COULOMB’S EARTH PRESSURE THEORY
Coulomb (1776) proposed that a condition of limit equilibrium exists through which a soil mass behind a vertical retaining wall will slip along a plane inclined an angle u to the horizontal. He then determined the slip plane by searching for the plane on which the maximum thrust acts. We begin consideration of Coulomb’s theory by reminding you of the basic tenets of limit equilibrium. The essential steps in the limit equilibrium method are
(1) selection of a plausible failure mechanism,
(2) determination of the forces acting on the failure surface, and
(3) use of equilibrium equations to determine the maximum thrust. Let us consider a vertical, frictionless wall of height Ho, supporting a soil mass with a horizontal surface (Figure). We are going to assume a dry, homogeneous soil mass and postulate that slip occurs on a plane inclined at an angle u to the horizontal. Since the soil is dry, g9 5 g. We can draw the free-body diagram as shown in Figure and solve for Pa using statics, as follows:
We are using P rather than Pa because Pa is the limiting value.
The weight of the sliding mass of soil is
At limit equilibrium,
Solving for P, we get
To fi nd the maximum thrust and the inclination of the slip plane, we use calculus to differentiate Equation with respect to u:
Coulomb failure wedge.
which leads to
Substituting this value of u into Equation , we get
This is the same result obtained earlier from considering Mohr’s circle. The solution from a limit equilibrium method is analogous to an upper bound solution because it gives a solution that is usually greater than the true solution. The reason for this is that a more effi cient failure mechanism may be possible than the one we postulated. For example, rather than a planar slip surface we could have postulated a circular slip surface or some other geometric form, and we could have obtained a maximum horizontal force lower than for the planar slip surface. For the Rankine active and passive states, we considered the stress states and obtained the distribution of lateral stresses on the wall. At no point in the soil mass did the stress state exceed the failure stress state, and static equilibrium is satisfi ed. The solution for the lateral forces obtained using the Rankine active and passive states is analogous to a lower bound solution—the solution obtained is usually lower than the true solution because a more effi cient distribution of stress could exist. If the lower bound solution and the upper bound solution are in agreement, we have a true solution, as is the case here. Poncelet (1840), using Coulomb’s limit equilibrium approach, obtained expressions for Ka and Kp for cases where wall friction (d) is present, the wall face is inclined at an angle h to the vertical, and the backfi ll is sloping at an angle b. With reference to Figure, KaC and KpC are