Transportation Research Record 1031 5
useful. The stochastic model described previously the NCDOT have been major factors in making this can be used to develop a better understanding of paper possible. L.G. Grimes' editorial assistance is rockfall motion. Because of the low cost of computer also deeply appreciated. time, it is possible to perform several simulation runs with different random numbers to generate a distribution of the predicted rockfall motion at a REFERENCES given slope segment. Depending on the number of replicates, certain levels of confidence can be 1. A.M. Ritchie. Evaluation of Rockfall and Its Con achieved. These results can be used as references to trol. In Highway Research Record 17, HRB, Na determine catchment area width and rock wall loca tional Research Council, Washington, D.C., 1963, tion. This model could also be used to evaluate the pp. 13-28. type of material that should be placed in the catch 2. D'Appolonia Consulting Engineers, Inc. Rockfall ment area to optimize the investment. Analyses. North Carolina Department of Transpor Simulation is a powerful tool for evaluating tation and Highway Safety, Raleigh, 1979. existing or proposed facilities because it can be 3. D.R. Morton. Slope Stability Problems and Recom used to generate data at low costs. However, extreme mendations for Remedial Treatment Along I-40 in caution should be exercised when using any model. the Pigeon River Gorge. North Carolina Department The results generated for this project should be of Transportation and Highway Safety, Raleigh, verified as appropriate in other areas before they 1981. are relied on. As I-40 construction continues, more data will be collected to refine the ROCKSIM model.
The op1n1ons, findings, and conclusions expressed in ACKNOWLEDGMENTS this paper are those of the author and not neces sarily those of the North Carolina Department of The author is indebted to many who have contributed Transportation. to, and encouraged the development of, this paper. The advice and assistance of John Ledbetter and Publication of this paper sponsored by Committee on Naariman Abar of the Soils and Foundation Section of Engineering Geology.
Stability Charts for Geotextile-Reinforced Walls
DOV LESHCHINSKY and JOHN C. VOLK
ABSTRACT
The results of a mathematical approach to estimate the shear failure resistance of a geotextile-retained soil wall are presented in this paper. The analytical method used is based on a limiting-equilibrium approach combined with variational extremization, and it satisfies all equilibrium requirements. The analytically derived failure mechanism consists of a log-spiral slip surface and reinforcing geotextile sheets positioned orthogonally to the radii defining it. A closed-form solution is obtained that provides complete insight into the problem's behavior. The results indicate that (a) as the geotextile tensile strength increases, the extent of the critical slip surface increases, (b) as the geotextile strength in creases, the compressive stress over the er i tical slip surface also increases, (c) as the geotextile strength increases, the magnitude and extent of tensile normal stress that tends to develop near the top decreases, and (d) when fric tional soil is concerned, the strength of the geotextile at the bottom is mobi lized the most. The end products are design charts that can easily be applied to a particular problem. The charts indicate that reinforcement may significantly increase the stability of a wall (or slope) depending on the geotextile's tensile strength, the soil's strength properties, and the inclination of the structure face. 6 Transportation Research Record 1031
In recent years, geotextiles and allied products have been utilized in the construction of retained soil walls. The geotextile sheets are used to wrap compacted soil in layers, thereby producing a stable composite structure. It has been shown that such a construction technique has the potential for a cost ef fective substitute for conventional retaining '• structures (1,2). Geotextile-retained soil walls, referred to In- this paper as geotextile-reinforced walls, resemble the popular sandbag walls that have been in use for decades. Contrary to sandbag walls, however, geotextile-reinforced walls can be con eolextllc structed to significant dimensions because of the Sheels geotextile's high strength, its durability, and a simple mechanized construction procedure. A major H consideration in the design of a geotextile-rein forced wall is its overall resistance to shear fail ure that develops through its composite structure. The objective of this paper is to present a simple and consistent analytical tool that is capable of addressing this complex aspect of the problem. A comprehensive analytical approach to this prob (l) lem is the finite-element method (3). The soil-geo (x,,0.0) textile interaction, however, is -highly nonlinear and, therefore, a complicated constitutive relation ship must be employed in this analysis (_!). As a FIGURE 1 Notation and schematic presentation of the consequence, the application of the f ini te-elemen t problem. method to standard problems might be too involved and impractical at the present time. There are currently numerou5 simplified anu.lysia methods that have been developed to assess the sta bility of geotextile-reinforced earth structures. Some are an extension of conventional retaining wall theories (5,6) and others are derived from simpli fied slope- ;tabili ty analysis methods ANALYSIS OUTLINE and it can carry, in its planar direction, only a tensile force of known magnitude tj. This force, Figure l shows a gravity wall on the verge of col howevei: , is inclined a t an unknown angle a · and lapse. The wall is composed of soil wrapped in rein is acting at the geotextile's intersection witli the forcing geotextile sheets. The soil possesses unit slip surface (see Figure 2). weight y, cohesion c, and friction angle $. The To attain the state of limiting equilibrium height of the geotextile-reinforced wall is H and (i.e., verge of failure), the shear resistance the general inclination of its face is i. A poten developed over the slip surface must be fully mobi- tial slip surface, expressed by an unknown function 1 ized. This condition, however, rarely occurs. r (8), initiates at the crest level [point (2)] and Therefore, the common concept of limited mobiliza exits the soil mass at the toe level [point (l)]. tion of shear resistance is adopted. As a result, a Note that the slip surface is defined relative to a iiiiii stable system can now realize the limiting equilib polar coordinate system that has its origin at some rium state. By utilizing Coulomb's failure criterion unknown point (Xe, Ycl. As a consequence, points and scaling down the components that strictly resist (1) and (2) are defined by the two unknown variables collapse (i.e., T and tj) so that limit equilib denoted as 81 and 82. The potential failure rium is artificially attained, the following equa surface is acted on by two unknown distributed tions result: stresses: normal [0(8)], and shear [T(B)]. It is as assumed that geotextile j has zero flexural rigidity Tm= (c + al/l)/F, (I) Leshchinsky and Volk 7 (2) and Garber (11). Equation 3 represents a log-spiral slip surface a nd Equation 6 indicates that the geo textile is orthogonal to the radius vector defining where it at its intersection with the slip surface. The failure mechanism described by these two equations Fs an unknown reduction factor termed is shown in Figure 2. Note that this mechanism is "factor of safety"i identical to rigid body rotational failure used in Tm and tm the mobilized soil shear strength limit analysis where the velocity is opposing the and mobilized geotextile tensile geotextile tensile force (_!2). strength, respectivelyi and The geotextile tensile strength is not explicitly ljl = tan¢>. expressed in Equations 3-5, indicating that the For a given problem, the minimum value of Fs is nature o .f R and S is not affected by tj. The for c e t., however, appears in the equ i l ibrium equation s sought. When F s is a minimum, it is then the con t6gether with R and Sati sfaction of e quilibrium ventional factor of safety commonly used in limit s. implies that the constants A and B, defining S and equilibrium analysis. In the physical sense, Fs R, d epend on tj . As a consequence, the slip sur quantifies the margin of safety with respect to f ace location and magnitude of the normal stress are available shear strength. Note, however, that Fs linked to tj. is applied equally to the soil (cohesion and fric tion) strength and the geotextile tensile strength, and that its value is constant for the entire slip surface. This is consistent with the averaging nature of the limit-equilibrium approach. It should be pointed out that because a composite structure is being analyzed, which is made up of materials possibly exhibiting different stress strain character is tics, the materials' strength parameters (Equations 1 and 2) should be determined on the basis of a limiting strain value that is identical for all materials. In most cases, however, this limiting value should correspond to the soil's shear deformation at failure rather than to the geo textile' s tear strength, which may occur at rela tively large elongation. Note that similar con siderations are frequently used in determining the strength parameters of a layered soil profile for the purpose of limiting-equilibrium slope stability analysis. Further discussion of this important topic, however, is beyond the scope of the present work. It can be verified (13) that by satisfying all equilibrium equations for---a potentially sliding body (e.g., for which Equations 1 and 2 are valid) while extremizing Fs by using variational techniques, the following results are obtained: (-~mPJ R= Aexp (3) FIGURE 3 Virtual rotation of a rigid body. (4) S= Acos/l+ 2Nm/l+ B for i/lm =O (5) In considering the virtual rotation of a rigid body combined with the derived failure mechanism, it (6) can be shown that the geotextiles do not tend to stretch uniformly unless the log-spiral degenerates into a circle (see Figure 3). Because of the slip where surface geometry, the geotextile at the lowest ele vation, which is denoted as 1 in Figure 3, will A and B unknown constantsi elongate the most and therefore it is assumed that R r/H is the nondimensional representation this geotextile will reach its tensile strength t1 of the potential slip surface r, de first. From a strict limiting equilibrium viewpoint, fined relative to a polar coordinate once t1 is fully developed, a collapse resembling system having its origin at an unknown a row of dominoes falling down will occur; that is, point (Xe, Ycl where Xe = Xc/H and Ye = all other geotextiles will fail one after the other Yc/H (see Figure 2) i in an upward orderly manner. It implies that as geo (tan¢>) /F s 1 textile 1 is approaching its strength t 1 , all c/(FsyH) i other geotextiles are not necessarily on the verge cr/yH is the nondimensional represen of collapse and therefore their tensile force at tation of the normal stress distribu that instance needs to be specified. In assuming tion cr(B) over R(B) i and that up to the elongation that corresponds to t 1 , the angle defining the intersection the load-elongation relationship is linear and by of geotextile j and the slip surface R. utilizing the mechanism shown in Figure 3, the ad justed tensile force relative to t 1 can be written Equations 3-5 were previously obtained by Baker as B Transportation Research Record 1031 where (7) n the number of equally spaced geotextiles, where t1 is the geotextile's actual tensile Tm the nondimensional tensile strength of the :;tLealgth, fully di::vt:lUfn;:..'.i if1 Lh~ y~uL~xL.i.lt:! c:tL tli~ l yt:!vL~xtile placed ac che coe elevation, and lowest elevation; and tj is the tensile force Tm = the nondimensional equivalent tensile developed in geotextile j at the same instant that strength. t1 is fully mobilized. At this time, it is convenient to convert the All the results presented here are based on force per unit length tj into the following non analysis of walls reinforced by 10 horizontal and dimensional expression: equally spaced geotextiles (n = 10). Some walls re inforced by 2 or 40 geotextiles were also analyzed. 2 1 1 2 Tm i = (I /yH )(t;/F,) = { exp -w "' (Pg; -P" ) I /yH } (t 1 /F,) When the equivalent strength Tm was the same for 1->l>m (Pg; - Pg JI n = 2, 10, or 40, the generated results exhibited =Tm exp 1 (8) 1 insignificant differences. It can therefore be con cluded that, in effect, the results presented here where Tm. is the nondimensional tensile force of geo are general and applicable (when appropriate) for textile j. any number of horizontal and equally spaced geotex It can be verified (ll,) that for given i, ~m' Tml' t iles. This conclusion is limited to n > 2 where and elevations of all geotextiles (i.e., Bg;l, the the first geotextile is placed at the toe elevation. following unknown constants exist: Xe, Ye, A; B, Nm Fiyure~ 4-7 show cypical discribucions of tensile forces developed in a set of geotextiles. Note that B1 , B2, and Tm.(j = 2,3 ••• n). The necessary equations the soil strength parameters are substituted by a can be obtainea by essentially foUowing a procP.nnrP. single parameter Ac~' defined for a system on developed by Baker (~ • These equations consist of the verge of collapse (i.e., for a given Tml as 1. Three equations of limiting equilibrium; (IO) 2. Two equations representing geometrical bound ary conditions, i.e., Y(X1) = O and Y(X2) = l; Figures 4-7 show that for a highly cohesive soil 3 ~ Two equations resulting from extreJTiization of {e.g., Ac$ - .LVVJ 1 wut:::L.t::: th~ developed slip R(B) and S(B) at the boundaries [transversality surface closely resembles a circular arc, all geo conditions (11)); and textiles are equally mobilized. In the case of a 4. Equati'On B multiplied by (n - 1) where n is highly frictional soil (e.g., Ac~= 0.1), how the number of geotextile sheets. ever, there is a significant decrease in the geotex tiles' tensile strength mobilization as their eleva For a given i, ~m' Tml' and geotextiles' elevation tion approaches the crest. This decrease is more (i.e., Bg.l, the required nondimensional cohesion Nm pronounced as the inclination of the wall's face de creases. can be delermined by solving these equations. Figures B and 9 show the effect of the geotextile tensile strength on the potential slip surface. Note RESULTS that (a) the larger the Tm, the deeper the slip nurface regardless of th@ soil strength parameters; '!'he analysis outlined before was used to gain some (b) the effect of Tm increases as i decreases insight into the geotextiles' reinforcing effects (this is particularly true when cohesive soil is in and to generate design charts. It appears that the volved), and (c) the more cohesive the soil, the deeper the slip surface. following definition of equivalent tensile strength Figures 10 and 11 show some typical nondimen is useful when a number of geotextiles are involved: sional normal stress distributions. It is interest ing to note that as the critical slip surface is (9) shifted by the geotextile reinforcement, there is Tm 10 EOBBlm, lm 0.89 Im, 9 = Tm - o 10 \c 020H Tm Tm,= IQ= 001 FIGURE 4 Typical distribution of geotextile tensile force for the case i = 90 degrees and Tm = .10 for highly frictional soil. 9 Ac ~ Tm,= f{f = 001 FIGURE 5 Typical distribution of geotextile tensile force for the case i = 90 degrees and Tm = .10 for highly cohesive soil. I.OOH 0.80H O.l>OH Tm• OJO Acct> - 0 .1 0.40H 0.20H Tm,•~~ • 0.01 FIGURE 6 Typical distribution of geotextile tensile force for the the case i =60 degrees for highly frictional soil. lOOH .. c O.BOH 0.60H Tm= 010 Ac Tm 0.20H Tm,=10=001 FIGURE 7 Typical distribution of geotextile tensile force for the case i =60 degrees for highly cohesive soil. 10 Transportation Research Record 1031 f-ccp=IOO l.OOHf 0.80H O.GOH FIGURE 8 Typical effect of geotextile tensile strength on slip Tm= 0 00 surface with a wall inclination of i = 90 de~ees. I.OOH 080H /, c'I>= 100 0.60H /, c;= 0 I .Ac~~ 100 I.OOH[ 080 H 0 .60H 0.40H FIGURE 11 Typical effect of geotextile reinforcement on normal stress distribution with a wall inclination of i = 60 degrees. FIGURE 9 Typical effect of geotextile tensile strength on slip carried out by following the procedure used by Baker (16). surfar.P. with ~ w~ll inclination of i = 60 degrees. Figures 12-14 show design charts for walls in clined at 60, 75, a nd 90 degrees. These charts can be used to determine the factor of safety of a given reinforced wall or, alternatively , be used to design IOOH r----ir---?-~~~~ a wall. Further elaboration about the application of 080H the design charts is given in the next section. Note that all the charts possess the same unique point on the ordinate <~m = 0). At this point Nm e q ual s 0.1810 and the critical slip surfac e is infi nitely deep . The d esign charts indicate t hat for deep failure (slip surface passing below the toe) , as ~m decreases, the corresponding value of Nm converges to 0.1810 for any Tm greater than a cer tain value. This implies that under some conditions, the geotextile strength has very little or no effect I.OOH ·~~~~tr---;fhW<" on the stability. It should be stated that this 080H phenomenon occurs together with an increase in the Ac Wail Inclination i=60° 0.30 1 Critical Sli p Surface Passing. ct>m=tan c~ 0 ,15 0.10 0.05 0 0 0.00 0 5 10 15 20 25 30 35 40 45 Wall Inclination i= 75° 0.30 1 Critical Sli p Surface Passing: cp m =tan caF~ 0.15 0.10 0.05 0.00 0 5 10 15 20 25 30 35 40 45 Wall Inc lination l=90° 0.30 'T"'------Crit ica l Sl ip Sur face Passing : ~--~I Thr01Jgf'J the toe ~-___.I Be low t h e toe 1 _g_ Nm=1H Fs n t, Tm "''i"H'2 Fs 0 5 10 15 20 25 30 35 40 45 FIGURE 14 Design chart for 90-degree wall. results are not then valid). Figure 15 shows some Figures 16 through 18 provide all the information basi c par ameters that are used to define a slip sur needed to determine the slip surface trace for walls face exiting at the toe (i.e., x1 = Yl = 0.0). inclined at i = 60, 75, and 90 degrees. Once $m, For this c ase , t he parame ters used are x2, Y2 = B , Nm' a nd Tm are deter mi ned f o r a g i ve n problem, and 6 2 wh e re e2 is the a ng le of the tangent 62 a nd x2 (f or wh i ch x1 = y1 = o.o and y2 = H) can to the s lip surface a t cx2 , Y2 ). From the geome be e val uated f rom the correspondi ng figure (i.e., try of the problem, the following can be obtained: Figures 16-18). By combining Equations 11-13 with the preceding data, five equations will result that (i J) contain the following unknowns: A, Xe, Yer 81, and e2• By solving these five equations, the slip surface It can be verifie d t hat the following parametric can be fully defined. As a consequence, a check can equations, relating the pola r and Cartesian coordi then be made to determine whether the predicted slip nate s ystem, exist: surface can be physically developed for the given problem. X = Xe + Aexp(- .P m p) sin/3 (12) ILLUSTRATIVE EXAMPLE Y = Y - Aexp(-.P mP)cos/3 (13) 0 Consider the following design problem: A vertical geotextile-reinforced gravity wall is to be con structed to B = 20 ft high. The soil properties are ·y = 120 lb/ft J, ~ c: 35 degrees, and c = 275 lb/ ft•. The required tensile strength of the geo textiles, so that Fs = 1.5, is sought. For Fs = 1.5, the mobilized soil strength parameters are $m =tan- : (tan,/Fsl = 25 degrees and Nm= c/(FsyH) =0.076. By using Figure 10, it follows that the required Tm equals 0.10. The selection of 20 equally spaced geo textile sheets n = 20 (1 per foot) will result in a 2 specified strength of t1 = (yH ) (FsTm)/n = 360 lb/ ft. By using Figure 18 (Ac$ = 0.076/tan 25 degrees : O .163) and sol vi ng Equations 11-13, the following slip surface is predicted: x 1 = y 1 = 0.0, x 2 = 0.66 H = 13.2 ft, y 2 = H = 20 ft 62 = 65 degrees, 82 = 40 degrees, xc = 30.8 ft, Ye = 72.0 ft, A = 94.6 ft, and 8 1 = 23 degrees. Let us assume now that the wall was eventually constructed by using n = 30 equally spaced geotex iiiii tile sheets each possessing a tensile strength of 300 lb/ft (t1 = 300 lb/ft). The factor of safety needs to be determined for this case. If it is as sumed that Fs = 2, then $m equals 19.3 degrees and can be expressed as FIGURE 15 The parameters used in Figures 16-18 to define the slip surface. Leshchinsky and Volk 13 3.0 90 Wall Inclination i-60° P,;r t, 2.8 Tm= 1H F;"" 86 s e;n,B;+ IP~ 82 x,/H .__ >---~ ...... ~-..,;. "' e, / 78 1.:::::::_._ r-- ~ v ~ -- r-...... ~~ ,_._ ...... / 2.2 ~ 74 ~- ,.. -,__ ,~_.._- I'-.. R::'...... _ i,.-1-- ,_ - r-..- - 1-.,_ F-- ,..._ ,__ ,_,._ r-.... V'. ..____ r-..' ...... t- I-'- · ~ ·o..oo- 2.0 ~ 0,02 70 1-- - -.... "" ,...._ ...... I'-.. "" t-- k~~ ...- ()()4 I--- ' ~ l7_ .. ...___ -.... OO• I'-.. ii' t---. J k [?k: i.-- 1.8 ,___ " -.... 0.01 66 I-- - v. ~ - ...... L;t> 1 _ ... ,..~ .....- r-- ,.._ L/< .... !>~ ~ 1°1 .... --- ~ i.- / ... "r- MV; P- OJ 2 62 1.6 v~ '~ v i..- "' ~ [;< r- ,/ ~ v c-1- - ~u >- Tm=Of20 k~ i,...i..-"" - v )~ ~ I 1.4 / k: - S8 / ,I,/ ~ Kt>. I / [,, L.-- --0.11- ....-i- /,/[,, ~ ._ rn I ~ OISS 1.2 ~ ~ ~ - _._ 54 '-- 017 -0.14- ,_ / /_,i,. "',~ ' - I i..--1-.. v [/,,[,, _on- ... ~ v-- !'-...... 11 1 1.0 __ 010 ~ ~ so --o.oe.- i..--"" /' .:::::- .__- .__ - o~ ~~ i..--'-" .-o.ott- ... ~ 0.8 ~ 46 ==oo ~ '-"' ~0 01· '-"' ... .. Y, = H ~ 0 oo• 0.6 x, = y, = 0 42 0.4 38 0.2 34 0.0 JO 0.01 0.1 10 100 /\crp = J_ c 7H tan rp FIGURE 16 Parameters defining the slip surface where i = 60 degrees. By using Figure 14, it follows that the require d ing the common procedures for conventional retaining Nm is 0 .106; the computed Nm, based on the given walls. cohesion and the assumed Fs, is represented as COMPARISON OF RESULTS Nm = c/(F;yH) = 0.057 A new Fs must be assumed and the procedure is re The results presented here are based on an approach peated until the assumed Fs produces a required that is equivalent to the upper bound theorem of Nm (see Figure 10) equal to the actual Nm, indi plasticity. For a safe design, however, the lower cating that all strength components are indeed bound generally is sought. Hence, to determine equally mobilized. It can be verified that for the whether the method is useful for design purposes, given problem, Fs = 1.63. some of the results must be compared with pre d i c tions pro duced by an extension of a limiting-equi It is interesting t o note that an unreinforced librium approach that is considered to be conserva slope, inclined at i = 53 degrees, and possessing tive when applied to unreinforced slopes. identical soil (i.e., = 35 degrees, c = 275 Christie and El-Hadi (7) and Christie (18) modi 2 lb/ft , and y = 120 lb/ft'), will have the fied the ordinary method of slices (Felle nius same factor of safety (i.e., F s = 1.63) as this method) so that it would account for reinforcing vertical reinforced wall. This has clear eny l11ee1 ln•J geotextile sheets. Although the geotextile tensile and economical implications in design and cons truc strength is mobilized by uneven deformations of the tion of stabilized embankments. It should be pointed soil, Christie and El-Hadi conse rvatively assumed out, however, that if the designed wall is used to that the geotextile tensile force remains horizontal retain backfill soil, its stability against over at the slip surface and considered the tensile force turning and base slide has to be checked by utiliz- action as a uniform distributed force (per unit 11111 ..... ~ 3.0 90 3.0 90 1..-- Wall Inclination i=75. l,/ Wall Inclination i=90' _ _n_ t, 2.8 - I fl t Bb 2.8 Tm-7H' Fg Bb !-.. Tm=m f. r--.. s 1-- 1... ) " "' r-- .... e;=p ;+ 111 11 0.0 I 30 0.0 -r _30 0.01 0.1 10 100 0.01 0.1 10 100 \ 1 c "Ac 11111 Leshchinsky and Volk 15 --- Chri sti e (1982 ) height) rather than as a line load <2l· Christie ap plied the safety factor equally to the geotextile THI S WO R K tensile strength and soil strength (~) , an approach Slip Surface Passing : that is also used in this work. 0----0 Be lo w the Toe (C ritical ) By using this modified Fellenius method, Christie ~ -6. Through the Toe (Cr it ical ) analyzed a 66-ft high embankment with 45 degree 0- --o Through th e Toe (Forc ed) side-slopes, subjected to undrained ( { } surface (a) is forced to exit at the toe and (b) is T=0.4 mathematically the critical one. Figure 19 shows that for the particular problem analyzed (i = 45 degrees, c = 0), the safety factors generated by this work are smaller (i.e., more critical) , than Chris tie's, especially as the geo textile tensile strength or the friction angle in creases. Figure 20, for which THIS W O R K C.Ul· Slip Surface Pa ssing : 0- - --o Below th e To e (Criti cal ) CONCLUSIONS D---0 Through th e Toe (Forced ) Results of a mathematical approach to stability of geotextile-retained soil walls are presented. The analytical approach is based on limiting equilibrium 3.0 = 45° combined with variational extremization. Because of } T=06 the generality of the analysis, the results can also Fs ¢ =00 be applied to a wide range of problems of similar -~ 2.5 T -,, H' } T=04 nature (e.g., the stability of geotextile-reinforced earth slopes). The analysis results indicate that the critical slip surface is log-spiral and that on 2.0 the verge of failure, each geotextile sheet is orthogonal to the radius vector defining its inter section with the critical failure surface. To ac count for several geotextiles in a consistent man 1.5 ner, a methodology that employs a rigid body ,,,,,,,.,,. rotation was introduced. Conclusions made on the basis of figures obtained 1.0 1-~-L+-----,OO--t--c:_~+-'7::r~...!...../ Fo r any T (Failure surface through a closed-form solution, are as follows: extends infinitely) 1. When a granular soil is concerned, there is a significant decrease in the mobilized strength of the geotextiles as their elevation approaches the crest. o. o------"'"--~--~-~ 2. As the geotextile strength increases, the ex 0.00 0.05 0.10 0.15 0 . 20 0.25 tent of the critical slip surface increases. This effect is more pronounced as the wall face inclina c N= tion decreases. 'Y H 3. As the geotextile tensile strength increases (a) the compressive stress normal to the critical FIGURE 20 Comparison of results where rp = 0. slip surface increases, and (b) the magnitude and 16 Transportation Research Record 1031 extent of tensile stress, typically acting near the of Fabrics in Geotechnics, Vol. 1, Paris, crest, decreases. France, 1977, pp. 99-103. 8. R.T. Murray. Fabric Reinforcement of Embank The end products of this work are design charts ments and Cuttings. Proc., 2nd International Conference on Geotextiles, Vol. 3, Las Vegas, To verify whether the presented approach is ac Nev., 1982, pp. 707-713. ceptable at all, its results were compared with pre 9. T. S. Ingold. An Analytical Study of Geotextile dictions based on a conservative extension of the Reinforced Embankments. Proc., 2nd Interna ordinary method of slices as applied to a problem tional Conference on Geotextiles, Vol. 3, Las possessing similar characteristics to reinforced Vegas, Nev., 1982, pp. 683-688. walls. The results compared favorably. The applica 10. R.A. Jewell. A Limit-Equilibrium Design Method tion of the variational approach, however, is for Reinforced Embankments on Soft Foundations. easier. It should be noted that a closed-form solu P roe., 2nd International Conference on Geotex tion can also be assembled for various types of tiles, Vol. 3, Las Vegas, Nev., 1982, pp. 671- loading conditions such as pore water pressures and 676. surcharge loads. As a consequence, similar stability 11. R. Baker and M. Garber. Variational Approach to charts may be developed for these conditions. Slope Stability. Proc., 9th International Con Furthermore, the analysis can be modified so that ference on Soil Mechanics and Foundation the geotextile spacing and strength can be specified Engineering, Vol. 2, Tokyo, 1977, pp. 9-12. individually in accordance with the investigated 12. R. Baker and M. Garber. Theoretical Analysis of problem_ the Stability of Slopes. Geotechnique 28. No. 4, 1978, pp. 395-411. 13. D. Leshchinsky. Geotextile Reinforced Ea~th. REFERENCES Parts I and II, Department of Civil Engineer ing, University of Delaware, Newark, Del., July 1. J .R. Bell and J.E. Steward. Construction and 1984, Research Reports CE 84-44/45. Observations of Fabric Retained Soil walls. 14. D. Leshchinsky, R. Baker, and M.L. Silver. Proc., International Conference on the use of Three Dimensional Analysis of Slope Stability. Fabrics in Geotechnics, Vol. 1, Paris, France, International Journal for Numerical and Ana 1977, pp. 123-128. lytical Methods in Geomechanics, Vol. 9, No. 3, 2. G.E. Douglas. 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