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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
BB
cc
σσ
3.5.3ˆ HcHB
cc ψψ
4.5.322
ˆ HB
H
B
5.5.322ˆ
2 HQ
B
H
B
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
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
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
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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.
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MIR publishers, Moscow, 1977.
10. Pars, L.A., An Introduction to the calculus of variations, Hienemann
Educational Books, London, 1962.
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Edition, McGraw Hill Book Co; 1971.
12. Terzaghi, K., Theoretical Soil Mechanics, John Wiley and Sons Inc., New
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Standard publishers, Delhi, 2003.
82
14. Robert, W., Calculus of variations: With Applications to Physics and
Engineering, McGraw Hill Book Co, USA, 1952.
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17. Meyerhoff, G.G. “Ultimate Bearing capacity of Foundations”
Geotechnique, London, England, vol. 2, 1951, pp 301-322.
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Edition, McGraw Hill Book Co., Singapore, 1985.
19. Casuba, G. Prabhakar Narayan, Vijay, P. Bhatkar, and Ramamurthy, T.,
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Geotechnical Engineering Division, Proc. of the American society of
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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.
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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.
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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.
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Innovation, Enugu, 2007.
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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