Lateral Earth Pressure: 13 At-Rest, Rankine, and Coulomb Retaining structures such as retaining walls, basement walls, and bulkheads commonly are encountered in foundation engineering as they support slopes of earth masses. Proper design and construction of these structures require a thorough knowledge of the lateral forces that act between the retaining structures and the soil masses being retained. These lateral forces are caused by lateral earth pressure. This chapter is devoted to the study of the various earth pressure theories. 13.1 At-Rest, Active, and Passive Pressures Consider a mass of soil shown in Figure. 13.1a. The mass is bounded by a frictionless wall of height AB. A soil element located at a depth z is subjected to a vertical effective pressure, œ œ so, and a horizontal effective pressure,sh. There are no shear stresses on the vertical and œ œ horizontal planes of the soil element. Let us define the ratio of sh to so as a nondimen- sional quantity K, or œ sh K ϭ œ (13.1) so Now, three possible cases may arise concerning the retaining wall: and they are described Case 1 If the wall AB is static—that is, if it does not move either to the right or to the left œ of its initial position—the soil mass will be in a state of static equilibrium. In that case, sh is referred to as the at-rest earth pressure, or sœ ϭ ϭ h K Ko œ (13.2) so ϭ where Ko at-rest earth pressure coefficient. Case 2 If the frictionless wall rotates sufficiently about its bottom to a position of AЈB (Figure 13.1b), then a triangular soil mass ABCЈ adjacent to the wall will reach a state of 424 13.1 At-Rest, Active, and Passive Pressures 425 AЈ A ⌬ A At-rest pressure La Active pressure CЈ Ј Ј so z so z sЈ ϭ s Ј K sЈ ϭ s Ј H Ko o h a o h ϭ Ј ϩ Ј Ј ϭ Ј ϩ Ј Ј tf c s tan f tf c s tan f B B (a) (b) A AЉ ⌬ Lp Passive pressure CЉ z Ј ϭ Ј Kpso sh Ј so ϭ Ј ϩ Ј Ј tf c s tan f B (c) Figure 13.1 Definition of at-rest, active, and passive pressures (Note: Wall AB is frictionless) plastic equilibrium and will fail sliding down the plane BCЈ. At this time, the horizontal œ ϭ œ effective stress,sh sa, will be referred to as active pressure. Now, sœ sœ ϭ ϭ h ϭ a K Ka œ œ (13.3) so so ϭ where Ka active earth pressure coefficient. Case 3 If the frictionless wall rotates sufficiently about its bottom to a position AЉB (Figure 13.1c), then a triangular soil mass ABCЉ will reach a state of plastic 426 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Ј Earth pressure, sh Ј Passive pressure, sp Ј At-rest pressure, sh Ј Active pressure, sa ⌬ ⌬ Wall tiltLa Lp Wall tilt H H Figure 13.2 Variation of the magnitude of lateral earth pressure with wall tilt ⌬ ⌬ Table 13.1 Typical Values of La /H and Lp/H ⌬ ⌬ Soil type La /H Lp /H Loose sand 0.001–0.002 0.01 Dense sand 0.0005–0.001 0.005 Soft clay 0.02 0.04 Stiff clay 0.01 0.02 equilibrium and will fail sliding upward along the plane BCЉ. The horizontal effective œ ϭ œ stress at this time will be sh sp, the so-called passive pressure. In this case, œ sœ s ϭ ϭ h ϭ p K Kp œ œ (13.4) so so ϭ where Kp passive earth pressure coefficient Figure 13.2 shows the nature of variation of lateral earth pressure with the wall ⌬ ⌬ ϭ Ј ⌬ ⌬ ϭ Љ tilt. Typical values of La /H ( La A A in Figure 13.1b) and Lp /H ( Lp A A in Figure 13.1c) for attaining the active and passive states in various soils are given in Table 13.1. AT-REST LATERAL EARTH PRESSURE 13.2 Earth Pressure At-Rest The fundamental concept of earth pressure at rest was discussed in the preceding sec- tion. In order to define the earth pressure coefficient Ko at rest, we refer to Figure 13.3, 13.2 Earth Pressure At-Rest 427 A Јϭ so gz z s Јϭ K gz H h o ϭ Ј ϩ Ј Ј tf c s tan f Figure 13.3 B Earth pressure at rest which shows a wall AB retaining a dry soil with a unit weight of g. The wall is static. At a depth z, ϭ œ ϭ Vertical effective stress so gz ϭ œ ϭ Horizontal effective stress sh Kogz So, sœ ϭ h ϭ Ko œ at-rest earth pressure coefficient so For coarse-grained soils, the coefficient of earth pressure at rest can be estimated by using the empirical relationship (Jaky, 1944) ϭ Ϫ ¿ Ko 1 sin f (13.5) where fЈϭdrained friction angle. While designing a wall that may be subjected to lateral earth pressure at rest, one must take care in evaluating the value of Ko. Sherif, Fang, and Sherif (1984), on the basis of their laboratory tests, showed that Jaky’s equation for Ko [Eq. (13.5)] gives good results when the backfill is loose sand. However, for a dense, compacted sand backfill, Eq. (13.5) may grossly underestimate the lateral earth pressure at rest. This underestimation results because of the process of compaction of backfill. For this reason, they recommended the design relationship g ϭ 1 Ϫ 2 ϩ c d Ϫ d Ko 1 sin f 1 5.5 (13.6) gd1min2 ϭ where gd actual compacted dry unit weight of the sand behind the wall ϭ 1 2 gd1min2 dry unit weight of the sand in the loosest state Chapter 3 428 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb The increase of Ko observed from Eq. (13.6) compared to Eq. (13.5) is due to over- consolidation. For that reason, Mayne and Kulhawy (1982), after evaluating 171 soils, rec- ommended a modification to Eq. (13.5). Or ϭ 1 Ϫ ¿ 21 2 sinf¿ Ko 1 sin f OCR (13.7) where OCR ϭ overconsolidation ratio œ preconsolidation pressure, sc ϭ œ present effective overburden pressure, so Equation (13.7) is valid for soils ranging from clay to gravel. For fine-grained, normally consolidated soils, Massarsch (1979) suggested the following equation for Ko: PI 1% 2 K ϭ 0.44 ϩ 0.42 c d (13.8) o 100 For overconsolidated clays, the coefficient of earth pressure at rest can be approxi- mated as ϭ 1 Ko 1overconsolidated2 Ko 1normally consolidated2 OCR (13.9) Figure 13.4 shows the distribution of lateral earth pressure at rest on a wall of height H retaining a dry soil having a unit weight of g. The total force per unit length of the wall, Po, is equal to the area of the pressure diagram, so ϭ 1 2 Po 2 KogH (13.10) ϭ Ј ϩ Ј Ј tf c s tan f Unit weight ϭ g H H 3 KogH Figure 13.4 Distribution of lateral earth pressure at-rest on a wall 13.3 Earth Pressure At-Rest for Partially Submerged Soil 429 13.3 Earth Pressure At-Rest for Partially Submerged Soil Figure 13.5a shows a wall of height H. The groundwater table is located at a depth H1 below the ground surface, and there is no compensating water on the other side of the wall. Յ œ ϭ For z H1, the lateral earth pressure at rest can be given as sh Kogz. The variation of œ Ն sh with depth is shown by triangle ACE in Figure 13.5a. However, for z H1 (i.e., below the groundwater table), the pressure on the wall is found from the effective stress and pore water pressure components via the equation ϭ œ ϭ ϩ ¿1 Ϫ 2 Effective vertical pressure so gH1 g z H1 (13.11) Јϭ ϭ where g gsat gw the effective unit weight of soil. So, the effective lateral pressure at rest is œ ϭ œ ϭ 3 ϩ ¿1 Ϫ 24 sh Koso Ko gH1 g z H1 (13.12) œ The variation of sh with depth is shown by CEGB in Figure 13.5a. Again, the lat- eral pressure from pore water is ϭ 1 Ϫ 2 u gw z H1 (13.13) The variation of u with depth is shown in Figure 13.5b. A ϭ H1 Unit weight of soil g z Groundwater table C E I H KogH1 ϩ H2 Saturated unit weight of soil ϭ g Ј sat sh u F B G JK ϩ Ј Ko(gH1 g H2) g H2 (a) (b) H1 ϭ KogH1 H2 sh Figure 13.5 ϩ Ј ϩ Distribution of earth Ko(gH1 g H2) g H2 pressure at rest for partially (c) submerged soil 430 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Ն Hence, the total lateral pressure from earth and water at any depth z H1 is equal to ϭ œ ϩ sh sh u ϭ 3 ϩ ¿1 Ϫ 24 ϩ 1 Ϫ 2 Ko gH1 g z H1 gw z H1 (13.14) The force per unit length of the wall can be found from the sum of the areas of the pressure diagrams in Figures 13.5a and 13.5b and is equal to (Figure 13.5c) ϭ 1 2 ϩ ϩ 1 1 ¿ ϩ 2 2 Po 2 KogH1 KogH1H2 2 Kog gw H2 (13.15) ⎧ ⎪ ⎨ ⎪ ⎩ ⎧ ⎪ ⎨ ⎪ ⎩ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Area Area Areas ACE CEFB EFG and IJK Example 13.1 Figure 13.6a shows a 15-ft-high retaining wall.