# COULOMB, RANKINE, OR OTHER PRESSURES?

Two classic lateral earth pressure theories (Coulomb and Rankine) were presented at failed stage in backfill soils, and these two theories are still popular among practicing geotechnical engineers. Several questions may arise at this stage:

1. Is it the engineer’s preference to use either solution?
2. Are there any rules to select either solution?
3. Are there any limitations on those theories?

In order to answer these questions, first, distinct differences between Coulomb theory and Rankine theory are listed:

– Rankine theory assumes that all the backfill soils are in a state of plastic equilibrium (failure) as seen in Figures 1 and 2; while Coulomb theory assumes that failure occurs only along a failure surface in the backfill and along the wall face, as seen in Figures 3 and 4, and that the inside of a failed wedge could be solid (non-failed).

– In Rankine theory, due to the plastic equilibrium of all the soil elements, the distribution of the lateral pressure is a linearly increasing function (triangle distribution), while Coulomb’s theory assumes its triangular distribution without any assurance.

– Rankine pressure is applied normal to the boundary (wall) face, while Coulomb pressure is applied with the δ angle (wall friction angle) inclined from the normal to the wall face.

Now consider typical earth pressure problems in Figure 5: (a) gravity retaining wall, (b) cantilever retaining wall, (c) basement wall, (d) geosynthetic reinforced earth, and (e) bridge abutment. Among them, obviously, case (c) uses at-rest lateral earth pressure (K0) since no movement of the basement wall of this stable structure is anticipated.
Case (a) and case (e) may be the Coulomb case since the back faces of the wall may become sliding planes. Meanwhile, case (b) and case (d) will be Rankine’s case since the wall face will not be sliding surfaces. In case (b) and case (d), Rankine’s lateral earth pressure is applied on imaginary vertical planes (shown with dotted lines).
Case (a) and case (e) need further attention. Both could be a Coulomb’s case. However, anticipated failure modes are different. In case (a), the wall most likely fails by a rotation of the wall about the base of the wall, while case (e) may be a failure mode of rotation about the top due to restriction of top movement due to the bridge structure.

Wall movement mode (rotation about the top, rotation about the base, and translational) makes pressure distribution different. Figure 6 demonstrates potential pressure distribution differences according to the different wall movement modes.
In Figure 6(a), initial backfill soil elements are modeled with equal parallelograms.
In Figure 6(b), the wall moves in a translational way. In this case, a solid soil wedge like the one observed in Coulomb’s model would be formed in the backfill soil and the wall face would become a sliding surface. Inside the wedge, initial parallelogram elements still maintain the original shapes. In Figure 6(c), the wall rotates about the base. Most likely, all the backfill soil elements of the failed section deform to more skewed parallelograms as seen. This implies that all the elements in the failed zone become plastic (failed) as in the case of Rankine theory.
However, in Figure 6(c), the back face of the wall may be a failure surface, so the Coulomb’s solution with a triangle earth pressure distribution may be the most appropriate solution.
Based on these observations of backfill soils, lateral earth pressure distributions are predicted in Figure 6(d). At-rest pressure (K0) from no wall movement (Figure 6a) is seen with a dotted line. Since Figure 6(c) is similar to the Rankine’s pressure distribution, the pressure distribution will be triangular shaped.
Figure 6(b) could be the Coulomb’s condition since the solid failure wedge will be formed in the backfill. However, the distribution will be hardly triangular shaped as Coulomb assumed. In fact, the non-yielded soil elements—in particular, at the upper part of the wedge—form arches between the wall face and the failure plane in the backfill. Arching stress will be higher at the upper section of the backfill since more elements of soils are involved to form arches. Accordingly, the distribution will be the one shown as “arching active” in Figure 6(d).
This illustration suggests that Coulomb’s triangular pressure distribution assumption is not always true, and thus the point of application of the thrust could be different from ⅓H from the base of the wall. Readers can refer to the literature (e.g., Fang and Ishibashi 1986) on the effect of wall movement modes on the lateral earth pressures.

Estimation of lateral earth pressure is a very important practice in many foundation designs. Basic theories of Coulomb and Rankine, which are widely used by engineers at present, were presented in this chapter. Engineers should be aware of those limitations and the different assumptions behind those theories.