however, to give a succinct description of the method by which the general formula is evolved. 2 In fig. 87, let the angle NOM (=) represent the limiting angle of repose, and the semicircle N2 N No, the locus of the point N, as in fig. 85. Through O draw the line OXY, making the angle MOY= a, the obliquity of the conjugate pressures, and cutting the semicircle in X and Y. Then the limits of the ratio of the intensities of the conjugate pressures are OY Ο Χ and Ο Χ The angle & may have any value between zero and p. In the former limit, which is the case when the conjugate pressures are perpendicular to each other, and become principal stresses, O X Y coincides with O N2 No and is the minimum value of When the obliquity is ON, O No ( 1 - sin 2 2 the greatest possible, such that a = 9, the points N, and N, coalesce in N, and the limit of the ratio of the conjugate pressures becomes unity. For any intermediate position in which α = XO M, the limiting ratio (2) of the conjugate pressures may be determined as follows:-Draw S M perpendicular to X Y, and join M X, MY, each line making the angle with X Y. Now, as the stresses are inclined to one another at the angle a, the intensity of the vertical pressures in the case of earth work will be equal to the weight of a unit column multiplied by cos a. It will be seen that when the surface of the ground is horizontal a = 0, cos a = 1, and as previously demonstrated. wh2 1- sin P= For a surface sloping upwards at the angle of repose, a = and (17) A D According to Professor Rankine, the line of action of the resultant force is always parallel to the surface of the ground. A modification of the theory, due to Dr. Scheffler, determines the direction of the earth thrust as inclined to the horizontal at a constant angle, identical with the angle of repose. In this way, although the total amount of the thrust is greater by Scheffler's hypothesis (being as E G to EF, fig. 88), yet, except in one instance, the overturning effect is less, owing to the nearer approach of the line of thrust to the vertical. The one exception is the case in which the surface of the ground has an inclination to the horizontal, and then the two theories lead to the same result. G E F B C Fig. 88. Another modification, due to Professor Reilly, takes into consideration the batter, or inclination to the vertical, of the back of the wall. MX at an angle, OM X N In fig. 89, the point X is determined by drawing 2 B. = Then the total thrust is measured graphically by to When the back of the wall is vertical, B = 0, and the equation reduces which agrees with Rankine's result for similar conditions. The direction of the resultant is constant at an angle 7 to the horizontal, such that y = ẞ+ λ, the last-named angle being deduced from the equation— sin 2 = sin sin 26 (19) It will be observed that in none of the foregoing expressions is any account taken of the friction exerted by the particles against the back of the wall—a factor which tends to resist displacement. In fact, the assumed conditions only hold good at a suitable distance from the wall beyond the range of its frictional influence. A formula has been devised by Professor Boussinesq to cover this defect. If be the angle of friction between the wall and the earth, and x the horizontal distance from the face of the wall, the following expressions are given by him for the intensity of horizontal and vertical pressure for values sin of a less than h:- 1 + sin At the face of the wall x = O, and the expressions become— (21) 1 + tan + sin Coulomb's Theorem.—What is practically the same formula as that enunciated by Rankine has been developed by MM. Prony and Coulomb, on somewhat different lines, as follows: In fig. 90, C E is the line of repose. Were the wedge of earth, D C E, a solid mass it would have no tendency to slide down the plane, CE, the frictional resistance between the two surfaces being sufficient to counteract movement. Evidently, then, if the earth yield at all, it must do so by fracturing along some other plane, the position of which remains to be determined. Meanwhile, assume a position, C F. Through the centre of gravity of the wedge, DCF, draw K O, vertically, to represent its weight, W. Draw LO, making an angle, o, with the normal to the plane, C F, to represent the ultimate reaction of the plane, and L K a horizontal line through K. Then the pressure on the back of the wall is It is now necessary to find the angle which gives the greatest possible value to P. Take the variable factors in the preceding expression, differentiate, and equate to zero. and, therefore, since the sines of supplementary angles are equal, whence it is evident that the greatest thrust is obtained when the line of rupture, CF, bisects the complement, D C E, of the angle of repose. In this case, which is a variant, in form only, of Rankine's expression, since There are, in fact, several different methods of arriving at the same result. For instance, without using the angle of friction, as in the preceding investigation, take the forces acting at the point, O, in fig. 91, and resolve them along the plane of rupture, C F. Then equate them for equilibrium. The coefficient of friction being tan, we have Chaudy's Theorem.*-The undoubtedly excessive values attributed to earth pressure, in the preceding investigations, have led a French engineer to approach the problem from a fresh standpoint, and to evolve a solution which, despite its complexity, yields results more in accordance with practical observation. M. Chaudy starts with the postulate that a pressure, Q, applied to the surface of a mass of earth causes an oblique thrust, P, and the object of his investigation is to find the amount of this thrust, and the angle at which *Mémoires et Comptes Rendus des Travaux de la Société des Ingénieurs Civils de France, Bulletin de Decembre, 1895. |