## 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.

10

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.

13

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