequently the representation of the bearing capacity by the
three factors Nq, Nc, and N૪ is appropriately possible. Finally, the classical
relation between Nc and Nq is again found by the LE approach and is therefore
independent of the constitutive law of the soil medium.
TABLE OF CONTENTS
Title Page
Certification
Approval
Dedication
Acknowledgement
Abstract
Table of Content
Notations
List of Tables
List of Figures
CHAPTER: INTRODUCTION
1.1Background of Study
1.2Research Problem
1.3Objectives
1.4Significance of Study
1.5Scope and Limitations
CHAPTER TWO: LITERATURE REVIEW
2.1 Historical Background
2.1.1 Analytical Methods for Determining Ultimate Bearing capacity of
foundations
2.2 Basic principles of variational calculus
2.2.1 Necessary Condition for Extremum
2.2.2 Euler-Lagrangian Equation
2.3 Basis for parametric representation
2.4 The lagrangian multiplier Approach
2.5 Isoperimetric problems
2.6 Variational Nature of Soil stability problems
2.6.1 Variational Formulation of stability problems
2.6.2 Conditions to solving bearing capacity problems of footing on slope
2.6.3 Failure modes of soil Foundation
2.7 Conclusion
CHAPTER THREE: MATHEMATICAL DERIVATIONS AND
SOLUTION
3.1 Statement of problem
3.2 Fundamental Assumptions
3.3 Limitations
vii
3.4 Boundary conditions
3.5 Non-dimensional parametric representation
3.6 Construction of Euler-Lagrangian Intermediate function for the problem
3.7 Formulation of Euler-lagrangian Differential Equation for the problem
3.8 Co-ordinate Transformation and General Solution
3.8.1 Co-ordinate Transformation
3.8.2 Solution of Resulting Differential Equation
3.8.3 Solution of Transversality condtion
3.8.4 Determination of constants of Integration
3.9 Bearing capacity determination
CHAPTER FOUR: RESULTS AND DISCUSSIONS
CHAPTER FIVE: CONCLUSION AND RECOMMENDATION
References
Appendixes
viii
CHAPTER ONE
INTRODUCTION
Many of the problems encountered in soil Mechanics and Foundation
Engineering Designs are the extreme-value type. These problems include the
stability of sloppy soil, the bearing capacity of foundations on horizontal,
adjacent to sloppy soil and on sloppy soil, the limiting forces (active-Pa and
passive Pp) acting on retaining structures like retaining walls, dams, sheet pile
walls and others.
All problems of the types mentioned above can be solved within the
framework of the limiting equilibrium (LE) approach. This approach which
considers the overall stability of a “test body” bounded by soil surface [y(x)]
and ship surface [y(x)] is based on the following three concepts [1].
(a) Satisfaction of failure criteria S = f (d) along the ship surface, y(x)
over which t (x) and d (x) constitute the shear and normal stresses
distribution.
(b) Satisfaction of all equilibrium equations for the test body (vertical,
horizontal and rotational equilibria).
(c) Extremization of the factor S with respect to two unknown functions
y(x) and d (x). Thus S is considered to be function of these (y (x) and
d (x) functions.
The extreme value is defined as;
Sex = Extr S [y (x) , (x) ]- – – – – – – – – – – – – – – – – – – – – 1.1 d
However, the determination of the bearing capacity of soil and
associated critical rupture surface and normal stress condition along the surface
remains one of the most important problems of engineering soil mechanics.
Several approaches to this problem have evolved over the years.
2
One of the early sets of bearing capacity equations was proposed by
Terzaghi. These equations by Terzaghi used shape factors noted when the
limitations of the equation were discussed. These equations were produced
from a slightly modified bearing capacity theory developed by Prandtl from the
theory of plasticity to analyze the punching of rigid base into a softer (soil)
material [2].
Another method which has been widely used, though equally
misleading, involves the determination of the bearing capacity by the plate
loading test at given work site. No doubt, the size of the plate vis-a-vise the
prototype physical footing lack accurate correlation. Besides, the significant
depth of pressure influence is usually not specified in the code [3].
The analytical methods of prediction of the ultimate bearing capacity of
soils originated from prandtl [4] plastic equilibrium theory, developed
originally for the analysis of failure in a block of metal under a long narrow
loading.
Accordingly, Prandtl identified zones in the metal at failure as follows:
(a) A wedge zone under the loaded area pressing the material downward
as a unit.
(b) Two zones of all-radial failure planes bounded by a logarithmic spiral
curve.
(c) Two triangular zones forced by pressure upward and outward as two
independent units.
Although the experimental behaviour of loaded soil is not in close
agreement with prandtl’s model, the mechanism of failure of most soils permits
the utilization of prandtl’s ultimate stress equations for the calculation of the
bearing capacity of cohesive soils of known C and ø under narrow footings.
The solution advanced by Prandtl is of course only a particular solution
for which the width of the strip and its position below ground surface are
3
neglected and the unit weight r, assumed to be zero i.e. for weightless
materials.
Although efforts were made by other researchers like Hansen,
Meyerhorf, Vesic etc [4] to present more encompassing and dependable
solution, it was Terzaghi [2] who developed the first rational and practical
approach to this problem. The method involves three determinant factors i.e.
(a) the soil unit weight, r.
(b) the effect of surcharge, q or applied load Q.
(c) the strength parameters of the soil, therefore, it is more
comprehensive than any other approach before it.
Terzaghi had expressed his result in simple super possible form such that
contributions to bearing capacity from different soil and loading parameters are
summed. These contributions are expressed with three bearing capacity factors
with respect to the effect of cohesion, unit weight and surcharge thus Nc, Nr,
and Nq.
Meyerhoff [2] had also used a technique similar to that of Terzaghi’s
approximate solutions. By including shape and depth factors for plastic
equilibrium of footing and assuming failure mechanism, like Terzaghi, he
expressed results with bearing capacity factors. It has been generally agreed
that the bearing capacity obtained by Terzaghi’s method are conservative, and
experiments on model and full-scale footings been to substantiate this for
cohesionless soil [5].
However, rigorous treatment of the bearing capacity problems have been
based on the theory of plasticity. Such treatments have involved a solution of
the boundary value problem for the soil-foundation system, and have therefore
been very complicated [1]. Consequently, complete mathematical solutions
have been obtained for a few very idealized cases for instance, frictionless and
weightless materials. Besides, the available information with respect to the
4
nature of soil plasticity indicate the necessity of utilizing a non-associative flow
rule as material model. In that case, even a numerical solution of the boundary
value problem becomes almost intractable.
The difficulties so far outlined in the forgoing have further accentuated
the need to utilize the considerations of the overall stability (limiting
equilibrium) in order to evaluate the ultimate foundation load. The use of the
stability approach, however, requires a for knowledge of the shape of the
critical rupture surface as well as the distribution of the normal stresses of
failure along this surface.
Hitherto, none of the above two parameters has been mathematically
quantified and so bearing capacity calculations have been based on various
assumed rupture lines and normal stress distributions. The existing methods
therefore, differ from one another in the assumptions about the character of the
functions y(x) and d(x). Most of the assumptions are motivated by the available
plasticity solutions for idealized cases. The resulting solutions, therefore,
contain errors of unknown magnitude.
Usually, the straight line, the circular arc, and the logarithmic spiral are
the widely assumed character of y(x) (failure surface). The form of d(x)
(normal stress distribution) is either assumed directly or introduced indirectly
by assumption regarding the nature of the interaction between sections of
sliding mass. However, if the aforementioned assumptions regarding y(x) may
be validated by some experimental observations, what about the popular
assumptions regarding d(x) which are considerably arbitrary? Again the
existing methods are poorly argued! [1]. As a result, one cannot apply them
with sufficient confidence. Above all, one cannot conclude in any specific case
which one of the methods is most justified.
The foregoing further accentuates the need for a more accurate and
encompassing formulation based on limiting equilibrium conditions and free
5
from assumptions with respect to the rupture surface and normal stress
distribution along it. Several attempts have been made in this directions but it
was Akubuiro [1] who tried to use variational calculus to evolve an equation
for the rupture surface with a basic assumption that the soil surface is
horizontal which is been criticized because in real life, no surface is horizontal.
The present work therefore attempts to advance the solution to the
stability problem further by formulating the stability equations using the
limiting equilibrium conditions, transcribing the problem as a minimization
problem in the calculus of variations and then determining the normal stress
distribution along the failure surface with the basic assumption that the
foundation is on a slope. With the normal stress distribution at failure and the
rupture surface mathematically defined, coupled with Feda’s [6] semi-empirical
equations, the equation of the bearing capacity of the soil is formulated from
determinable parameters of the soil by completely solving the resulting
equations using the techniques of calculus of variations.
The critical stress distribution must satisfy the requirement that the ratio
of the shearing strength of the soil along the surface of sliding and the shearing
stress tending to produce the sliding must be a minimum [7]. Hence the
determination of the critical stress distribution belongs to the category of
maxima and minima (extreme-value) problems.
On the other hand, the calculus of variations is an advanced
generalization of the calculus of maxima and minima, in which the maxima and
minima of functionals are studied instead of functions. A functional here is
technically defined to mean a correspondence which assigns a definite (real)
number to each function (or curve) belonging to some class [8]. Thus a
functional is a kind of function where the otherwise independent variable is
itself a function (or curve).
6
The decision to use the theory of calculus of variations as the analytical
test here is predicated on the basis of the fact that the problem of determining
the critical normal stress distribution d(x) along the rupture surface is a
minimization problem which can therefore be advantageously transcribed as a
problem of calculus of variations.
1.1 Historical Background of Study
The calculus of variations has ranked for nearly three centuries among
the most important branches of mathematical analysis. It can be applied with
great power to a wide range of problems in pure and applied mathematics, and
can also be used to express the fundamental principles of both applied
mathematics and mathematical physics in unusually simple and elegant forms
[9].
In general, the history of the subject has been conveniently divided into
four different periods by Pars [10], thus:
(i) In its earliest period; ideas of variational calculus emerged from
Newton’s formulation of the problem of the solids of revolutions
having minimum resistance when rotated through the air of density l .
The physical hypothesis of the Newton’s problem was to find a curve
joining the point A, (origin) with coordinates (O, O) with B, (any
other point in first quadrant) with coordinates (x>0, y>0), such that in
rotating the curve about ox, the resulting solid of revolution shall
suffer the least possible resistance when it moves to the left through
the air at a steady speed. For the resistance, Newton gave the formula
as;
R = 2prv2 ò ySin2ydy – – – – – – – – – – – – – – – – 1.1
Where R = resistance suffered
7
ρ = density of air
v = sped of projectile
tan ψ = y1 = i.e. slope of curve
By omitting the positive multiplier, the integral [10] to be minimized is
( )
ò ( ) – – – – – – – – – – – –
+
= x dx
y
I Y Y
0 1 2
1 3
1.2
1
The brachistochrone problem presents yet another classical example of
the early variational calculus problems [11] and [9]. Under it, the shape of a
smooth wire joining A to B is determined such that a bead sliding on the wire
under gravity and starting from A with a given speed reaches B in the shortest
possible time. The curve is found to lie in the vertical plane through A and B
when the axis OY is taken vertically and ox, the energy level. The speed of the
bead at any point on the wire is (2gy) and the time for the journey from A to B
along the curve y = ø (x) is;
( ( ) ) ò – – – – – – – – – – – – – –
+
= x
x
dx
Y
Y
gy
T 1 1.3
2
1 1 2
The brachistochrome problem becomes to minimize the integral.
( ( ) ) ò – – – – – – – – – – – – – –
+
= x
x
dx
Y
I 1 Y 1.4 1 2
(ii) The second stage in the development of the theory of calculus of
variations heralds the emergence of a systematic and fairly more
elaborate procedures with broad-based applicability. It was the era of
Euler and Lagrange.
In minimizing the integral equation.
= ò – – – – – – – – – – – – – – – – x
x
I F [x, f (x), f 1 (x)] 1.5
8
Euler had formulated a famous differential equation.
FY 1 [x, (x), 1 (x)] = FY [x, (x), 1 (x)] – – – – – – – – – -1.6
dx
d f f f f
The Euler’s equation must be satisfied by any minimizing curve.
Y = f (x) – – – – – – – – – – – – – – – – – – – – – – – – – – -1.7
(iii) In the third period of development of the variational calculus,
distinctions between conditions necessary for a minimizing curve and
conditions sufficient to ensure a minimum emerged clearly [11].
(iv) Among the prominent contributors in recent developments of the
study are Hilbert, Bolza, Bliss, Tonelli etc.
In general, the principal steps in the progress of the calculus of variations
during recent past may be characterized as follows [9].
(a) A critical revision of the foundations and demonstrations of the older
theory of the first and second variations according to the modern
requirements of vigour, by weierstrass, Erdmann, Du-Bois-Ray mond,
schefer, and Schwarz. The result of this revision was a charper
formulation of the problems, vigorous proofs for the first three necessary
conditions, and a vigorous proof of the sufficiency of these conditions
for what is now called a “weak” extremum.
(b) Weierstrass extension of the theory of the first and second variations to
the case where the curves under consideration are given in parametricrepresentation
[9]. This was a major advance of great importance for all
geometrical applications of the calculus of variations; for the older
method implied- for geometrical problems-a rather artificial restrictions.
(c) Weierstrass discovery of the fourth necessary condition and his
sufficiency proof for a so-called “strong” extremum, which gave for the
first time a complete solution by means of an entirely new method based
upon what is now known as “weierstrass construction”.
9
(d) Kneser’s theory, which is based upon an extension of certain theories of
geodesiecs to extremals in general. This new method furnishes likewise
a complete system of sufficient conditions and goes beyond weierstrass
theory.
(e) Hilbert’s a priori existence proof for an extremum of definite integral-a
discovery of far reaching importance in both calculus of variations and
general theory of functions.
1.2 Research Problem
The determination of bearing capacity of foundation soils on slope and
corresponding stress distribution, d(x) along the failure surface lies the problem
of this research work. Only few researchers have seen this as a problem
because they in this area of study accept the erroneous assumption that all
foundations rest on a horizontal soil condition.
The present of therefore attempts to advance the solution to the stability
problem by formulating the stability equations using the limiting equilibrium
conditions.
1.3 Objectives
So far, the determination of the bearing capacity of soil and associated
critical rupture surface and normal stress condition along the surface remains
one of the most important problems of engineering soil mechanics. However
the objective of this research work is to basically determine the bearing
capacity of foundation soil on slope and its associated critical stress distribution
along this plane of failure by employing a more mathematical approach to
finding solutions to this problem.
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1.4 Significance of Study
This research work is very important in that it has given researchers a
wide range of knowledge towards attaining to solutions to the problem of
determining the bearing capacity of foundation soils on slope and also
identified areas for future research. It is considered that a successful
implementation of this approach will be most useful in that both the shape of
the critical rupture surface, the distribution of the normal stress at failure along
it and the bearing capacity can now be evaluated from measurable parameters
of the soil without making empirical, sometimes misleading assumptions as has
been the case hitherto.
1.5 Scope and Limitations
Bearing capacity of footings on soils are usually calculated by super
position method suggested by Terzaghi [12] in which the contributions to the
bearing capacity from different soils are summed up, for different loading
parameters. These contributions are expressed in three bearing capacity factors
Nc, Nr and Nq representing the effects due to cohesion C, soil unit weight r and
surface loading (surcharge) q, respectively. These parameters N are all
functions of the internal frictional angle, ø. It is known that this quasi-empirical
approach assumes that the effect of the various contributions, are directly super
possible, whereas in actual fact, soil behaviour is non-linear and thus super
position does not strictly hold for general soil bearing capacity.
Meyerhoff has obtained by technique similar to Terzaghi’s approximate
solutions [13] to the plastic equilibrium of footing (deep and shallow) by
assuming failure mechanism for the footing and like Terzaghi, expressed result
in form of bearing capacity factors.
It has, however, been experimentally found, using models and full scale
footings [6] on cohesionless soils, that the bearing capacity obtained using
Terzaghi’s method falls short of the actual Qo = Q/c = 0. No work, however,
11
has been done (experimentally or analytically) on soils with both cohesion and
friction for the purpose of checking the validity of Terzaghi’s superposition
approach.
The present work bridges this gap, first a mathematical technique is
developed for determining the critical normal stress distribution along the
rupture line using only determinable strength parameters of the soil.
Second, the bearing capacity of the footing on sloppy soils with both C
and ø is formulated. The result of the bearing capacity formulation is expressed
in terms of bearing capacity factors. However, the bearing capacity factors, N
are here determined by a method different from Terzaghi’s and Meyerhoff’s
approaches [13, 22]. The N-factors are compared with these of the Terzaghi’s
and Meyerhoff’s solution. Variations within admissible limits are explained
based on the variations in the basic theories governing both analyses.
The basic mechanism applied here is that the soil footing system is
assumed to satisfy conditions of horizontal, vertical and moment equiolibria.
Thus, the ultimate load functional and the various constraining integral
equations are generated from first principles.
The results which are expressed as follows:
( )
( )
( , , , , ) 0 1.10
;
, , , , 0 1.9
;
, , , , 1.8
1
1
1
= – – – – – – – – – – – – – – – – – – – –
= – – – – – – – – – – – – – – – – – – –
= – – – – – – – – – – – – – – – – –
M y y c
For the horizontal equilibrium and
L y c dx
For the vertical equilibrium
Q K y y c dx
d y
d y
d y
For the rotational equilibrium; and then used to generate the Lagrangian
intermediate auxiliary functions.
This is then shown to belong to the class of variational problems of the
isoperimetric type. By introducing non-dimensional parameters, the solution is
12
constructed using Lagrangian undetermined multipliers. The criticals
d (x) and y(x) are then determined by subjecting the auxiliary function to;
(a) systems of Euler Differential equations,
(b) the integral constraint equations,
(c) set of boundary conditions at the end points, and
(d) the variational boundary condition (condition of transversality), and
finally solving using polar coordinate transformations.
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