!I'RANSPORTATION RESEARCH RECORD JJ29 63 Passive Lateral Earth Pressure Development Behind Rigid Walls S. BANG AND H. T. KIM An analytical solution procedure is described to estimate the the top or at the toe of the wall, depending upon the location developed passive lateral earU1 pressures behind a vertical rigid of rotation center, experiences a sufficient amount of retaining wall rotating about it toe or top into a mass of deformation (limiting deformation) to achieve a limiting cohesion less soil. Various stages of wall rotation, from an at-rest passive condition. The full passive state occurs when the state to an initial passive state to a full passive state, are entire zone of soil elements from the top to the toe of the wall considered in the analysis. A condition of failure defined by u modified Mohr-Coulomb criterion and equil.ibrium conditions is in the limiting passive condition. are used to obtain the necessary equations for solution. The When the retaining wall rotates about its toe, the initial development of friction along the wall surface at various tage passive state will be developed initially at the top of the wall, of wall rotation i also taken into account in the analy is. Finally, whereas the wall rotation about its top produces the initial the res ults predjcted by the developed method of analysis are passive state at the toe of the wall first. The original concept compared with those obtained from the experimental model of this approach and the theoretical formulation and the tests on loose and dense sand. The comparisons show good numerical procedures applied to the solution of active lateral agreements at various stages of wall movement. earth pressure development have already been described by the authors (6). T he estimation and prediction of the lateral ea rth pressure The analytical method developed has been applied to developmelll ha ve bee n among the mo t important aspects in estimate the passive lateral earth pressures behind a rigid geotechnical engi ne ering. The development of active lateral retaining wall experiencing rotations about its toe and top at earth pressures in particular has received a considerable various stages of rotation. The results are compared with amount of attention, because a majority of retaining structure those of experimental model tests and show in general good are des igned based on the active lateral earth pressure due to agreements, indicating the effectiveness of the proposed the tend ency of outward movement. However, design of method of analysis. many geotechni cal structure require co n ~ id era ti o n of pa sive latera l earth pressures. Several a nalytica l and experiment al studies have been made in the pa t to in ve tigate the THEORETICAL BACKGROUND magnitude a nd di stribution of pa ivc lateral ea rth pres ure. developed behind the retai ning wa ll s (/-5). These studie ha ve Full details of the formulation have been presented by Bang been helpful for understanding the mechanism of the and Kim (6) previously; therefore only a brief summary of the development of passive lateral ea rth pressures. However. principal features is given in the following paragraphs. most of the analytical studies fa il to provide adequate The fundamental equations governing the behavior of the comparisons with experimental model test res ult . T his system are those for two-dimensional plane-strain equilibrium failure may be in part due to the uncertainties associated with (7). the variations of the soil strength and the wall friction with respect to the magnitude of wall movement. A need for an analytical solution that takes into account the variation of material properties at various stages of wall movement (1) therefore has been realized. This paper presents an analytical solution method that where y indicates the unit weight of the soil. The center of the describes the transition of the pa siv e lateral earth pres ure coordinate is located at the top of the retaining wall with x from a n at-rest state to an initial passive state to a full pa ive and z axes being taken positive toward the backfill and tate behind a ve rtical rigid wall rotating a bout its toe or top downward, respectively. It is obvious from Equation I that into a mas of cohcsionlcss soil. The at-res t state i defin ed as an additional equation is necessary to solve for the three a stage of no wall movement. The initial passive state refers to unknown stresses. Using any familiar failure criterion for this a stage of wall rotation when only the soil element either at purpose will lead to the solution at the limiting state, that is, at failure. Sokolovskii (8) solved this problem with a Mohr­ Coulomb failure criterion in 1965. In this paper, however, to Department of Civil Engineering, South Dakota School of Mines describe the transition of the passive stresses from the at-rest and Technology, Rapid City, S. Dak. 57701-3995. to the full passive state (the limiting state), a relationship 64 TRANSPORTATION RESEARCH RECORD 1129 between the major and minor principal stresses has been a = (a 1. + a 3)/ 2 at any depth z, and assumed: B = rotation angle from the x-axis to the direction of a 1 measured clockwise positive (Figure 2). Principal stress ratio= (1 - sin t/J)/(l +sin t/J) (2) x In Equation 2, the angle t/J describes the slope of the line tangent to the stress Mohr's circle (Figure l). Note that ifthe angle t/J equals -<I> (the internal soil friction angle), Equation 2 reduces to Rankine's passive lateral earth pressure expression. T z FIGURE 2 Orientation of pseudoslip lines. Substitution of Equations 4 into Equations I leads to the 0 following pair of differential equations (6): da + 2atant/J(z)dB = y (dz+ tant/J(z)dx) 0 + a [ot/J(z)/ oz] dx (5) FIGURE 1 Mohr-Coulomb stress relationship. da - 2atant/J(z)dB = y (dz - tant/J(z)dx) - a [ot/J(z)/ oz ]dx (6) The angle t/J is assumed to vary in magnitude from ¢ 0 to -</>,where <l>o indicates the inclination angle relating CT 1 and Equations 5 and 6 contain the four unknowns x, z, a, and a 3 in the at-rest state. Note that a negative magnitude of friction angle is used for the purpose of developing a general 8. Additional two equations for the necessary solutions can formulation, that is, a positive friction angle describes the be obtained from the geometry of the slope ofpseudoslip lines transition of active lateral pressures. The value of <l>o can be as shown in Figure 2. As can be seen from the figure, the easily obtained if the at-rest lateral earth pressure coefficient slopes can be expressed as K 0 is known. dz/ dx = tan(B ± µ) (7) ¢ 0 =sin-I [(I - KQJ/(J + KOJ] (3) whereµ indicates a rotation angle from the direction of CT 1 to the pseudoslip lines. It is obvious from the Mohr-Coulomb By varying the angle t/J from ¢ 0 to -</>, Equation 2 could stress circle relationship (Figure I) that represent both the at-rest and full passive states. However, when the wall experiences movements other than translational, µ = rr/4 - t/J(z)/2 (8) the resulting rotation may produce different stress ratios at different depths. In other words, a portion of the backfill soil Equation 5,8 can be olved simultaneously with ap­ may experience deformations exceeding the limiting value, propriate boundary condition ·. A backward finite difference whereas the remaining portion may not. The former case method ha been applied for the · lution. The resulting would then achieve the t/J = -<I> state and the latter case the expressions are as follows: <1> 0 > t/J > -<I> tale. Therefore, the angle t/J describing the relationship between the major and minor principal stresses z .. - z . = (x . - x . ~tan(B. - µ . ) (9) may have to be described as a function of the depth z. l,j I· 1J I,) J- 1jJ I-1J J- 1J' Based on this assumption and Mohr's circle relationship, three unknown stresses can be expressed as (10) ax= a[l + sint/J(z)cos28] CT z = a[ I - sint/J(z)cos28] r xz = asint/J(z)sin28 (4) -a.1 rx .. -x.1.)o·i./ozl·1 · where I- J\ I,] I- ,j 'I' 1- J (11) Bang and Kim 65 ( <T i,J - <T i,j-1) + 2u i,J-1 ( (} iJ - (} i,j-l)tanl/J i,j-1 For l.O < f3 :::; 2.0, within a zone already in the limiting passive condition [O :::; z :::; (/3 - !)HJ, = y[(z;J- z;,1_1) + (x;,1 - xiJ- 1)tanl/J;,J-i1 l/J(z) = -</J (T .. J .. + /,)-1(X I..,) - x 1..,)- 1)al/1 I az l,j- I (12) Equations 9-12 completely describe recurrence formulas for al/J(z)/az = 0 (14) the determination of Lhe pseudoslip line coordina.tc x1J and Within a zone not yet in the limiting passive condition zfJ' the pseudoslip line lope (81J± µ;)•and the a ociated average stre (u ;) in term of previous values at coordinates [(/3 - l)H :::; z:::; HJ, i-1 J and i,j-1. The solution process starts from the backfill ground surface whose coordinates and stress values are l/J(z) = ¢ 0 - (</J + </Jo)(/3 - z/ H) known and proceeds to the back face of the wall (8) . However, the detailed solution steps require a description al/l(z)/az = (¢ + ¢ 0)/ H (15) of the function l/J(z) and its derivative, which define the transition of lateral earth pressures from the at-rest to the full 2.