By both Rankine's and Coulomb's theory, one or the other of which is generally used when any theory is employed, when the back of the wall is vertical and the upper surface of the earth is level, the usual case in practice, the resultant thrust is perpendicular to the back of the wall, which seems to be inconsistent with the theory of a wedge of earth sliding down the back of the wall. Further, ex perience and experiments with the pressure of grain in bins show that the pressure against the side of the bin greatly influences the amount of the resultant pressure; and hence it is safe to conclude that the friction against the back of the wall is an important factor in the stability of a retaining wall. Including the friction on the back of the wall materially increases the theoretical stability of the wall.
Some authors, in deducing the formula for lateral pressure, con sider the wall as replacing a similar mass of earth, and then assume that because the pressure across any plane in a homogeneous mass of earth is normal to that plane, the resultant pressure will be normal to the back of a wall. But whether the earth is thrown in loosely or is rammed in behind a retaining wall, the settlement or the com pression causes it to slide down the back of the wall and develop friction; and hence the state of stress between the compressible earth and the unyielding wall is entirely different from that between any two adjacent portions of the imaginary homogeneous mass of indefinite extent. Experience uniformly shows that earth, whether deposited loosely or rammed behind a retaining wall, always settles, and hence friction against the back of the wall is always developed. This friction may disappear if the earth shrinks by drying out, in which case it is probable that enough cohesion is developed to render the earth self-supporting.
Again, Weyrauch's formula, of which several others are special cases, is deduced for a wall leaning away from the earth to be sup ported, and is claimed to be perfectly general; and yet, if applied to a wall leaning toward the earth to be supported, it gives a lateral thrust which increases with the backward inclination of the wall.
Nearly all of the theories are inconsistent when applied to special cases. For example, according to Rankine's theory* for a vertical wall, and for earth standing at a slope of 1 I to 1, and for a level top surface, E = 0.28 (j; w h'); and for a surcharge at the angle of repose, E = 0.83 (i w h'). The last result is practically three times
the first; and if the back of the wall be considered to lean away from the earth supported, the value of E for a surcharge as above is four times that for a level top surface. These results are contrary to reason, to ordinary experience, and to careful experiments (see 1008), which shows that the theory is fundamentally wrong. Rankine, who proposed this theory, said of it: "For want of precise experimental data, its practical utility is doubtful." The preceding examples illustrate that most, if not all, theories are logically self-contradictory, either in their fundamental assump tions or in their application to special cases. These inconsistencies crop out in one place in one theory and in another place in another theory, which shows that the underlying assumptions are inconsistent.
1011. Most of the theories are at variance with experiments and experience. For example, all theories agree that for a level earth surface and a wall with a vertical back, the pressure of the earth against the wall AC, Fig. 112, is equal to the pressure of the prism ACE sliding down the perfectly smooth plane, CE, which bisects the angle between the back of the wall and the natural slope, CD; whereas "experi ments show that the lateral pressure of the prism ACE between two boards AC and CE against AC is quite as much when the board EC is at the slope of repose, 11 to 1, as when it is at half that angle; and there was hardly any difference whether the board was hori zontal or at a slope of 1 to 1, or at any intermediate slope." * Sir Benjamin Baker used Coulomb's formula, equation 4, page 493, to interpret indirect experiments, and regarded the theoretical pressure of the earth as that of a liquid having a weight per cubic unit = w tan' (45° — 1 0). If w = 100 lb. per cu. ft., and & = 34° (natural slope 11 : 1), then the theoretical pressure against the back of the wall is that of a liquid weighing 28 lb. per cu. ft. He found for his different practical examples that the pressure producing over turning was equal to that of a liquid weighing from 7.4 to 11 lb. per cu. ft.; and comparing these with the corresponding theoretical pressure, he found the factor of safety to vary from 2.1 to 3, and concluded that a wall which by, Coulomb's formula was on the point of overturning has a factor of safety of at least two.* One of the author's students experimented with fine shot, which appears to fulfill the fundamental assumptions of this theory, and found that the observed resistance was 1.97 times that computed by Coulomb's formula.* The uncertainties of the fundamental assumptions and the questionableness of a portion of the mathematical process are sufficient explanation of the difference between theory and practice.