Lateral “Pressures” and the Retaining Wall Problem

Lateral “Pressures” and the Retaining Wall Problem

To further set the stage, let us look at the same retaining wall problem that Coulomb considered as a young civil engineer in charge of various construction projects on the island of Martinique in the 1760s. In order to address this problem, Coulomb first had to develop from scratch the concepts of slip surfaces and limit analysis along with his law of dry frictional resistance and cohesion.** Unfortunately, the information in his 1776 paper, while not obscure, is so condensed that it must be studied with great attention and expanded to be appreciated. Coulomb was elected to the academy in 1781 and never returned to soil mechanics and the retaining wall problem, so his solution comes down to us as a few algebraic equations and two small figures* with a minimum of explanation. It took nearly 200 years and thousands of pages of text to recognize that his work includes all the concepts and methodology needed for analysis and design of gravity retaining walls.

If we consider a vertical line (the wall) embedded in a MC halfspace (Figure 12.4a), the weight of the soil, B_z = γ, introduces the elastic stress field:

insights into such a short space that, according to Poncelet, “during the next 40 years the attention of engineers seemed not to focus on any of them.” Perhaps because the presentation of this paper in 1773 was his introduction to the French Academy of Sciences, Coulomb felt intimidated as “only a practicing engineer” and kept throwing in fundamental insight after insight with less and less explanation to the point where the power of the whole obscures the beauty of the individual parts. Certainly his introduction in which he states:

If I dare to present it (this paper) to this Academy it is only because the feeblest endeavors are kindly welcomed by it when they have useful objective . . .

would suggest he suffered at this time from the same “engineering inferiority syndrome” that Hooke developed a century earlier trying to deal with Newton’s intolerable egocentricity and wholesale theft of Hooke’s ideas in the name of science. In fact, Coulomb amplifies this paranoia (probably justified) with further selfdenigration noting that

… while great men will be carried to the top of the edifice (science) where they can mark out and construct the upper stories, ordinary artisans (engineers) who are scattered through the lower stories …. should seek only to perfect that which cleverer hands have created.

Unfortunately, such nonsense is as prevalent a view today as it was in Hooke’s and Coulomb’s time. * Coulomb’s figure (1776).

jacking force being gradually reduced and the wall rotating about its base or translating slightly away from the soil so that the shear strength could be mobilized to help support the soil (Figure 12.4c). However, at some point when the shear strength is fully developed on some slip surface, the material would slide downward and outward if the jacking force were reduced further. The jacking force or the force necessary to just prevent failure (sliding of the wedge), is the active force, P_A. If it is assumed that there is no wall friction then the stress field [Equation (12.4)] is linear, the slip surface is straight,

If instead the wall is jacked into the soil, Figure 12.4d, the passive case develops where the friction mobilized in the soil again resists the motion but, in this case, adds to the jacking force necessary to cause failure. Again, if there is no wall friction, the slip surface will be straight and Equation (12.3a) applies with σ₃ =γz = σ_z (line ml in Figure 12.5 for c > 0, ϕ > 0 ). Thus the passive

force P_P just necessary to cause the failure wedge to slide upward is

The amount of plastic strain necessary to fully mobilize the shear strength and fully develop the slip surface corresponding to either P_A or P_P is uncertain. Yield may occur first at some point on the eventual slip surface and then spread with continuing small movements of the wall until the complete mechanism develops. In soils (to some extent rock and concrete) the issue is further complicated in that not only does the shear strength depend on the confining pressure, but also on the soil’s density, the water content, and even its history. Moreover, dense soils (and concrete) have a peak strength and then a lower residual value at large strains (a strainsoftening material). Thus, the complete active and passive curves for lateral load vs. deflection shown in Figure (12.4e) are only approximate as far as values for the wall rotation ψ are concerned.

Eighty years later, Rankine,* considering the retaining wall problem without regard to Coulomb’s solution, again assumed a frictionless wall interface but also neglected cohesion. In rederiving Coulomb’s equations he introduced coefficients of lateral pressure:

noting that, for such a special case of a purely frictional material like sand, lateral pressures are like fluid pressure (a heavy fluid) increasing linearly with depth as shown in Figure 12.5 lines Ac and Ah. Thus σ_{h =} σ_x = Krz and the total active or passive force on the wall will be: