Rankine's formula may he produced in the same manner as Poncelet's by assuming the pressure parallel to the upper slope.
Thus in Fig. 61 if the angle made by P with the normal to the «all be equal to i, we find which may be transformed 1 into Rankine's formula as given above. TVeyrauch's theory is practically the same as Rankine's although produced in a different way.
all of the ordinary formulas for earth pressure, the effect of cohesion is neglected. Experiments indicate that this effect is not sufficient to affect very materially the actual pressure upon a wall. It causes the earth to break off and slide upon a concave surface in stead of a plane surface. At the upper surface of the earth, the coke ' Wm. Cain, Practical Designing of Retaining walls, 1914, p. 103.
siocr is sufficient to overcome the lateral thrust and cause the earth to stand in a vertical position, while as the lateral thrust increases with the depth, the cohesion becomes relatively less important and the surface of rupture flattens out. When earth is placed behind a wall after it is constructed cohesion is probably negligible at first, although after the earth has become compacted may develop in some cases so that practically no pressure comes against the wall. It is so uncertain that no reliance should be placed upon it in designing walls.
Value of Theories.—On account of the variable nature of the material, it is evident that estimates of earth pressures are only rough approximations to the actual pressures. The material assumed as possessing uniform friction and without cohesion does not exist in practice. The general laws developed, however, do give rational methods of reaching reasonable estimates upon which safe designs may be based.
Experiments upon sand pressures, and experience with walls in use, indicate that Coulomb's use of horizontal earth pressures, or Rankine's thrust parallel to earth surface, where the surface is near the horizontal, give thrusts much greater than those actually produced upon walls with vertical backs. For such walls, the use of the Poncelet's formulas, taking into account the friction of the earth on the back of the wall, give results which seem to agree fairly well with experiment and experience.
For walls leaning forward, so that considerable weights of earth rest upon them, Rankine's formulas may be applied to find the thrust upon the vertical section through the earth at the inner edge of the base of the wall. This thrust, combined with the weight of earth resting upon the wall, gives the thrust against the wall.
124. Computation of Earth Thrusts.—When the back of a wall is nearly vertical, the thrust may usually be taken as making the angle of friction with a normal to the surface of the wall, as assumed in the theory of Poncelet. For such walls the thrust may be obtained from the formula already given: If we place values of Q may be tabulated for various slopes and angles of friction as shown in Table \VII. The of 1' obtained by this method tire supposed to act against the wall at a distance h . 3 above the base, and make the angle of friction with the normal to the \va11.
When a mass of earth rests upon a wall, as in a wall with sloping back or a reinforced concrete wall, the formula of Rankine for pres sure upon a vertical section may he applied. This pressure combined with the weight of the earth resting upon the wall gives the thrust against the wall.
The value of the pressure upon the vertical section is given by Rankine's formula: Values of K corresponding to various values of i and 4 are tabu lated in Table XVIII, thus greatly reducing the labor of computing the pressures. E as computed from this formula is supposed to act at a distance S/3 from the bottom of the section and to be parallel to the upper surface of the earth. S in this formula is the height of the earth section and not the height of the wall.
Angle of Friction.—In order to be able to apply any of the formulas for determining earth pressures, it is necessary to know the weight per unit volume and the angle of friction of the earth. These vary with the kind of material to be filled behind the wall and its condition as to compactness and moisture.