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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.


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; S  Extr S y (x) , (x)                      1.1 ex 
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.
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
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 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
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).
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
2 1.1 2 2 R  v  ySindy                
Where R = resistance suffered
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


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