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lass CIVIL ENGINEERING REFERENCE MANUAL <br />C. COHESIONLESS SAND (c = 0) The horizontal pressure depends on the coeJj'icient of <br />The maximum slope angle for cohesionless sand is the earth pressure at rese, k°, which varies from 0.4 to 0.5 <br />angle oC internal friction. ~. for untamped sand. <br />phorrzoncal = k°PH 10.58 <br />k° ~ 1 - sin p 10.59 <br />23 EARTH PRESSURE THEORIES R° = zk°pHa <br />The general equation for horizontal active earth pressure <br />is: <br />phorizoncal = pvertical tan`/1 4~ - ~1~ <br />-2ctanl9`5°-z I/ 10.51 <br />c in equation 10.51 is the soil's cohesion. <br />Pvertlcal can be due to surcharge. externally applied <br />loads, or the soil's own mass. <br />If ~ = 0°, as in the limiting case for saturated clay, then <br />phonzoncal = pvertical - 2C 10.52 <br />If c = 0, as in the limiting case for drained sand, <br />phorizonra] = pvertrcal I tangy 145° - 211 10.53 <br />Equations 10.57 and 10.58 apply only co a sand deposit <br />of infinite depth and extent. For sand that is com- <br />pressed or tensioned (as i^ around a retaining wall) the <br />reactions are given by equ(atio1ns 10.60 and 10.61. <br />RA = (phori:ontal) 1 ~J = zkapH2 10.60 <br />Rp = ~kppH' 10.6] <br />R,t and Rp aze horizontal if the soil above the heel and <br />toe is horizontal. (See figure 10.19.) <br />B. WEDGE THEORIES <br />The Rankine theory is based on infinite. cohesionless <br />soil. It also requires that the soil above the heel be <br />level. Modifications can be made to lift these restrc- <br />tions, as well as to allow a water table above the foun- <br />dation base. Several modifications are known as wedge <br />theories. Coulomb's earth-pressure theory is one such <br />wedge theory. <br />The quantit}• in brackets in equation 10.53 is known as <br />the coefjicienl of active earth pressure. <br />1 -sin td <br />kA =tan'- 45° - ~ = 10.54 <br />l+sin~ <br />The general equation for horizontal passive earth pres- <br />sure is: <br />phonzontal =pvertical tang// 45° t 9 <br />+ 2c Lan (45° + ~ I 10.55 <br />The coe,~cient of passive earth pressure for sand is <br /> <br />1 <br />kp = kA = tanz 45° + 2) 1 ± sin d 10.56 <br />.A. THE R.4NKINE THEORY <br />If it is assumed that the backfil] soil is dn~, cohesionless <br />sand, then the Rankine theory can be used. At any <br />depth. H, the vertical pressure is <br />pvertical = PH 10.57 <br />The wedge methods are based on the observation that <br />retaining walls fail when the active soil shear. Al- <br />though the shear plane is actually a slightly curved sur- <br />face, it is assumed to be linear (line +-• in fio re 10.15). <br />However, since the actual shear plane is not known in <br />advance, several trial planes need to be taken. This is <br />known as the trial wedge method. <br />24 SLOPED AND BROKEN SLOPE <br />BACKFILL <br />It is possible to derive equations for the active force with <br />a sloped backs]], as shown in figure 10.16. However, the <br />complexity of these equations usually makes a graphical <br />solution a better choice. <br />With sloped or broken slope backfill, the active force is <br />not. horizontal. Appendix A and appendix B provide a <br />method of evaluating the horizontal and vertical earth <br />pressure. Notice that kn and k„ have units of lbf/ftz <br />per foot of wall. Soil density is not used. <br />R,t,n = 2khHz 10.62 <br />RA,,, = Zk„Hz 10.63 <br />1O It is appropriate to use the at-rest soil case whenever the <br />wall does not move. Bridge abutments and basement wa1Ls aze <br />example where movement is essentially nonexistent. <br />PROFESSIONAL PUBLICATIONS INC. • P.O. Box 199, $an Carlos, CA 94070 <br />