Theories of Lateral Pressure

THEORIES OF LATERAL PRESSURE Or EARTH. The numerous theories of the lateral pressure of earth may be divided into two classes, viz.: 1. The first class consists of those theories that assume that when a retaining wall fails, a prism of earth severs its connection from the bank and slides on a plane surface called the plane of rup ture. The first theory of this class was proposed by Coulomb in 1773; and it has since been elaborated by Poncelet (1840) and Scheffler (1857). and ingenious graphical solutions have been pro posed by Moh and von Ott. This theory is frequently used; and is usually called Coulomb's theory, but sometimes the "theory of the prism of maximum pressure." 2. The theories of the second class are founded upon what is called the principle of conjugate pressures, whereby the differential equations representing the equilibrium of a particle in the interior of the supported earth are first established, and then by integration the total resultant earth pressure is deduced. This theory was proposed by Rankine in 1858, and has since been elaborated by Levy, Winkler, Mohr, and Weyrauch (1878); and ingenious graphical solutions have been proposed by Culmann, Greene, Scheffler, von Ott, and Winkler. This theory is usually called Rankine's, but some times "the theory of conjugate stresses." The several theories of the lateral pressure of earth will be considered under three heads, viz.: (1) theories for the amount of the pressure; (2) theories for the direction of the pressure; and (3) theories for its point of application.

Theories for the Amount of the Lateral Pressure. Although it is fre quently claimed that the two classes of theories are essentially different in their fundamental assumptions, and although the mathematical processes employed in the two cases are entirely different, the formulas for both classes of theories are only special cases of a single general equation, as will now be shown.

In Fig. 108, AB is the back of a wall which makes an angle 0 with the horizontal; BC is the natural slope, which makes an angle with the horizontal; BM is the plane of rupture, which makes an unknown angle x with the hori zontal; 0 is any point in the supported earth; Ti' is the weight of the prism ABM; OL is perpendicular to AB, and ON is perpendicular to BM. The force W is resolved into two components E and R,

the former making an unknown angle z with the normal to the back of the wall and the latter an angle 4) with the normal to the plane of rupture.

h =

the vertical height of the wall; E = the pressure of the earth against the back of the wall, the angle between E and the normal to the back of the wall being z; w e the weight pf a cubic unit of the earth; W = the weight of the earth prism, per unit of length of the wall, causing the maximum lateral pressure; 6 = the angle between the back of the wall and the horizontal; = the angle of repose of earth, i.e., the angle between the natural slope and the horizontal; x = the unknown angle between the plane of rupture and the horizontal; z = the unknown angle between the resultant earth pressure and the normal to the back of the wall; It is assumed that the earth prism ABM, Fig. 108, is in equilib rium under the action of three forces: (1) the weight of the mass: (2) the resultant reaction of the wall, which is equal and opposite to E; and (3) the resultant reaction of the plane BM, which is equal and opposite to R. Under these assumptions, The angle WOR = x — 4, and the angle WRO = 6 + z — x + ˘. Substituting these values in equation 1 gives To find the greatest thrust of the earth against the wall, differ entiate equation 2 with reference to E and x, and find the maximum value of E. To differentiate equation 2, all of the terms containing x must first be reduced to the form cot (0 — x). This transforma tion and the subsequent differentiation and reduction are too long to be presented here. The maximum value of E is Equation 3 is a general formula for the maximum lateral pressure of earth against a retaining wall in terms of z, the unknown angle between the resultant earth pressure and the normal to the back of the wall.* Obviously equation 3 is limited to values of 8 not greater than 0, and to values of 0 greater than 0. The angle 8 may be either plus or minus; and 0 may be more or less than 90°. A trial will show that E increases with both 0 (the angle of the back of the wall with the horizontal) and 8 (the angle of the surcharge).