ONYELOWE, KENNEDY CHIBUZOR

PG/M.ENG/08/49285

PG/M. Sc/09/51723

BEARING CAPACITY AND CRITICAL NORMAL STRESS

DISTRIBUTION OF SOILS BY METHOD OF VARIATIONAL

CALCULUS

MACHANICAL ENGINEERING

A THESIS SUBMITTED TO THE DEPARTMENT OF MACHANICAL ENGINEERINGN,

FACULTY ENGINEERING, UNIVERSITY OF NIGERIA , NSUKKA

Webmaster

Digitally Signed by Webmaster’s Name

DN : CN = Webmaster’s name O= University of Nigeria, Nsukka

OU = Innovation Centre

2011

1

BEARING CAPACITY AND CRITICAL NORMAL STRESS DISTRIBUTION OF SOILS BY

METHOD OF VARIATIONAL CALCULUS

BY

ONYELOWE, KENNEDY CHIBUZOR

PG/M.ENG/08/49285

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT

FOR THE AWARD OF THE DEGREE OF MASTER OF ENGINEERING

IN SOIL MECHANICS AND FOUNDATION ENGINEERING

SIGNATURE OF AUTHOR __________________

STUDENT

CERTIFIED BY _________________

PROJECT SUPERVISOR

ACCEPTED BY ______________________

HEAD, DEPARTMENT OF

CIVIL ENGINEERING

i

CERTIFICATION

This is to certify that this work, Bearing Capacity and Critical Normal

Stress Distribution of Soils by Method of Variational Calculus was carried out

by me Onyelowe, Kennedy Chibuzor.

__________________ __________________

Signature Date

ii

APPROVAL

This work, Bearing Capacity and Critical Normal Stress Distribution of

Soils by Method of Variational Calculus is hereby approved as a satisfactory

project for the award of the degree of Master of Engineering (M.Eng) Soil

Mechanics and Foundation Engineering in Civil Engineering Department,

University of Nigeria, Nsukka.

________________________ ________________

Engr. Prof. J.C. Agunwamba Date

(Supervisor)

________________________ _________________

Engr. Prof. J.O. Eze-Uzoamaka Date

(Supervisor)

________________________ _________________

External Examiner Date

________________________ _________________

Engr. Prof. J. C. Ezeokonkwo Date

(Acting Head of Department)

iii

DEDICATION

This work is dedicated to the cosmic and the cause of research.

iv

ACKNOWLEDGEMENT

My sincere gratitude goes to my supervisors, Engr. Prof. J. C.

Agunwamba and Engr. Prof. J. O. Ezeuzoamaka and other members of staff of

the Department of Civil Engineering, University of Nigeria, Nsukka for their

time and contributions channeled towards ensuring that this programme

worked. Also to my wife who staked a fortune and prayed unrelentlessly to the

realization of this work.

I am not forgetting my parents, Mr. and Mrs. S.U. Onyelowe, my

brothers, Arc. D.C.B. Onyelowe and Livinson E. Onyelowe for their

encouragement and moral support. My grand councilor, Frater K.L. Ikeata is

not left out for his spiritual support. My profound gratitude goes to each one of

them.

Onyelowe K.C.

v

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, min, 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 < min (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.

vi

TABLE OF CONTENTS

Title Page

Certification

Approval

Dedication

Acknowledgement

Abstract

Table of Content

Notations

List of Tables

List of Figures

CHAPTER: INTRODUCTION 1.1 Background of Study

1.2 Research Problem

1.3 Objectives

1.4 Significance of Study

1.5 Scope 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

NOTATIONS

The following symbols are used in this work;

A (ө) = auxiliary function

D = integration constant

C = cohesion

c = non-dimensional cohesion

Hcc ˆ = reduced non-dimensional cohesion

e = eccentricity

H = depth of foundation embedment

H = non-dimensional depth

K,L,M = auxiliary functions

K(ө) = auxiliary function

Nc, Nq, N૪ = bearing capacity factors

o = initial reduced non-dimensional load at c = 0

Q = foundation load

Q = non-dimensional load

= reduced non-dimensional load

q = bearing capacity

)(r = equation of rupture surface in polar co-ordinates

),( r = polar coordinate system (non-dimension radius and angle)

S = lagrange’s intermediate function

(x, y) = cartesian coordinate system

(x0, y0), (x1, y1) = coordinates of end points of y(x)

yx, = non-dimensional coordinate system

1100 ,,, yxyx = non-dimensional coordinates of end points.

ix

rr yx , = non-dimensional coordinates of center of polar

coordinate system

Hi = horizontal component of soil

Y = vertical component of soil

)(xy = equation of rupture surface

B = width of foundation

dx

dy1tan = slope of y(x)

21, = lagrange’s undetermined multipliers

)(x = normal stress distribution

)()( orx = non-dimensional stress

H

= reduced non-dimensional stress

1 = reduced non-dimensional normal stress at end point

[r(ө), ө1]

)(x = shear stress distribution

= angle of internal friction

tan = internal friction parameter

M.S = Meyertroff’s solution

V.S = variational solution

H.S = Hansen’s solution

x

LIST OF TABLES

Table 4.1: Semi-empirical equation results

Table 4.2: Bearing capacity factors from variational solution

Table 4.3: Bearing capacity factors by Hansen’s solution

Table 4.4: Bearing capacity factors by Meyerhoff’s solution

Table 4.5: Bearing capacity factors from the present work and Akubuiro’s

work at zero slope.

Table 4.6: Computed values of ө0 and ө1

Table 4.7: Bearing capacity values by variational solution and Myerhoff’s

solution

xi

LIST OF FIGURES

Fig. 2.1: Earth pressure conditions immediately below a foundation

Fig. 2.2: Foundation failure rotation about one edge

Fig. 2.6.2a: Footing on a slope

Fig. 2.6.2b: Footing on a horizontal surface

Fig. 2.6.3a: Local shear failure pattern

Fig. 2.6.3b: Punching shear failure pattern

Fig. 2.6.3c: General shear failure pattern

Fig. 3.1: Foundation buried in sloppy soil mass

Fig. 3.2: Calculation scheme

Fig. 4.1: Plot of Bearing capacity factors Nc versus ø (V.S /M.S/ H.S)

Fig. 4.2: Bearing capacity factor Nc: Akubuiro/New Equation

1

CHAPTER I

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 () along the ship surface, y(x)

over which )(x and (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 (x). Thus S is considered to be function of these (y (x) and

(x) functions.

The extreme value is defined as;

1.1)(,)( xxySExtrSex

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 (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 (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 (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 (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 .

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;

1.12 22 dyySinvR

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

x

dxy

YYI

0 21

31

2.11

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;

x

xdx

Y

Y

gyT 3.1

1

2

121

The brachistochrome problem becomes to minimize the integral.

x

xdx

Y

YI 4.1

121

(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.

x

xxxxFI 5.1)](),(,[ 1

8

Euler had formulated a famous differential equation.

6.1)](),(,[)](),(,[ 111 xxxFYxxxFYdx

d

The Euler’s equation must be satisfied by any minimizing curve.

7.1)( xY

(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 parametric-

representation [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, (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:

10.10,,,,

;

9.10,,,,

;

8.1,,,,

1

1

1

cyyM

andmequilibriuhorizontaltheFor

dxcyL

mequilibriuverticaltheFor

dxcyyKQ

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

)()( xyandx 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

CHAPTER TWO

LITERATURE REVIEW

2.1 Analytical Methods for Determining Ultimate Bearing Capacity of

Foundations

The ultimate bearing capacity of a foundation is given the symbol qu and

there are various analytical methods by which it can be evaluated. As will be

discussed, some of these approaches are not all that suitable but they still form

a very useful introduction to the study of the bearing capacity of a foundation

[5].

2.1.1 Earth Pressure Theory

Consider an element of soil under a foundation (fig. 2.1). The vertical

downward pressure of the footing, qu, is a major principal stress causing a

corresponding Rankine active pressure, P. For particles beyond the edge of the

foundation this lateral stress can be considered as a major principal stress (i.e.

passive resistance) with its corresponding vertical minor principal stress ɤz

(weight of the soil). [5].

1.21

1

Sin

SinqP u

Fig. 2.1: Earth pressure conditions immediately below a foundation.

P

૪z qu

P

Also p = ૪z 2.21

1

Sin

Sin

qu = ૪z 3.21

12

Sin

Sin

14

This is also the formula for the ultimate bearing capacity, qu. It will be

seen that it is not satisfactory for shallow footings because when z = 0, qu = 0

[5]. Bell’s development of the Rankine solution for C – ø soils gives the

following equation.

2.1.2 Slip Circle Methods

With slip circle methods the foundation is assumed to fail by rotation

about some slip surface, usually taken as the arc of a circle. Almost all

foundation failures exhibit rotational effects and Fellenins (1927) showed that

the center of rotation is slightly above the base of the foundation and to one

side of it. He found that in a cohesive soil the ultimate bearing capacity for a

surface footing is [5].

6.225.5 cqu

To illustrate the method we consider a foundation failing by rotation

about one edge and founded at a depth z below the surface of the soil (fig. 2.2).

Fig. 2.2: Foundation failure rotation about one edge.

qu = ૪z 4.21

12

1

12

1

132

Sin

Sinc

Sin

Sinc

Sin

Sin

0For

qu = ૪z + 4c ------------------------------------------------------------------2.5

footingsurfaceforcqor u 4

B

Z D O

B

qu

L

•

C

15

Disturbing moment about 0.

7.222

2

qlBBxLBxqu

Resisting moment about 0.

Cohesion along cylindrical sliding

Surface = cπLB

Moment = cπLB2 …………………………….2.8

Cohesion along CD = czL

Moment = czLB2 …………………………….2.9

Weight of soil above foundation level = r2LB

Moment = ……………………………….2.10

2.1.3 Plastic Failure Theories

(a) Prandtl’s analysis: Prandtl (1921) was interested in the plastic failure of

metals and one of his solutions (for the penetration of a punch into

metal) can be applied to the case of a foundation penetrating down wards

into a soil with no attendant rotation. The analysis gives solutions for

various values of ø, and for a surface footing with ø = 0, Prandtl

obtained [5].

czLBcLBqLB

ei 22

π2

. ૪zLB2

2

B

czcqu

2π2 ૪z

cB

zc

π2π1π2

11.216.032.0128.6

cB

zc

૪z

૪z

16

19.21.5 cqu

(b) Terzaghi’s analysis: working on similar lines to Prandtl’s analysis,

Terzaghi (19 + 3) produced a formula for qu which allows for the effect

of cohesion and friction between the base of the footing and soil and is

also applicable to shallow (Z /B ≤ 1) and surface foundations. His

solution for a strip footing is [5].

cNcqu ૪z 5.0qN ૪ 20.2rBN

The coefficients depend upon the soils angle of shearing resistance. It

can be seen that Rankine’s theory does not give satisfactory results and that, for

variable subsoil conditions, equation 2.20 can be used but not sufficient for

foundations on slope where there is a reduction in Nc and Nq factors.

2.2 Basic Principles of Variational Calculus

The calculus of variations deals with the problem of maxima and minima

[8] and [9]. But while in the ordinary theory of maxima and minima, the

problem is to determine those values of the independent variables for which a

given function of these variables take a maximum or minimum value, in

calculus of variations, definite integrals involving one or more unknown

functions are considered and it is required to determine those unknown

functions that the definite integrals shall take a maximum or minimum value.

The definite integrals here are called functions.

In effect, we define a functional as a correspondence which assigns

definite (real) number to each function (or curve) belong to some class. It is

therefore a function where the otherwise independent variable is itself a

function (or curve). The functional expressed as follows:

2

121.2,,

x

xdx

dxdy

yxFI

17

For instance, a well-defined quantity-a number;

When x1 and x2 have definite numerical values

When the integrand f is a given function of the argument x, y, dy/dx and

When y is a given function of x.

The first problem of the calculus of variations involves comparison of

the values assumed by equation 2.2.1, when different choices of y as a function

of x as substituted into the integrand of equation 2.2.1. What is sought

specifically is the particular function of y = (y(x) which gives the equation its

maximum and minimum value.

Generalization of the first problem is effected in many ways. For

example:

The integrand of Eq. 2.2.1 may be replaced by a function of several

dependent variables, with respect to which a maximum or minimum of

the definite integral is sought.

The functions with respect to which the minimization or maximization is

sought or carried out may be required to satisfy some certain subsidiary

conditions.

Equation 2.2.1 may be replaced by a multiple integral whose minimum

or maximum is sought with respect to one or more functions of the

independent variables of integrals. For example, we seek to minimize the

double integral.

Ddxdy

dy

dw

dx

dwwyxf 2.2.2,,,,

carried out over a fixed domain D of the xy plane with respect to function w =

w (x, y).

The techniques of solving the problems of maximizing or minimizing

and related definite integral are intimately interwoven with those of solving the

18

problems of maxima and minima that are encountered in elementary

differential calculus:

If, for example, we seek to determine the values for which the function y

= g(x) achieves a maximum or minimum, we seek the derivative (dy/dx) =

g1(x), set g’(x) = 0 and solve for x. The roots of this equation – the only values

for which y = g(x) can possibly achieve a maximum or minimum – do not,

however, necessarily designated the locations of minima or maxima. The

condition g’(x) = 0 is merely a necessary condition for minimum or maximum.

The conditions of sufficiency involve derivatives of higher order than the first.

The vanishing of g1(x) for a given value of x implies merely that the

curve representing y = g(x) has a horizontal tangent at that value of x. A

horizontal tangent on the other hand may imply one of three circumstances:

minimum, maximum or horizontal inflexion. We call any of these an extremum

of y = g(x).

In general, however, the treatment of problems in calculus of variations

is analogous to treatment of maximum and minimum problems through the use

of first derivatives, while quite often we derive a set of boundary conditions for

a minimum or maximum and rely upon geometric or physical intuition to

establish the applicability of our solution.

2.2.1 Necessary Condition for Extremum

Now consider the function F (x, y, z) which is a function of a set of

admissible functions ø(x). The fundamental problem of the calculus of

variations is to find among the admissible functions ø(x), the one that realizes

an extremum of the functional.

2

1

1.2.2)(),(, 'x

xxxxFJ

Geometrically, we speak of admissible curve k, instead of admissible

function ø(x). The admissible curve k would be defined by;

19

.2.2.2)()( xxy

and the basic variational problem becomes that of finding from the admissible

curves the one that extremises the functional J, i.e.

k

dxyyxfJ 3.2.2,, 1

Y(x) is defined which gives J an extremum value [14].

7.2.20)()(

6.2.2)(),(),(

5.2.2)(),(

,0,

4.2.2),()(

21

xxand

xxoyxY

xyxoY

forthatsuch

xyxY

where η(x) is a function having a continuous first derivative and vanishes at the

end points. With the foregoing specifications, in any neighbourhood.

8.2.2),(),,(, 12

1

dxxyxyxFJx

x

the necessary condition for the integral J to have an extremum is that J should

be independent of α in the first order for all η(x) [15] i.e.

9.2.20 odx

dy

Consequently, if the function f(x) is differentiable at the extremum point

αo, then its differential is zero at this point:

10.2.20)(.. odJei

From the foregoing, we define as critical points, the points at which the

necessary condition for an extremum of the function J is fulfilled. Similarly, the

point (αo) at which dJ (αo) = 0 is defined as stationary point of the function J. It

is however noted that the conditions for the critical points and stationary points

of a given function J are equivalent, thus:

11.2.2)0(0)0( ood

dJdJ

The existence of a critical point does not therefore guarantee the

existence of an extremum of the function.

20

2.2.2 The Euler-Lagrangian Equation

The basic thrust of the Euler-Lagrangian equation is stated in analytical

terms as [9]. Given that there exists a twice differentiable function Y = y(x)

satisfying the conditions y(x1) = y1, Y(x2) = Y2, and which renders the

functional

12.2.2),,( 12

1

dxyyxfJx

x

a minimum, what is the differential equation satisfied by y(x)? The constants

x1, x2, y1, y2 are supposedly given and f is a function of the arguments x, y, y1

which is twice differential with respect to any, or any combination of them

[14].

We denote the function that extremizes equation (2.2.12) by y(x) and

proceed to form the one parameter family of “comparison” functions y(α, x)

defined by;

13.2.2)(),(),( xxoyxY

Where η(x) is an arbitrary differentiable function for which

14.2.20)()( 21 x and α is the parameter of the family. Now

replacing y and Y1 in Eq. 2.2.12 by y(x) and y1(x) respectively, we form the

integral 15.2.2),,( 12

1

dxyyxfJx

xwhere for a given function

η(x), the above integral is clearly a function of the parameter α.

The argument Y1 is given, through eqn 2.2.13 by

16.2.2)(),(),( 1111 xxoYxYY certainly, the integral Eqn 2.2.15

is minimum at α = o and is equivalent to replacing Y and Y1 respectively with

Y(x) and Y1(x). Also from elementary calculus [16], the necessary condition

for a minimum is that the vanishing of the first derivative of J with respect to α

must hold for α = o, thus;

21

18.2.2

,,

17.2.2)(

1

1

1

1

1

1

2

1

2

1

2

1

dxY

f

Y

f

dxY

Y

Fy

y

f

dxxyyf

JJ

x

x

x

x

x

x

Since setting α = 0 is equivalent to replacing (Y, Y1) by (Y(x), Y1(x) ), we

have according equation 2.2.18.

19.2.20)( 1

1

1 2

1

dx

Y

f

Y

foJ

x

x

Integrating by parts, the second term in the integral we obtain

20.2.2)( 1

1

2

11 2

1

dxY

f

dx

d

Yf

x

x

Yf

oJx

x

As a result of equation 2.2.14.

22.2.20

0

)0(

21.2.20

1

1

1

2

2

1

dxY

f

dx

d

Y

fJand

x

x

Y

f

x

x

The only way the above equation 2.2.20 can equal zero since (x) is zero

only at end points is for the function.

23.2.201

Yf

dx

d

Y

f

This is the Euler-Lagrangian differential equation which is the necessary

condition for the function J to have an extremum.

22

2.3 Basis for Parametric Representation

We proceed to show, however, that the extremizing relationship between

a pair of variables x and y is the same, whether the solution is derived under the

assumption that Y is a single-valued function of x or that a more general

parametric representation is required to express the relationship between x and

y. This we do by showing that the solution of the Euler Lagrange equation

derived on the basis of the assumption of the single-valuedness of Y as a

function of x satisfies also the system of Euler-Lagrangian equation derived on

the basis of the parametric relationship between x and y [17].

Under the assumption that Y is a single – valued function of x, the

functional to be minimized is given as

1.2.2),,( 12

1

dxYyxfJx

x

Where y is required to have the values Y1 and Y2 at x = x1 and x = x2

respectively. If instead we use the parametric representation x = x(t) and y =

y(t), where x(ti) = xi, y(ti) = yi for i =1, 2, the integral (2.2.1) is transformed

through the relationships.

3.3.2,,

2.3.2

1.3.2

2

1

1

dtxx

yyxfJ

dtxdx

x

y

dx

dyY

t

t

But the Euler-Lagrangian equation corresponding to equation 2.2.1 is

23.201

y

f

dx

d

y

f

The system of Euler-langrangian equations associated with e.g. 2.3.3 is,

if we write.

23

by

g

dt

d

y

g

ax

g

dt

d

x

g

bx

yY

axyyxfyxyxg

5.3.20

5.3.20

4.3.2

4.3.2),,(),,,(

1

1

From equation 2.3.4, we obtain

by

fyf

x

y

y

fxf

x

g

axx

f

x

g

6.3.2

6.3.2

1

1

21

From equation 2.3.2, we have, after introducing to 2.3.6.

7.3.211

1

1

1

x

f

Y

f

dx

d

Yf

YxY

fYf

dx

dx

x

g

dt

d

Furthermore, differentiating equations 2.3.4 gives:

bY

f

xY

fx

Y

g

axy

f

Y

g

8.3.21

8.3.2

11

Thus according to equations 2.3.1 and 2.3.2 9.3.21

Y

f

dx

dx

y

g

dt

d

combining the above result with 2.3.8a; and 2.3.7 with 2.3.6a gives the

following pair of equations:

11.3.2

10.3.2

1

1

Y

f

dx

d

y

fx

y

g

dt

d

y

g

Y

f

dx

d

y

fy

x

g

dt

d

x

g

From this result, we conclude that any relationship, single-valued or not,

that satisfies the Euler-Lagrangian Equation 2.2.1 derived on the basis of an

24

assumed single-valued solution y = y(x) – satisfies also the system 2.3.5 which

derivation requires no assumption of single valuedness of y as a function of x.

2.4 The Lagrangian Multiplier Approach

Lagrange sought the conditional extremum of the function fi (x1, x2----

xn) subject to the constraints ψi (x1, x2---- xn) i = 1,2,3 ----m, using

undetermined multipliers λ [9]. Suppose that the function F (x1, x2,------ xn) and

ψi (x1, x2,------ xn) have continuous partial derivatives of the first order in the

domain D, and also that m ≥ n, and the rank of the matrix

1.4.2,2,1,,2,1,

njjmi

xj

i is equation to m at

every point of D. the function f (x1, x2---- xn) is called the objective function,

while the function ψi (x1, x2,------ xn), the constraints.

A new function known as the auxiliary function is formed as follows

2.4.2,,,,, 211

1

212121

n

m

i

nmn xxxixxxfxxx

where λi are unknown constants known as Lagrangian undetermined

multipliers.

The auxiliary function which is now assumed to be a function of mtn

unknowns is then investigated for extremum as a function of mtn variables. A

strict maximum for ø signifies a conditional maximum for the auxiliary

function gives a conditional minimum for the objective function. So the

problem of conditional extremum is reduced to formation of an auxiliary

function which is subsequently investigated for an absolute extremum. The

constants λi’s, the Lagrangian multipliers are evaluated together with the

minimizing (or maximizing) values of xis’ by means of the following set of

equations.

25

4.4.2

3.4.2,),,(

21

21

n

ini

xxx

cxxxf

2.5 Isoperimetric Problems

This are problems in which the functions eligible for the extremization

of a given definite integral or functional are required to conform with certain

restrictions that are added to the usual continuity requirements and possible and

point conditions. In particular, the additional restrictions lie in the prescription

of the values of certain auxiliary definite integrals [14, 18].

The best known example of the isoperimetric problem consists of finding

a curve y = y(x), for which the functional 1.5.2),,(),( 1

21

2

1

dxYyxfJx

xεε

has an extremum, where the admissible curves satisfy the boundary conditions

bYxY

aYxY

2.5.2)(

2.5.2)(

22

11

and are such that another functional 3.5.2),,(),( 1

21

2

1

dxYyxgKx

xεε

takes a fixed value 1.

To solve the problem, the assumption is made that the functions f and g

defining the functionals (2.5.1) and (2.5.3) have continuous first and second

derivatives in (x1, x2) for arbitrary values of y and y1. Applying the lagrangian

undetermined multipliers, we define the function

4.5.2),(),(),( 212121 εελεεεε KJI

6.5.2*

5.5.2),,(*),( 1

21

2

1

gffwhere

dxYyxfIx

x

λ

εε

Now if we define Y(x) as a two parameter family

7.5.2)()()()( 2211 xxxYxy ηεηε in which η1(x) and η2(x) are

arbitrary differentiable functions for which η1(x1) = η2(x2) = 0 = η2(x1) = η1(x2)

26

---2.5.8 then we can immediately see that y(x1) = y(x1) = y1 -----2.5.9a, y(x2)

= y(x2) = y2 -----2.5.9b as prescribed for all values of the parameters ε1 and

ε2.Thus equations 2.5.1 and 2.5.2 now become

10.5.2),,(),( 1

21

2

1

dxYyxfJx

xεε

13.5.20

12.5.20

11.5.2),,(),(

21

21

1

21

2

1

εε

εε

εε

where

II

extremumanFor

dxYyxgKx

x

Introducing 2.5.7 into 2.5.4, 2.5.5, it follows

2,1

15.5.2**

14.5.2**

1

1

2

1

2

1

ifor

dxY

fi

Y

f

dxY

Y

fi

Y

fI

x

x

i

x

xi

ηη

ηε

Setting ε1 = ε2 = 0, so that according to (2.5.7), (y, y1) is replaced by (y, y1), we

have

16.5.2** 1

1

2

1

dxi

Y

fi

Y

f

oi

I x

x

Integrating by parts, the second part of the integrand (2.5.16) and using

equation 2.5.8, we obtain 17.5.20**1

2

1

dx

Y

f

dx

d

Y

fi

x

x

since ni = 0 only at the end points x1 and x2, then the only way the integrand of

equation 2.5.17 can equal zero is by 18.5.20**1

Y

f

dx

d

Y

f.

Equation 2.5.18 is therefore the Euler-Lagrangian equation which must

be satisfied by the function y(x) which extremizes 2.5.1 under restriction that

2.5.3 be maintained at a prescribed value.

27

2.6 Variational Nature of Soil Stability Problems

2.6.1 Variational Formulation of Stability Problems

The calculus of variations has been stated to deal with problems of

maxima and minima. But while in the ordinary theory of maxima and minima,

the problem is to determine those values of the independent variables for which

a given function of these variables takes a maximum or minimum value, in the

calculus of variations, definite integrals and functions defined by differential

equations involving one or more unknown functions are considered and the

problem becomes to determine these unknown functions that the integral shall

take up maximum or minimum values [9, 25].

On the other hand, the term “stability” in soil mechanics has been widely

used to describe the strength of the soil in its own natural or artificial form of

structure to withstand its own weight as well as the various external forces that

would influence equilibrium conditions [19]. Concerning various soil structure,

early studies have focused on earth slope structure problems i.e. earth dams,

natural slopes, road embankments etc.

Most of the problems associated with these are of the extreme value

types in which it is required to find the extreme (maximum or minimum) value

S of some parameter S while other parameters defining the problems are

assumed known. According to the character of the problem, S may be one of

the following parameters Q – external load, F – factor of safety with respect to

strength, Xp, Xp – coordinates of point of application of Q; P – the direction of

Q, - normal stress distribution, M – an external moment etc.

From the foregoing, we see that if the soil stability problem could be

accurately formulated as an extremization problem, it automatically falls into

the category of variational problem solvable by application of the appropriate

calculus.

28

In conlomb’s application of the elementary theory of maxima and

minima for stability of soil slope, for instance, the worst failure mechanism was

been sought, which mobilizes the entire soil strength and so possesses the

lowest factor of safety with respect to failure angle θ, thus, [20].

1.6.20 dt

df

It was at Kinson [21] who further demonstrated the use of calculus of

extreme as the mathematical tool for the energy method of stability analysis. In

his approach, the soil structure collapse load was determined using the potential

energy function as the function to be minimized with respect to the

displacement, thus

2.6.20 du

dy

However, here, in the calculus of variations approach, we use functionals

rather than functions. The functionals whose extreme are sought do not, unlike

in the ordinary calculus of maxima and minima, depend on the independent

variable or finite number of independent variables within a certain region, but

instead, they are functions of functions, the latter belonging to a class of perfect

by smooth functions, i.e. continuous functions with continuous derivatives of

any order. This presupposes that only the failure mode(s) treatable in stability

problem can be appropriately formulated and so are only considered.

29

2.6.2 Conditions to solving Bearing Capacity Problems of Footing on

slope

A special problem that may be encountered occasionally is that of a

footing located on or adjacent to a slope as shown below fig 2.6.2a [2].

d

g

D B

Q

Where r = roeθtanø

f

r ro

a

c 45 + ø/2 45 - ø/2

E

Fig. 2.6.2a; Footing on a slope

f

D B

Q

Fig. 2.6.2b: Footing on horizontal surface

c

a

d

q = ૪D

e

α α

b

30

2.6.2.1 Meyerhoff’s Approach

From the fig. 2.6.2a, it can be seen that the lack of soil on the slope side

of the footing will tend to reduce the stability of the footing. J.E. Bowles

developed as follows:

(a) Develop the exit point E for a footing as shown in fig. 2.6.2a. The

angle of exit is taken as 45 – ø/2 since the slope line is a principal

plane.

(b) Compute a reduced Nc based on the failure surface ade = Lo of fig.

2.6.2b and the failure surface adE = LI of fig. 2.6.2a to obtain.

3.6.20

11

L

LNN cc

(c) Compute a reduce Nq based on the ratio of area ecfg = A0 of fig.

2.6.2b to the equivalent area Efg = A1 of fig. 2.6.2a to obtain the

following.

4.6.20

11

A

ANN qq

However the effect on Nr is so insignificant that it was ignored [2],

therefore equations 2.6.3 and 2.6.4 are standards that must be achieved at the

end of this work.

2.6.3 Failure modes of soil Foundation

It is known from observation of foundations subjected to load that

bearing capacity failure occurs as a shear failure of the soil supporting the

footing [5]. The three principal shear failure modes under foundations have

been described as local shear failure, punching shear failure and general shear

failure [22, 13, 4, 5]. In the local shear failure, the failure is characterized by a

pattern which is clearly defined only immediately below the foundation (fig.

2.6.3a).

31

This pattern consist of a wedge and ship surfaces which start at the edges

of the footing just as in the case of general shear failure mode. This is visible

tendency towards soil bulging on the sides of the footing.

The punching, shear failure mode is characterized by a failure pattern

that is not easily observed fig. 2.6.3b. As the load increases, the vertical

movement of the footing is accompanied by a compression of the soil

immediately under wealth.

Continued penetration of the footing is made possible by vertical shear

around the footing perimeter. The soil outside this region remains relatively

I

II

III

II

III

Fig. 2.6.3a: Local shear failure pattern

Fig. 2.6.3b: Punching shear failure pattern

32

uninvolved and there are practically no movements of the soil on the sides of

the footing.

Contrasting sharply with these is the general shear failure mode which is

characterized by the existence of well-defined failure pattern consisting of

continuous slip surfaces from one edge of the footing to the ground surface

(Fig. 2.6.3c).

In stress – controlled conditions under which most foundations operate,

failure is sudden and catastrophie. Similarly, unless the structure prevents the

footing from rotating, failure is also accompanied by substantial filing of the

foundation [23].

The mode of failure that can be expected in any particular case of a

foundation depends on a number of factors among which include the relative

compressibility of the soil, the particular geometrical and loading conditions,

depth of footing, degree of saturation etc.

Of these modes, only the general shear failure mode can be appropriately

formulated and treated as a stability problem; and so is the failure mode

assumed in the present work (i.e. the general shear mode).

Fig. 2.6.3c: General shear failure pattern

33

2.7 Conclusion

In using the variational calculus, the philosophy of the limiting equation

(LE) is used to generate the functional [24] Sex 1.7.2)](),([ xxyS in

which the critical value of any of the parameters being sought is determined by

extremization of the functional Sex with respect to the two unknown functions

y(x) and (x).

The normal stress distribution associated with the least factor of safety or

critical foundation load is the critical stress distribution. Due to the nature of

limiting equilibrium formulation a solution such as that determined from the

above consideration will be independent of the details of any particular

constitutive model and shall therefore realistically reflect the present state of

uncertainty with respect to soil behaviour such solution shall be sought among

a class of perfectly smooth continuous functions with continuous derivatives of

any order.

34

CHAPTER THREE

MATHEMATICAL DERIVATIONS AND SOLUTION

3.1 Statement of Problem

A shallow strip foundation of width B is buried in soil mass of slope β at

a depth of H as shown in fig. 3.1.

The soil mass is of semi-infinite extent and is homogeneous and

isotropic. It has an effective unit weight r, and shear strength parameters C and

ø (the cohesion and angle of internal friction respectively).

Fig. 3.2: Calculation scheme (S = arc length along y(x) and θ = tan-1 (dy/dx)

B

H

Q

Fig. 3.1: Foundation buried in sloppy soil mass

B

Xo Q-qBcosβ

X1

y(x)

y

૪H = q

(a)

β

θ

τ

σ

S(x)

x

35

From fig. 3.2a, we define Q as the ultimate foundation load; and the foundation

soil is in a state of limiting equilibrium (LE) as soon as the foundation load

equals Q. Y(x) represents the equation of the failure surface in a two-

dimensional plane and σ(x), the normal stress distribution. The effect of the

over burden height of soil H above the foundation level is represented by ૪H, ૪

being the soil unit weight.

The problem presented in the foregoing is formalized by finding from

first principles, the expression for the critical normal stress distribution σ(x)

which, along with the expression for the critical rupture surface Y(x)

determined from an earlier work [28], when substituted into the integral

expression for the minimum allowable (bearing) foundation load Q, will bring

the system described in fig. 3.2 to a state of limiting equilibrium. The

formulation is made without any priori assumption as all derivations are made

right from first principles.

A mass of soil such as the one in fig. 3.2 is considered to be in a state of

limiting equilibrium if:

(1) Coulomb’s yield condition is satisfied along a potential rupture line Y(x)

that smoothly connects one edge of the footing to the ground surface,

thus

1.1.3tan)()( xCx

where τ(x) and σ(x) are the shear and normal stress distributions along Y(x)

respectively.

(2) The three equations of equilibrium-vertical, horizontal and rotational

equilibrium-are satisfied for the sliding mass, thus:

(a) For vertical equilibrium, we have:

n

i

Fiv1

2.1.30

36

For vertical component of an equivalent force Fi which replaces the system of n

forces in fig. 3.2. Resolving therefore all forces in the vertical direction and

summing for vertical equilibrium, are have

In the limit ds 0 and x 0, we have

On simplification, we have

(b) Similarly, for horizontal equilibrium,

n

i

ihF1

6.1.30

Fih = horizontal component of force Fi. Resolving all forces horizontally, we

have

n

i

sd1

7.1.30cossin

In the limit as ds 0, we have

8.1.30sincos ss

d

(c) For rotational equilibrium, about x0 we have

n

i

iM1

9.1.30

n

ii

n

ii

n

i

yhxv FF111

.. ૪y.x dx

s

n

i

dsqBQ1

cosσsinτcos ૪ydx

+ ૪ 0xi dH -----------------------------------------------------------------3.1.3

cosqBQ s

cosσsinτ ds + 1X

X o

૪y dx + 1X

X o

૪Hdx = 0 -------------3.1.4

cosqBQ s

cosσsinτ ds + 1X

X o

૪(y + H)dx = 0 -------------3.1.5

37

n

i 1

૪ 10.1.30. dxxH

In the limit dx 0, ds 0, then

ss

dxy cossinsincos

1

0

X

X૪ 11.1.30 xdxHy

in which X0 and X1 are the end points y(x), s = the arc length along y(x) and α

arc tan (dy/dx).

From equation 3.1.5,

1

0

cossincosX

Xs

sdqBQ ૪ ,0)( xdHY

we have on rearrangement

1

0

cossincos

X

X

ss

dqBQ ૪ 12.1.3 xdHy

In the limit as dQ Qmin, it is intended to determine the equation of the

function σ(x), the critical normal stress distribution without any prior

assumption. In fact, the functions minimizing Q are those of y(x) and σ(x). If

y(x), the rupture surface, is taken as a logarithmic spiral curve, the present

problem could be restated thus: Find the equation of the critical normal stress

distribution σ(x) along y(x) and which minimizes the functional Q defined by

the integral equation 3.1.12 and subject to two integral constraint equations

3.1.8 and 3.1.11.

If the appropriate expression for σ(x) is determined, that coupled with

that for y(x), it is therefore possible to easily use equation 3.1.12 to determine

minimum Q (i.e. the critical Q) identified with the bearing capacity.

The curves y(x) and σ(x) for which the limiting equilibrium occurs at

minimum Q are termed the criticals. In the foregoing formulation, no

38

constitutive law beyond Coulomb’s yield criterion is included. Consequently,

no restrictions or constraints are placed on the character of the criticals, except

the overall equilibrium of the failing section. This means that the critical

normal stress distribution σ(x) determined as a result of the solution shall lead

to Qmin, the critical load which represents the smallest load that can lead to

failure.

Put succinctly and differently, for a soil with parameters C, ,r and

footing with geometry B, H, if σ(x) is less than critical σ(x) (C, ,૪,B,H), the

foundation will be stable regardless of the constitutive laws characterizing the

soil. For σ(x) > σ(x) (C, , ૪, B, H), the stability would depend upon the

constitutive character of the medium. σ(x) (C, , ૪, B, H) therefore represents

an upper bound solution.

Since both the conlomb yield criterion and the equilibrium conditions are

simultaneously satisfied, we proceed thus:

Introduce equation 3.1.1 into equations 3.1.5, 3.1.8 and 3.1.11, we have

(a) For equation 3.1.5,

ss

dxxcqBQ cos)(sintan)(cos

1

0

X

X

૪ 13.1.3)( xx dHy

Making Q the subject of formula, and letting ψ = tan , the frictional

coefficient of the soil, we have,

ss

dcqBQ cossincos

1

0

X

X

૪ 14.1.3 xdHy

(b) For equation 3.1.8,

39

tan

tan

0cossin

forc

cBut

dxs

16.1.30coscossin

15.1.30cossin

ss

ss

dc

dc

(c) For equation 3.1.11,

17.1.30)(

)cossin()sincos(

1

0

x

X

X

ss

dHyxy

dy

xcycs

cossintansincostan

1

0

X

X

sd ૪x adHy x 17.1.30)(

But ψ = tan ,

sd

sxcyc cossinsincos

1

0

X

X

૪x bdHy x 17.1.30)(

On rearranging, we have

ss

dxcyCosc sinsincossincos

1

0

X

X

૪x cdHy x 17.1.30)(

The foregoing formulation contains five parameters of the problem – C,

, ૪, B, H. In the following sections, a parametric transformation shall be

carried out using non-dimensional quantities to reduce the number of

parameters, a design to give series of advantages in both the construction of the

solution and the presentation of the results.

40

3.2 Fundamental Assumptions

(i) The soil mass under study is homogenous and isotropic that is to say that

the properties of any soil element are assumed the same as the properties

of the whole soil mass, irrespective of location or orientation of the soil

element. The soil possesses a unique effective unit weight r and effective

shear strength parameters; c, the cohesive strength and , the angle of

internal friction.

(ii) the general solution is sought in the class of perfectly smooth functions.

A smooth function here refers to function with continuous derivatives of

any order. This valid for both the rupture surface and the normal stress

distribution along it.

(iii) Coulomb’s law is strictly valid;

1.2.3tan c

(iv) On the imminence of failure, the failure mechanism satisfies the basic

conditions of equilibrium thus vertical, horizontal and rotational

simultaneously thus

(a) for vertical equilibrium,

2.2.3;0)(1

vFin

i

Where Fi(v) = vertical component if force Fi.

(b) for horizontal equilibrium,

3.2.3;0)(1

hFin

i

Where Fi(v) = horizontal component of force Fi.

(c) for rotational equilibrium,

4.2.3;01

i

n

i

M

Where Mi = moment of force Fi about specified position.

41

(v) The foundation base is assumed to be smooth and this means that the

angle of internal friction α for the zone of elastic equilibrium is

expressed as (45o + /2).

(vi) Failure of foundation is assumed to take place by the general shear mode

(fig. 2.6.3c) and is characterized by the existence of well-defined failure

pattern which consists of footing to the ground surface. Failure is then

accompanied by substantial rotation of the foundation and the final soil

collapse occurs only on one side of the foundation.

(vii) The ground surface is assumed to be sloppy and the overburden pressure

at foundation level is equivalent to a surcharge load q1o = rH cos .

(viii)The load exerted on foundation is assumed to be vertical and

symmetrical.

(ix) All soil elements are in plastic equilibrium at any potential rupture

surface [22]. Similarly, the forces acting on each element of the mass are

in equilibrium. The stresses within this zone of plasticity (the rupture

surface) are those which produce failure. The strains, defying the stress-

strain law are now indefinite. As usual, the state of stress in plastic

equilibrium can be approximated by Mohr-Coulomb yield criterion

which is based on Mohr strength hypothesis which states thus:

(a) The strength along any plane is determined by limiting combinations of

normal stress and shear stress on that plane.

(b) A combination of failure will occur along planes of which the limiting

equilibrium are reached. The documented cases of bearing capacity

failures indicate that usually the following three factors (separately or in

combination) are the cause of the failure [27].

(1) There was a overestimation of the shear strength of the underlying soil.

(2) The actual structural load at the time of the bearing capacity failure was

greater than that assumed during the design phase.

42

(3) The site was subjected to alteration, such as the construction of an

adjacent excavation which resulted in a reduction in support and a

bearing failure.

(x) Foundation is assumed to be shallow in which case H ≤ B.

A condition of failure therefore exists on any plane when the shear stress

τ on that plane attains a maximum value τmax; where the value of τmax is a

function of the normal stress σf acting on the plane at failure, and of the

material properties ,, ji along that plane in question. This implies that

failure will occur along that plane which first satisfies the functional relation.

5.2.3,,,max jiff

for isotropic materials, the Mohr-coulomb strength criterion can be written in

terms of the principal stress as:

6.2.3,,,,2

1312

131 jiff

In which θ is the angle between the major failure plane and plane normal to

major principal axis. The Mohr-Coulomb yield criterion is a special case of the

Mohr hypothesis in which the functional relation 3.2.6 is of the explicit form.

7.2.3tan nc

Where c is the cohesive strength, is the angle of internal friction of soil. For

all elements of the soil on the rupture surface.

8.2.3tan)()( xcx

Where τ(x) is the distribution of shear stress on rupture surface. Σ(x) is the

distribution of normal stress on rupture surface. c and are soil strength

parameters.

43

3.4 Boundary Conditions

Variational problems deal with two types of boundary conditions:

(a) Fixed and points such as ox (fig. 3.2)

(b) End points that can slide along a prescribed curve. Their position is

determined in such a way as to assure on external value of the functional.

Since such points are not known in advance, a variational boundary

condition known as the transverslaity condition has to be satisfied.

For the general shear mode of failure, the function y(x) has to satisfy the

following end conditions in order to comply with it:

2.4.30)(

1.4.30)(

00

12

xxyy

xxyy

Using these conditions, we simplify the following expressions thus:

dxxyxxy

c

x

x

y

byd

axdxd

ydxdy

x

x

x

x

x

x

3.4.30

3.4.3

3.4.3

3.4.3

01

0

1

1

1

0

1

0

1

0

axdxd

ydyxdyy

Similarly

xdy

x

x

x

x

x

x

5.4.3

4.4.30

1

0

1

0

1

0

1

1

c

x

x

y

bydy

x

x

5.4.3

5.4.3

0

1

2

21

1

0

44

6.4.30

5.4.302

1

1

0

2

1

2

1

0

xdyy

dxxyxxy

x

x

Finally, with regards to the parent problem, we notice that the location of

the end points X1 is not known in advance. This therefore demands the

application of the condition of transversality. In this particular case, the

appropriate form of this condition is given as [8]

8.4.3ˆ

7.4.30

11

1

1

1

1

1

xxwhere

xx

s

y

syS

3.5 Non-dimensional Parametric Representation

To appropriately reduce the problem parameters to analytically

manageable number and hence advantageously construct the solution, it is most

convenient to introduce a set of non-dimensional parameters.

Define therefore the following non-dimensional parameters as follows

[32]

1.5.3,, B

HH

B

yy

B

xx

2.5.3,,21

B

QQ

BB

cc

σσ

3.5.3ˆ HcHB

cc ψψ

4.5.322

ˆ HB

H

B

5.5.322ˆ

2 HQ

B

H

B

QQ

૪ ૪ ૪

૪

૪

૪

45

The problem is now presented in terms of c, σ and Q. Now from the

geometry of the rupture surface (fig. 3.2), it is easy to see that

7.5.3tan

6.5.3cos

1

dx

dyy

dxds

From the definition of the non-dimensional parameters

8.5.3,, BHHByyBxx

cc ૪ σσ ,B ૪ QQB , 2B ૪ 9.5.3

The parameters are then used to transform the problem equations 3.1.14,

3.1.16, and 3.1.17c as follows

(a) for equation 3.1.14, we have

Q ૪ dsSinCCosHbs

sincos

1

0

x

x

૪ 14.1.3 dxHy

Introducing the appropriate parameter equations 3.5.8 and 3.5.9, we have

૪ QB2 ૪ 1

0

cos

x

x

HBB c sincos ૪ sinB

1

0cos

x

x

dx

૪ 10.2.3 dxBHBy

૪ QB2 ૪ cos2HB ૪

cos

sinsincos1

0

dxcB

x

x

-૪ 11.5.31

0

dxHyB

x

x

12.5.3 xBdBxddxBut

Substituting 3.5.12 into 3.5.11

46

૪ QB2 ૪ cos2 HB ૪

cos

sinsincos1

0

2 xdcB

x

x

-૪ 13.5.31

0

2 xdHyB

x

x

Dividing all through by rB2, we have

14.5.3

cossinsincoscos

1

0

1

0

xdHy

xdcHQ

x

x

x

x

16.5.3tan

15.5.31

15.5.3

15.5.3tan

11

1

1

yy

cyxd

yd

xBd

yBd

bBxd

Byd

dx

dyy

aydx

dyNow

From Eqn 3.5.14, we get

19.5.31cos

18.5.3

tantancos

1

0

1

0

1

0

1

0

11

xdHydxycyHQ

xdHy

xdcbHQ

x

x

x

x

x

x

x

x

Now cos β = H/B fig. 3.1 and fig. 3.2 and from equation 3.5.1 B

HH

20.5.3cos H

Substitute equation 3.5.20 into 3.5.19 thus

47

21.5.311

0

1

0

112

dxHydxycyHQ

x

x

x

x

Now, further treatment of equation 3.5.3, 3.5.4 and 3.5.5 results in the

following:

24.5.32ˆ

23.5.3ˆ

22.5.3ˆ

HQQ

H

Hcc

Substituting therefore equations 3.5.22, 3.5.23 and 3.5.24 into equation 3.5.21,

we have

25.5.3ˆ1ˆ1

0

1

02

11

xdHydxyHcyH

x

x

x

xH

Q

Further expanding, we get

30.5.3ˆˆ

29.5.3ˆ1ˆ

28.5.3ˆ1ˆ

27.5.3ˆ1ˆ

11

1

11

112

1

0

1

0

1

0

1

0

1

0

1

0

xdycxdyyH

dxyycy

dxHyHycy

xdHyxdHycyHQ

x

x

x

x

x

x

x

x

x

x

x

x

ψσ

ψσ

ψσ

ψσ

31.5.3ˆ1ˆ 111

0

1

0

xdycxdyy

x

x

x

x

ψσ

By invoking the result of equation 3.4.4, equation 3.5.31 becomes

32.5.31 121

0

xdyyHQ

x

x

ψσ

(b) For equation 3.1.16, we have

16.1.30coscossin1

0

dsc

x

x

48

Introducing the non-dimensional parameters of equations 3.5.8 and 3.5.9, we

obtain

1

0

x

x

σ ૪ cB cossin ૪ cosB 33.5.30cos

Bdx

૪ 34.5.30cos

coscossin1

0

2

xdcB

x

x

૪ 35.5.30tan1

0

2 xdcB

x

x

Dividing although by ૪B2, we get

36.5.30tan1

0

xdc

x

x

Introducing equations 3.5.22 and 3.5.23 into 3.5.36, we get

40.5.30ˆ

39.5.30ˆ

38.5.30ˆ

37.5.30ˆtan

11

11

1

1

0

1

0

1

0

1

0

dxcyHy

xdHcHyHy

xdHcyH

xdHcH

x

x

x

x

x

x

x

x

42.5.30ˆ

41.5.30ˆ

11

11

1

0

1

0

1

0

1

0

xdyHxdcy

dxyHdxcy

x

x

x

x

x

x

x

x

ψσ

ψσ

Invoking the result of equation 3.4.4, the equation 3.5.42 simplifies to

43.5.30ˆ11

0

xdcy

x

x

ψσ

(c) For equation 3.1.17c, we have

49

xcyCoscs

sinsincossincos

1

0

x

x

ds ૪ 0 dxHyx ------------------- c17.1.3

Introducing the non-dimensional parameters of equations 3.5.8 and 3.5.9 into

3.1.17c gives us:

σ1

0

x

x

૪ cB sincos ૪ cosB By σ ૪ cB sincos

૪ sinB cos

dxxB

1

0

x

x

૪ 44.5.30 xdBHByxB

૪ xcycB

x

x

sinsincoscossincos1

0

2

cos

dx+ ૪ 45.5.30

1

0

2 xdHyxB

x

x

૪ xdxcycB

x

x

tantan1tan1

0

2

૪ 46.5.301

0

2 xdHyxB

x

x

By using equations 3.5.7 and 3.5.8 and making necessary substitutions;

૪ xBdxycyycyB

x

x

1112 11

0

ψσψσ

+ ૪ 47.5.301

0

2 xBdHyxB

x

x

૪ xdxycyycyB

x

x

1112 11

0

ψσψσ

50

+ ૪ 48.5.301

0

2 xdHyxB

x

x

Dividing although by rB2, we have

49.5.30

1

1

0

1

0

111

xdHyx

dxxycyycy

x

x

x

x

Introducing the results of equation 3.5.9 into equation 3.5.49 gives

50.5.30

ˆ1ˆˆˆ

1

0

1

0

111

xdHyxxd

xyHcyHyHcyH

x

x

x

x

Expansion of e.g. 3.5.50 gives

51.5.30

ˆˆˆˆˆˆ

1

0

1

0

11

dxxHyx

xyHycyHHyyHcyHHy

x

x

x

x

ψψψσσψψσψσ

Simplifying

54.5.30ˆˆ

53.5.30

ˆˆˆ

52.5.30

ˆˆˆˆˆˆ

1111

11

1111

1

0

1

0

1

0

1

0

xdyHxyyxycyyxyxy

xdyxxHxHyyH

yxycyyxyxy

togethertermslikeTaking

xdxHyx

xdxycxyyxxycyyHyyy

x

x

x

x

x

x

x

x

ψσ

ψψσ

ψσσψσψσ

51

Rearranging e.g. 3.5.54 for easy handling

56.5.30

ˆˆ

55.5.30

ˆˆ

1

111

1

111

1

0

1

0

1

0

1

0

xdyyH

xdxyyxycyyxyxy

xdyyH

xdxyyxycyyxyxy

x

x

x

x

x

x

x

x

But the result of equation 3.4.6

57.5.30ˆˆ

56.5.3

0

111

1

1

0

1

0

xdyxyxycyyxyxy

becomesequationsoand

xdyy

x

x

x

x

The basic five parameters of the problem (c, , ૪, H, B) enter into the

system of equations represented by 3.5.32, 3.5.43 and 3.5.57 in the combination

of ψ and c only. Thus the transformation into non-dimensional parameters has

effectively and advantageously reduced the number of problem parameters

from five to two.

3.6 Construction of Euler-Lagrangian Intermediate Function for the

Problem

Consider the stability function of equation 3.5.32 given as

32.5.31ˆ 121

0

dxyyHQ

x

x

Denote the integrand by U which is now a function of σ, y, y1, and c. This

implies that

52

)coscos(

2.6.3ˆ,,,,ˆ

1.6.31ˆ,,,,

22

12

11

1

0

HHbut

xdcyyUHQ

and

yyCyyU

x

x

For the stability function of equation 3.5.43 representing the horizontal

equilibrium state, the equation is

43.5.30ˆˆ 11

0

xdcy

x

x

the integrand is denoted by V, then v which is now a function of σ, y1, ψ, and c

becomes

4.6.30ˆ,,,

3.6.3ˆˆ,,,

1

11

1

0

xdcyVand

cycyV

x

x

Similarly, denote by W, the integrand of the stability function representing the

moment equilibrium and given in equation 3.5.57 as

57.5.30ˆˆ 1111

0

xdyxyxycyyxyxy

x

x

Obviously W is a function of σ, xandcyy ˆ,,, 1 and so

6.6.30ˆ,,,,,ˆ

5.6.3ˆˆˆ,,,,,ˆ

1

1111

1

0

cxyyW

yxyxycyyxyxycxyyW

x

x

ψσ

ψσψσ

The solution of the foregoing variational problem will now be

constructed using the method of lagrange’s immediate multipliers. In line with

this is defined an intermediate function S according [9].

7.6.321 WVUS

which is seen to incorporate the load function U and the necessary constraints

V and W. ,, 21 are Lagrange’s undetermined multipliers. Replacing U, V,

53

and W with their appropriate expressions from equations 3.6.1, 3.6.3 and 3.6.5,

we have

8.6.3ˆˆ

ˆˆ1ˆ

111

2

1

1

1

yxyxycyyxyxy

cyyyS

ψσλ

ψσλψσ

The equation 3.6.8 for s integrates the load (objective) function with the

constraints. It is the functional which itself is a function of two functions )(xy ,

the rupture surface and )(ˆ x , the normal stress distribution on the rupture

surface.

In the subsequent section, S is immunized with respect to the functions

)(xy and )(ˆ x by subjection to the appropriately constructed Euler-Lagrange’s

differential equation. The determination of the expressions for the critical

normal stress distribution )(ˆ x and the critical rupture surface )(xy and which

ultimately results from the minimization of S thus the main thrust of the

bearing capacity problem.

3.7 Formulation of Euler-Lagrang Differential Equation

The criticals )(ˆ x and )(xy must necessarily satisfy

(a) system of Euler differential equation in S

(b) the integral constraints of equation S 3.5.43 and 3.5.57

(c) the set of boundary conditions at the end points 10 xandx

For the differential equation, Euler had theorized that for a functional of one

function )]([ xy

1.7.3...,,,,)( 11

0

dxyyyyFyJ n

x

x

54

The appropriate differential equation is [8, 9].

n

n

n

n

nn

y

FFy

y

FFy

y

FFywhen

Fydx

dyF

dx

dFy

dx

dFy

3.7.3

2.7.30)1(

1

1

2

21

Thus the Euler differential equation of equation 3.7.2 becomes

4.7.30)1(2

2

1

nn

nn

y

F

dx

d

y

F

dx

d

y

F

dx

d

y

F

In the same light, for a functional of two functions y(x) and z(x),

6.7.30)1(

0)1(

5.7.3,,,,,,,,,)(),(

2

2

1

2

2

1

111

0

nn

nn

nn

nn

nn

x

x

y

F

dx

d

z

F

dx

d

z

F

dx

d

z

F

y

F

dx

d

y

F

dx

d

y

F

dx

d

y

F

isequationaldifferentiEulerofsystemtheand

dxzyzyzyzyxFxxyJ

For the particular case of the formulated problem, we have that the

functional S is a function of two variables incorporating first order differential.

The appropriate Euler’s differential equation for the present problem may

therefore be written as

8.7.30

7.7.30ˆˆ

1

1

y

s

dx

d

y

s

s

dx

ds

further, bringing the condition of transversality, i.e., the variational boundary

condition of equation 3.4.7

55

0ˆ

ˆ

1

1

1

1

1

xx

s

y

syS

Now since S in equation 3.6.8 does not depend on ̂ , then equation S 3.7.7,

3.7.8 and 3.4.7 simplify to

11.7.30

10.7.30

9.7.30ˆ

1

1

1

1

xxy

syS

y

s

xd

d

y

s

s

Thus the problem reduces to that of solving the two differential equations,

Equations 3.7.9 and 3.7.10, subject to the fulfillment of the two integral

constraints Equation 3.5.43 and 3.5.57, the geometrical boundary conditions,

equations 3.4.1 and 3.4.2 and transversality condition equation 3.7.11.

3.8 Co-ordinate Transformation and General Solution

3.8.1 Co-ordinate Transformation

From equation 3.6.8, we discover that S is linear in σ, and so equation

3.7.9 is independent in σ, and is a first order differential equation in y only. It is

solved independent of Euler’s second equation 3.7.10. The solution which is

found elsewhere [28] results into an expression for the critical rupture surface

which is found to be logarithmic spiral curve.

Following a rigorous process and using polar coordinate system, the

expression for the critical normal stress distribution σ(x) is obtained by a

complete solution of equation 3.7.6. it is found convenient to introduce the

following coordinate transformation.

2.8.3sin

1.8.31

cos

2

1

2

ry

rx

56

when (r, θ) is a polar coordinate system centered around the point (xr, yr):

5.8.31

*

]29[*

4.8.3

3.8.31

1

2

1

2

d

xdd

yd

ondifferentiofruleChainxd

d

d

yd

xd

ydyBut

y

x

r

r

Now y (eq 3.8.2) is a function of two variables r and θ. So we use product rule

thus:

8.8.3cossin

coscos

7.8.3sincos

sinsin

6.8.3)(

)()(

)()(),(

d

rdr

d

rd

d

dr

d

xd

similarly

d

rdr

d

rd

d

dr

d

yd

thatso

dx

xduxV

dx

xdVxUxVxU

dx

d

(Note that λ1 and λ2 are constants and so result to zero on differentiation).

Thus, by introducing the results of equations 3.8.7 and 3.8.8 into

equation 3.8.5,

θθ

θθ

θθ

θ

θ

sin

1sincos

1*

1

1

rd

rdCos

d

rdry

havewe

d

xdd

ydy

57

10.8.31

*]29[*

9.8.3

sincos

sincos

1

θ

θ

σθ

θ

σσ

θθ

θθθ

d

xdd

d

dx

d

d

d

similarly

rd

rdd

rdr

and introducing equation 3.8.8 into 3.8.10 takes us to

11.8.3

sincos

1*1

rd

rdd

d

The solution of the Euler first differential equation has already been

dealt with. The result was obtained by introducing the definition of S (eq 3.6.8)

into the Eulers first differential equation, Equation 3.7.9 and using the

coordinate transformation equations 3.8.1, 3.8.2 and 3.8.9. A resulting first

order differential equation

13.8.3.)(

12.8.31

)(

00

err

obtaintorforsolvedwas

d

rd

r

in which (ro, θo) are the constants of integration that may be conveniently taken

as polar coordinate of point (ro, yo).

The above equation, equation 3.8.13 is identified as the equation of a

logarithmic spiral curve and which is the shape of the critical rupture surface.

To solve the Euler second equation to obtain the normal stress distribution )(ˆ x

on the critical rupture surface, we introduce the definition of S in the Euler’s

second equation, Equation 3.7.10, thus:

58

18.8.3ˆˆˆ

17.8.3ˆˆ

16.8.3ˆˆ)(ˆˆˆ

15.8.3ˆˆˆ1

14.8.3ˆˆˆ1

8.6.3ˆˆ

ˆˆ1ˆ

Re

2

1

22221

1

1

2221

211

22

1

22

1

2

1111

2

1

1

1

cyyxy

s

dx

d

xcyx

xcyxy

s

xcy

xcyy

s

yxyxycyyxyxy

cyyyS

call

But Euler’s second equation is recalled thus

10.7.301

y

s

dx

d

y

s

Substituting equations 3.8.15 and 3.8.18 into 3.7.10 results into:

19.8.30ˆˆˆ

ˆˆˆˆ1

2

1

22

221

1

22

1

22

cy

yxxcy

Simplifying

20.8.30ˆˆ2ˆ21 221

1

222 yxxc

Transforming polar-coordinate – wise by introducing the expressions for y and

x from equations 3.8.1 and 3.8.2, we have

21.8.301

sincos

1cosˆ2ˆ21

2

2221

1

2

222

rr

rc

59

28.8.30sincos1ˆ

cosˆ2ˆ2

27.8.30sincos1

cosˆ2ˆ2

26.8.30sincoscosˆ2ˆ2

25.8.30sincosˆcosˆ2ˆ2

24.8.30sincosˆ1cosˆ2ˆ21

23.8.30sincosˆ1cosˆ2ˆ21

22.8.30sincosˆ1

cosˆ221

1

1

2222

22

1

222

1221

1

222

r

d

dxd

drc

r

d

dxd

drc

rdx

drc

rrc

rrrc

rrrc

rr

rc

Introducing the expression for d

dx from equation 3.8.8 into equation 3.8.28, we

obtain

29.8.30sincos

sincos

1ˆcosˆ2ˆ2

r

rd

drd

drc

From equation 3.8.12, we see that

30.8.3

rd

dr

Introduce equation 3.8.30 into equation 3.8.29 thus

33.8.30ˆ

cosˆ2ˆ2

32.8.30sincos

sincosˆcosˆ2ˆ2

31.8.30sincos

sincosˆcosˆ2ˆ2

d

drc

r

r

d

drc

rr

r

d

drc

substituting the expression for r(θ), equation 3.8.13 into 3.8.33 results to

60

35.8.30cosˆ2ˆ2

ˆ

34.8.30ˆ

cosˆ2ˆ2

0

0

0

0

ercd

d

d

derc

Rearranging equation 3.8.35, thus

36.8.30ˆ2cosexpˆ2ˆ

00 crd

d

We can clearly see that it is a first order linear differential equation in ̂ . This

is solved by procedure of separation of variables.

3.8.2 Solution of the Resulting Differential Equation

If we rearrange the differential equation, equation 3.8.36, we get

37.8.3ˆ2cosˆ2

ˆ0

0

cerd

d

Equation 3.8.37 is a first order linear non-homogeneous differential equation

and which we solve by separating the variable thus [29]

41.8.3..)(ˆ

40.8.32cos

39.8.32)(

38.8.32)(

0

0

Bdgee

Then

cerg

dfh

fLet

hh

substituting Equations 3.8.38, 3.8.39 and 3.8.40 into equation 3.8.41 takes us:

48.8.3cos3sin91

cos

47.8.3cos

46.8.3cos

45.8.32cos

44.8.32cos

43.8.32)cos

42.8.32cos)(ˆ

2

33

23

0

22232

0

22232

0

23

0

2

23

0

2

0

22

0

0

0

0

0

0

edeBut

Bec

deer

Beec

deer

Bedecedeer

Bdecdeere

Bdcedeere

Bdceree

61

(see Appendix 1)

substituting equation 3.8.48 into 3.8.47 gives

50.8.3cos3sin91

49.8.3cos3sin91

)(ˆ

2

20

2

2

3

0

0

20

Bece

r

Bece

er

codifying and rearranging, we get

52.8.3cos3sin91

)(

51.8.3)()(

2

2

0

0

eAwhere

cBeAr

B = integrating constant.

Now for a case where ψ = tan = 0; i.e. frictionless soil, we substitute this

zero value into equation 3.8.45 before performing the integration thus:

bBcr

aBdcdr

substitute

Bedecedeer

53.8.302sin

53.8.32cos)(

0

45.8.32cos)(

0

0

22232

00

3.8.3 Solution of Transversality condition (Variational Boundary

Condition)

The expression of the variational boundary condition is given in equation

3.7.11 as

0

11

1

1

xxy

syS

Applying the definition of S of equation 3.6.8 here,

54.8.3ˆˆ

ˆˆ1ˆ

11

2

1

1

1

yxyxycyyxyxy

cyyyS

ψσλ

ψσλψσ

62

8.6.3)(Re0

55.8.3ˆˆˆˆˆ

54.8.3ˆˆˆˆˆ

)(ˆ)(ˆ)(ˆˆˆ

1

1

1

1

1

1

1

1

1

21

22211

cally

syS

xcyxyyyy

sy

xcyx

xcyxy

s

σψσλσλψσ

σψσλσλψσ

λσλσψλσλψσ

This implies, further necessary substitution with equations 3.6.8 and 3.8.55 and

expanding

56.8.30ˆˆ

ˆˆˆ

ˆˆˆˆ

ˆˆˆˆˆˆ

1

2

1

2

1

2

1

1

1

2

1

22

1

22

1

2

211

1

1

1

yxcyy

xyyyyx

yxcycyyxyx

ycyyy

Further simplification yields

57.8.30ˆˆˆˆˆˆ222211 yxycxycy

Introduce into equation 3.8.57, the expressions for the coordinate

transformation of x and y from equation 3.8.1 and 3.8.2, we get.

58.8.31

cossinˆ

cossinˆ

sinˆˆˆsinˆ

22

12

2121

211

2

1

rr

rcCr

rcr

Simplifying

60.8.3coscossinsincossinˆ

59.8.30coscossinsin

cossinsinsin

1

2

2222

2

11

2

2

222

2

1

rrCrrr

rrr

rcrrr

63

Dividing although by rλ2 cos θ

64.8.30sin

63.8.30

62.8.3tan1

tansin

61.8.31tan

tansin

ˆ

sintan1tanˆ

2

111

11

2

1

2

1

2

1

ryimpliesThis

atyyBy

cr

cr

rC

66.8.3tan1

tan

tan1

sintansinˆˆ

65.8.3sin

1

1

1111

1

2

1

1

crcr

Thus

r

3.8.4 Determination of Integration Constant (D)

The integration constant D of equation 3.8.53b is determined by

pursuing the fact that the critical )(ˆ x determined from the solution of the Euler

equation, Equation 3.7.10 must also satisfy the condition of transversality (i.e.

the variational boundary condition), Equation 3.7.10 at any point on the critical

rupture surface. This realized, we therefore apply the solution of )(ˆ x

equations 3.8.51 and 3.8.53 (for ψ = 0, and ψ ≠ 0 respectively) to the end point

(r1, θ1) and on comparing the result with the solution, equation 3.8.66, for the

variational boundary condition, we can solve for B, thus:

Now, the solution for the Euler second equation is

51.8.30;ˆ

)(ˆ 2

)(0

cDeAr

bDcr 35.8.30;ˆ2sin)(ˆ0

64

52.8.3cos3sin91 2)(

0

θψθψ

ψθθ

θ

eAwhere

B = integration constant

Now if )(ˆ x on equations 3.8.51 and 3.8.53b is applied to the end condition,

we obtain

68.8.30;2sin)(ˆ

67.8.30;)(ˆ

1101

2

)(011

1

cDr

cDeAr

If these are simultaneously compared with the )(ˆ1 Equation 3.8.66

resulting from the solution of the transversality condition, we get as follows:

)(0

1

)(0

1

11

10

1

12

1

12

10

1

11

1

1

1

1

)tan1(

70.8.3)tan1(

tanˆˆtanˆ

)(tan1

tan

69.8.3tan1

tan)(

66.8.3tan1

tan)(

Arc

Arccc

Arcc

De

ccDeArThus

c

71.8.30;tan1

ˆ1

1

1

2

)(0

1

2

eArec

D

similarly for the case where ψ = 0; we have

75.8.3sin2tanˆ

74.8.3sinˆ2tanˆ

73.8.3tanˆ2sin

0

72.8.3tan1

tanˆˆ2sin

1011

1011

1110

1

1110

rcD

rccD

ccDr

tousleadsngsubstituti

ccDr

65

3.9 Bearing Capacity Determination

asc

HQQdefineNow 0ˆ

2

0

the bearing capacity of cohesionless soil (c = 0), [27] from the result of semi-

empirical work [6], it has been found that the ratio 0

2

Q̂

HQ

which depends on both and c is of the relation

1.9.3ˆ)(1ˆ

0

2

ckQ

HQ

where k( ), the slope of the relation depends on , the internal frictional

angel.

2.9.3ˆˆ)(ˆ00

2 QckQHQ

Introducing the definitions of the non-dimensional parameter of equation 3.5.3

i.e. Hcc ˆ into 3.9.2

substituting 3.5.1 into 3.9.4 gives

Recall equation 3.5.2 and substitute into equation 3.9.6;

4.9.3ˆ)(ˆ)(ˆ

3.9.3ˆ)(ˆ

000

00

2

HQkcQkQ

HcQkQHQ

૪

6.9.3)(ˆ)(ˆ

5.9.3)(ˆ)(ˆ

2

2

000

000

2

B

H

B

HQk

B

cQkQQ

B

HQk

B

cQkQHQ

ψφφ

ψφφ

૪

૪

B

H

B

HQk

B

QQkQ

B

Q

B

QQ

2

0002

2

ˆ)(ˆ)(ˆ

2.5.3

ψφφ

૪

૪

૪

66

Multiply although by B૪/2 thus

Now denote

N૪ 9.9.32

)(0 φQ

11.9.3)(5.02

)()(5.0

10.9.3)(2

)()(

0

0

ψψφφ

φφ

φ

cNQk

Nq

NrkQ

kNc

Nr, Nc and Nq therefore define the bearing capacity factors. Substituting

equations 3.9.9, 3.9.10 and 3.9.11 with equation 3.9.8, we have

cNcq ૪ HNq ૪ BN ૪ 12.9.3

Equation 3.9.12 is identical to Terzaghi, Meyerhoff and Hansen’s bearing

capacity equation (12, 17, 31). Thus the superposition principles and analytical

approach taken by Terzaghi, Meyerhoff and Hensen is here by derived by the

use of variational analysis.

8.9.32

2

)()(5.0

2

)()(

2

7.9.322

ˆ)(

2

ˆ)(

2

ˆ

2

0

00

2

000

BQ

QkHH

Qkc

B

Qq

HHQk

cQk

BQ

B

Q

૪

૪

૪

૪

૪

67

CHAPTER FOUR

RESULTS AND DISCUSSIONS

Taking the Meyerhoff and Hansen equation for the bearing capacity

given by:

5.0)( qNqcNcq m ૪ BN ૪, 1.4

by Meyerhoff [2, 5] and for Hansen, he included inclination factors ic, iq and ir

this q by Hansen is given as;

5.0)( SqdqiqqNqScdciccNcq H ૪ BN ૪ S ૪ d ૪ i 2.4]13,2[

But for strip footing, shape, depth and inclination factor are taken as 1.

However, Meyerhoff found that bearing capacity factors could be

variously calculated according as (2, 5)

4.4cot1

3.42

45tan tan2

φ

φ φ

qc

o

q

NN

eN

N૪ 5.44.1tan1 φqN

and Hansen in his own work found that bearing capacity factors could also be

calculated according as [2, 3]

7.4cot1

6.42

45tan tan2

φ

φ φ

qc

o

q

NN

eN

N૪ 8.4tan18.1 φqN

Recall Equations 3.9.9, 3.9.10 and 3.9.11 the variational solution results

into the following equations

N૪ 9.9.32

)(0 φQ

)(φkNc N૪ 10.9.3

68

11.9.3)(5.0 ψcq NN

From the semi-empirical equation, equation 3.9.1, values of k ( ) and

Qo( ) are tabulated with respect to given (internal frictional angles, thus [6]

Table 4.1: Semi-empirical equation results

K( ) Qo( )

0 1.2 x 103 8.6 x 10-3

5 18.25 8.1 x 10-1

10 7.70 2.50

15 4.24 2.80

20 2.51 13.62

25 1.54 32.61

30 1.26 62.49

35 0.65 185.0

40 0.51 284.2

45 0.45 596.0

With the above data, the values of the learning capacity factors N૪, Nc

and Nq, are calculated using equations 3.9.9, 3.9.10 and 3.9.11 as derived using

the variational analysis. The results of the calculations are tabulated below.

69

Table 4.2: Bearing capacity factors from variational solution

N૪ =

2

)(φoQ

rc NkN )(φ cq NN 5.0

0 0.0043 5.16 0.5

5 0.405 7.405 1.15

10 1.25 9.62 2.19

15 2.89 12.29 3.79

20 6.81 17.03 6.70

25 16.31 25.11 12.20

30 31.25 39.37 23.23

35 92.50 59.20 41.95

40 142.10 72.47 61.31

45 298.00 134.10 134.60

Simultaneously using equation 4.6 – 4.8 for the bearing capacity factors as

derived by Hansen, the values of Nr, Nc and Nq are calculated, and tabulated

below

70

Table 4.3: Bearing Capacity factors by Hansen’s solution

N૪ Nc Nq

0 0.0 5.7 1.0

5 0.09 7.45 1.57

10 0.09 9.76 2.47

15 1.42 12.56 3.94

20 3.54 17.60 6.40

25 8.11 25.25 10.66

30 18.08 39.46 18.40

35 40.69 59.87 32.29

40 95.41 72.70 64.18

45 240.85 134.56 134.85

Also using equations 4.3 – 4.5 derived by Meyerhoff to calculate the

bearing capacity factor Nr, Nc and Nq, we have as tabulated below

Table 4.4: Bearing capacity factors by Meyerhoff

N૪ Nc Nq

0 0.0 5.70 1.0

5 0.1 7.45 1.6

10 0.4 9.70 2.5

15 1.1 12.50 3.9

20 2.9 17.50 6.4

25 6.8 25.40 10.7

30 15.7 39.44 18.4

35 41.69 59.79 32.30

40 93.60 72.60 64.1

45 262.30 134.50 134.70

71

A comparison of the function Nc( ) as obtained in the approaches by

Meyerhoff and Hansen with the classical solutions for footings is shown below

in fig 4.1. The variational solution corresponds with values smaller as a result

of the effect of the slope.

Since the rotation between Nc and Nq is linear, no separate comparison

is needed for Nq. Also the effect of the slope on bearing capacity factors does

not affect the unit weight of soil r, however Nr was not affected or the effect is

so insignificant that it is ignored.

72

* * *

* *

*

*

*

*

*

*

5 10 15 20 25 30 35 40 45 50

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

Variational solution V.S

Meyerhoff’s solution M.S

Hansen’s solution H.S

Nc

0

Fig. 4.1: Bearing Capacity factor Nc: VS / MS / HS

73

Furthermore, the results of the present work are compared with the

results of Akubiro’s work by equating β = 0, in equation 3.5.18.

We have,

Q = 1

0

x

x

[ (1 + ψ ) + + ] d ̶ 1

0

x

x

( + ] d ……………………..4.9.

Solving equation 4.9 gives [1]

Nc = k( ) ( ) + 1 …………………………………………………4.10.

For clarity, the table below shows the values of bearing capacity factor Nc on

slopy ground and at zero slope i.e. β = 0.

Table 4.5: Bearing Capacity factors from the present work and

Akubiro’s work at zero slope.

Nc (new equation) Nc(Akubiro’s equation)

0 5.16 6.16

5 7.405 8.405

10 9.62 10.62

15 12.29 13.29

20 17.03 18.03

25 25.11 26.11

30 39.37 40.37

35 59.20 60.20

40 72.47 73.47

45 134.10 135.10

74

From the above graph it is deduced that at zero slope (i.e. at β = 0), the

new equation is equal to Akubiro’s equation.

*

* *

*

*

*

*

*

* *

10

20

*

30

*

40

*

50

*

60

*

70

*

80

*

90

100

110

120

130

140

150

160

5 10 15

Akubuiro’s result

20

New equation

25 30

Nc

35

0 40

Fig. 4.2: Bearing Capacity factor Nc: Akubiro/New equation

45 50

* *

*

75

In determining the critical normal stress distribution according to the

variational solution of equation 3.8.51 and 3.8.54 for 0̂ and supporting

equations, equations 3.8.52 for A and 3.8.71 and 3.8.75 for B, it is necessary to

define the ranges of validity of the angle , defining the polar coordinate

system.

If we express the geometrical boundary condition of equations 3.4.1 and

3.4.2 in terms of polar coordinates by introducing equations 3.8.3 and 3.8.2 and

then the results into equations 3.4.1 and 3.4.2, we obtain the following

expressions:

4.8.3

3.8.31

2.8.3sin

1.8.31

sin

2.4.30

1.4.30,Re

2

1

2

2

1

2

00

1

1

r

r

y

x

ry

rx

xxyy

xxyycall

Introducing Equations 3.8.3 and 3.8.4 respectively into equations 3.8.1 and

3.8.2 gives

13.9.3sin

12.9.3cos

r

r

yry

xrx

Introducing equations 3.9.12 and 3.9.13 into equations 3.4.1 and 3.4.2 gives

16.9.30expsin

15.9.30sin

14.9.30cos

10001

000

000

r

r

r

yry

yry

xrx

From equations 3.9.14 and 3.9.15 rearranged,

19.9.3][expexpsin

18.9.3sin

17.9.3cos

01110

00

00

yr

ry

rx

r

r

76

Equation 3.9.19 shows that the relation between 1 and 0 is of the form

expsin

20.9.310

fwhere

ff

This function is only positive is the range

21.9.30 10 and

except θ = 1 , , in which 0)()0()( 1 fff

The function also reaches maximum value at .2 The range of 0 , 1 are

therefore:

23.9.3

22.9.30

02

21

Based on the foregoing, the following ranges of θ are calculated for various

values of (internal frictional angle).

Table 4.6: Computed values of 10 and

0 020 :

211 0:

0 02 210

5 048.1 48.10 1

10 0396.1 39.10 1

15 031.1 31.10 1

20 022.1 22.10 1

25 013.1 13.10 1

30 0047.1 047.10 1

35 096.0 96.00 1

40 0872.0 872.00 1

45 07854.0 7854.00 1

77

Precise values for θ0 and θ1 are obtained for various values by

choosing θ0 within the range tabulated and using equation 3.9.19 modified by

the satisfaction of equation 3.9.21, the corresponding θ1 value is obtained.

It is convenient to take r0 to equal the foundation width; i.e. r0 = B. For a

foundation of total width 2.40m, r0 = 2.40m. It must be pointed out that the

radius r0 that defines the point (r0 , θ0) of the polar coordinates of the point (x0,

y0) would vary with the cohesive strength for various soils. Consequently, from

equation 3.8.13.

13.8.30

0

err

the shape of the critical surface r(θ) depends on both c and also. This result is

different from that of the classical solutions in which the shape of the critical

rupture surface is independent of cohesion.

Example

A footing 2.25m square is located at a depth of 1.5m in soil, the cohesive

strength being zero. The unit weight of the soil is 18kN/m3 and the saturated

unit weight is 20kN/m3. If the unit weight of water is 9.8kN/m3, determine the

bearing capacity when;

(a) the water table is well below foundation level,

(b) the water table is at the surface; for the various internal frictional angles

and compare with the results obtained using the Meyerhoff.

Solution

In using the equation derived for strip footing to calculate the value of

the bearing capacity for square footing, the contributions by the surcharge and

cohesion are multiplied by 0.4 and 1.3 respectively instead of 0.5 and 1.0 thus

rHNqcNcrBNrq 3.14.0

For the case of a cohesionless soil i.e. c = 0, therefore q = 0.4rBNr + rHNq case

a: 3/18,5.1,25.2 mkNrmHmB and the bearing capacity factors in

tables the following results are obtained.

78

Table 4.7: Bearing capacity values by V.S and M.S

q (Variational solution) q (Meyerhoff solution)

kN/m3 kN/m2

0 13.57 27.00

5 37.00 44.82

10 79.38 73.98

15 149.15 123.12

20 291.22 219.78

25 593.62 399.06

30 1133.46 751.14

35 2631.15 1547.48

40 3957.39 3247.02

45 8461.8 7886.16

From the foregoing, it is seen that the Meyerhoff’s solution under-

estimates the bearing capacity. The variational solution is obtained in the class

of perfectly smooth y(x) and σ(x) functions. A class of functions which permits

discontinuities in derivatives from a certain order and higher is not

accommodated in this formulation.

However, Meyerhoff and Hansen’s solution is built by combining three

plastic zones (active, passive and radical). At the points of junction of different

zones, there exists a discontinuity of the second order and higher. This is why

Meyerhoff and Hansen’s solution are lower than the variational solution.

79

CHAPTER FIVE

CONCLUSIONS AND RECOMMENDATIONS

5.1 Conclusion

The relevance of the study has been fully demonstrated. First, a new

approach to the computation of the bearing capacity factors is hereby advanced.

The factors Nr, Nq and Nc are hereby computed from the semi-empirical

values of θ0 obtained for cohesionless soils c = 0. Good agreement is evidently

noticed when compared with those of Meyerhoff and Hansen’s factors. Second,

the superposition principle that was assumed by Meyerhoff is derived here by

variational calculus. Consequently, the representation of bearing capacity by

the three factors Nr, Nc and Nq is justified.

Third, the function Nc( ) and Nq( ) correspond rather closely with

those obtained from the plasticity theory. Besides, the classical relation

between Nc and Nq is hereby recovered by the variational approach. It

therefore independent of the constitutive law of the soil mass.

Fourth, a cursory equation for the computation of the critical normal

stress distribution on the rupture surface from determinable strength parameters

of the soil is also evolved. Since the analysis here admits the logarithmic spiral

curve for the critical rupture surface, the magnitude of the normal stress

distribution on the surface is rightly observed to vary with position on the

curve.

This result is evidently an improvement over the normal stress equation

suggested by De Beer [31] and given by

whichin

qq φσ sin134

100

21

0 qNqcNcq ૪BN૪ and q = ૪H

valuetconsatoitswhichand tanlim 0σ

80

5.2 Recommendations

The present analysis considers the general case of concentric and vertical

loading. Sometimes, however, the loading is eccentric with respect to the center

of the foundation. Also it is possible to have inclined loading of a foundation

system.

Each of these cases requires a new approach to the analysis. For the

former, the integral constraint equation for the rotational equilibrium (Eq.

3.1.11) is no more identical to zero, rather it is equated to Q.e, where e is the

eccentricity. Similarly, the equation for the geometrical boundary condition

now becomes, lex /

The analysis is then carried out with these modifications.

For the latter case of inclined loading, all that would be necessary to

repeat the analysis using a coordinate system that is inclined in the direction of

the loading and resolve forces appropriately in the various directions.

It therefore recommended that further work be carried out in this area

and taken into account the foregoing modifications. Such work would

obviously be worthwhile in view of the novelty of the approach and

recognizing the encouraging results emanating from the present work.

81

REFERENCES

1. Akubuiro, Edwin Onwukwe “Bearing capacity and critical normal stress

Distribution of soils by Method of variational calculus” Unpublished

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2. Bowles J.E., Foundation Analysis and Design, 5th Edition, McGraw-Hill,

London, 1997.

3. Chukwueze, H.O. Advacned Engineering soil Mechanics, ABic Books and

Equipment Limited, Enugu, 1990.

4. Garg, S.K., Soil Mechanics and Foudnation Engineering, 6th Edition, Khanna

publishers, Delhi, 2005.

5. Smith, G.N. and Smith, Ian G.N., Elements of soil Mechanics, 7th Edition,

Blaokwell Science Ltd, UK, 1998.

6. Feda, J. “Research on the Bearing capacity of Loose soil”, Proceeding, 5th

International Conference on soil Mechanics and Foundation

Engineering, vol. 1, 1961. pp. 635.

7. Terzaghi, K., and Peek, R.B., Soil Mechanics in Engineering practice, 2nd

Edition, John Wiley and Sons Inc., New York 1948.

8. Swokowski, E.W., Calculus, 5th Edition, P.W.S-kent Publishing Co., Boston,

1991.

9. Elsgolts, L., Differential Equations and the calculus of variations, 6th Edition,

MIR publishers, Moscow, 1977.

10. Pars, L.A., An Introduction to the calculus of variations, Hienemann

Educational Books, London, 1962.

11. Pipes and Harvill, Applied Mathematics for Engineers and Physicists, 3rd

Edition, McGraw Hill Book Co; 1971.

12. Terzaghi, K., Theoretical Soil Mechanics, John Wiley and Sons Inc., New

York, 1943.

13. Arora, K.R., Soil Mechanics and Foundation Engineering, 6th Edition,

Standard publishers, Delhi, 2003.

82

14. Robert, W., Calculus of variations: With Applications to Physics and

Engineering, McGraw Hill Book Co, USA, 1952.

15. Acho, T.M. “Lectures on calculus of variations” unpublished monograph,

Department of Mathematics, UNN, 1990.

16. Stroud, K.A., Engineering Mathematics: Programmes and problems, 4th

Edition, Macmillan press Ltd, London, 1995.

17. Meyerhoff, G.G. “Ultimate Bearing capacity of Foundations”

Geotechnique, London, England, vol. 2, 1951, pp 301-322.

18. Whylie, C. and Barret, L.C. Advanced Engineering Mathematics, 5th

Edition, McGraw Hill Book Co., Singapore, 1985.

19. Casuba, G. Prabhakar Narayan, Vijay, P. Bhatkar, and Ramamurthy, T.,

“Non-Local variational Method in stability Analysis” Journal of The

Geotechnical Engineering Division, Proc. of the American society of

Civil Engineers, ASCE, vol. 108, No. GT II, Nov. 1982, pp 1143-1457.

20. Bolton, M. A Guide to soil Mechanics, 2nd Edition Macmillan, London,

1973.

21. Atkinson, H.H., Foundation and shapes, Mchraw Hill Book Co., Ltd., 1981.

22. Murthy, V.N.S., Geotechnical Engineering: Principles and Practice of Soil

Mechanics and Foundation Engineering, Marcel Dekker, Inc. New York,

2002.

23. Hon Yim, Ko and Ronald, S.F., “Bearing capacities by Plasticity Theory”,

Journal of Soil Mechanics and Foundation Division, Proc. ASCE, vol.

99. No. SM I, Jan. 1973, pp 23-43.

24. Wilfred Kaplan, Advanced calculus, 5th Edition, publishing House of

Electronics Industry, Tokyo, 2004.

25. Moser, J., “Lecture Notes on variational calculus” Zurich, September 1988.

26. Ike, C.C. Principles of soil Mechanics, De-Adroit Innovation, Enugu, 2006.

27. Day, R.W. Geotechnical and Foundation Engineering: Design and

Construction, 1st Edition, McGraw Hill, London, 1999.

83

28. Ike, C.C. “Critical Rupture Surface Equations by Method of Calculus of

variations” Unpublished M.Engr. Thesis, Department of Civil

Engineering, UNN, 1979.

29. Agunwamba, J.C., Engineering Mathematical Analysis, De-Adroit

Innovation, Enugu, 2007.

30. Ike, C.C. Advanced Engineering Analysis, 3rd Edition, De-Adriot

Innovation, Enugu, 2009.

31. Das, B.M., Shallow Foundations: Bearing Capacity and settlement, 4th

Edition, CRC press, London, 1999.

84

APPENDIX A

Perform the integration

θθψθ de cos3

The above integration is carried out by parts.

Let )1(cos,3 ddveu

)5(

)4(cos

)3(sincos

)2(3

3

3

3

3

vdvuvudvBut

udvdeNow

ddvv

dedu

ed

du

Substituting

)6(sin3sinsin3sin 3333 deedeeudv

integrating again the cycle function

)7(cos3cossin

cos;3

sin

:sin

333

3

3

3

deede

vedu

ddveuLet

de

Substitute (7) into (6), then

)9(cos3sin91

cos

cos3sincos3sincos91

cos9cos3sin

)8(cos3cos3sincos

2

33

33332

3233

3333

θψθψ

θθ

θψθθψθθθψ

θθψθψθ

θθψθψθθθ

ψθψθ

ψθψθψθψθ

ψθψθψθ

ψθψθψθψθ

ede

eeede

deee

deeede